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LINEAR PROGRAMMING PROJECT
VPAVITHRA SUKANYAH VK RIZWANA SULTANA SHILPA JAIN
INTRODUCTION
bull Modern technological advance growth ofscientific techniques
bull Operations Research (OR) recent additionto scientific tools
bull OR new outlook to many conventionalmanagement problems
bull Seeks the determination of best (optimum)course of action of a decision problem underthe limiting factor of limited resources
bull Operational Research can be considered asbeing the application of scientific method byinter-disciplinary teams to problemsinvolving the control of organized systems so as to provide solutionswhich best serve the purposes of theorganization as a whole
WHAT IS OR
CHARACTERISTIC NATURE OF OR
Inter-disciplinary team approachbull Systems approachbull Helpful in improving the quality of solutionbull Scientific methodbull Goal oriented optimum solutionbull Use of modelsbull Require willing executivesbull Reduces complexity
PHASES TO OR
bull Judgment phasendash Determination of the problemndash Establishment of the objectives and valuesndash Determination of suitable measures of effectiveness
bull Research phasendash Observation and data collectionndash Formulation of hypothesis and modelsndash Observation and experimentation to test the hypothesisndash Prediction of various results generalization consideration of alternative method
bull Action phasendash Implementation of the tested results of the model
METHODOLOGY OF OR
bull Formulating the problembull Constructing the model
bull Deriving the solutionndash Analytical methodndash Numerical methodndash Simulation methodbull Testing the validity
bull Controlling the solutionbull Implementing the result
PROBLEMS IN OR
bull Allocationbull Replacementbull Sequencingbull Routingbull Inventorybull Queuingbull Competitivebull Search
OR TECHINIQESbull Linear programming
bull Waiting line or queuing theorybull Inventory control planningbull Game theorybull Decision theorybull Network analysisndash Program Evaluation and Review
Techniquendash Critical Path Method (CPM) etcbull Simulationbull Integrated production models
SIGNIFICANCE OF OR
Provides a tool for scientific analysisbull Provides solution for various business problemsbull Enables proper deployment of resourcesbull Helps in minimizing waiting and servicing costsbull Enables the management to decide when to buy and how much to buybull Assists in choosing an optimum strategybull Renders great help in optimum resource allocationbull Facilitates the process of decision makingbull Management can know the reactions of the integrated
business systemsbull Helps a lot in the preparation of future managers
LIMITATIONS OF OR
The inherent limitations concerning mathematicalexpressions
bull High costs are involved in the use of OR techniquesbull OR does not take into consideration the intangible factorsbull OR is only a tool of analysis and not the complete decision-making processbull Other limitations ndash Bias ndash Inadequate objective functions ndash Internal resistance ndash Competence ndash Reliability of the prepared solution
INTRODUCTION TO LINEAR PROGRAMMING
Today many of the resources needed as inputs to operations are in limited supplyOperations managers must understand the impact of this situation on meeting their objectivesLinear programming (LP) is one way that operations managers can determine how best to allocate their scarce resources
Linear programmingWe use graphs as useful modeling abstractions to help us develop computational solutions for a wide variety of problemsA linear program is simply another modeling abstraction (tool in your toolbox)Developing routines that solve general linear programs allows us to embed them in sophisticated algorithmic solutions to difficult problems (eg like we did for TSP)The cutting edge algorithmic solutions to many problems use the ideas from mathematical programming linear programming forming the foundation
BASIC CONCEPT OF LP PROGRAM
Objective functionConstraintsOptimizationSolution of lppFeasible solutionOptimal solution
LP PROBLEMS IN OM PRODUCT MIX
ObjectiveTo select the mix of products or services
that results in maximum profits for the planning period
Decision VariablesHow much to produce and market of each
product or service for the planning period
ConstraintsMaximum amount of each product or
service demanded Minimum amount of product or service policy will allow Maximum amount of resources available
Objective function the linear functions which is to be optimized ie maximized or minimized this may be expressed in linear expressionSolution of Lpp
The set of all the values of the variable x1x2helliphellipxn which satisy the constraints is called the solution of Lpp Feasible solution The set of all the values of the variable x1x2helliphellipxn which satisy the constraints and also the non negative conditions is called the feasible solution of lpp
Recognizing LP Problems
Characteristics of LP Problems in OMA well-defined single objective must be statedThere must be alternative courses of actionThe total achievement of the objective must be constrained by scarce resources or other restraintsThe objective and each of the constraints must be expressed as linear mathematical functions
Steps in Formulating LP Problems
Define the objective (min or max)Define the decision variables (positive) Write the mathematical function for the objectiveWrite a 1- or 2-word description of each constraintWrite the right-hand side (RHS) of each constraintWrite lt = or gt for each constraintWrite the decision variables on LHS of each constraintWrite the coefficient for each decision variable in each constraint
Linear Programming
An optimization problem is said to be a linear program if it satisfied the following properties There is a unique objective function Whenever a decision variable appears in
either the objective function or one of theconstraint functions it must appear only as a power term with an exponent of 1possibly multiplied by a constant
LP Problems in General
Units of each term in a constraint must be the same as the RHSUnits of each term in the objective function must be the same as ZUnits between constraints do not have to be the sameLP problem can have a mixture of constraint types
No term in the objective function or in anyof the constraints can contain products ofthe decision variables
The coefficients of the decision variables inthe objective function and each constraintare constant
The decision variables are permitted toassume fractional as well as integer values
Examples of lpp
We are already familiar with the graphical representation of equations and inequations here we describe the application of linear equations and inequations in solving different kinds of problems The examples are stated belowExample 1
Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum
Step1 we have to choose the positive two numbers Let the 2 positive numbers be x and y this x and y are decision variables
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
VPAVITHRA SUKANYAH VK RIZWANA SULTANA SHILPA JAIN
INTRODUCTION
bull Modern technological advance growth ofscientific techniques
bull Operations Research (OR) recent additionto scientific tools
bull OR new outlook to many conventionalmanagement problems
bull Seeks the determination of best (optimum)course of action of a decision problem underthe limiting factor of limited resources
bull Operational Research can be considered asbeing the application of scientific method byinter-disciplinary teams to problemsinvolving the control of organized systems so as to provide solutionswhich best serve the purposes of theorganization as a whole
WHAT IS OR
CHARACTERISTIC NATURE OF OR
Inter-disciplinary team approachbull Systems approachbull Helpful in improving the quality of solutionbull Scientific methodbull Goal oriented optimum solutionbull Use of modelsbull Require willing executivesbull Reduces complexity
PHASES TO OR
bull Judgment phasendash Determination of the problemndash Establishment of the objectives and valuesndash Determination of suitable measures of effectiveness
bull Research phasendash Observation and data collectionndash Formulation of hypothesis and modelsndash Observation and experimentation to test the hypothesisndash Prediction of various results generalization consideration of alternative method
bull Action phasendash Implementation of the tested results of the model
METHODOLOGY OF OR
bull Formulating the problembull Constructing the model
bull Deriving the solutionndash Analytical methodndash Numerical methodndash Simulation methodbull Testing the validity
bull Controlling the solutionbull Implementing the result
PROBLEMS IN OR
bull Allocationbull Replacementbull Sequencingbull Routingbull Inventorybull Queuingbull Competitivebull Search
OR TECHINIQESbull Linear programming
bull Waiting line or queuing theorybull Inventory control planningbull Game theorybull Decision theorybull Network analysisndash Program Evaluation and Review
Techniquendash Critical Path Method (CPM) etcbull Simulationbull Integrated production models
SIGNIFICANCE OF OR
Provides a tool for scientific analysisbull Provides solution for various business problemsbull Enables proper deployment of resourcesbull Helps in minimizing waiting and servicing costsbull Enables the management to decide when to buy and how much to buybull Assists in choosing an optimum strategybull Renders great help in optimum resource allocationbull Facilitates the process of decision makingbull Management can know the reactions of the integrated
business systemsbull Helps a lot in the preparation of future managers
LIMITATIONS OF OR
The inherent limitations concerning mathematicalexpressions
bull High costs are involved in the use of OR techniquesbull OR does not take into consideration the intangible factorsbull OR is only a tool of analysis and not the complete decision-making processbull Other limitations ndash Bias ndash Inadequate objective functions ndash Internal resistance ndash Competence ndash Reliability of the prepared solution
INTRODUCTION TO LINEAR PROGRAMMING
Today many of the resources needed as inputs to operations are in limited supplyOperations managers must understand the impact of this situation on meeting their objectivesLinear programming (LP) is one way that operations managers can determine how best to allocate their scarce resources
Linear programmingWe use graphs as useful modeling abstractions to help us develop computational solutions for a wide variety of problemsA linear program is simply another modeling abstraction (tool in your toolbox)Developing routines that solve general linear programs allows us to embed them in sophisticated algorithmic solutions to difficult problems (eg like we did for TSP)The cutting edge algorithmic solutions to many problems use the ideas from mathematical programming linear programming forming the foundation
BASIC CONCEPT OF LP PROGRAM
Objective functionConstraintsOptimizationSolution of lppFeasible solutionOptimal solution
LP PROBLEMS IN OM PRODUCT MIX
ObjectiveTo select the mix of products or services
that results in maximum profits for the planning period
Decision VariablesHow much to produce and market of each
product or service for the planning period
ConstraintsMaximum amount of each product or
service demanded Minimum amount of product or service policy will allow Maximum amount of resources available
Objective function the linear functions which is to be optimized ie maximized or minimized this may be expressed in linear expressionSolution of Lpp
The set of all the values of the variable x1x2helliphellipxn which satisy the constraints is called the solution of Lpp Feasible solution The set of all the values of the variable x1x2helliphellipxn which satisy the constraints and also the non negative conditions is called the feasible solution of lpp
Recognizing LP Problems
Characteristics of LP Problems in OMA well-defined single objective must be statedThere must be alternative courses of actionThe total achievement of the objective must be constrained by scarce resources or other restraintsThe objective and each of the constraints must be expressed as linear mathematical functions
Steps in Formulating LP Problems
Define the objective (min or max)Define the decision variables (positive) Write the mathematical function for the objectiveWrite a 1- or 2-word description of each constraintWrite the right-hand side (RHS) of each constraintWrite lt = or gt for each constraintWrite the decision variables on LHS of each constraintWrite the coefficient for each decision variable in each constraint
Linear Programming
An optimization problem is said to be a linear program if it satisfied the following properties There is a unique objective function Whenever a decision variable appears in
either the objective function or one of theconstraint functions it must appear only as a power term with an exponent of 1possibly multiplied by a constant
LP Problems in General
Units of each term in a constraint must be the same as the RHSUnits of each term in the objective function must be the same as ZUnits between constraints do not have to be the sameLP problem can have a mixture of constraint types
No term in the objective function or in anyof the constraints can contain products ofthe decision variables
The coefficients of the decision variables inthe objective function and each constraintare constant
The decision variables are permitted toassume fractional as well as integer values
Examples of lpp
We are already familiar with the graphical representation of equations and inequations here we describe the application of linear equations and inequations in solving different kinds of problems The examples are stated belowExample 1
Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum
Step1 we have to choose the positive two numbers Let the 2 positive numbers be x and y this x and y are decision variables
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
INTRODUCTION
bull Modern technological advance growth ofscientific techniques
bull Operations Research (OR) recent additionto scientific tools
bull OR new outlook to many conventionalmanagement problems
bull Seeks the determination of best (optimum)course of action of a decision problem underthe limiting factor of limited resources
bull Operational Research can be considered asbeing the application of scientific method byinter-disciplinary teams to problemsinvolving the control of organized systems so as to provide solutionswhich best serve the purposes of theorganization as a whole
WHAT IS OR
CHARACTERISTIC NATURE OF OR
Inter-disciplinary team approachbull Systems approachbull Helpful in improving the quality of solutionbull Scientific methodbull Goal oriented optimum solutionbull Use of modelsbull Require willing executivesbull Reduces complexity
PHASES TO OR
bull Judgment phasendash Determination of the problemndash Establishment of the objectives and valuesndash Determination of suitable measures of effectiveness
bull Research phasendash Observation and data collectionndash Formulation of hypothesis and modelsndash Observation and experimentation to test the hypothesisndash Prediction of various results generalization consideration of alternative method
bull Action phasendash Implementation of the tested results of the model
METHODOLOGY OF OR
bull Formulating the problembull Constructing the model
bull Deriving the solutionndash Analytical methodndash Numerical methodndash Simulation methodbull Testing the validity
bull Controlling the solutionbull Implementing the result
PROBLEMS IN OR
bull Allocationbull Replacementbull Sequencingbull Routingbull Inventorybull Queuingbull Competitivebull Search
OR TECHINIQESbull Linear programming
bull Waiting line or queuing theorybull Inventory control planningbull Game theorybull Decision theorybull Network analysisndash Program Evaluation and Review
Techniquendash Critical Path Method (CPM) etcbull Simulationbull Integrated production models
SIGNIFICANCE OF OR
Provides a tool for scientific analysisbull Provides solution for various business problemsbull Enables proper deployment of resourcesbull Helps in minimizing waiting and servicing costsbull Enables the management to decide when to buy and how much to buybull Assists in choosing an optimum strategybull Renders great help in optimum resource allocationbull Facilitates the process of decision makingbull Management can know the reactions of the integrated
business systemsbull Helps a lot in the preparation of future managers
LIMITATIONS OF OR
The inherent limitations concerning mathematicalexpressions
bull High costs are involved in the use of OR techniquesbull OR does not take into consideration the intangible factorsbull OR is only a tool of analysis and not the complete decision-making processbull Other limitations ndash Bias ndash Inadequate objective functions ndash Internal resistance ndash Competence ndash Reliability of the prepared solution
INTRODUCTION TO LINEAR PROGRAMMING
Today many of the resources needed as inputs to operations are in limited supplyOperations managers must understand the impact of this situation on meeting their objectivesLinear programming (LP) is one way that operations managers can determine how best to allocate their scarce resources
Linear programmingWe use graphs as useful modeling abstractions to help us develop computational solutions for a wide variety of problemsA linear program is simply another modeling abstraction (tool in your toolbox)Developing routines that solve general linear programs allows us to embed them in sophisticated algorithmic solutions to difficult problems (eg like we did for TSP)The cutting edge algorithmic solutions to many problems use the ideas from mathematical programming linear programming forming the foundation
BASIC CONCEPT OF LP PROGRAM
Objective functionConstraintsOptimizationSolution of lppFeasible solutionOptimal solution
LP PROBLEMS IN OM PRODUCT MIX
ObjectiveTo select the mix of products or services
that results in maximum profits for the planning period
Decision VariablesHow much to produce and market of each
product or service for the planning period
ConstraintsMaximum amount of each product or
service demanded Minimum amount of product or service policy will allow Maximum amount of resources available
Objective function the linear functions which is to be optimized ie maximized or minimized this may be expressed in linear expressionSolution of Lpp
The set of all the values of the variable x1x2helliphellipxn which satisy the constraints is called the solution of Lpp Feasible solution The set of all the values of the variable x1x2helliphellipxn which satisy the constraints and also the non negative conditions is called the feasible solution of lpp
Recognizing LP Problems
Characteristics of LP Problems in OMA well-defined single objective must be statedThere must be alternative courses of actionThe total achievement of the objective must be constrained by scarce resources or other restraintsThe objective and each of the constraints must be expressed as linear mathematical functions
Steps in Formulating LP Problems
Define the objective (min or max)Define the decision variables (positive) Write the mathematical function for the objectiveWrite a 1- or 2-word description of each constraintWrite the right-hand side (RHS) of each constraintWrite lt = or gt for each constraintWrite the decision variables on LHS of each constraintWrite the coefficient for each decision variable in each constraint
Linear Programming
An optimization problem is said to be a linear program if it satisfied the following properties There is a unique objective function Whenever a decision variable appears in
either the objective function or one of theconstraint functions it must appear only as a power term with an exponent of 1possibly multiplied by a constant
LP Problems in General
Units of each term in a constraint must be the same as the RHSUnits of each term in the objective function must be the same as ZUnits between constraints do not have to be the sameLP problem can have a mixture of constraint types
No term in the objective function or in anyof the constraints can contain products ofthe decision variables
The coefficients of the decision variables inthe objective function and each constraintare constant
The decision variables are permitted toassume fractional as well as integer values
Examples of lpp
We are already familiar with the graphical representation of equations and inequations here we describe the application of linear equations and inequations in solving different kinds of problems The examples are stated belowExample 1
Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum
Step1 we have to choose the positive two numbers Let the 2 positive numbers be x and y this x and y are decision variables
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
bull Operational Research can be considered asbeing the application of scientific method byinter-disciplinary teams to problemsinvolving the control of organized systems so as to provide solutionswhich best serve the purposes of theorganization as a whole
WHAT IS OR
CHARACTERISTIC NATURE OF OR
Inter-disciplinary team approachbull Systems approachbull Helpful in improving the quality of solutionbull Scientific methodbull Goal oriented optimum solutionbull Use of modelsbull Require willing executivesbull Reduces complexity
PHASES TO OR
bull Judgment phasendash Determination of the problemndash Establishment of the objectives and valuesndash Determination of suitable measures of effectiveness
bull Research phasendash Observation and data collectionndash Formulation of hypothesis and modelsndash Observation and experimentation to test the hypothesisndash Prediction of various results generalization consideration of alternative method
bull Action phasendash Implementation of the tested results of the model
METHODOLOGY OF OR
bull Formulating the problembull Constructing the model
bull Deriving the solutionndash Analytical methodndash Numerical methodndash Simulation methodbull Testing the validity
bull Controlling the solutionbull Implementing the result
PROBLEMS IN OR
bull Allocationbull Replacementbull Sequencingbull Routingbull Inventorybull Queuingbull Competitivebull Search
OR TECHINIQESbull Linear programming
bull Waiting line or queuing theorybull Inventory control planningbull Game theorybull Decision theorybull Network analysisndash Program Evaluation and Review
Techniquendash Critical Path Method (CPM) etcbull Simulationbull Integrated production models
SIGNIFICANCE OF OR
Provides a tool for scientific analysisbull Provides solution for various business problemsbull Enables proper deployment of resourcesbull Helps in minimizing waiting and servicing costsbull Enables the management to decide when to buy and how much to buybull Assists in choosing an optimum strategybull Renders great help in optimum resource allocationbull Facilitates the process of decision makingbull Management can know the reactions of the integrated
business systemsbull Helps a lot in the preparation of future managers
LIMITATIONS OF OR
The inherent limitations concerning mathematicalexpressions
bull High costs are involved in the use of OR techniquesbull OR does not take into consideration the intangible factorsbull OR is only a tool of analysis and not the complete decision-making processbull Other limitations ndash Bias ndash Inadequate objective functions ndash Internal resistance ndash Competence ndash Reliability of the prepared solution
INTRODUCTION TO LINEAR PROGRAMMING
Today many of the resources needed as inputs to operations are in limited supplyOperations managers must understand the impact of this situation on meeting their objectivesLinear programming (LP) is one way that operations managers can determine how best to allocate their scarce resources
Linear programmingWe use graphs as useful modeling abstractions to help us develop computational solutions for a wide variety of problemsA linear program is simply another modeling abstraction (tool in your toolbox)Developing routines that solve general linear programs allows us to embed them in sophisticated algorithmic solutions to difficult problems (eg like we did for TSP)The cutting edge algorithmic solutions to many problems use the ideas from mathematical programming linear programming forming the foundation
BASIC CONCEPT OF LP PROGRAM
Objective functionConstraintsOptimizationSolution of lppFeasible solutionOptimal solution
LP PROBLEMS IN OM PRODUCT MIX
ObjectiveTo select the mix of products or services
that results in maximum profits for the planning period
Decision VariablesHow much to produce and market of each
product or service for the planning period
ConstraintsMaximum amount of each product or
service demanded Minimum amount of product or service policy will allow Maximum amount of resources available
Objective function the linear functions which is to be optimized ie maximized or minimized this may be expressed in linear expressionSolution of Lpp
The set of all the values of the variable x1x2helliphellipxn which satisy the constraints is called the solution of Lpp Feasible solution The set of all the values of the variable x1x2helliphellipxn which satisy the constraints and also the non negative conditions is called the feasible solution of lpp
Recognizing LP Problems
Characteristics of LP Problems in OMA well-defined single objective must be statedThere must be alternative courses of actionThe total achievement of the objective must be constrained by scarce resources or other restraintsThe objective and each of the constraints must be expressed as linear mathematical functions
Steps in Formulating LP Problems
Define the objective (min or max)Define the decision variables (positive) Write the mathematical function for the objectiveWrite a 1- or 2-word description of each constraintWrite the right-hand side (RHS) of each constraintWrite lt = or gt for each constraintWrite the decision variables on LHS of each constraintWrite the coefficient for each decision variable in each constraint
Linear Programming
An optimization problem is said to be a linear program if it satisfied the following properties There is a unique objective function Whenever a decision variable appears in
either the objective function or one of theconstraint functions it must appear only as a power term with an exponent of 1possibly multiplied by a constant
LP Problems in General
Units of each term in a constraint must be the same as the RHSUnits of each term in the objective function must be the same as ZUnits between constraints do not have to be the sameLP problem can have a mixture of constraint types
No term in the objective function or in anyof the constraints can contain products ofthe decision variables
The coefficients of the decision variables inthe objective function and each constraintare constant
The decision variables are permitted toassume fractional as well as integer values
Examples of lpp
We are already familiar with the graphical representation of equations and inequations here we describe the application of linear equations and inequations in solving different kinds of problems The examples are stated belowExample 1
Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum
Step1 we have to choose the positive two numbers Let the 2 positive numbers be x and y this x and y are decision variables
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
CHARACTERISTIC NATURE OF OR
Inter-disciplinary team approachbull Systems approachbull Helpful in improving the quality of solutionbull Scientific methodbull Goal oriented optimum solutionbull Use of modelsbull Require willing executivesbull Reduces complexity
PHASES TO OR
bull Judgment phasendash Determination of the problemndash Establishment of the objectives and valuesndash Determination of suitable measures of effectiveness
bull Research phasendash Observation and data collectionndash Formulation of hypothesis and modelsndash Observation and experimentation to test the hypothesisndash Prediction of various results generalization consideration of alternative method
bull Action phasendash Implementation of the tested results of the model
METHODOLOGY OF OR
bull Formulating the problembull Constructing the model
bull Deriving the solutionndash Analytical methodndash Numerical methodndash Simulation methodbull Testing the validity
bull Controlling the solutionbull Implementing the result
PROBLEMS IN OR
bull Allocationbull Replacementbull Sequencingbull Routingbull Inventorybull Queuingbull Competitivebull Search
OR TECHINIQESbull Linear programming
bull Waiting line or queuing theorybull Inventory control planningbull Game theorybull Decision theorybull Network analysisndash Program Evaluation and Review
Techniquendash Critical Path Method (CPM) etcbull Simulationbull Integrated production models
SIGNIFICANCE OF OR
Provides a tool for scientific analysisbull Provides solution for various business problemsbull Enables proper deployment of resourcesbull Helps in minimizing waiting and servicing costsbull Enables the management to decide when to buy and how much to buybull Assists in choosing an optimum strategybull Renders great help in optimum resource allocationbull Facilitates the process of decision makingbull Management can know the reactions of the integrated
business systemsbull Helps a lot in the preparation of future managers
LIMITATIONS OF OR
The inherent limitations concerning mathematicalexpressions
bull High costs are involved in the use of OR techniquesbull OR does not take into consideration the intangible factorsbull OR is only a tool of analysis and not the complete decision-making processbull Other limitations ndash Bias ndash Inadequate objective functions ndash Internal resistance ndash Competence ndash Reliability of the prepared solution
INTRODUCTION TO LINEAR PROGRAMMING
Today many of the resources needed as inputs to operations are in limited supplyOperations managers must understand the impact of this situation on meeting their objectivesLinear programming (LP) is one way that operations managers can determine how best to allocate their scarce resources
Linear programmingWe use graphs as useful modeling abstractions to help us develop computational solutions for a wide variety of problemsA linear program is simply another modeling abstraction (tool in your toolbox)Developing routines that solve general linear programs allows us to embed them in sophisticated algorithmic solutions to difficult problems (eg like we did for TSP)The cutting edge algorithmic solutions to many problems use the ideas from mathematical programming linear programming forming the foundation
BASIC CONCEPT OF LP PROGRAM
Objective functionConstraintsOptimizationSolution of lppFeasible solutionOptimal solution
LP PROBLEMS IN OM PRODUCT MIX
ObjectiveTo select the mix of products or services
that results in maximum profits for the planning period
Decision VariablesHow much to produce and market of each
product or service for the planning period
ConstraintsMaximum amount of each product or
service demanded Minimum amount of product or service policy will allow Maximum amount of resources available
Objective function the linear functions which is to be optimized ie maximized or minimized this may be expressed in linear expressionSolution of Lpp
The set of all the values of the variable x1x2helliphellipxn which satisy the constraints is called the solution of Lpp Feasible solution The set of all the values of the variable x1x2helliphellipxn which satisy the constraints and also the non negative conditions is called the feasible solution of lpp
Recognizing LP Problems
Characteristics of LP Problems in OMA well-defined single objective must be statedThere must be alternative courses of actionThe total achievement of the objective must be constrained by scarce resources or other restraintsThe objective and each of the constraints must be expressed as linear mathematical functions
Steps in Formulating LP Problems
Define the objective (min or max)Define the decision variables (positive) Write the mathematical function for the objectiveWrite a 1- or 2-word description of each constraintWrite the right-hand side (RHS) of each constraintWrite lt = or gt for each constraintWrite the decision variables on LHS of each constraintWrite the coefficient for each decision variable in each constraint
Linear Programming
An optimization problem is said to be a linear program if it satisfied the following properties There is a unique objective function Whenever a decision variable appears in
either the objective function or one of theconstraint functions it must appear only as a power term with an exponent of 1possibly multiplied by a constant
LP Problems in General
Units of each term in a constraint must be the same as the RHSUnits of each term in the objective function must be the same as ZUnits between constraints do not have to be the sameLP problem can have a mixture of constraint types
No term in the objective function or in anyof the constraints can contain products ofthe decision variables
The coefficients of the decision variables inthe objective function and each constraintare constant
The decision variables are permitted toassume fractional as well as integer values
Examples of lpp
We are already familiar with the graphical representation of equations and inequations here we describe the application of linear equations and inequations in solving different kinds of problems The examples are stated belowExample 1
Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum
Step1 we have to choose the positive two numbers Let the 2 positive numbers be x and y this x and y are decision variables
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
PHASES TO OR
bull Judgment phasendash Determination of the problemndash Establishment of the objectives and valuesndash Determination of suitable measures of effectiveness
bull Research phasendash Observation and data collectionndash Formulation of hypothesis and modelsndash Observation and experimentation to test the hypothesisndash Prediction of various results generalization consideration of alternative method
bull Action phasendash Implementation of the tested results of the model
METHODOLOGY OF OR
bull Formulating the problembull Constructing the model
bull Deriving the solutionndash Analytical methodndash Numerical methodndash Simulation methodbull Testing the validity
bull Controlling the solutionbull Implementing the result
PROBLEMS IN OR
bull Allocationbull Replacementbull Sequencingbull Routingbull Inventorybull Queuingbull Competitivebull Search
OR TECHINIQESbull Linear programming
bull Waiting line or queuing theorybull Inventory control planningbull Game theorybull Decision theorybull Network analysisndash Program Evaluation and Review
Techniquendash Critical Path Method (CPM) etcbull Simulationbull Integrated production models
SIGNIFICANCE OF OR
Provides a tool for scientific analysisbull Provides solution for various business problemsbull Enables proper deployment of resourcesbull Helps in minimizing waiting and servicing costsbull Enables the management to decide when to buy and how much to buybull Assists in choosing an optimum strategybull Renders great help in optimum resource allocationbull Facilitates the process of decision makingbull Management can know the reactions of the integrated
business systemsbull Helps a lot in the preparation of future managers
LIMITATIONS OF OR
The inherent limitations concerning mathematicalexpressions
bull High costs are involved in the use of OR techniquesbull OR does not take into consideration the intangible factorsbull OR is only a tool of analysis and not the complete decision-making processbull Other limitations ndash Bias ndash Inadequate objective functions ndash Internal resistance ndash Competence ndash Reliability of the prepared solution
INTRODUCTION TO LINEAR PROGRAMMING
Today many of the resources needed as inputs to operations are in limited supplyOperations managers must understand the impact of this situation on meeting their objectivesLinear programming (LP) is one way that operations managers can determine how best to allocate their scarce resources
Linear programmingWe use graphs as useful modeling abstractions to help us develop computational solutions for a wide variety of problemsA linear program is simply another modeling abstraction (tool in your toolbox)Developing routines that solve general linear programs allows us to embed them in sophisticated algorithmic solutions to difficult problems (eg like we did for TSP)The cutting edge algorithmic solutions to many problems use the ideas from mathematical programming linear programming forming the foundation
BASIC CONCEPT OF LP PROGRAM
Objective functionConstraintsOptimizationSolution of lppFeasible solutionOptimal solution
LP PROBLEMS IN OM PRODUCT MIX
ObjectiveTo select the mix of products or services
that results in maximum profits for the planning period
Decision VariablesHow much to produce and market of each
product or service for the planning period
ConstraintsMaximum amount of each product or
service demanded Minimum amount of product or service policy will allow Maximum amount of resources available
Objective function the linear functions which is to be optimized ie maximized or minimized this may be expressed in linear expressionSolution of Lpp
The set of all the values of the variable x1x2helliphellipxn which satisy the constraints is called the solution of Lpp Feasible solution The set of all the values of the variable x1x2helliphellipxn which satisy the constraints and also the non negative conditions is called the feasible solution of lpp
Recognizing LP Problems
Characteristics of LP Problems in OMA well-defined single objective must be statedThere must be alternative courses of actionThe total achievement of the objective must be constrained by scarce resources or other restraintsThe objective and each of the constraints must be expressed as linear mathematical functions
Steps in Formulating LP Problems
Define the objective (min or max)Define the decision variables (positive) Write the mathematical function for the objectiveWrite a 1- or 2-word description of each constraintWrite the right-hand side (RHS) of each constraintWrite lt = or gt for each constraintWrite the decision variables on LHS of each constraintWrite the coefficient for each decision variable in each constraint
Linear Programming
An optimization problem is said to be a linear program if it satisfied the following properties There is a unique objective function Whenever a decision variable appears in
either the objective function or one of theconstraint functions it must appear only as a power term with an exponent of 1possibly multiplied by a constant
LP Problems in General
Units of each term in a constraint must be the same as the RHSUnits of each term in the objective function must be the same as ZUnits between constraints do not have to be the sameLP problem can have a mixture of constraint types
No term in the objective function or in anyof the constraints can contain products ofthe decision variables
The coefficients of the decision variables inthe objective function and each constraintare constant
The decision variables are permitted toassume fractional as well as integer values
Examples of lpp
We are already familiar with the graphical representation of equations and inequations here we describe the application of linear equations and inequations in solving different kinds of problems The examples are stated belowExample 1
Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum
Step1 we have to choose the positive two numbers Let the 2 positive numbers be x and y this x and y are decision variables
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
METHODOLOGY OF OR
bull Formulating the problembull Constructing the model
bull Deriving the solutionndash Analytical methodndash Numerical methodndash Simulation methodbull Testing the validity
bull Controlling the solutionbull Implementing the result
PROBLEMS IN OR
bull Allocationbull Replacementbull Sequencingbull Routingbull Inventorybull Queuingbull Competitivebull Search
OR TECHINIQESbull Linear programming
bull Waiting line or queuing theorybull Inventory control planningbull Game theorybull Decision theorybull Network analysisndash Program Evaluation and Review
Techniquendash Critical Path Method (CPM) etcbull Simulationbull Integrated production models
SIGNIFICANCE OF OR
Provides a tool for scientific analysisbull Provides solution for various business problemsbull Enables proper deployment of resourcesbull Helps in minimizing waiting and servicing costsbull Enables the management to decide when to buy and how much to buybull Assists in choosing an optimum strategybull Renders great help in optimum resource allocationbull Facilitates the process of decision makingbull Management can know the reactions of the integrated
business systemsbull Helps a lot in the preparation of future managers
LIMITATIONS OF OR
The inherent limitations concerning mathematicalexpressions
bull High costs are involved in the use of OR techniquesbull OR does not take into consideration the intangible factorsbull OR is only a tool of analysis and not the complete decision-making processbull Other limitations ndash Bias ndash Inadequate objective functions ndash Internal resistance ndash Competence ndash Reliability of the prepared solution
INTRODUCTION TO LINEAR PROGRAMMING
Today many of the resources needed as inputs to operations are in limited supplyOperations managers must understand the impact of this situation on meeting their objectivesLinear programming (LP) is one way that operations managers can determine how best to allocate their scarce resources
Linear programmingWe use graphs as useful modeling abstractions to help us develop computational solutions for a wide variety of problemsA linear program is simply another modeling abstraction (tool in your toolbox)Developing routines that solve general linear programs allows us to embed them in sophisticated algorithmic solutions to difficult problems (eg like we did for TSP)The cutting edge algorithmic solutions to many problems use the ideas from mathematical programming linear programming forming the foundation
BASIC CONCEPT OF LP PROGRAM
Objective functionConstraintsOptimizationSolution of lppFeasible solutionOptimal solution
LP PROBLEMS IN OM PRODUCT MIX
ObjectiveTo select the mix of products or services
that results in maximum profits for the planning period
Decision VariablesHow much to produce and market of each
product or service for the planning period
ConstraintsMaximum amount of each product or
service demanded Minimum amount of product or service policy will allow Maximum amount of resources available
Objective function the linear functions which is to be optimized ie maximized or minimized this may be expressed in linear expressionSolution of Lpp
The set of all the values of the variable x1x2helliphellipxn which satisy the constraints is called the solution of Lpp Feasible solution The set of all the values of the variable x1x2helliphellipxn which satisy the constraints and also the non negative conditions is called the feasible solution of lpp
Recognizing LP Problems
Characteristics of LP Problems in OMA well-defined single objective must be statedThere must be alternative courses of actionThe total achievement of the objective must be constrained by scarce resources or other restraintsThe objective and each of the constraints must be expressed as linear mathematical functions
Steps in Formulating LP Problems
Define the objective (min or max)Define the decision variables (positive) Write the mathematical function for the objectiveWrite a 1- or 2-word description of each constraintWrite the right-hand side (RHS) of each constraintWrite lt = or gt for each constraintWrite the decision variables on LHS of each constraintWrite the coefficient for each decision variable in each constraint
Linear Programming
An optimization problem is said to be a linear program if it satisfied the following properties There is a unique objective function Whenever a decision variable appears in
either the objective function or one of theconstraint functions it must appear only as a power term with an exponent of 1possibly multiplied by a constant
LP Problems in General
Units of each term in a constraint must be the same as the RHSUnits of each term in the objective function must be the same as ZUnits between constraints do not have to be the sameLP problem can have a mixture of constraint types
No term in the objective function or in anyof the constraints can contain products ofthe decision variables
The coefficients of the decision variables inthe objective function and each constraintare constant
The decision variables are permitted toassume fractional as well as integer values
Examples of lpp
We are already familiar with the graphical representation of equations and inequations here we describe the application of linear equations and inequations in solving different kinds of problems The examples are stated belowExample 1
Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum
Step1 we have to choose the positive two numbers Let the 2 positive numbers be x and y this x and y are decision variables
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
PROBLEMS IN OR
bull Allocationbull Replacementbull Sequencingbull Routingbull Inventorybull Queuingbull Competitivebull Search
OR TECHINIQESbull Linear programming
bull Waiting line or queuing theorybull Inventory control planningbull Game theorybull Decision theorybull Network analysisndash Program Evaluation and Review
Techniquendash Critical Path Method (CPM) etcbull Simulationbull Integrated production models
SIGNIFICANCE OF OR
Provides a tool for scientific analysisbull Provides solution for various business problemsbull Enables proper deployment of resourcesbull Helps in minimizing waiting and servicing costsbull Enables the management to decide when to buy and how much to buybull Assists in choosing an optimum strategybull Renders great help in optimum resource allocationbull Facilitates the process of decision makingbull Management can know the reactions of the integrated
business systemsbull Helps a lot in the preparation of future managers
LIMITATIONS OF OR
The inherent limitations concerning mathematicalexpressions
bull High costs are involved in the use of OR techniquesbull OR does not take into consideration the intangible factorsbull OR is only a tool of analysis and not the complete decision-making processbull Other limitations ndash Bias ndash Inadequate objective functions ndash Internal resistance ndash Competence ndash Reliability of the prepared solution
INTRODUCTION TO LINEAR PROGRAMMING
Today many of the resources needed as inputs to operations are in limited supplyOperations managers must understand the impact of this situation on meeting their objectivesLinear programming (LP) is one way that operations managers can determine how best to allocate their scarce resources
Linear programmingWe use graphs as useful modeling abstractions to help us develop computational solutions for a wide variety of problemsA linear program is simply another modeling abstraction (tool in your toolbox)Developing routines that solve general linear programs allows us to embed them in sophisticated algorithmic solutions to difficult problems (eg like we did for TSP)The cutting edge algorithmic solutions to many problems use the ideas from mathematical programming linear programming forming the foundation
BASIC CONCEPT OF LP PROGRAM
Objective functionConstraintsOptimizationSolution of lppFeasible solutionOptimal solution
LP PROBLEMS IN OM PRODUCT MIX
ObjectiveTo select the mix of products or services
that results in maximum profits for the planning period
Decision VariablesHow much to produce and market of each
product or service for the planning period
ConstraintsMaximum amount of each product or
service demanded Minimum amount of product or service policy will allow Maximum amount of resources available
Objective function the linear functions which is to be optimized ie maximized or minimized this may be expressed in linear expressionSolution of Lpp
The set of all the values of the variable x1x2helliphellipxn which satisy the constraints is called the solution of Lpp Feasible solution The set of all the values of the variable x1x2helliphellipxn which satisy the constraints and also the non negative conditions is called the feasible solution of lpp
Recognizing LP Problems
Characteristics of LP Problems in OMA well-defined single objective must be statedThere must be alternative courses of actionThe total achievement of the objective must be constrained by scarce resources or other restraintsThe objective and each of the constraints must be expressed as linear mathematical functions
Steps in Formulating LP Problems
Define the objective (min or max)Define the decision variables (positive) Write the mathematical function for the objectiveWrite a 1- or 2-word description of each constraintWrite the right-hand side (RHS) of each constraintWrite lt = or gt for each constraintWrite the decision variables on LHS of each constraintWrite the coefficient for each decision variable in each constraint
Linear Programming
An optimization problem is said to be a linear program if it satisfied the following properties There is a unique objective function Whenever a decision variable appears in
either the objective function or one of theconstraint functions it must appear only as a power term with an exponent of 1possibly multiplied by a constant
LP Problems in General
Units of each term in a constraint must be the same as the RHSUnits of each term in the objective function must be the same as ZUnits between constraints do not have to be the sameLP problem can have a mixture of constraint types
No term in the objective function or in anyof the constraints can contain products ofthe decision variables
The coefficients of the decision variables inthe objective function and each constraintare constant
The decision variables are permitted toassume fractional as well as integer values
Examples of lpp
We are already familiar with the graphical representation of equations and inequations here we describe the application of linear equations and inequations in solving different kinds of problems The examples are stated belowExample 1
Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum
Step1 we have to choose the positive two numbers Let the 2 positive numbers be x and y this x and y are decision variables
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
OR TECHINIQESbull Linear programming
bull Waiting line or queuing theorybull Inventory control planningbull Game theorybull Decision theorybull Network analysisndash Program Evaluation and Review
Techniquendash Critical Path Method (CPM) etcbull Simulationbull Integrated production models
SIGNIFICANCE OF OR
Provides a tool for scientific analysisbull Provides solution for various business problemsbull Enables proper deployment of resourcesbull Helps in minimizing waiting and servicing costsbull Enables the management to decide when to buy and how much to buybull Assists in choosing an optimum strategybull Renders great help in optimum resource allocationbull Facilitates the process of decision makingbull Management can know the reactions of the integrated
business systemsbull Helps a lot in the preparation of future managers
LIMITATIONS OF OR
The inherent limitations concerning mathematicalexpressions
bull High costs are involved in the use of OR techniquesbull OR does not take into consideration the intangible factorsbull OR is only a tool of analysis and not the complete decision-making processbull Other limitations ndash Bias ndash Inadequate objective functions ndash Internal resistance ndash Competence ndash Reliability of the prepared solution
INTRODUCTION TO LINEAR PROGRAMMING
Today many of the resources needed as inputs to operations are in limited supplyOperations managers must understand the impact of this situation on meeting their objectivesLinear programming (LP) is one way that operations managers can determine how best to allocate their scarce resources
Linear programmingWe use graphs as useful modeling abstractions to help us develop computational solutions for a wide variety of problemsA linear program is simply another modeling abstraction (tool in your toolbox)Developing routines that solve general linear programs allows us to embed them in sophisticated algorithmic solutions to difficult problems (eg like we did for TSP)The cutting edge algorithmic solutions to many problems use the ideas from mathematical programming linear programming forming the foundation
BASIC CONCEPT OF LP PROGRAM
Objective functionConstraintsOptimizationSolution of lppFeasible solutionOptimal solution
LP PROBLEMS IN OM PRODUCT MIX
ObjectiveTo select the mix of products or services
that results in maximum profits for the planning period
Decision VariablesHow much to produce and market of each
product or service for the planning period
ConstraintsMaximum amount of each product or
service demanded Minimum amount of product or service policy will allow Maximum amount of resources available
Objective function the linear functions which is to be optimized ie maximized or minimized this may be expressed in linear expressionSolution of Lpp
The set of all the values of the variable x1x2helliphellipxn which satisy the constraints is called the solution of Lpp Feasible solution The set of all the values of the variable x1x2helliphellipxn which satisy the constraints and also the non negative conditions is called the feasible solution of lpp
Recognizing LP Problems
Characteristics of LP Problems in OMA well-defined single objective must be statedThere must be alternative courses of actionThe total achievement of the objective must be constrained by scarce resources or other restraintsThe objective and each of the constraints must be expressed as linear mathematical functions
Steps in Formulating LP Problems
Define the objective (min or max)Define the decision variables (positive) Write the mathematical function for the objectiveWrite a 1- or 2-word description of each constraintWrite the right-hand side (RHS) of each constraintWrite lt = or gt for each constraintWrite the decision variables on LHS of each constraintWrite the coefficient for each decision variable in each constraint
Linear Programming
An optimization problem is said to be a linear program if it satisfied the following properties There is a unique objective function Whenever a decision variable appears in
either the objective function or one of theconstraint functions it must appear only as a power term with an exponent of 1possibly multiplied by a constant
LP Problems in General
Units of each term in a constraint must be the same as the RHSUnits of each term in the objective function must be the same as ZUnits between constraints do not have to be the sameLP problem can have a mixture of constraint types
No term in the objective function or in anyof the constraints can contain products ofthe decision variables
The coefficients of the decision variables inthe objective function and each constraintare constant
The decision variables are permitted toassume fractional as well as integer values
Examples of lpp
We are already familiar with the graphical representation of equations and inequations here we describe the application of linear equations and inequations in solving different kinds of problems The examples are stated belowExample 1
Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum
Step1 we have to choose the positive two numbers Let the 2 positive numbers be x and y this x and y are decision variables
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
SIGNIFICANCE OF OR
Provides a tool for scientific analysisbull Provides solution for various business problemsbull Enables proper deployment of resourcesbull Helps in minimizing waiting and servicing costsbull Enables the management to decide when to buy and how much to buybull Assists in choosing an optimum strategybull Renders great help in optimum resource allocationbull Facilitates the process of decision makingbull Management can know the reactions of the integrated
business systemsbull Helps a lot in the preparation of future managers
LIMITATIONS OF OR
The inherent limitations concerning mathematicalexpressions
bull High costs are involved in the use of OR techniquesbull OR does not take into consideration the intangible factorsbull OR is only a tool of analysis and not the complete decision-making processbull Other limitations ndash Bias ndash Inadequate objective functions ndash Internal resistance ndash Competence ndash Reliability of the prepared solution
INTRODUCTION TO LINEAR PROGRAMMING
Today many of the resources needed as inputs to operations are in limited supplyOperations managers must understand the impact of this situation on meeting their objectivesLinear programming (LP) is one way that operations managers can determine how best to allocate their scarce resources
Linear programmingWe use graphs as useful modeling abstractions to help us develop computational solutions for a wide variety of problemsA linear program is simply another modeling abstraction (tool in your toolbox)Developing routines that solve general linear programs allows us to embed them in sophisticated algorithmic solutions to difficult problems (eg like we did for TSP)The cutting edge algorithmic solutions to many problems use the ideas from mathematical programming linear programming forming the foundation
BASIC CONCEPT OF LP PROGRAM
Objective functionConstraintsOptimizationSolution of lppFeasible solutionOptimal solution
LP PROBLEMS IN OM PRODUCT MIX
ObjectiveTo select the mix of products or services
that results in maximum profits for the planning period
Decision VariablesHow much to produce and market of each
product or service for the planning period
ConstraintsMaximum amount of each product or
service demanded Minimum amount of product or service policy will allow Maximum amount of resources available
Objective function the linear functions which is to be optimized ie maximized or minimized this may be expressed in linear expressionSolution of Lpp
The set of all the values of the variable x1x2helliphellipxn which satisy the constraints is called the solution of Lpp Feasible solution The set of all the values of the variable x1x2helliphellipxn which satisy the constraints and also the non negative conditions is called the feasible solution of lpp
Recognizing LP Problems
Characteristics of LP Problems in OMA well-defined single objective must be statedThere must be alternative courses of actionThe total achievement of the objective must be constrained by scarce resources or other restraintsThe objective and each of the constraints must be expressed as linear mathematical functions
Steps in Formulating LP Problems
Define the objective (min or max)Define the decision variables (positive) Write the mathematical function for the objectiveWrite a 1- or 2-word description of each constraintWrite the right-hand side (RHS) of each constraintWrite lt = or gt for each constraintWrite the decision variables on LHS of each constraintWrite the coefficient for each decision variable in each constraint
Linear Programming
An optimization problem is said to be a linear program if it satisfied the following properties There is a unique objective function Whenever a decision variable appears in
either the objective function or one of theconstraint functions it must appear only as a power term with an exponent of 1possibly multiplied by a constant
LP Problems in General
Units of each term in a constraint must be the same as the RHSUnits of each term in the objective function must be the same as ZUnits between constraints do not have to be the sameLP problem can have a mixture of constraint types
No term in the objective function or in anyof the constraints can contain products ofthe decision variables
The coefficients of the decision variables inthe objective function and each constraintare constant
The decision variables are permitted toassume fractional as well as integer values
Examples of lpp
We are already familiar with the graphical representation of equations and inequations here we describe the application of linear equations and inequations in solving different kinds of problems The examples are stated belowExample 1
Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum
Step1 we have to choose the positive two numbers Let the 2 positive numbers be x and y this x and y are decision variables
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
LIMITATIONS OF OR
The inherent limitations concerning mathematicalexpressions
bull High costs are involved in the use of OR techniquesbull OR does not take into consideration the intangible factorsbull OR is only a tool of analysis and not the complete decision-making processbull Other limitations ndash Bias ndash Inadequate objective functions ndash Internal resistance ndash Competence ndash Reliability of the prepared solution
INTRODUCTION TO LINEAR PROGRAMMING
Today many of the resources needed as inputs to operations are in limited supplyOperations managers must understand the impact of this situation on meeting their objectivesLinear programming (LP) is one way that operations managers can determine how best to allocate their scarce resources
Linear programmingWe use graphs as useful modeling abstractions to help us develop computational solutions for a wide variety of problemsA linear program is simply another modeling abstraction (tool in your toolbox)Developing routines that solve general linear programs allows us to embed them in sophisticated algorithmic solutions to difficult problems (eg like we did for TSP)The cutting edge algorithmic solutions to many problems use the ideas from mathematical programming linear programming forming the foundation
BASIC CONCEPT OF LP PROGRAM
Objective functionConstraintsOptimizationSolution of lppFeasible solutionOptimal solution
LP PROBLEMS IN OM PRODUCT MIX
ObjectiveTo select the mix of products or services
that results in maximum profits for the planning period
Decision VariablesHow much to produce and market of each
product or service for the planning period
ConstraintsMaximum amount of each product or
service demanded Minimum amount of product or service policy will allow Maximum amount of resources available
Objective function the linear functions which is to be optimized ie maximized or minimized this may be expressed in linear expressionSolution of Lpp
The set of all the values of the variable x1x2helliphellipxn which satisy the constraints is called the solution of Lpp Feasible solution The set of all the values of the variable x1x2helliphellipxn which satisy the constraints and also the non negative conditions is called the feasible solution of lpp
Recognizing LP Problems
Characteristics of LP Problems in OMA well-defined single objective must be statedThere must be alternative courses of actionThe total achievement of the objective must be constrained by scarce resources or other restraintsThe objective and each of the constraints must be expressed as linear mathematical functions
Steps in Formulating LP Problems
Define the objective (min or max)Define the decision variables (positive) Write the mathematical function for the objectiveWrite a 1- or 2-word description of each constraintWrite the right-hand side (RHS) of each constraintWrite lt = or gt for each constraintWrite the decision variables on LHS of each constraintWrite the coefficient for each decision variable in each constraint
Linear Programming
An optimization problem is said to be a linear program if it satisfied the following properties There is a unique objective function Whenever a decision variable appears in
either the objective function or one of theconstraint functions it must appear only as a power term with an exponent of 1possibly multiplied by a constant
LP Problems in General
Units of each term in a constraint must be the same as the RHSUnits of each term in the objective function must be the same as ZUnits between constraints do not have to be the sameLP problem can have a mixture of constraint types
No term in the objective function or in anyof the constraints can contain products ofthe decision variables
The coefficients of the decision variables inthe objective function and each constraintare constant
The decision variables are permitted toassume fractional as well as integer values
Examples of lpp
We are already familiar with the graphical representation of equations and inequations here we describe the application of linear equations and inequations in solving different kinds of problems The examples are stated belowExample 1
Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum
Step1 we have to choose the positive two numbers Let the 2 positive numbers be x and y this x and y are decision variables
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
INTRODUCTION TO LINEAR PROGRAMMING
Today many of the resources needed as inputs to operations are in limited supplyOperations managers must understand the impact of this situation on meeting their objectivesLinear programming (LP) is one way that operations managers can determine how best to allocate their scarce resources
Linear programmingWe use graphs as useful modeling abstractions to help us develop computational solutions for a wide variety of problemsA linear program is simply another modeling abstraction (tool in your toolbox)Developing routines that solve general linear programs allows us to embed them in sophisticated algorithmic solutions to difficult problems (eg like we did for TSP)The cutting edge algorithmic solutions to many problems use the ideas from mathematical programming linear programming forming the foundation
BASIC CONCEPT OF LP PROGRAM
Objective functionConstraintsOptimizationSolution of lppFeasible solutionOptimal solution
LP PROBLEMS IN OM PRODUCT MIX
ObjectiveTo select the mix of products or services
that results in maximum profits for the planning period
Decision VariablesHow much to produce and market of each
product or service for the planning period
ConstraintsMaximum amount of each product or
service demanded Minimum amount of product or service policy will allow Maximum amount of resources available
Objective function the linear functions which is to be optimized ie maximized or minimized this may be expressed in linear expressionSolution of Lpp
The set of all the values of the variable x1x2helliphellipxn which satisy the constraints is called the solution of Lpp Feasible solution The set of all the values of the variable x1x2helliphellipxn which satisy the constraints and also the non negative conditions is called the feasible solution of lpp
Recognizing LP Problems
Characteristics of LP Problems in OMA well-defined single objective must be statedThere must be alternative courses of actionThe total achievement of the objective must be constrained by scarce resources or other restraintsThe objective and each of the constraints must be expressed as linear mathematical functions
Steps in Formulating LP Problems
Define the objective (min or max)Define the decision variables (positive) Write the mathematical function for the objectiveWrite a 1- or 2-word description of each constraintWrite the right-hand side (RHS) of each constraintWrite lt = or gt for each constraintWrite the decision variables on LHS of each constraintWrite the coefficient for each decision variable in each constraint
Linear Programming
An optimization problem is said to be a linear program if it satisfied the following properties There is a unique objective function Whenever a decision variable appears in
either the objective function or one of theconstraint functions it must appear only as a power term with an exponent of 1possibly multiplied by a constant
LP Problems in General
Units of each term in a constraint must be the same as the RHSUnits of each term in the objective function must be the same as ZUnits between constraints do not have to be the sameLP problem can have a mixture of constraint types
No term in the objective function or in anyof the constraints can contain products ofthe decision variables
The coefficients of the decision variables inthe objective function and each constraintare constant
The decision variables are permitted toassume fractional as well as integer values
Examples of lpp
We are already familiar with the graphical representation of equations and inequations here we describe the application of linear equations and inequations in solving different kinds of problems The examples are stated belowExample 1
Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum
Step1 we have to choose the positive two numbers Let the 2 positive numbers be x and y this x and y are decision variables
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
Linear programmingWe use graphs as useful modeling abstractions to help us develop computational solutions for a wide variety of problemsA linear program is simply another modeling abstraction (tool in your toolbox)Developing routines that solve general linear programs allows us to embed them in sophisticated algorithmic solutions to difficult problems (eg like we did for TSP)The cutting edge algorithmic solutions to many problems use the ideas from mathematical programming linear programming forming the foundation
BASIC CONCEPT OF LP PROGRAM
Objective functionConstraintsOptimizationSolution of lppFeasible solutionOptimal solution
LP PROBLEMS IN OM PRODUCT MIX
ObjectiveTo select the mix of products or services
that results in maximum profits for the planning period
Decision VariablesHow much to produce and market of each
product or service for the planning period
ConstraintsMaximum amount of each product or
service demanded Minimum amount of product or service policy will allow Maximum amount of resources available
Objective function the linear functions which is to be optimized ie maximized or minimized this may be expressed in linear expressionSolution of Lpp
The set of all the values of the variable x1x2helliphellipxn which satisy the constraints is called the solution of Lpp Feasible solution The set of all the values of the variable x1x2helliphellipxn which satisy the constraints and also the non negative conditions is called the feasible solution of lpp
Recognizing LP Problems
Characteristics of LP Problems in OMA well-defined single objective must be statedThere must be alternative courses of actionThe total achievement of the objective must be constrained by scarce resources or other restraintsThe objective and each of the constraints must be expressed as linear mathematical functions
Steps in Formulating LP Problems
Define the objective (min or max)Define the decision variables (positive) Write the mathematical function for the objectiveWrite a 1- or 2-word description of each constraintWrite the right-hand side (RHS) of each constraintWrite lt = or gt for each constraintWrite the decision variables on LHS of each constraintWrite the coefficient for each decision variable in each constraint
Linear Programming
An optimization problem is said to be a linear program if it satisfied the following properties There is a unique objective function Whenever a decision variable appears in
either the objective function or one of theconstraint functions it must appear only as a power term with an exponent of 1possibly multiplied by a constant
LP Problems in General
Units of each term in a constraint must be the same as the RHSUnits of each term in the objective function must be the same as ZUnits between constraints do not have to be the sameLP problem can have a mixture of constraint types
No term in the objective function or in anyof the constraints can contain products ofthe decision variables
The coefficients of the decision variables inthe objective function and each constraintare constant
The decision variables are permitted toassume fractional as well as integer values
Examples of lpp
We are already familiar with the graphical representation of equations and inequations here we describe the application of linear equations and inequations in solving different kinds of problems The examples are stated belowExample 1
Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum
Step1 we have to choose the positive two numbers Let the 2 positive numbers be x and y this x and y are decision variables
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
BASIC CONCEPT OF LP PROGRAM
Objective functionConstraintsOptimizationSolution of lppFeasible solutionOptimal solution
LP PROBLEMS IN OM PRODUCT MIX
ObjectiveTo select the mix of products or services
that results in maximum profits for the planning period
Decision VariablesHow much to produce and market of each
product or service for the planning period
ConstraintsMaximum amount of each product or
service demanded Minimum amount of product or service policy will allow Maximum amount of resources available
Objective function the linear functions which is to be optimized ie maximized or minimized this may be expressed in linear expressionSolution of Lpp
The set of all the values of the variable x1x2helliphellipxn which satisy the constraints is called the solution of Lpp Feasible solution The set of all the values of the variable x1x2helliphellipxn which satisy the constraints and also the non negative conditions is called the feasible solution of lpp
Recognizing LP Problems
Characteristics of LP Problems in OMA well-defined single objective must be statedThere must be alternative courses of actionThe total achievement of the objective must be constrained by scarce resources or other restraintsThe objective and each of the constraints must be expressed as linear mathematical functions
Steps in Formulating LP Problems
Define the objective (min or max)Define the decision variables (positive) Write the mathematical function for the objectiveWrite a 1- or 2-word description of each constraintWrite the right-hand side (RHS) of each constraintWrite lt = or gt for each constraintWrite the decision variables on LHS of each constraintWrite the coefficient for each decision variable in each constraint
Linear Programming
An optimization problem is said to be a linear program if it satisfied the following properties There is a unique objective function Whenever a decision variable appears in
either the objective function or one of theconstraint functions it must appear only as a power term with an exponent of 1possibly multiplied by a constant
LP Problems in General
Units of each term in a constraint must be the same as the RHSUnits of each term in the objective function must be the same as ZUnits between constraints do not have to be the sameLP problem can have a mixture of constraint types
No term in the objective function or in anyof the constraints can contain products ofthe decision variables
The coefficients of the decision variables inthe objective function and each constraintare constant
The decision variables are permitted toassume fractional as well as integer values
Examples of lpp
We are already familiar with the graphical representation of equations and inequations here we describe the application of linear equations and inequations in solving different kinds of problems The examples are stated belowExample 1
Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum
Step1 we have to choose the positive two numbers Let the 2 positive numbers be x and y this x and y are decision variables
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
LP PROBLEMS IN OM PRODUCT MIX
ObjectiveTo select the mix of products or services
that results in maximum profits for the planning period
Decision VariablesHow much to produce and market of each
product or service for the planning period
ConstraintsMaximum amount of each product or
service demanded Minimum amount of product or service policy will allow Maximum amount of resources available
Objective function the linear functions which is to be optimized ie maximized or minimized this may be expressed in linear expressionSolution of Lpp
The set of all the values of the variable x1x2helliphellipxn which satisy the constraints is called the solution of Lpp Feasible solution The set of all the values of the variable x1x2helliphellipxn which satisy the constraints and also the non negative conditions is called the feasible solution of lpp
Recognizing LP Problems
Characteristics of LP Problems in OMA well-defined single objective must be statedThere must be alternative courses of actionThe total achievement of the objective must be constrained by scarce resources or other restraintsThe objective and each of the constraints must be expressed as linear mathematical functions
Steps in Formulating LP Problems
Define the objective (min or max)Define the decision variables (positive) Write the mathematical function for the objectiveWrite a 1- or 2-word description of each constraintWrite the right-hand side (RHS) of each constraintWrite lt = or gt for each constraintWrite the decision variables on LHS of each constraintWrite the coefficient for each decision variable in each constraint
Linear Programming
An optimization problem is said to be a linear program if it satisfied the following properties There is a unique objective function Whenever a decision variable appears in
either the objective function or one of theconstraint functions it must appear only as a power term with an exponent of 1possibly multiplied by a constant
LP Problems in General
Units of each term in a constraint must be the same as the RHSUnits of each term in the objective function must be the same as ZUnits between constraints do not have to be the sameLP problem can have a mixture of constraint types
No term in the objective function or in anyof the constraints can contain products ofthe decision variables
The coefficients of the decision variables inthe objective function and each constraintare constant
The decision variables are permitted toassume fractional as well as integer values
Examples of lpp
We are already familiar with the graphical representation of equations and inequations here we describe the application of linear equations and inequations in solving different kinds of problems The examples are stated belowExample 1
Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum
Step1 we have to choose the positive two numbers Let the 2 positive numbers be x and y this x and y are decision variables
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
Objective function the linear functions which is to be optimized ie maximized or minimized this may be expressed in linear expressionSolution of Lpp
The set of all the values of the variable x1x2helliphellipxn which satisy the constraints is called the solution of Lpp Feasible solution The set of all the values of the variable x1x2helliphellipxn which satisy the constraints and also the non negative conditions is called the feasible solution of lpp
Recognizing LP Problems
Characteristics of LP Problems in OMA well-defined single objective must be statedThere must be alternative courses of actionThe total achievement of the objective must be constrained by scarce resources or other restraintsThe objective and each of the constraints must be expressed as linear mathematical functions
Steps in Formulating LP Problems
Define the objective (min or max)Define the decision variables (positive) Write the mathematical function for the objectiveWrite a 1- or 2-word description of each constraintWrite the right-hand side (RHS) of each constraintWrite lt = or gt for each constraintWrite the decision variables on LHS of each constraintWrite the coefficient for each decision variable in each constraint
Linear Programming
An optimization problem is said to be a linear program if it satisfied the following properties There is a unique objective function Whenever a decision variable appears in
either the objective function or one of theconstraint functions it must appear only as a power term with an exponent of 1possibly multiplied by a constant
LP Problems in General
Units of each term in a constraint must be the same as the RHSUnits of each term in the objective function must be the same as ZUnits between constraints do not have to be the sameLP problem can have a mixture of constraint types
No term in the objective function or in anyof the constraints can contain products ofthe decision variables
The coefficients of the decision variables inthe objective function and each constraintare constant
The decision variables are permitted toassume fractional as well as integer values
Examples of lpp
We are already familiar with the graphical representation of equations and inequations here we describe the application of linear equations and inequations in solving different kinds of problems The examples are stated belowExample 1
Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum
Step1 we have to choose the positive two numbers Let the 2 positive numbers be x and y this x and y are decision variables
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
Recognizing LP Problems
Characteristics of LP Problems in OMA well-defined single objective must be statedThere must be alternative courses of actionThe total achievement of the objective must be constrained by scarce resources or other restraintsThe objective and each of the constraints must be expressed as linear mathematical functions
Steps in Formulating LP Problems
Define the objective (min or max)Define the decision variables (positive) Write the mathematical function for the objectiveWrite a 1- or 2-word description of each constraintWrite the right-hand side (RHS) of each constraintWrite lt = or gt for each constraintWrite the decision variables on LHS of each constraintWrite the coefficient for each decision variable in each constraint
Linear Programming
An optimization problem is said to be a linear program if it satisfied the following properties There is a unique objective function Whenever a decision variable appears in
either the objective function or one of theconstraint functions it must appear only as a power term with an exponent of 1possibly multiplied by a constant
LP Problems in General
Units of each term in a constraint must be the same as the RHSUnits of each term in the objective function must be the same as ZUnits between constraints do not have to be the sameLP problem can have a mixture of constraint types
No term in the objective function or in anyof the constraints can contain products ofthe decision variables
The coefficients of the decision variables inthe objective function and each constraintare constant
The decision variables are permitted toassume fractional as well as integer values
Examples of lpp
We are already familiar with the graphical representation of equations and inequations here we describe the application of linear equations and inequations in solving different kinds of problems The examples are stated belowExample 1
Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum
Step1 we have to choose the positive two numbers Let the 2 positive numbers be x and y this x and y are decision variables
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
Steps in Formulating LP Problems
Define the objective (min or max)Define the decision variables (positive) Write the mathematical function for the objectiveWrite a 1- or 2-word description of each constraintWrite the right-hand side (RHS) of each constraintWrite lt = or gt for each constraintWrite the decision variables on LHS of each constraintWrite the coefficient for each decision variable in each constraint
Linear Programming
An optimization problem is said to be a linear program if it satisfied the following properties There is a unique objective function Whenever a decision variable appears in
either the objective function or one of theconstraint functions it must appear only as a power term with an exponent of 1possibly multiplied by a constant
LP Problems in General
Units of each term in a constraint must be the same as the RHSUnits of each term in the objective function must be the same as ZUnits between constraints do not have to be the sameLP problem can have a mixture of constraint types
No term in the objective function or in anyof the constraints can contain products ofthe decision variables
The coefficients of the decision variables inthe objective function and each constraintare constant
The decision variables are permitted toassume fractional as well as integer values
Examples of lpp
We are already familiar with the graphical representation of equations and inequations here we describe the application of linear equations and inequations in solving different kinds of problems The examples are stated belowExample 1
Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum
Step1 we have to choose the positive two numbers Let the 2 positive numbers be x and y this x and y are decision variables
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
Linear Programming
An optimization problem is said to be a linear program if it satisfied the following properties There is a unique objective function Whenever a decision variable appears in
either the objective function or one of theconstraint functions it must appear only as a power term with an exponent of 1possibly multiplied by a constant
LP Problems in General
Units of each term in a constraint must be the same as the RHSUnits of each term in the objective function must be the same as ZUnits between constraints do not have to be the sameLP problem can have a mixture of constraint types
No term in the objective function or in anyof the constraints can contain products ofthe decision variables
The coefficients of the decision variables inthe objective function and each constraintare constant
The decision variables are permitted toassume fractional as well as integer values
Examples of lpp
We are already familiar with the graphical representation of equations and inequations here we describe the application of linear equations and inequations in solving different kinds of problems The examples are stated belowExample 1
Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum
Step1 we have to choose the positive two numbers Let the 2 positive numbers be x and y this x and y are decision variables
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
LP Problems in General
Units of each term in a constraint must be the same as the RHSUnits of each term in the objective function must be the same as ZUnits between constraints do not have to be the sameLP problem can have a mixture of constraint types
No term in the objective function or in anyof the constraints can contain products ofthe decision variables
The coefficients of the decision variables inthe objective function and each constraintare constant
The decision variables are permitted toassume fractional as well as integer values
Examples of lpp
We are already familiar with the graphical representation of equations and inequations here we describe the application of linear equations and inequations in solving different kinds of problems The examples are stated belowExample 1
Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum
Step1 we have to choose the positive two numbers Let the 2 positive numbers be x and y this x and y are decision variables
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
No term in the objective function or in anyof the constraints can contain products ofthe decision variables
The coefficients of the decision variables inthe objective function and each constraintare constant
The decision variables are permitted toassume fractional as well as integer values
Examples of lpp
We are already familiar with the graphical representation of equations and inequations here we describe the application of linear equations and inequations in solving different kinds of problems The examples are stated belowExample 1
Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum
Step1 we have to choose the positive two numbers Let the 2 positive numbers be x and y this x and y are decision variables
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
Examples of lpp
We are already familiar with the graphical representation of equations and inequations here we describe the application of linear equations and inequations in solving different kinds of problems The examples are stated belowExample 1
Find two positive numbers such that whose sum is atleast 15 and whose difference is at the most 7 such that the product is maximum
Step1 we have to choose the positive two numbers Let the 2 positive numbers be x and y this x and y are decision variables
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
Step 2 our objective is to minimize the product x y
Let z=xy we have to maximize zStep3 we have the following conditions on the variables as x and ystep 4 x+ygt=15 x-ylt=7 xygt0 as the linear constraints the mathemetical constraint of this equation is to maximize the objective function z=xy
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
PROBLEMS1 A producer wants to maximise revenues producing two goods x
1and x2 in the market Market prices of goods are 10 and 5 respectively Production of x 1and x2 requires 25 and 10 units of skilled labour and total endowment of skilled labour is1000 Similarly production of x1 and x2 also requires 20 and 50 units of unskilled labour and whose total endowment is 1500 How much should this firm produce x 1and x 2 in order to maximise the total revenue
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
SOLUTION
Max R =10x1 + 5x2
Subject toSkilled labour constraint 25x1 +10x2lt=1000 Unskilled labour constraint 20x1 +50x2 lt=1500 Non-negativity constraints x1 x2 gt=0
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
Pounds of each alloy needed per frameAluminum Alloy
Steel Alloy Deluxe 2 3 deluxe
Professional 4 2
Example LP Formulation
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
Define the objectiveMaximize total weekly profitDefine the decision variables
x1 = number of Deluxe frames produced weeklyx2 = number of Professional frames produced weeklyWrite the mathematical objective function
Max Z = 10x1 + 15x2
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
Write a one- or two-word description of each constraint
Aluminum availableSteel availableWrite the right-hand side of each constraint100 80Write lt = gt for each constraintlt 100lt 80
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
Write all the decision variables on the left-hand side of each constraint
x1 x2 lt 100x1 x2 lt 80Write the coefficient for each decision in each constraint+ 2x1 + 4x2 lt 100+ 3x1 + 2x2 lt 80
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
LP in Final FormMax Z = 10x1 + 15x2
Subject To2x1 + 4x2 lt 100 ( aluminum constraint)3x1 + 2x2 lt 80 ( steel constraint)x1 x2 gt 0 (non-negativity constraints
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
x
20
15
0
(55)
0
152
203
subject to
920 max
yx
yx
yx
yxyx
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
20
15
0
(55)
Examplegraphical method
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
Examplegraphical method
0
152
203
subject to
920 max
yx
yx
yx
yxyx
x
y
20
15
0
(55)
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
THE SIMPLEX METHOD
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
So far we find an optimal point by searchingamong feasible intersection points1048698 The search can be improved by starting withan initial feasible point and moving to aldquobetterrdquo solution until an optimal one is found1048698 The simplex method incorporates bothoptimality and feasibility tests to find theoptimal solution(s) if one exists
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
An optimality test shows whether anintersection point corresponds to a value of theobjective function better than the best valuefound so far1048698 A feasibility test determines whether theproposed intersection point is feasible1048698 The decision and slack variables are separatedinto two nonoverlapping sets which we callthe independent and dependent sets
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
THE SIMPLEX METHOD
Transform Linear Program into a systemof linear equations using slack variables
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
0
152
203
subject to
920 max
yx
yx
yx
yxyx
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
0100920
1501012
2000113
0
0920
152
203
21
2
1
ssyx
Pyx
syx
syx
THE SIMPLEX METHOD
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
Start from the vertex (x=0 y=0)Move to the next vertex that increases profit as much as possible
0100920
1501012
2000113
THE SIMPLEX METHOD
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
0100920
1501012
2000113
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient
has larger magnitude than the y coefficient)bull Compute check ratios to find pivot row (smallest
ratio)
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
Basic Idea Start from a vertex (x=0 y=0) Move to next vertex that increases profit as much as possible
bull At (00) P = 0bull Increasing x can increase P the most (x coefficient has
larger magnitude than the y coefficient)bull Compute check ratios to findpivot row (smallest ratio)bull Pivot around the element inboth pivot column and row
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
0100920
1501012
2000113
x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
1
203
1
1
syx
syx
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
0100920
1501012
3200031311x y s1 s2 P RHS
Pivoting means solve for that variableThen substitute into the other equations
3
20
3
1
3
11 syx
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
340010320370
350132310
3200031311
x y s1 s2 P RHS
3
20
3
1
3
11 syx
Pivoting means solve for that variableThen substitute into the other equations
THE END
THE END