chap02-LinearProgramming

1Chapter 2 - Linear Programming: Model

Formulation & Graphical Solution

Chapter 2

Linear Programming: Model Formulation and Graphical Solution

Introduction to Management Science

9th Edition

by Bernard W. Taylor III

© 2007 Pearson Education

Chapter Topics

Chapter 2 - Linear Programming: Model Formulation & Graphical Solution

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Model Formulation

A Maximization Model Example

Graphical Solutions of Linear Programming Models

A Minimization Model Example

Irregular Types of Linear Programming Models

Characteristics of Linear Programming Problems


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Objectives of business decisions frequently involve maximizing profit or minimizing costs.

Linear programming is an analytical technique in which linear algebraic relationships represent a firm’s decisions, given a business objective, and resource constraints.

Steps in application:

Identify problem as solvable by linear programming.

Formulate a mathematical model of the unstructured problem.

Solve the model.

Implementation

Linear Programming: An Overview

• Decision variables - mathematical symbols representing levels of activity of a firm.

• Objective function - a linear mathematical relationship describing an objective of the firm, in terms of decision variables - this function is to be maximized or minimized.

• Constraints – requirements or restrictions placed on the firm by the operating environment, stated in linear relationships of the decision variables.

• Parameters - numerical coefficients and constants used in the objective function and constraints.


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Model ComponentsModel Components


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Summary of Model Formulation Steps

Step 1 : Clearly define the decision variables

Step 2 : Construct the objective function

Step 3 : Formulate the constraints

• Product mix problem - Beaver Creek Pottery Company

• How many bowls and mugs should be produced to maximize profits given labor and materials constraints?

• Product resource requirements and unit profit:


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Resource Requirements

Product Labor

(hr/unit) Clay

(lb/unit) Profit

($/unit) Bowl 1 4 40 Mug 2 3 50

LP Model FormulationA Maximization Example (1 of 3)


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Resource 40 hrs of labor per dayAvailability: 120 lbs of clay

Decision x1 = number of bowls to produce per day

Variables: x2 = number of mugs to produce per day

Objective Maximize Z = $40x1 + $50x2

Function: Where Z = profit per day

Resource 1x1 + 2x2 40 hours of laborConstraints: 4x1 + 3x2 120 pounds of clay

Non-Negativity x1 0; x2 0 Constraints:


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Complete Linear Programming Model:

Maximize Z = $40x1 + $50x2

subject to: 1x1 + 2x2 40

4x2 + 3x2 120

x1, x2 0


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A feasible solution does not violate any of the constraints:

Example x1 = 5 bowls

x2 = 10 mugs

Z = $40x1 + $50x2 = $700

Labor constraint check:

1(5) + 2(10) = 25 < 40 hours, within constraint

Clay constraint check:

4(5) + 3(10) = 70 < 120 pounds, within constraint

Feasible SolutionsFeasible Solutions


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An An infeasible solutioninfeasible solution violates at least one of the violates at least one of the constraints:constraints:

Example xExample x11 = 10 bowls = 10 bowls

xx22 = 20 mugs = 20 mugs

Z = $1400Z = $1400

Labor constraint check:Labor constraint check:

1(10) + 2(20) = 50 > 40 hours, violates constraint1(10) + 2(20) = 50 > 40 hours, violates constraint

Infeasible Solutions

• Graphical solution is limited to linear programming models containing only two decision variables (can be used with three variables but only with great difficulty).

• Graphical methods provide visualization of how a solution for a linear programming problem is obtained.


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Graphical Solution of LP Models


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Coordinate AxesGraphical Solution of Maximization Model (1 of 12)


subject to: 1x1 + 2x2 40 4x2 + 3x2 120

x1, x2 0

Figure 2.2 Coordinates for Graphical Analysis


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Labor ConstraintGraphical Solution of Maximization Model (2 of 12)


subject to: 1x1 + 2x2 40 4x2 + 3x2 120

x1, x2 0

Figure 2.3 Graph of Labor Constraint


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Labor Constraint AreaGraphical Solution of Maximization Model (3 of 12)


subject to: 1x1 + 2x2 40 4x2 + 3x2 120

x1, x2 0

Figure 2.4 Labor Constraint Area


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Clay Constraint AreaGraphical Solution of Maximization Model (4 of 12)


subject to: 1x1 + 2x2 40 4x2 + 3x2 120

x1, x2 0

Figure 2.5 Clay Constraint Area


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Both ConstraintsGraphical Solution of Maximization Model (5 of 12)


subject to: 1x1 + 2x2 40 4x2 + 3x2 120

x1, x2 0

Figure 2.6 Graph of Both Model Constraints


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Feasible Solution AreaGraphical Solution of Maximization Model (6 of 12)


subject to: 1x1 + 2x2 40 4x2 + 3x2 120

x1, x2 0

Figure 2.7 Feasible Solution Area


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Objective Function Solution = $800Graphical Solution of Maximization Model (7 of 12)


subject to: 1x1 + 2x2 40 4x2 + 3x2 120

x1, x2 0

Figure 2.8 Objection Function Line for Z = $800


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Alternative Objective Function Solution LinesGraphical Solution of Maximization Model (8 of 12)


subject to: 1x1 + 2x2 40 4x2 + 3x2 120

x1, x2 0

Figure 2.9 Alternative Objective Function Lines


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Optimal SolutionGraphical Solution of Maximization Model (9 of 12)


subject to: 1x1 + 2x2 40 4x2 + 3x2 120

x1, x2 0

Figure 2.10 Identification of Optimal Solution


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Optimal Solution CoordinatesGraphical Solution of Maximization Model (10 of 12)


subject to: 1x1 + 2x2 40 4x2 + 3x2 120

x1, x2 0

Figure 2.11 Optimal Solution Coordinates


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Extreme (Corner) Point SolutionsGraphical Solution of Maximization Model (11 of 12)


subject to: 1x1 + 2x2 40 4x2 + 3x2 120

x1, x2 0

Figure 2.12 Solutions at All Corner Points


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Optimal Solution for New Objective FunctionGraphical Solution of Maximization Model (12 of 12)


subject to: 1x1 + 2x2 40 4x2 + 3x2 120

x1, x2 0

Figure 2.13 Optimal Solution with Z = 70x1 + 20x2


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LP Model FormulationA Minimization Example (1 of 7)

Chemical Contribution

Brand Nitrogen (lb/bag)

Phosphate (lb/bag)

Super-gro 2 4

Crop-quick 4 3

Two brands of fertilizer available - Super-Gro, Crop-Quick.Two brands of fertilizer available - Super-Gro, Crop-Quick.

Field requires at least 16 pounds of nitrogen and 24 pounds Field requires at least 16 pounds of nitrogen and 24 pounds of phosphate.of phosphate.

Super-Gro costs $6 per bag, Crop-Quick $3 per bag.Super-Gro costs $6 per bag, Crop-Quick $3 per bag.

Problem: How much of each brand to purchase to minimize Problem: How much of each brand to purchase to minimize total cost of fertilizer given following data ?total cost of fertilizer given following data ?


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LP Model FormulationA Minimization Example (2 of 7)

Decision Variables: x1 = bags of Super-Gro

x2 = bags of Crop-Quick

The Objective Function:Minimize Z = $6x1 + 3x2

Where: $6x1 = cost of bags of Super-Gro $3x2 = cost of bags of Crop-Quick

Model Constraints:2x1 + 4x2 16 lb (nitrogen constraint)4x1 + 3x2 24 lb (phosphate constraint)x1, x2 0 (non-negativity constraint)


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LP Model Formulation and Constraint GraphA Minimization Example (3 of 7)

Minimize Z = $6x1 + $3x2

subject to: 2x1 + 4x2 16 4x2 + 3x2 24

x1, x2 0

Figure 2.16 Graph of Both Model Constraints


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Feasible Solution AreaA Minimization Example (4 of 7)


subject to: 2x1 + 4x2 16 4x2 + 3x2 24

x1, x2 0

Figure 2.17 Feasible Solution Area


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Optimal Solution PointA Minimization Example (5 of 7)


subject to: 2x1 + 4x2 16 4x2 + 3x2 24

x1, x2 0

Figure 2.18 Optimum Solution Point

• For some linear programming models, the general rules do not apply.

• Special types of problems include those with:

– Multiple optimal solutions

– Infeasible solutions

– Unbounded solutions


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Irregular Types of Linear Programming ProblemsIrregular Types of Linear Programming Problems


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Objective function is parallel to to a constraint line.

Maximize Z=$40x1 + 30x2

subject to: 1x1 + 2x2 40 4x2 + 3x2 120 x1, x2 0Where:x1 = number of bowlsx2 = number of mugs

Figure 2.20 Example with Multiple Optimal Solutions

Multiple Optimal SolutionsBeaver Creek Pottery Example


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An Infeasible ProblemAn Infeasible Problem

Every possible solution violates at least one constraint:

Maximize Z = 5x1 + 3x2

subject to: 4x1 + 2x2 8 x1 4 x2 6 x1, x2 0

Figure 2.21 Graph of an Infeasible Problem


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Value of objective function increases indefinitely:


subject to: x1 4 x2 2 x1, x2 0

An Unbounded ProblemAn Unbounded Problem

Figure 2.22 Graph of an Unbounded Problem

• A linear programming problem requires a decision - a choice amongst alternative courses of action.

• The decision is represented in the model by decision variables.

• The problem encompasses a goal, expressed as an objective function, that the decision maker wants to achieve.

• Constraints exist that limit the extent of achievement of the objective.

• The objective and constraints must be definable by linear mathematical functional relationships.


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Characteristics of Linear Programming ProblemsCharacteristics of Linear Programming Problems

• Proportionality - The rate of change (slope) of the objective function and constraint equations is constant.

• Additively - Terms in the objective function and constraint equations must be additive.

• Divisibility -Decision variables can take on any fractional value and are therefore continuous as opposed to integer in nature.

• Certainty - Values of all the model parameters are assumed to be known with certainty (non-probabilistic).


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Properties of Linear Programming ModelsProperties of Linear Programming Models


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Problem StatementExample Problem No. 1 (1 of 3)

Hot dog mixture in 1000-pound batches.

Two ingredients, chicken ($3/lb) and beef ($5/lb).

Recipe requirements:

at least 500 pounds of chicken

at least 200 pounds of beef

Ratio of chicken to beef must be at least 2 to 1.

Determine optimal mixture of ingredients that will minimize costs.


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Step 1:

Identify decision variables.

x1 = lb of chicken in mixture (1000 lb.)

x2 = lb of beef in mixture (1000 lb.)

Step 2:

Formulate the objective function.


where Z = cost per 1,000-lb batch $3x1 = cost of chicken $5x2 = cost of beef

SolutionExample Problem No. 1 (2 of 3)


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SolutionExample Problem No. 1 (3 of 3)

Step 3:

Establish Model Constraints x1 + x2 = 1,000 lb x1 500 lb of chicken x2 200 lb of beef x1/x2 2/1 or x1 - 2x2 0 x1, x2 0

The Model: Minimize Z = $3x1 + 5x2

subject to: x1 + x2 = 1,000 lb x1 50 x2 200 x1 - 2x2 0 x1,x2 0

Example Problem No. 2 (1 of 3)


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Solve the following model graphically:


subject to: x1 + 2x2 10

6x1 + 6x2 36

x1 4

x1, x2 0

Step 1: Plot the constraints as equations

Figure 2.23 Constraint Equations



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6x1 + 6x2 36

x1 4

x1, x2 0

Step 2: Determine the feasible solution space

Figure 2.24 Feasible Solution Space and Extreme Points


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6x1 + 6x2 36

x1 4

x1, x2 0

Step 3 and 4: Determine the solution points and optimal solution

Figure 2.25 Optimal Solution Point

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