McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved.

6S6S

Linear Programming

6S-2

Learning ObjectivesLearning Objectives

Describe the type of problem that would lend itself to solution using linear programming

Formulate a linear programming model from a description of a problem

Solve linear programming problems using the graphical method

Interpret computer solutions of linear programming problems

Do sensitivity analysis on the solution of a linear programming problem

6S-3

Used to obtain optimal solutions to problems that involve restrictions or limitations, such as: Materials Budgets Labor Machine time

Linear ProgrammingLinear Programming

6S-4

Linear programming (LP) techniques consist of a sequence of steps that will lead to an optimal solution to problems, in cases where an optimum exists

Linear ProgrammingLinear Programming

6S-5

Objective Function: mathematical statement of profit or cost for a given solution

Decision variables: amounts of either inputs or outputs

Feasible solution space: the set of all feasible combinations of decision variables as defined by the constraints

Constraints: limitations that restrict the available alternatives

Parameters: numerical values

Linear Programming ModelLinear Programming Model

6S-6

Linearity: the impact of decision variables is linear in constraints and objective function

Divisibility: noninteger values of decision variables are acceptable

Certainty: values of parameters are known and constant

Nonnegativity: negative values of decision variables are unacceptable

Linear Programming Linear Programming AssumptionsAssumptions

6S-7

1.Set up objective function and constraints in mathematical format

2.Plot the constraints

3.Identify the feasible solution space

4.Plot the objective function

5.Determine the optimum solution

Graphical Linear ProgrammingGraphical Linear Programming

Graphical method for finding optimal solutions to two-variable problems

6S-8

Objective - profitMaximize Z=60X1 + 50X2

Subject toAssembly 4X1 + 10X2 <= 100 hours

Inspection 2X1 + 1X2 <= 22 hours

Storage 3X1 + 3X2 <= 39 cubic feet

X1, X2 >= 0

Linear Programming ExampleLinear Programming Example

6S-9

Assembly Constraint4X1 +10X2 = 100

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Linear Programming ExampleLinear Programming Example

6S-10

Linear Programming ExampleLinear Programming Example

Add Inspection Constraint2X1 + 1X2 = 22

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6S-11

Add Storage Constraint3X1 + 3X2 = 39

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AssemblyStorage

Inspection

Feasible solution space

Linear Programming ExampleLinear Programming Example

6S-12

Add Profit Lines

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Z=900

Z=600

Linear Programming ExampleLinear Programming Example

6S-13

The intersection of inspection and storage Solve two equations in two unknowns

2X1 + 1X2 = 223X1 + 3X2 = 39

X1 = 9X2 = 4Z = $740

SolutionSolution

6S-14

Redundant constraint: a constraint that does not form a unique boundary of the feasible solution space

Binding constraint: a constraint that forms the optimal corner point of the feasible solution space

ConstraintsConstraints

6S-15

Solutions and Corner PointsSolutions and Corner Points

Feasible solution space is usually a polygon Solution will be at one of the corner points

Enumeration approach: Substituting the coordinates of each corner point into the objective function to determine which corner point is optimal.

6S-16

Surplus: when the optimal values of decision variables are substituted into a greater than or equal to constraint and the resulting value exceeds the right side value

Slack: when the optimal values of decision variables are substituted into a less than or equal to constraint and the resulting value is less than the right side value

Slack and SurplusSlack and Surplus

6S-17

Simplex: a linear-programming algorithm that can solve problems having more than two decision variables

Simplex MethodSimplex Method

6S-18

Figure 6S.15

MS Excel Worksheet for MS Excel Worksheet for Microcomputer ProblemMicrocomputer Problem

6S-19

Figure 6S.17MS Excel Worksheet SolutionMS Excel Worksheet Solution

6S-20

Range of optimality: the range of values for which the solution quantities of the decision variables remains the same

Range of feasibility: the range of values for the fight-hand side of a constraint over which the shadow price remains the same

Shadow prices: negative values indicating how much a one-unit decrease in the original amount of a constraint would decrease the final value of the objective function

Sensitivity AnalysisSensitivity Analysis