6s-1

McGraw-Hill/IrwinOperations Management, Seventh Edition, by William J. StevensonCopyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.

Linear Programming

Chapter 6 Supplement

Linear Programming

6s-2

McGraw-Hill/IrwinOperations Management, Seventh Edition, by William J. StevensonCopyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.

Linear Programming

• 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 Programming

6s-3

McGraw-Hill/IrwinOperations Management, Seventh Edition, by William J. StevensonCopyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.

Linear Programming

Linear Programming Model

• Objective: the goal of an LP model is maximization or minimization

• 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

6s-4

McGraw-Hill/IrwinOperations Management, Seventh Edition, by William J. StevensonCopyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.

Linear Programming

Linear Programming Assumptions

• 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

6s-5

McGraw-Hill/IrwinOperations Management, Seventh Edition, by William J. StevensonCopyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.

Linear Programming

Graphical Linear Programming

• Set up objective function and constraints in mathematical format

• Plot the constraints

• Identify the feasible solution space

• Plot the objective function

• Determine the optimum solution

6s-6

McGraw-Hill/IrwinOperations Management, Seventh Edition, by William J. StevensonCopyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.

Linear Programming

Linear Programming Example

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PlotConstraint 1X1 + 3X2 = 12

6s-7

McGraw-Hill/IrwinOperations Management, Seventh Edition, by William J. StevensonCopyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.

Linear Programming

Linear Programming Example

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AddConstraint 24X1 + 3X2 = 24

Constraint 1X1 + 3X2 = 12

Solution space

6s-8

McGraw-Hill/IrwinOperations Management, Seventh Edition, by William J. StevensonCopyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.

Linear Programming

Linear Programming Example

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Z x x

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6s-9

McGraw-Hill/IrwinOperations Management, Seventh Edition, by William J. StevensonCopyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.

Linear Programming

MS Excel worksheet for microcomputer problem

6s-10

McGraw-Hill/IrwinOperations Management, Seventh Edition, by William J. StevensonCopyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.

Linear Programming

MS Excel worksheet solution

6s-11

McGraw-Hill/IrwinOperations Management, Seventh Edition, by William J. StevensonCopyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.

Linear Programming

• 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

Constraints

6s-12

McGraw-Hill/IrwinOperations Management, Seventh Edition, by William J. StevensonCopyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.

Linear Programming

Slack and Surplus

• 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

6s-13

McGraw-Hill/IrwinOperations Management, Seventh Edition, by William J. StevensonCopyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.

Linear Programming

Simplex Method

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

• Tableau: One in a series of solutions in tabular form, each corresponding to a corner point of the feasible solution space

6s-14

McGraw-Hill/IrwinOperations Management, Seventh Edition, by William J. StevensonCopyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.

Linear Programming

Sensitivity Analysis

• 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