Operations management by stevenson 9th student slides supplement 6

Supplement 6
Linear Programming

McGraw-Hill/Irwin

Copyright © 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved.


Supplement 6: Learning Objectives
• You should be able to:
– Describe the type of problem that would be appropriately solved
using linear programming
– Formulate a linear programming model
– Solve simple 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

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Linear Programming (LP)
• LP
– A powerful quantitative tool used by operations and other
manages to obtain optimal solutions to problems that involve
restrictions or limitations

• Applications include:
– Establishing locations for emergency equipment and
personnel to minimize response time

– Developing optimal production schedules
– Developing financial plans
– Determining optimal diet plans

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Model Formulation
1. List and define the decision variables (D.V.)
– These typically represent quantities

2. State the objective function (O.F.)
– It includes every D.V. in the model and its contribution to profit
(or cost)

3. List the constraints
– Right hand side value
– Relationship symbol (≤, ≥, or =)
– Left Hand Side
• The variables subject to the constraint, and their coefficients
that indicate how much of the RHS quantity one unit of the
D.V. represents

4. Non-negativity constraints

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Computer Solutions
• MS Excel can be used to solve LP problems using its Solver routine

– Enter the problem into a worksheet
– You must designate the cells where you want the optimal values
for the decision variables

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Computer Solutions
• Click on Tools on the top of the worksheet, and in the
drop-down menu, click on Solver
• Begin by setting the Target Cell
– This is where you want the optimal objective function value to be
recorded
– Highlight Max (if the objective is to maximize)
– The changing cells are the cells where the optimal values of the
decision variables will appear

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Computer Solutions
• Add the constraint, by clicking add
– For each constraint, enter the cell that contains the left-hand
side for the constraint
– Select the appropriate relationship sign (≤, ≥, or =)
– Enter the RHS value or click on the cell containing the value

• Repeat the process for each system constraint

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Computer Solutions
• For the nonnegativity constraints, enter the range of
cells designated for the optimal values of the decision
variables
– Click OK, rather than add
– You will be returned to the Solver menu

• Click on Options
– In the Options menu, Click on Assume Linear Model
– Click OK; you will be returned to the solver menu

• Click Solve

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Solver Results
• The Solver Results menu will appear
– You will have one of two results

• A Solution
– In the Solver Results menu Reports box
» Highlight both Answer and Sensitivity
» Click OK

• An Error message
– Make corrections and click solve

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Solver Results
• Solver will incorporate the optimal values of the decision variables
and the objective function into your original layout on your
worksheets

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Sensitivity Analysis
• Sensitivity Analysis
– Assessing the impact of potential changes to the numerical
values of an LP model
– Three types of changes

• Objective function coefficients
• Right-hand values of constraints
• Constraint coefficients

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O.F. Coefficient Changes
• A change in the value of an O.F. coefficient can cause a
change in the optimal solution of a problem
• Not every change will result in a changed solution
• Range of Optimality
– The range of O.F. coefficient values for which the

optimal values of the decision variables will not change

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Basic and Non-Basic Variables
• Basic variables
– Decision variables whose optimal values are non-zero
• Non-basic variables
– Decision variables whose optimal values are zero
– Reduced cost

• Unless the non-basic variable’s coefficient increases by
more than its reduced cost, it will continue to be nonbasic

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RHS Value Changes
• Shadow price
– Amount by which the value of the objective function would
change with a one-unit change in the RHS value of a constraint
– Range of feasibility

• Range of values for the RHS of a constraint over which
the shadow price remains the same

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Binding vs. Non-binding Constraints
• Non-binding constraints
– have shadow price values that are equal to zero
– have slack (≤ constraint) or surplus (≥ constraint)
– Changing the RHS value of a non-binding constraint (over its range of
feasibility) will have no effect on the optimal solution

• Binding constraint
– have shadow price values that are non-zero
– have no slack (≤ constraint) or surplus (≥ constraint)
– Changing the RHS value of a binding constraint will lead to a change in
the optimal decision values and to a change in the value of the objective
function

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