Linear Programming

Chapter 19

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19-2

You should be able to:LO 19.1 Describe the type of problem that would lend

itself to solution using linear programmingLO 19.2 Formulate a linear programming model from a

description of a problemLO 19.3 Solve simple linear programming problems

using the graphical methodLO 19.4 Interpret computer solutions of linear

programming problemsLO 19.5 Do sensitivity analysis on the solution of a

linear programming problem

Chapter 19: Learning Objectives

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In order for LP models to be used effectively, certain assumptions must be satisfied: Linearity

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

DivisibilityNoninteger values of decision variables are

acceptable Certainty

Values of parameters are known and constant Nonnegativity

Negative values of decision variables are unacceptable

LP Assumptions

LO 19.1

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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

Model Formulation

LO 19.2

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Graphical LPGraphical LP

A method for finding optimal solutions to two-variable problems

Procedure1. Set up the objective function and the constraints in

mathematical format2. Plot the constraints3. Identify the feasible solution space

The set of all feasible combinations of decision variables as defined by the constraints

4. Plot the objective function5. Determine the optimal solution

LO 19.3

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Computer Solutions

LO 19.4

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In Excel 2010, click on Tools on the top of the worksheet, and in that 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

Computer Solutions

LO 19.4

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Add a 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

Computer Solutions

LO 19.4

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For the non-negativity constraints, check the checkbox to Make Unconstrained Variables Non-Negative

Select Simplex LP as the Solving MethodClick Solve

Computer Solutions

LO 19.4

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Computer Solutions

LO 19.4

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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

LO 19.4

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Answer Report

LO 19.4

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Sensitivity Report

LO 19.5

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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 OptimalityThe range of O.F. coefficient values for which the

optimal values of the decision variables will not change

O.F. Coefficient Changes

LO 19.5

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Shadow priceAmount by which the value of the objective

function would change with a one-unit change in the RHS value of a constraint

Range of feasibilityRange of values for the RHS of a constraint over

which the shadow price remains the same

RHS Value Changes

LO 19.5