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Linear Programming
Chapter 19
Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.
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
19-3
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
19-4
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
19-5
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
19-6
Computer Solutions
LO 19.4
19-7
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
19-8
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
19-9
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
19-10
Computer Solutions
LO 19.4
19-11
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
19-12
Answer Report
LO 19.4
19-13
Sensitivity Report
LO 19.5
19-14
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
19-15
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