Chapter 7 - Linear Programming: Computer Solution and Sensitivity Analysis 1 Chapter 7 Linear Programming: Computer Solution and Sensitivity Analysis Introduction

Chapter 7 - Linear Programming: Computer Solution and Sensitivity Analysis 1

Chapter 7

Linear Programming: ComputerSolution and Sensitivity Analysis

Introduction to Management Science

8th Edition

by

Bernard W. Taylor III


Chapter Topics

Computer SolutionMicrosoft Excel’s “Solver” module

QM for Excel

QM for Windows

Sensitivity AnalysisDone “by hand”

Using Microsoft Excel’s “Solver” module

QM for Excel

QM for Windows


Early linear programming used lengthy manual mathematical solution procedure called the Simplex Method (See CD-ROM Module A).

Steps of the Simplex Method have been programmed in software packages designed for linear programming problems.

Many such packages available currently.

Used extensively in business and government.

Text focuses on Excel Spreadsheets and QM for Windows.

Computer Solution


Beaver Creek Pottery ExampleExcel Spreadsheet – Data Screen (1 of 5)

Exhibit 7.1


Beaver Creek Pottery Example“Solver” Parameter Screen (2 of 5)

Exhibit 7.2 Click on “Options” and then on

“Assume Linear Models” !

Click on “Options” and then on

“Assume Linear Models” !


Exhibit 7.3

Beaver Creek Pottery ExampleAdding Model Constraints (3 of 5)


Exhibit 7.4

Beaver Creek Pottery ExampleSolution Screen (4 of 5)

Running “Solver” will iterativelyvary B10 and B11 to reduce theslack in G10 & G11.

Running “Solver” will iterativelyvary B10 and B11 to reduce theslack in G10 & G11.


Beaver Creek Pottery ExampleAnswer Report (5 of 5)

Exhibit 7.5

This report is generated only when

one clicks on “Options” and then on

“assume Linear Models” when setting

up the “Solver” calculation !

This report is generated only when

one clicks on “Options” and then on

“assume Linear Models” when setting

up the “Solver” calculation !


Linear Programming Problem Standard Form

Standard form requires all variables in the constraint equations to appear on the left of the inequality (or equality) and all numeric values to be on the right-hand side.

Examples:

x3 x1 + x2 must be converted to x3 - x1 - x2 0

x1/(x2 + x3) 2 becomes x1 2 (x2 + x3)and then x1 - 2x2 - 2x3 0


Beaver Creek Pottery ExampleQM for Windows – Data Screen (1 of 3)

Exhibit 7.6


Beaver Creek Pottery ExampleModel Solution Screen (2 of 3)

Exhibit 7.7


Beaver Creek Pottery ExampleGraphical Solution Screen (3 of 3)

Exhibit 7.8


Beaver Creek Pottery ExampleSensitivity Analysis (1 of 4)

Sensitivity analysis determines the effect on optimal solutions of changes in parameter values of the objective function and constraint equations.

Changes may be reactions to anticipated uncertainties in the parameters or to new or changed information concerning the model.


Maximize Z = $40x1 + $50x2

subject to: 1x1 + 2x2 40 4x1 + 3x2 120

x1, x2 0

Figure 7.1Optimal Solution Point

Beaver Creek Pottery ExampleSensitivity Analysis (2 of 4)



subject to: 1x1 + 2x2 40 4x1 + 3x2 120

x1, x2 0

Figure 7.2Changing the x1 Objective Function Coefficient

Beaver Creek Pottery ExampleChange x1 Objective Function Coefficient (3 of 4)

Old optimal solution

New optimal solution

Changed from $40Changed from $40



subject to: 1x1 + 2x2 40 4x1 + 3x2 120

x1, x2 0

Figure 7.3Changing the x2 Objective Function Coefficient

Beaver Creek Pottery ExampleChange x2 Objective Function Coefficient (4 of 4)

Changed from $50Changed from $50

Old optimal solution


The sensitivity range for an objective function coefficient is the range of values over which the current optimal solution point will remain optimal.

The sensitivity range for the xi coefficient, i.e., of ci, the coefficient of xi in the objective function, Z:

Z = c1 x1 + c2 x2 + c3 x3 + …

The sensitivity range is the interval within which a single coefficient, one at a time, can be varied without changing the optimal value of the objective function. (Although there do appear multiple alternate optimal values.)

Objective Function CoefficientSensitivity Range (1 of 3)


objective function Z = $40x1 + $50x2 sensitivity range for:

x1: 25 c1 66.67 x2: 30 c2 80

Figure 7.4Determining the Sensitivity Range for c1

Objective Function CoefficientSensitivity Range for c1 and c2 (2 of 3)

Slope = – c1/c2Slope = – c1/c2

– 66.67/50 = – 4/3– 66.67/50 = – 4/3

– 25/50 = – 1/2– 25/50 = – 1/2


Minimize Z = $6x1 + $3x2

subject to: 2x1 + 4x2 164x1 + 3x2 24

x1, x2 0

sensitivity ranges: 4 c1 0 c2 4.5

Objective Function CoefficientFertilizer Cost Minimization Example (3 of 3)

Figure 7.5Fertilizer Cost Minimization Example


Exhibit 7.9

Objective Function Coefficient RangesExcel “Solver” Results Screen (1 of 3)


Exhibit 7.10

Objective Function Coefficient RangesBeaver Creek Example Sensitivity Report (2 of 3)

This report is generated only when one clicks on “Options” and then on “assume Linear Models” when setting up the “Solver” calculation !


Exhibit 7.11

Objective Function Coefficient RangesQM for Windows Sensitivity Range Screen (3 of 3)


Changes in Constraint Quantity ValuesSensitivity Range (1 of 4)

The sensitivity range for a right-hand-side value is the range of values over which the quantity values can change without changing the solution variable mix, including slack variables.


Changes in Constraint Quantity ValuesIncreasing the Labor Constraint (2 of 4)


subject to: 1x1 + 2x2 40 4x1 + 3x2 120

x1, x2 0

Figure 7.6Increasing the Labor Constraint Quantity

Consider the Labor Constraintchanging from q1=40 to q1=60


Changes in Constraint Quantity ValuesSensitivity Range for Labor Constraint (3 of 4)

Sensitivity range for:30 q1 80 hr

Figure 7.7Determining the Sensitivity Range for Labor Quantity

The “maximal” and the “minimal”values of the Labor ConstraintQuantity:

The “maximal” and the “minimal”values of the Labor ConstraintQuantity:


Changes in Constraint Quantity ValuesSensitivity Range for Clay Constraint (4 of 4)

Sensitivity range for: 60 q2 160 lb

Figure 7.8Determining the Sensitivity Range for Clay Quantity

The amount of change in Zfor a unit change in a constraintquantity is that quantity’s dual(shadow) price.

The amount of change in Zfor a unit change in a constraintquantity is that quantity’s dual(shadow) price.


Exhibit 7.12

Constraint Quantity Value Ranges by ComputerExcel Sensitivity Range for Constraints (1 of 2)


Exhibit 7.13

Constraint Quantity Value Ranges by ComputerQM for Windows Sensitivity Range (2 of 2)


Changing individual constraint parameters

One at a time!

Adding new constraints

…or modifying the existing ones by changing the relatione.g., by changing ‘≤’ into ‘=’ to enforce saturation (no slack)

Adding new variables

Makes the problem more complicated, but also more versatile

Other Forms of Sensitivity AnalysisTopics (1 of 4)


Other Forms of Sensitivity AnalysisChanging a Constraint Parameter (2 of 4)


subject to: 1x1 + 2x2 40 4x1 + 3x2 120

x1, x2 0

Figure 7.9Changing the x1 Coefficient in the Labor Constraint


Adding a new constraint to Beaver Creek Model: 0.20x1+ 0.10x2 5 hours for packaging Original solution: 24 bowls, 8 mugs, $1,360 profit

Exhibit 7.13

Other Forms of Sensitivity AnalysisAdding a New Constraint (3 of 4)


Adding a new variable to the Beaver Creek model, x3, a third product, cups

Maximize Z = $40x1 + 50x2 + 30x3

subject to:

x1 + 2x2 + 1.2x3 40 hr of labor

4x1 + 3x2 + 2x3 120 lb of clay

x1, x2, x3 0

Solving model shows that change has no effect on the original solution (i.e., the model is not sensitive to this change).

Other Forms of Sensitivity AnalysisAdding a New Variable (4 of 4)


Defined as the marginal value of one additional unit of resource.

Also defined (earlier) as: “the amount of change in the objective function for a unit change in a constraint quantity.”

That is, by allotting one extra unit of a resource/constraint quantity (denoted qi), the objective function changes by the shadow price or that resource.

The sensitivity range for a constraint quantity value is also the range over which the shadow price is valid.

Shadow Prices (Dual Values)


Maximize Z = $40x1 + $50x2 subject to:

x1 + 2x2 40 hr of labor4x1 + 3x2 120 lb of clay

x1, x2 0

Exhibit 7.14

Excel Sensitivity Report for Beaver Creek PotteryShadow Prices Example (1 of 2)

For every additional hour of labor,the profit will increase by $16.Labor can be increased from 40 to40 + 40 = 80, before the solution(x1,x2) changes.


Excel Sensitivity Report for Beaver Creek PotterySolution Screen (2 of 2)

Exhibit 7.15


Two airplane parts: no.1 and no. 2.

Three manufacturing stages: stamping, drilling, milling.

Decision variables: x1 (number of part no.1 to produce) x2 (number of part no.2 to produce)

Model: Maximize Z = $650x1 + 910x2

subject to: 4x1 + 7.5x2 105 (stamping,hr) 6.2x1 + 4.9x2 90 (drilling, hr) 9.1x1 + 4.1x2 110 (finishing, hr) x1, x2 0

Example ProblemProblem Statement (1 of 3)


Maximize Z = $650x1 + $910x2

subject to: 4x1 + 7.5x2 105 6.2x1 + 4.9x2 90 9.1x1 + 4.1x2 110 x1, x2 0

s1 = 0, s2 = 0, s3 = 11.35 hr

Coefficient sensitivity:485.33 c1 1,151.43

Quantity sensitivity:89.10 q1 137.76

Figure 7.10Graphical Solution

Example ProblemGraphical Solution (2 of 3)


Example ProblemExcel Solution (3 of 3)

Exhibit 7.16


Documents

Chapter 7 - Linear Programming: Computer Solution and Sensitivity Analysis 1 Chapter 7 Linear Programming: Computer Solution and Sensitivity Analysis Introduction