What-If Analysis for Linear Programming

Chapter 5: Hillier and Hillier

Agenda

Define What is What-If Analysis The Importance of What-If Analysis Discuss the Effect of Changing One

Coefficient in the Objective Function Discuss the Effect of Changing Two

Coefficients in the Objective Function Discuss the Effect of Single Changes in a

Constraint

What-If Analysis

Its an analysis that examines what happens to your optimal decision when the assumptions of your model change or are different.– In practice, what-if analysis consists of

changing a particular set of parameters in the objective function or the constraints to see what happens to the optimal solution.

Importance of What-If Analysis

In application, many aspects of the model are based on estimations which cannot be determined precisely.

What-if analysis is used to examine the model to understand how sensitive it is to the parameters in the model.– By knowing how sensitive the model is to the

parameters, you will know which parameters you should spend the most time on trying to estimate correctly.

Importance of What-If Analysis

What-if analysis can be broken-up into two major types.– Sensitivity analysis is when you examine the changes

in the parameters of the model to see what happens to the optimal solution.

– The second type of analysis examines when you look at different assumptions that affect more than just the parameters.

• This is usually done by changing the objective function and constraints in fundamental ways beyond changing the parameters.

Effect of Changing One Coefficient in the Objective Function

By changing a parameter in the objective function, you are affecting the slope of the objective function which has the possibility of changing your optimal solution.

What-if analysis examines how much of a parameter shift can be sustained before changing the optimal solution.

Effect of Changing One Coefficient in the Objective Function Cont.

There are two ways to examine how the change in the parameter will affect the optimal solution.– The first way is to solve the problem with the

new parameter multiple times.– The second method is to use Solver’s

Sensitivity Report to understand what parameter changes would affect the optimal solution.

Solving the Excel Model Multiple Times with Multiple Parameters

Whenever you change a parameter in the model you must tell Excel to resolve the problem by going to Solver.

When doing this type of sensitivity analysis, you want to change the parameters in a way that will allow you to find the key points quickly.– You could use some form of divide and conquer to find

the key changing points.– You could establish a particular interval to help find the

sensitive points.

Solver Table

Solver Table is a tool developed by the textbook authors to solve the model multiple times using different parameters.– The current version on your disk may not be

operable.– How would you go about finding an operable

version?

Solver Table Cont.

Solver Table can change up to two parameters at a time.

In class activity: Build a sensitivity chart for changing the prices of windows.– Examine prices that range from $100 to $1000.– Use the Solver Table to find the price of

windows that changes the optimal solution from 2,6 to 4,3.

Solver’s Sensitivity Report

Solver has another way of finding the parameters that will change the optimal solution.

This is done by using Solver’s Sensitivity Report.

To get the Sensitivity Report, you need to highlight the report after you have used Solver.

Solver’s Sensitivity Report Cont.Microsoft Excel 9.0 Sensitivity ReportWorksheet: [Wyndor.xls]WyndorReport Created: 6/19/2002 9:53:42 AM

Adjustable CellsFinal Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease$C$12 Units Produced Doors 2 0 300 450 300$D$12 Units Produced Windows6 0 500 1E+30 300

ConstraintsFinal Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease$E$7 Plant 1 Used 2 0 4 1E+30 2$E$8 Plant 2 Used 12 150 12 6 6$E$9 Plant 3 Used 18 100 18 6 6

Analyzing the Sensitivity Report

To find the range of the variable before the optimal solution will change, you can use the Solver information in the following way.– The bottom end of the range on the coefficient

is:• Objective coefficient – Allowable Decrease

– The upper end of the range of the coefficient is:• Objective coefficient + Allowable Increase

Analyzing the Sensitivity Report Cont.

In the Wyndor example the price of the doors could increase to $750 or decrease to $0 before the optimal solution would change.

In the Wyndor example the price of the windows could increase an infinite amount or decrease to $200 before the optimal solution would change.

Sensitive Parameters

A parameter is considered a sensitive parameter if small changes lead to a change in the optimal solution.– These parameters are the ones you will focus on

to make sure you have them as close to correct as possible.

Effect of Simultaneous Changes in the Objective Function Coefficients

In many cases, more than one parameter is uncertain.

In this case you would like to know what would happen to your optimal solution when multiple parameters are different than what you expected.

Typically, this analysis can be done by changing multiple parameters at once and seeing what happens to the optimal solution.

Excel Side Note

You can represent a solution set in a single cell by placing an & in front of the variable you want to add.– For example:="("&C12&", "&D12&")” gives

(2, 6) in the same cell.

The 100 Percent Rule for Simultaneous changes in Objective Function Coefficients

This is a rule that tells you how much of each constraint is allowed to change simultaneously before the optimal might change.

This rule says that if the sum of the proportions of parameter change divided by allowable changes in absolute value terms of all the coefficients does not exceed 100%, then the original optimal solution was still be optimal.– If it changes by more than 100%, you cannot be sure.

Calculating a Percentage Change

The percentage change for a value from the 100% rule can be calculated as:

(New Value – Old Value) / Allowable Change For example: when 300 changes to 600 and

the allowable change is 900 you get a proportional change of (600-300)/900 which equals approximately 33.33%.

The Effect of Single Changes in a Constraint

This type of what-if analysis examines what happens to the optimal decision when a constraint coefficient changes.

To examine this issue, you can methodically change the parameter on the coefficient or you could use the Sensitivity Report from Solver.

Solver’s Sensitivity ReportMicrosoft Excel 9.0 Sensitivity ReportWorksheet: [Wyndor.xls]WyndorReport Created: 6/19/2002 9:53:42 AM

Adjustable CellsFinal Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease$C$12 Units Produced Doors 2 0 300 450 300$D$12 Units Produced Windows6 0 500 1E+30 300

ConstraintsFinal Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease$E$7 Plant 1 Used 2 0 4 1E+30 2$E$8 Plant 2 Used 12 150 12 6 6$E$9 Plant 3 Used 18 100 18 6 6

Shadow Price

The shadow price for a constraint is the rate at which the value of the objective function can be increased by increasing the right-hand side of the constraint by a small amount.– This amount tells you what effect a change in

the constraint will have on the objective function.

Allowable Range

The allowable range of a functional constraint is the range of values for this right-hand side over which this constraint’s shadow price remains valid.

The bottom end of the range is calculated by: – Constraint RH Side – Allowable Decrease

The upper end of the range is calculated by: – Constraint RH Side + Allowable Increase