The Theory of the Simplex MethodChapter 6: Hillier and LiebermanDr. Hurley’s AGB 328 Course

Terms to KnowDual, Primal, Primal-Dual Table, Weak

Duality Property, Strong Duality Property, Complementary Solutions Property, Complementary Solutions, Complementary Optimal Solutions Property, Complementary Optimal Solution, Symmetry Property, Duality Theorem, Complementary Basic Solutions Property, Complementary Basic Solution,

Terms to Know Cont.Complementary Slackness

Property, Complementary Slackness, Complementary Optimal Basic Solutions Property, Complementary Optimal Basic Solution, Primal Feasible, Dual Feasible, Sensible-Odd-Bizarre Method, Sensitivity Analysis, Sensitive Parameters, Reduced Cost, Allowable Range

Duality TheoryThe point of duality theory is that

a maximization problem has a corresponding minimization problem and vice versa

The coefficients in the maximization problem can be mapped to a set of coefficients in the minimization problem

Duality Theory—The Two Problems

Subject to: Subject to:

Subject to: Subject to:

Primal Dual

Duality Theory-The CorrespondenceThe coefficients in the objective

function of the primal are the coefficients of the RHS of the constraints of the dual

The RHS coefficients of the constraints of the primal are the coefficients in the objective function of the dual

The coefficients in the functional constraints are the same in the primal as the dual

Duality TheoremRelationship between the primal

and the dual:◦If one problem is bounded and has

feasible solutions, then so does the other

◦If one problem is unbounded with feasible solutions, then the other problem has no feasible solution

◦If one of the problems has no feasible solution, then the other problem is either unbounded or has no feasible solutions

Why Care About the DualSometimes the dual is easier and

quicker to solve than the primal problem

The optimal value of the dual can be seen in the shadow prices of the primal

Economic Interpretation of Duality represents the current

contribution to the objective function of the mix of resources that would be consumed if 1 unit of activity j were utilized

To make the best possible use of the resources, we want to have

Economic Interpretation of Duality Cont.

The yi’s represents the values of the resources bi’s

Hence in the resource constraint of the dual, the value of the xi’s usage of the resources should be greater than or equal to their corresponding value in the primal, i.e., the ci’s

Subject to:

Subject to:

Economic Interpretation of Duality Cont.We should expect that when xj >

0 (j = 1, 2, …, n), then = cj

◦Why?We should also expect that when

xn+i > 0 (i = 1, 2, …, m), then yi = 0◦Why?

Role of Duality Theory in Sensitivity AnalysisDuality theory can help when you

are conducting a sensitivity analysis on your model◦Suppose you have a change in the

coefficient of a non-basic variable in your model You can check the complementary basic

solution in the dual to see if the constraints in the dual are still satisfied If so, then the change in the coefficient will not

change the optimal decision

Role of Duality Theory in Sensitivity Analysis Cont.

◦Suppose you introduce a new variable to the model This is equivalent to introducing a new

constraint in the dual Why?

If the complementary basic solution in the dual satisfies the new constraint, the new variable does not meaningful change the dual and will end up being zero

Essence of Sensitivity AnalysisSensitivity analysis examines

how robust your model and its optimal solution are to changes in the assumptions of your model◦These assumptions may include

what variables are in the model, the value of the coefficient for each variable, the functional form of the objective function and the constraints, etc.

Essence of Sensitivity Analysis Cont.One important aspect in

sensitivity analysis is to identify any sensitive parameters◦The way to determine if a parameter

is sensitive is to ascertain how much it can change before your optimal decision changes

Different Cases to Sensitivity AnalysisCase 1: Changes in bi

Case 2a: Changes in coefficients of non-basic variables

Case 2b: Introduction of a new variable

Case 3: Changes in the coefficient of a basic variable

Case 4: Introduction of a new constraint

A Single Change to a bi or a c jTo see if a single change to the bi

or cj causes a change in your optimal solution, you can examine the allowable range of the parameter

Allowable RangeThe 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

The 100 Percent Rule for Simultaneous changes in RHS CoefficientsThis is a rule that tells you how much

of each constraint is allowed to change simultaneously before the shadow prices 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 shadow prices will be valid◦ If it changes by more than 100%, you

cannot be sure.

The 100 Percent Rule for Simultaneous changes in Objective Function CoefficientsThis 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 ChangeThe 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%

Sensitivity Analysis and Spreadsheet ModelsThere are three major ways you

can use to conduct a sensitivity analysis on your spreadsheet model◦You can solve the model multiple

different times using different variables for the coefficient(s) of interest

◦You can use Solver Table◦You can use the Sensitivity Report in

Solver

Solving the Excel Model Multiple Times with Multiple ParametersWhenever 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 TableSolver Table is a tool developed

by the textbook authors to solve the model multiple times using different parameters◦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.

Excel Side NoteYou 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.

Solver’s Sensitivity ReportSolver 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

Analyzing the Sensitivity ReportTo 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

Solver Report for WyndorMicrosoft Excel 14.0 Sensitivity ReportWorksheet: [Wyndor Glass.xls]WyndorReport Created: 10/20/2013 2:51:26 PM

Variable Cells

    FinalReduce

d ObjectiveAllowabl

eAllowabl

e

Cell Name Value CostCoefficie

nt IncreaseDecreas

e$C$12

Batches Produced Doors 2 0 3000 4500 3000

$D$12

Batches Produced Windows 6 0 5000 1E+30 3000

Constraints

    Final ShadowConstrain

tAllowabl

eAllowabl

e

Cell Name Value Price R.H. Side IncreaseDecreas

e$E$7 Plant 1 Used 2 0 4 1E+30 2$E$8 Plant 2 Used 12 1500 12 6 6$E$9 Plant 3 Used 18 1000 18 6 6

Analyzing the Sensitivity Report Cont.In the Wyndor example the price

of the doors could increase to $7500 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 $2000 before the optimal solution would change

Sensitive ParametersA 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

In-Class Activity

Given the model above:◦ Set-up the dual problem◦ Solve the dual using the tabular form and show that the

optimal of the dual is the shadow of the primal◦ Conduct a sensitivity analysis of the dual problem using

Excel Solver including explaining the allowable ranges, shadow prices

If all the objective coefficients in the dual decreased by 4%, can you definitely say whether the optimal would change without resolving the model

Subject to