ISM 206Lecture 4

Duality and Sensitivity Analysis

Announcements

Outline

1. Review Simplex Method

2. Sensitivity analysis:

How does the solution change as the parameters change?

•How much would we ‘pay’ for more resources

•What is the effect of changing parameters A, b, c

Sensitivity through parametric LP solving

2. Duality

What is duality and why does it matter?

The dual of a linear program

Sensitivity through duality

Simplex Method

• Objective: Max Z (first line of table)

• Rules:– Update basic set of variables– Keep looking at feasible basic solutions

• Keep positive x_j values• Zeros in top line correspond to basic variables• Identity matrix rows in table correspond to basic

– Aim: Top line is non-negative

The elements of the simplex tableu

• After any iteration, the coefficients of the slack variables in each equation immediately reveal how that equation has been obtained from the initial equations.

• The text talks about the ‘fundamental insight’:– After any iteration, the coefficients of the slack

variables in each equation immediately reveal how that equation has been obtained by the initial equations

Sensitivity Analysis

• Changes in b

• Changes in c

• Changes in A

• Introduction of a new variable

• Introduction of a new constraint

• Parametric Linear Programming

• All demonstrated in OR Tutor

Changes in b

• Handle by checking optimality conditions under previous basis

• How much could b change and still be optimal?– ranging

Changes in c

• Change will not affect feasibility!

• Different procedure when parameter being changed depends on basic or nonbasic variable– Called ranging again

Introducing a New variable

• Same as changing the coefficients to a nonbasic variable

Introducing a New Constraint

• Check feasibility of original optimal

• Add row to tableu and proceed as

Parametric LP

• Are there values of the parameter for which the problem has a solution?

• How do the objective and optimal x depend on the parameter?

0

subject to

)'(max

x

bAx

xdc

Questions and Break

The dual Linear Program

0

subject to

max

x

bAx

xcZ

0

subject to

min

y

cyA

ybW

Primal Dual

Dual Linear Programs

• The dual of a LP is another LP – Coefficients of primal objective = rhs of dual

constraints– Rhs of primal constraints = coeffs of dual

objective– Variable coefficients are the same

(transposed)

Translating between primal and dual

• The dual of the dual is the primal

• Weak duality theorem

• Strong duality theorem

• Complementary Slackness

• Optimality Conditions

• Interpretation of dual variables

• Dual Simplex algorithm