CVL609 Lec4 LP-II ver8.0.pptx - CVL609 Lecture 4 Linear...

CVL609 Lecture 4
Linear Programming II
Dr Arnold Yuan, PEng
Department of Civil Engineering

Agenda

Duality of LP and Duality Simplex Method

General Concept of Post-Optimality Analysis

Sensitivity/Scenario Analysis of LP

Remaining Issues in Simplex Method

Initialization

Unbounded feasible region (no solution)

Empty feasible region (no solution)

Multiple solutions

Learning Objectives
After this lecture, students are expected to be
able to

Express the duality form of a given LP

Choose the proper post-optimality analyses

Use the final simplex table, and duality
simplex if necessary, to perform sensitivity
and scenario analyses
Reference: Hillier and Liebermann (2015) chapter 6 &
8.1

Pt1 – Duality of LP

Review
Simplex Method

Decision variables and slack variables

Basic variables and nonbasic variables
Geometry Algebra
CP solution Solution of n simultaneous equations
Feasible solution Nonnegativeness of slack variables
Move from one CPF to an
adjacent CPF
Exchange a basic and non-basic pair
Optimality test Optimality test by checking the sign of coefficients in non-
basic variables
Determine moving direction:
Steepest direction
Determine entering variable:
The nonbasic variable with the greatest improve rate
Determine step size:
Avoid infeasibility
Determine leaving variable:
The basic variable with the minimum ratio to avoid negative
value
Standard form of LP model Augmented/Slack form of LP model

A little stretch…
Simplex Method
The simplex iteration is equivalent to identify the set
of nonbasic variables.
• Once the nonbasic variables are identified, the basic variables
can be solved by Gaussian elimination.
•
The identification of nonbasic variables is the same as the
identification of active constraints.
Example : Wyndows-Glass problem.

A Very Important Slide!!!
A Fundamental Insight of LP

Given y* and S* , the remaining simplex tableau can
be established.

Z* = y * b (Value of objective function)

y * A – c ≥ 0 (Optimality)

S * b ≥ 0 (Feasibility)

Example

Example (cont’d)
A I b
S*A S*
S*
b
y*
y*A-
c
y*
b
-c

Duality of LP
Primal Problem
Dual Problem

Duality of LP
Wyndow-Glass Example

Primal-Dual Relationships
(Primal Feasible)
(Dual Feasible)
(Dual Infeasible)
(Primal Infeasible)
Complementary solutions
property:
At each iteration, the simplex
method simultaneously identifies
a CPF solution for the primal
problem and a complementary
solution for the dual problem,
where . If x is not optimal for the
primal problem, then is not
feasible for the dual problem.