Post-optimal analysis of LPP

Post-Optimal AnalysisLinear optimization problem

Contents

Introduction to Post-Optimal Analysis

Changes Affection Feasibility

Changes in the right-hand side (b)

Addition of a new constraint

Changes Affecting Optimality

Changes in the Objective Coefficients (c_j)

Addition of new activity (x_j)

Post-Optimal Analysis

The sensitivity of the optimum solution

By determining the ranges for the different LP

parameters that keeps the optimum basic variable

unchanged

Deal with making changes in the parameters of the model

and finding the new optimum solution

These changes require periodic re-calculation of the

optimum solution and

The new computations are rooted in the use duality

and the primal-dual relationships

Post-Optimal Analysis

Example : Assembling 3 types of toys

Example : Assembling 3 types of toys

Optimum tableau for the primal is

Changes Affection Feasibility

Two possibilities which can affect feasibility

are

Changes in the right-hand side (b)

A new constraint is added

The dual problem uses exactly the same parameters as the primal problem, but in different location.

Primal and Dual Problems

Primal Problem Dual Problem

Max

s.t.

Min

s.t.

n

j

jj xcZ1

,

m

i

ii ybW1

,

n

j

ijij bxa1

,

m

i

jiij cya1

,

for for.,,2,1 mi .,,2,1 nj

for.,,2,1 mi

for .,,2,1 nj ,0jx ,0iy

RHS means

Changes in the right-hand side (b)

Changes in the right-hand side (b)

New RHS of the problem

The current basic variable remain feasible at the new values and

The optimum revenue is $1880

Changes in the right-hand side (b)

Changes in the right-hand side (b)

Addition of a new constraint

The new constraint for the operation 4 is

3x1 +3x2 + x3 ≤ 500

Addition of a new constraint

Addition of a new constraint

Addition of a new constraint The tableau shows the x7 = 500 which is not consistent with the values

of x2 and x3 in the rest of the table….

Reason: the basic variable x2 and x3 have not been substituted out in

the new constraint…

Addition of a new constraint

Application of the dual simplex method will produce the new optimum solution

x1 = 0, x2 = 90, x3 = 230, and z = $1370 (verify!)

Changes Affecting Optimality

The changes in the Objective

Coefficients

The addition of a new

economic activity (variable)

Changes in the Objective Coefficients

Changes in the Objective Coefficients

Affect only the optimality of the solution and require recomputing the

z-row coefficients (reduced costs):-

1) Compute the dual values using the Method 2

2) Substitute the new dual values in Formula 2, to determine the new

reduced costs (z-row coefficients).

Recapitulate :

Recapitulate :

Changes in the Objective Coefficients

Maximize z = 2x1 + 3x2 + 4x3 is the new objective

function then,

Changes in the Objective Coefficients

Maximize

z=

2x1 +

3x2 +

4x3

Changes in the Objective Coefficients

Maximize z = 6x1 + 3x2 + 4x3 is the new objective function then,

Changes in the Objective Coefficients

The optimum solution :- x1 = 102.5 x2 = 215 and z = $12270.50 (verify)

Addition of new activity

Addition of new activity A new activity signifies adding a new variable to the model

Intuitively, the new activity desirable only if it is profitable

Formula 2 will help in checking this

To compute the reduced cost of the new variable

Let x7 represents the new product in the TOYCO and

Revenue per new toy is $4

Operation 1 1 minute

Operation 2 1 minute

Operation 3 2 minutes The new column in the

coefficient matrix of the

constraints

4x_7 will be the extra term in the

primal objective function

Addition of new activity

Given that (y_1, y_2, y_3) = (1,2,0) are the optimal dual variable,

Reduce cost of x7 is

Addition of new activityNot Optimal but

feasible solution

The new optimum is obtained by letting x7 enter the basis and x6 must leave the basis…..

The new optimum is x1 = 0, x2 = 0, x3 = 125 and x7 = 210 with the revenue z = $1465

(verify)

Thanks for your attention