sensitivity analysis
TRANSCRIPT
Department of Futures Studies
Sensitivity Analysis or Post Optimal Analysis Lekshmi Krishna M.R(100609)
Renjini R(100611)MTECH – Technology ManagementDepartment of Futures StudiesUniversity of Kerala
Department of Futures Studies
Thanks to Prof Sreenivasan(IIT Chennai) for his lecture series in Operations Research!!
Department of Futures Studies
Introduction
Common theme of OR – search of optimal solution
An optimal solution for the original model may vary from ideal for the real problem
Additional analysis – Post optimal analysis
Department of Futures Studies
Post Optimal Analysis
Post Optimal Analysis - Analysis done after finding an optimal solution
Some times called What-if analysis
Addressing questions about what would happen to the optimal solution if
different assumptions are made about future conditions
Department of Futures Studies
Cont……
In part post optimality analysis involves conducting sensitivity analysis
To determine which parameter of the model are most critical in determining the solution - Sensitive parameter
Sensitive parametersThe parameters whose values cannot be changed
without changing the optimal solution
Department of Futures Studies
Sensitivity Analysis
Involves the effect on the optimal solution making changes in values of model parameters aij, bi & cj
Changing parameter in primal also effect corresponding values in dual problem
Department of Futures Studies
Procedure for sensitivity analysis
REVISION OF MODEL
REVISION OF FINALTABLEAU
CONVERSION TO PROPER FORM
FEASIBILITY TEST
OPTIMALITY TEST
REOPTIMIZATION
Department of Futures Studies
Different case to do Sensitivity AnalysisCase 1: Changes in basic variable
of objective function
case 2: Changes in the non basic variable of objective function
Case 3 : Change in RHS
Case 4 : Change in the coefficients of constrains
Department of Futures Studies
Consider a LPP
Objective function : Max 4X1 +3X2 + 5X3
Sub to X1+2X2+3X3 ≤ 9
2X1+3X2+X3 ≤ 12
X1,X2,X3 ≥ 0
Department of Futures Studies
Simplex table4 3 5 0 0
B Cb X1 X2 X3 X4 X5 XB 0
X4 0 1 2 3 1 0 9 3
X5 0 2 3 1 0 1 12 12CJ-ZJ 4 3 5 0 0 0
X3 5 1/3 2/3 1 1/3 0 3 0
X5 0 5/3 7/3 0 -1/3 1 9 27/5CJ-ZJ 7/3 -1/3 0 -5/3 0 15
X3 5 0 1/5 1 2/5 -1/5 6/5
X1 4 1 7/5 0 -1/5 3/5 27/5CJ-ZJ 0 -18/5 0 -6/5 -7/5 138/5
Department of Futures Studies
Solution : X1 = 27/5
X3 = 6/5
Z = 138/5
Department of Futures Studies
CASE 1 : Change in objective functionCoefficient of non-basic variable Max Z = 4X1+ 3X2 + 5X3 + 0S1 + 0S2
Lets take X2
Effect in optimality 1. RHS = -ve
2. Cj-Zj = +ve
Here only change in C2 – Z2
Department of Futures Studies
C2 – Z2 = C2 – ( 1+28/5)
= C2 – 33/5 ≤ 0 If C2 > 33/5 then C2-Z2 becomes +ve & C2
becomes incoming row again and need to do
iteration till we get the optimum solution
Department of Futures Studies
4 7 5 0 0
B Cb X1 X2 X3 X4 X5 XB 0
X3 5 0 1/5 1 2/5 -1/5 6/5 6
X1 4 1 7/5 0 -1/5 3/5 27/5 27/7
CJ-ZJ 0 2/5 0 -6/5 -7/5 138/5
X3 5 -1/7 0 1 3/7 -2/7 3/7
X2 7 5/7 1 0 -1/7 3/7 27/7
Cj-Zj -2/7 0 0 -8/7 -11/7 204/7
Department of Futures Studies
Solution : X2 = 27/7
X3 = 3/7
Z = 204/7
Department of Futures Studies
Case 2 : Change in objective function coefficient of basic variable Max Z = 4X1+ 3X2 + 5X3 + 0S1 + 0S2
Lets take C1
Effect in optimality 1. RHS = -ve
2. Cj-Zj = +ve
Here only change in Cj-Zj of all the values
Department of Futures Studies
C1 3 5 0 0
B Cb X1 X2 X3 X4 X5 RHS
5 X3 0 1/5 1 2/5 -1/5 6/5
C1 X1 1 7/5 0 -1/5 3/5 27/5
Department of Futures Studies
Evaluation C2-Z2
C2 –Z2 =
3 - (1+7C1/3)
3 -7C1/5 ≤ 0
C1 ≥ 10/7
If C1 < 10/7
(X2 enters)
C4-Z4
C4 – Z4 =
0-(2 – C1/5)
C1/5 – 2 ≤ 0
C1 ≤ 10
If C1 > 10
(X4 enters)
C5-Z5
C5 - Z5 =
0 – (-1+3C1/5)
1-3C1/5 ≤ 0
C1 ≥ 5/3
If C1 < 5/3
(X5 enters)
Department of Futures Studies
Suppose C1 = 12
b CB X1 X2 X3 X4 X5 XB 0
X3 5 0 1/5 1 2/5 -1/5 6/5 3
X1 12 1 7/5 0 -1/5 3/5 27/5
Cj-Zj 0 - 74/5 0 2/5 -31/5 354/5
X4 0 0 3/2 5/2 1 -1/2 3
X1 12 1 3/2 1/2 0 1/3 6
Cj-Zj 0 -15 -1 0 -6 72
Department of Futures Studies
Solution : X1 = 6
X4 = 3
Z = 72
Department of Futures Studies
Case:3 Right Hand Side ChangesObjective function : Max 4X1 +3X2 + 5X3
Sub to X1+2X2+3X3 ≤ 9
2X1+3X2+X3 ≤ 12
X1,X2,X3 ≥ 0
b = 9
12
Department of Futures Studies
Effect of that change only in RHS RHS = B b
B = 2/5 - 1/5 b1
-1/5 3/5 12
= (2b1-12)/5
(36-b1)/5
Department of Futures Studies
Evaluation
(2b1-12)/5 ≥ 0
2b1-12 ≥ 0
b1 ≥ 6
(36-b1)/5 ≥ 0
b1 ≤ 36
Hence the range of b1 should be
6 ≤ b1≤ 36
for the solution to be optimum
Department of Futures Studies
Suppose b1 =40
RHS = (2b1-12)/5
(36-b1)/5
= 68/5
-4/5
Department of Futures Studies
Substituting the new RHS values
4 3 5 0 0
b Cb X1 X2 X3 X4 X5 RHS
X3 5 0 1/5 1 2/5 -1/5 68/5
X1 4 1 7/5 0 -1/5 3/5 -4/5
Cj-Zj 0 -18/5 0 -6/5 -7/5 138/5
Primal infeasible
Here dual is feasible .Hence calculate 0 to do dual simplex iteration
0 - 6 -
Department of Futures Studies
Evaluate by taking X4 comes in and X1 goes out
4 3 5 0 0
b Cb X1 X2 X3 X4 X5 Xb
X3 5 1 3 1 0 1 12
X4 0 -5 -7 0 1 -1 4
Cj-Zj -1 -9 0 0 -1 48
Solution X3=12 X4=4
Z =48
Department of Futures Studies
Case 4 : Change in the constrain coefficient of a non basic variable Objective function : Max 4X1 +3X2 + 5X3
Sub to X1+2X2+3X3 ≤ 9
2X+ 3X2+X3 ≤ 12
X1,X2,X3 ≥ 0
P2 = 2 a
Department of Futures Studies
Evaluation
P2 = B P2
= 2/5 -1/5 2
-1/5 3/5 a
= (4-a)/5
(3a-2)/5
Here C2-Z2 changes
C2-Z2
= C2 – y P2
= 3 – [6/5 7/5] 2
a
= (3 – 7a)/5
a= 0(3-7a)/5
C2-Z2 =3/5
P2 = 4/5 -2/5
Department of Futures Studies
New simplex table4 3 5 0 0
b Cb X1 X2 X3 X4 X5 Xb 0
X3 5 0 4/5 1 2/5 -1/5 6/5 3/2
X1 4 1 -2/5 0 -1/5 3/5 27/5 -
Cj-Zj 0 3/5 0 -6/5 -7/5 138/5
X2 3 0 1 5/4 1/2 -1/4 3/2
X1 4 1 0 1/2 0 1/2 6
Cj-Zj 0 0 -3/4 -3/2 -3/4 57/2
X1 = 6 ; X2= 3/2 ; Z = 57/2
Department of Futures Studies
Conclusion Sensitivity analysis carried out by
mathematical programming systems are called ranging (RANGE Procedure)
Sensitivity analysis needs to be performed to investigate what happens when the estimates are wrong
Helps to find out the range of likely values of the sensitive parameters
Hence an important part of most linear programming study
Department of Futures Studies
Reference
Introduction to operations research
Frederick S Hillier & Gerald J Lieberman