Sensitivity Analysis
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SENSITIVITY ANALYSIS
The solution obtained by simplex or graphical method of LP is based on deterministic assumptions i.e. we assume complete certainty in the data and the relationships of a problem – namely prices are fixed, resources known, time needed to produce a unit exactly etc. However in the real world, conditions are seldom static i.e. they are dynamic. How can such discrepancy be handled? For example if a firm realizes that profit per unit is not Rs 5 as estimated but instead closer to Rs 5.5, how will the final solution mix and total profit change? If additional resources, such as 10 labor hours or 3 hours of machine time, should become available, will this change the problems answer? Such analyses are used to examine the effects of changes in these three areas:
Contribution rates for each variable – C FACTOR
Technological coefficients – A FACTOR
Available Resources – B FACTOR
This task is alternatively called sensitivity analysis. It is also called as post optimality analysis.
Sensitivity analysis often involves a series of what if? questions. What if the profit of product 1 increases by 10%? What if less money is available in advertising budget constraints? What if new technology will allow a product to be wired in one-third the time it used to take? So we can see that sensitivity analysis can be used to deal not only with errors in estimating input parameters to the LP model but also with managements experiments with possible future changes in the firm that may affect profits.
There are two approaches to determining how sensitive an optimal solution is to changes. The first is simply a trial and error approach, however we prefer the second approach of post optimality method i.e. after an LP problem has been solved, and we attempt to determine a range of changes in problem parameters that will not affect the optimal solution or change the variables in the solution. This is done without resolving the whole problem again. This is illustrated by the following example.
Consider an example of ABC Sound Company that makes CD players (called X1s) and stereo players (called X2s). Its LP Formulation is:
MAXIMIZE PROFIT = $50X1 + $120X2
Subject to : 2X1 + 4X2 <= 80 (hours of electricians time available)
3X1 + 1 X2 <= 60 (hours of audio technicians time available)
1) ITERATIONS:
Sensitivity Analysis Solution
Basic Variables
CD Players
Stereo Players
slack 1
slack 2
Quantity
Iteration 1
slack 1
slack 2
cj-zj
Iteration 2
Stereo Players
slack 2
-0.25
2,400.
cj-zj
2) LINEAR PROGRAMMING RESULTS:
Sensitivity Analysis Solution
CD Players
Stereo Players
Maximize
Electrician Hrs
Audio Tech Hours
Solution->
2,400.
3) SOLUTION LIST:
Sensitivity Analysis Solution
Variable
Status
Value
CD Players
NON Basic
Stereo Players
Basic
slack 1
NON Basic
slack 2
Basic
Optimal Value (Z)
2,400.
4) RANGE:
Sensitivity Analysis Solution
Variable
Value
Reduced Cost
Original Value
Lower Bound
Upper Bound
CD Players
-Infinity
Stereo Players
Infinity
Constraint
Dual Value
Slack/Surplus
Original Val
Lower Bound
Upper Bound
Electrician Hrs
Audio Tech Hours
Infinity
ANALYSES OF EFFECT OF CHANGE IN C-FACTOR:
1) For Non-Basic Objective Function co-efficient:
Our objective is to find out how sensitive problems optimal solution is to changes in the C-Factors not in the basis. The answer lies in Cj-Zj row of Iterations table. Since this is a maximization problem, the basis will not change unless the Cj-Zj value of one of the non basic variables becomes positive. That is, the current solution will be optimal as long as all numbers in the bottom row are less than or equal to zero.
Now Cj-Zj <= 0 is same as Cj <= Zj. Since X1s Cj value is Rs.50 and its Zj value is Rs.60, the current solution is optimal as long as the profit per CD does not exceed Rs.60, or does not increase by Rs.10. Similarly the C-Factor for S1 (per hour of electricians time) may increase from Rs.0 to Rs.30 without changing the current solution