Sensitivity Analysis

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 problem's answer? Such analyses are used to examine the effects of changes in these three areas:

1. Contribution rates for each variable - C FACTOR

2. Technological coefficients - A FACTOR

3. 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 management's 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 X1's) and stereo players (called X2's). Its LP Formulation is:

MAXIMIZE PROFIT = $50X1 + $120X2

Subject to : 2X1 + 4X2 <= 80 (hour's of electrician's time available)

3X1 + 1 X2 <= 60 (hours of audio technician's time available)

1) ITERATIONS:

Sensitivity Analysis Solution

Cj Basic Variables 50

CD Players 120

Stereo Players 0

slack 1 0

slack 2 Quantity

Iteration 1

0 slack 1 2. 4. 1. 0. 80.

0 slack 2 3. 1. 0. 1. 60.

zj 0. 0. 0. 0. 0.

cj-zj 50. 120. 0. 0.

Iteration 2

120 Stereo Players 0.5 1. 0.25 0. 20.

0 slack 2 2.5 0. -0.25 1. 40.

zj 60. 120. 30. 0. 2,400.

cj-zj -10. 0. -30. 0.

2) LINEAR PROGRAMMING RESULTS:

Sensitivity Analysis Solution

CD Players Stereo Players RHS Dual

Maximize 50. 120.

Electrician Hrs 2. 4. <= 80. 30.

Audio Tech Hours 3. 1. <= 60. 0.

Solution-> 0. 20. 2,400.

3) SOLUTION LIST:

Sensitivity Analysis Solution

Variable Status Value

CD Players NON Basic 0.

Stereo Players Basic 20.

slack 1 NON Basic 0.

slack 2 Basic 40.

Optimal Value (Z) 2,400.

4) RANGE:

Sensitivity Analysis Solution

Variable Value Reduced Cost Original Value Lower Bound Upper Bound

CD Players 0. 10. 50. -Infinity 60.

Stereo Players 20. 0. 120. 100. Infinity

Constraint Dual Value Slack/Surplus Original Val Lower Bound Upper Bound

Electrician Hrs 30. 0. 80. 0. 240.

Audio Tech Hours 0. 40. 60. 20. 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 problem's 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 X1's 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 electrician's time) may increase from Rs.0 to Rs.30 without changing the current solution