MS-326

Production Management

Sensitivity Analysis and Parametric

ProgrammingBy

Yousaf Ali khan

Department of Management Sciences

GIKI, Pakistan

Sensitivity Analysis and Parametric

ProgrammingUp until now, we have mostly considered LPproblems as static models, in the sense that the datafor the problem has been known and fixed.

However, when discussing shadow pricing, we raisedthe possibility of changes to the availability ofresources and how they might impact on the totalprofit.

Of course, profit information and other aspects of theproblem can also change. Therefore, users ofmathematical programming are interested in muchmore than just the actual optimal solution to a model.

Two Techniques

This chapter introduces two techniques for

studying how the optimal solution changes

when some of the data changes:

sensitivity analysis (or postoptimality analysis)

and

parametric programming.

Sensitivity Analysis

Example

T&C Manufacturing Ltd. is a furniture manufacturer oftables and chairs. For each product, two different typesof wood and various amounts of labour are required.Each table requires 1.5 board-metres (bd-m) of oak, 0.6bd-m of pine, and 4 labour-hours (l-h). Each chairrequires 0.6 bd-m of oak, 0.9 bd-m of pine, and 2 l-h.The manufacturer can sell all that is produced andmake a profit of 12 per table and 8 per chair. For theupcoming week, T&C Manufacturing has 45 bd-m ofoak, 30 bd-m of pine, and 80 l-h available. Themanager has hired you to find out how much of eachproduct should be made (and sold) in order tomaximise the profit for the week.

Example-ContinueMax 12xt + 8xcsubject to

1.5xt + 0.6xc 45 (oak)

0.6xt + 0.9xc 30 (pine)

4xt + 2xc 80 (labour)

xt; xc 0

In standard form, it isMax 12xt + 8xcsubject to

1.5xt + 0.6xc + so = 45 (oak)

0.6xt + 0.9xc + sp = 30 (pine)

4xt + 2xc + sl = 80 (labour)

xt; xc; so; sp; sl 0

Optimal Dictionary for T&C

Manufacturing

Changes to the Right-Hand Side

We observed in the previous section that the marginal value for labour was $2.50.

But this marginal value is valid only as long as the change in the b is "Small enough"

"Small enough means that the optimal basis does not change.

Using sensitivity analysis, we can determine exactly how much each bi can change without causing a change of basis.

Sensitivity Analysis for Labour

Resulting Dictionary

The resulting dictionary is

Bounds for

2. Changes to the Objective Function

Bounds for

Parametric Programming

Parametric programming can be used to study howthe optimal solution changes as the objective function

and/or the right-hand side change continuously

according to a parameter

The idea is that we can solve the LP problem for aninfinite number of different (but related) sets of data.

We illustrate this via two examples.

First Example of Parametric Programming

First Example of Parametric Programming