Engineering Optimization: Theory and Practice, Fourth Edition

4.5 Sensitivity or Postoptimality Analysis 209

that is,

xi=

∑m

j= 1

βijbj, i= 1 , 2 ,... , m (4.38)

Finally, the change in the optimal value of the objective function (f )due to the
changebican be obtained as

f=cTBXB=cTBB−^1 b=πTb=

∑m

j= 1

πjbj (4.39)

Suppose that the changes made inbi(bi) aresuch that the inequality (4.34) is violated
for some variables so that these variables become infeasible for the new right-hand-side
vector. Our interest in this case will be to determine the new optimal solution. This can
be done without reworking the problem from the beginning by proceeding according
to the following steps:

1.Replace thebiof the original optimal tableau by the new values,b

′

=B−^1 (b+

b)and change the signs of all the numbers that are lying in the rows in which

the infeasible variables appear, that is, in rows for whichb

′

i<.^0

2.Add artificial variables to these rows, thereby replacing the infeasible variables

in the basis by the artificial variables.

3.Go through the phase I calculations to find a basic feasible solution for the

problem with the new right-hand side.

4.If the solution found at the end of phase I is not optimal, we go through the

phase II calculations to find the new optimal solution.

The procedure outlined above saves considerable time and effort compared to the
reworking of the problem from the beginning if only a few variables become infea-
sible with the new right-hand side. However, if the number of variables that become
infeasible are not few, the procedure above might also require as much effort as the
one involved in reworking of the problem from the beginning.

Example 4.5 A manufacturer produces four products,A, B, C, andD, by using two
types of machines (lathes and milling machines). The times required on the two machines
to manufacture 1 unit of each of the four products, the profit per unit of the product, and
the total time available on the two types of machines per day are given below:

Time required per unit (min) for product: Total time available

Machine A B C D per day (min)

Lathe machine 7 10 4 9 1200

Milling machine 3 40 1 1 800

Profit per unit ($) 45 100 30 50

Find the number of units to be manufactured of each product per day for maximizing
the profit.
Note:This is an ordinary LP problem and is given to serve as a reference problem
for illustrating the sensitivity analysis.