Engineering Optimization: Theory and Practice, Fourth Edition

4.5 Sensitivity or Postoptimality Analysis 213

If thecjare changed tocj+ cj, the original optimal solution remains optimal, pro-
vided that the new values ofcj,cj′satisfy the relation

cj′=cj+ cj−

∑m

k= 1

(ck+ ck)

(m

∑

i= 1

aijβki

)

≥ 0

=cj+ cj−

∑m

k= 1

ck

(m

∑

i= 1

aijβki

)

≥ 0 ,

j=m+ 1 ,m+ 2 ,· · ·, n (4.43)

wherecj indicate the values of the relative cost coefficients corresponding to the
original optimal solution.
In particular, if changes are made only in the cost coefficients of the nonbasic
variables, Eq. (4.43) reduces to

cj+ cj≥ 0 , j=m+ 1 , m+ 2 ,... , n (4.44)

If Eq. (4.43) is satisfied, the changes made incj, cj, will not affect the optimal basis
andthe values of the basic variables. The only change that occurs is in the optimal
value of the objective function according to

f=

∑m

j= 1

xjcj (4.45)

and this change will be zero if only thecjof nonbasic variables are changed.
Supposethat Eq. (4.43) is violated for some of the nonbasic variables. Then it
is possible to improve the value of the objective function by bringing any nonbasic
variable that violates Eq. (4.43) into the basis provided that it can be assigned a nonzero
value. This can be done easily with the help of the previous optimal tableau. Since
some of thecj′are negative, we start the optimization procedure again by using the old
optimum as an initial feasible solution. We continue the iterative process until the new
optimum is found. As in the case of changing the right-hand-sidebi, the effectiveness
of this procedure depends on the number of violations made in Eq. (4.43) by the new
valuescj+ cj.
In some of the practical problems, it may become necessary to solve the opti-
mization problem with a series of objective functions. This can be accomplished
without reworking the entire problem for each new objective function. Assume that
the optimum solution for the first objective function is found by the regular proce-
dure. Then consider the second objective function as obtained by changing the first
one and evaluate Eq. (4.43). If the resultingcj′≥ , the old optimum still remains 0
as optimum and one can proceed to the next objective function in the same manner.
On the other hand, if one or more of the resultingcj′< , we can adopt the proce- 0
dure outlined above and continue the iterative process using the old optimum as the
starting feasible solution. After the optimum is found, we switch to the next objective
function.