Engineering Optimization: Theory and Practice, Fourth Edition

212 Linear Programming II: Additional Topics and Extensions

If the variables are not renumbered, Eq. (4.36) will be applicable fori=3 and

2 in the present problem withb 3 = 300 andb 2 = 00. From Eqs. (E 2 1 ) to (E 5 ) of

Example 4.5, the left-hand sides of Eq. (4.36) become

x 3 +β 33 b 3 +β 32 b 2 =^8003 + 154 00 ( 3 )− 151 ( 002 )=^500015

x 2 +β 23 b 3 +β 22 b 2 =^403 − 1501 00 ( 3 )+ 752 ( 002 )=^2500150

Since both these values are≥ 0 , the original optimal basisBremains optimal even

with the new values ofbi. The new values of the (optimal) basic variables are given

by Eq. (4.38) as

X′B=

{

x′ 3

x′ 2

}

=XB+XB=XB+B−^1 b

=

{ 800

3

40

3

}

+

[ 4

15 −

1

15

− 1501 752

]{

300

200

}

=

{ 1000

3

50

3

}

and the optimum value of the objective function by Eq. (4.39) as

fm′in=fmin+ f=fmin+cTBXB= −

28 , 000

3

+(− 30 − 100 )

{ 200

3

10

3

}

=−

35 , 000

3

Thus the new profit will be $35,000/3.

4.5.2 Changes in the Cost Coefficientscj

The problem here is to find the effect of changing the cost coefficients fromcj to

cj+ cjon the optimal solution obtained withcj. The relative cost coefficients cor-

responding to the nonbasic variables,xm+ 1 , xm+ 2 ,... , xnare given by Eq. (4.10):

cj=cj−πTAj=cj−

∑m

i= 1

πiaij, j=m+ 1 ,m+ 2 ,... , n (4.40)

where the simplex multipliersπiare related to the cost coefficients of the basic variables

by the relation

πT=cTBB−^1

that is,

πi=

∑m

k= 1

ckβki, i= 1 , 2 ,·· ·, m (4.41)

From Eqs. (4.40) and (4.41), we obtain

cj=cj−

∑m

i= 1

aij

(m

∑

k= 1

ckβki

)

=cj−

∑m

k= 1

ck

(m

∑

i= 1

aijβki

)

,

i=m+ 1 ,m+ 2 ,... , n (4.42)