Engineering Optimization: Theory and Practice, Fourth Edition

11.3 Stochastic Linear Programming 651

Machining time required per unit (min) Maximum time
available
Part I Part II per week (min)
Type of
machine Mean

Standard

deviation Mean

Standard

deviation Mean

Standard

deviation

Lathes a 11 = 10 σa 11 = 6 a 12 = 5 σa 12 = 4 b 1 = 2500 σb 1 = 005
Milling
machines a 21 = 4 σa 21 = 4 a 22 = 10 σa 22 = 7 b 2 = 2000 σb 2 = 004
Grinding
machines a 31 = 1 σa 31 = 2 a 32 = 1. 5 σa 32 = 3 b 3 = 450 σb 3 = 05

Profit per unit c 1 = 50 σc 1 = 02 c 2 = 100 σc 2 = 05

SOLUTION By defining new random variableshias

hi=

∑n

j= 1

aijxj−bi,

we find thathiare also normally distributed. By assuming that there is no correlation
betweenaij’s andbi’s, the means and variances ofhican be obtained from Eqs. (11.73)
and (11.77) as

h 1 =a 11 x 1 +a 12 x 2 −b 1 = 01 x 1 + 5 x 2 − 5002

h 2 =a 21 x 1 +a 22 x 2 −b 2 = 4 x 1 + 01 x 2 − 0002

h 3 =a 31 x 1 +a 32 x 2 −b 3 =x 1 + 1. 5 x 2 − 504

σh^21 =x 12 σa^211 +x^22 σa^212 +σb^21 = 36 x 12 + 61 x 22 + 502 , 000

σh^22 =x 12 σa^221 +x^22 σa^222 +σb^22 = 16 x 12 + 94 x 22 + 601 , 000

σh^23 =x 12 σa^231 +x^22 σa^232 +σb^23 = 4 x 12 + 9 x^22 + 5002

Assuming that the profits are independent random variables, the covariance matrix of
cjis given by

V=

[

Var(c 1 ) 0

0 Var(c 2 )

]

=

[

4 00 0

0 2500

]

and the variance of the objective function by

Var(f )=XTVX = 400 x^21 + 5002 x 22

Thus the objective function can be taken as

F=k 1 ( 05 x 1 + 001 x 2 )+k 2

√

400 x^21 + 5002 x^22

The constraints can be stated as

P[hi≤ ] 0 ≥pi= 0. 99 , i= 1 , 2 , 3