Introduction to mathematical programming
Introduction to mathematical programming
4th Edition
ISBN: 9780534359645
Author: Jeffrey B. Goldberg
Publisher: Cengage Learning
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Chapter 4.16, Problem 14P

Explanation of Solution

Consider the LP problem of HAL computer considering the projects that should be started. Data related to each of the project tabulated.

Consider the following variables x1,x2,........,x7,

xi={1      If project i is undertaken0      otherwise

HAL has the following goals

Goal 1:

The total NPV of all chosen should be at least $200 million. Thus the constraint becomes,

40x1+30x2+60x3+45x4+55x5+40x6+90x7200

Goal 2:

The average probability of success for all projects chosen should be at least 0.75. Thus the constraint becomes,

0.75x1+0.70x2+0.75x3+0.90x4+0.65x5+0.60x6+0.65x70.75

Goal 3:

The average growth rate of all projects chosen should be at least 15%. Thus the constraint becomes,

20x1+16x2+12x3+8x4+18x5+18x6+19x715

Goal 4:

The total cost of all chosen project should be at most $1billion. Thus the constraint becomes,

220x1+140x2+280x3+240x4+300x5+200x6+440x71000

Hence the LP is formulated as given below

Minimize, 0x1+0x2+0x3+0x4+0x5+0x6+0x7

Subjected to the constraints,

40x1+30x2+60x3+45x4+55x5+40x6+90x72000.75x1+0.70x2+0.75x3+0.90x4+0.65x5+0.60x6+0.65x70.7520x1+16x2+12x3+8x4+18x5+18x6+19x715220x1+140x2+280x3+240x4+300x5+200x6+440x71000

xi=0 or 1

From the above equations it is found that these set of constraints there is no feasible region. That is all constraints cannot be met. So assign a cost value incurred if any of the priorities or goal is not met. So, introduce the following deviational variables.

Si= Amount by which numerically under the ith goal

Si+= Amount by which numerically exceed the ith goal

Hence the constraints become,

40x1+30x2+60x3+45x4+55x5+40x6+90x7+S1+S1=2000.75x1+0.70x2+0.75x3+0.90x4+0.65x5+0.60x6+0.65x7+S2+S2=0.7520x1+16x2+12x3+8x4+18x5+18x6+19x7+S3+S3=15220x1+140x2+280x3+240x4+300x5+200x6+440x7+S4+S4=1000

Giving lowest priority to goal 4 and highest priority to goal 1, assign weight to each goal

P1>>P2>>P3>>P4

Now, the goal is to minimize the deviation from each goal

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Chapter 4 Solutions

Introduction to mathematical programming

Ch. 4.5 - Prob. 1PCh. 4.5 - Prob. 2PCh. 4.5 - Prob. 3PCh. 4.5 - Prob. 4PCh. 4.5 - Prob. 5PCh. 4.5 - Prob. 6PCh. 4.5 - Prob. 7PCh. 4.6 - Prob. 1PCh. 4.6 - Prob. 2PCh. 4.6 - Prob. 3PCh. 4.6 - Prob. 4PCh. 4.7 - Prob. 1PCh. 4.7 - Prob. 2PCh. 4.7 - Prob. 3PCh. 4.7 - Prob. 4PCh. 4.7 - Prob. 5PCh. 4.7 - Prob. 6PCh. 4.7 - Prob. 7PCh. 4.7 - Prob. 8PCh. 4.7 - Prob. 9PCh. 4.8 - Prob. 1PCh. 4.8 - Prob. 2PCh. 4.8 - Prob. 3PCh. 4.8 - Prob. 4PCh. 4.8 - Prob. 5PCh. 4.8 - Prob. 6PCh. 4.10 - Prob. 1PCh. 4.10 - Prob. 2PCh. 4.10 - Prob. 3PCh. 4.10 - Prob. 4PCh. 4.10 - Prob. 5PCh. 4.11 - Prob. 1PCh. 4.11 - Prob. 2PCh. 4.11 - Prob. 3PCh. 4.11 - Prob. 4PCh. 4.11 - Prob. 5PCh. 4.11 - Prob. 6PCh. 4.12 - Prob. 1PCh. 4.12 - Prob. 2PCh. 4.12 - Prob. 3PCh. 4.12 - Prob. 4PCh. 4.12 - Prob. 5PCh. 4.12 - Prob. 6PCh. 4.13 - Prob. 2PCh. 4.14 - Prob. 1PCh. 4.14 - Prob. 2PCh. 4.14 - Prob. 3PCh. 4.14 - Prob. 4PCh. 4.14 - Prob. 5PCh. 4.14 - Prob. 6PCh. 4.14 - Prob. 7PCh. 4.16 - Prob. 1PCh. 4.16 - Prob. 2PCh. 4.16 - Prob. 3PCh. 4.16 - Prob. 5PCh. 4.16 - Prob. 7PCh. 4.16 - Prob. 8PCh. 4.16 - Prob. 9PCh. 4.16 - Prob. 10PCh. 4.16 - Prob. 11PCh. 4.16 - Prob. 12PCh. 4.16 - Prob. 13PCh. 4.16 - Prob. 14PCh. 4.17 - Prob. 1PCh. 4.17 - Prob. 2PCh. 4.17 - Prob. 3PCh. 4.17 - Prob. 4PCh. 4.17 - Prob. 5PCh. 4.17 - Prob. 7PCh. 4.17 - Prob. 8PCh. 4 - Prob. 1RPCh. 4 - Prob. 2RPCh. 4 - Prob. 3RPCh. 4 - Prob. 4RPCh. 4 - Prob. 5RPCh. 4 - Prob. 6RPCh. 4 - Prob. 7RPCh. 4 - Prob. 8RPCh. 4 - Prob. 9RPCh. 4 - Prob. 10RPCh. 4 - Prob. 12RPCh. 4 - Prob. 13RPCh. 4 - Prob. 14RPCh. 4 - Prob. 16RPCh. 4 - Prob. 17RPCh. 4 - Prob. 18RPCh. 4 - Prob. 19RPCh. 4 - Prob. 20RPCh. 4 - Prob. 21RPCh. 4 - Prob. 22RPCh. 4 - Prob. 23RPCh. 4 - Prob. 24RPCh. 4 - Prob. 26RPCh. 4 - Prob. 27RPCh. 4 - Prob. 28RP
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