
Explanation of Solution
Using preemptive goal
Consider the case of building recreational facilities at Gotham city. Goal course, Swimming pool, Gymnasiums and Tennis Courts are the recreational facilities under consideration. The facilities can be built at any of the 6 locations.
Lets,
Since, Golf course can only be built at site 1 or 6, so
Since each facility can be built at only one site, add the following constraints.
The land available at site 2,3,4,5 is 70,80,95, and 120 respectively. Land required, building annual maintenance cost is tabulated below,
Site | Construction cost | Maintenance cost | Land required |
Golf | 340 | 80 | Not relevant |
Swimming | 300 | 36 | 29 |
Gymnasium | 840 | 50 | 38 |
Tennis curt | 85 | 17 | 45 |
Site | 1 | 2 | 3 | 4 | 5 | 6 |
Golf | 31 | NA | NA | NA | NA | 27 |
Swimming | NA | 25 | 21 | 32 | 32 | NA |
Gymnasium | NA | 37 | 29 | 28 | 38 | NA |
Tennis court | NA | 20 | 23 | 22 | 20 | NA |
a.
Priority set are given below,
Priority 1: Limit land use at each site to the land available. Thus the following constraints are added.
Priority 2: Construction costs should not exceed $1.2 million. Thus the following constraints are added.
Priority 3: User days should exceed 200,000. Thus the following constraints are added.
Priority 4: Annual maintenance cost should not exceed $200,000. Thus the following constraints are added.
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.
Therefore, the constraints become,
Now, the goal is to minimize the deviation from each goal. Hence if the left side of constraint was less than right hand side than
Hence the minimization equation is,
Minimize,
For each Priority assign a value of

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Chapter 4 Solutions
Introduction to mathematical programming
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