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,
xij={1 If facility, i is built in location j2 otherwise
Since, Golf course can only be built at site 1 or 6, so
x11+x16=1
Since each facility can be built at only one site, add the following constraints.
x22+x23+x24=1x32+x33+x34=1x42+x43+x44=1x52+x53+x54=1
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.
29x22+38x32+45x42≤7029x23+38x33+45x43≤8029x24+38x34+45x44≤9529x25+38x35+45x45≤120
Priority 2: Construction costs should not exceed $1.2 million. Thus the following constraints are added.
340x11+340x16+300x22+300x23+300x24+300x25840x32+840x33+840x44+840x3585x42+85x43+85x44+85x45≤1200
Priority 3: User days should exceed 200,000. Thus the following constraints are added.
31x11+27x16+25x22+21x23+32x24+32x2537x32+29x33+28x44+38x3520x42+23x43+22x44+20x45≥200
Priority 4: Annual maintenance cost should not exceed $200,000. Thus the following constraints are added.
80x11+80x16+36x22+36x23+36x24+36x2550x32+50x33+50x3517x42+17x43+17x44+17x45≤200
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
Therefore, the constraints become,
29x22+38x32+45x42+S1+−S1−=7029x23+38x33+45x43+S2+−S2−=8029x24+38x34+45x44+S3+−S3−=9529x25+38x35+45x45+S4+−S4−=120
340x11+340x16+300x22+300x23+300x24+300x25840x32+840x33+840x44+840x3585x42+85x43+85x44+85x45+S5+−S5−=1200
31x11+27x16+25x22+21x23+32x24+32x2537x32+29x33+28x44+38x3520x42+23x43+22x44+20x45+S6+−S6−=200
80x11+80x16+36x22+36x23+36x24+36x2550x32+50x33+50x3517x42+17x43+17x44+17x45+S7+−S7−=200
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 Si− is included in minimization equation. If left hand side is more than right hand side, then Si+ is included in minimizations equation.
Hence the minimization equation is,
Minimize, z=P1S1−+P1S2−+P1S3−+P1S4−+P2S5−+P3S6++P4S7−
For each Priority assign a value of P1=4,P2=3,P3=2,P1=1 for priority 1,2,3,4 respectively
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