Chapter 4.16, Problem 11P

Explanation of Solution

Using preemptive goal programming to achieve the 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,

${\text{x}}_{\text{ij}}\text{=}\{\begin{array}{l}\text{1 If facility, i is built in location j}\\ \text{2 otherwise}\end{array}$

Since, Golf course can only be built at site 1 or 6, so

${\text{x}}_{11}+{\text{x}}_{16}=1$

Since each facility can be built at only one site, add the following constraints.

$\begin{array}{l}{\text{x}}_{22}+{\text{x}}_{23}+{\text{x}}_{24}=1\\ {\text{x}}_{32}+{\text{x}}_{33}+{\text{x}}_{34}=1\\ {\text{x}}_{42}+{\text{x}}_{43}+{\text{x}}_{44}=1\\ {\text{x}}_{52}+{\text{x}}_{53}+{\text{x}}_{54}=1\end{array}$

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.

$\begin{array}{l}{\text{29x}}_{\text{22}}{\text{+38x}}_{\text{32}}{\text{+45x}}_{\text{42}}\le \text{70}\\ {\text{29x}}_{\text{23}}{\text{+38x}}_{\text{33}}{\text{+45x}}_{\text{43}}\le \text{80}\\ {\text{29x}}_{\text{24}}{\text{+38x}}_{\text{34}}{\text{+45x}}_{\text{44}}\le \text{95}\\ {\text{29x}}_{\text{25}}{\text{+38x}}_{\text{35}}{\text{+45x}}_{\text{45}}\le \text{120}\end{array}$

Priority 2: Construction costs should not exceed $1.2 million. Thus the following constraints are added.

$\begin{array}{l}{\text{340x}}_{11}{\text{+340x}}_{\text{16}}\text{+}\\ {\text{300x}}_{\text{22}}{\text{+300x}}_{\text{23}}{\text{+300x}}_{\text{24}}+{\text{300x}}_{\text{25}}\\ {\text{840x}}_{\text{32}}{\text{+840x}}_{\text{33}}{\text{+840x}}_{\text{44}}{\text{+840x}}_{35}\\ {\text{85x}}_{\text{42}}{\text{+85x}}_{\text{43}}{\text{+85x}}_{\text{44}}+85{\text{x}}_{\text{45}}\le 1\text{200}\end{array}$

Priority 3: User days should exceed 200,000. Thus the following constraints are added.

$\begin{array}{l}{\text{31x}}_{11}{\text{+27x}}_{\text{16}}\text{+}\\ {\text{25x}}_{\text{22}}{\text{+21x}}_{\text{23}}{\text{+32x}}_{\text{24}}+{\text{32x}}_{\text{25}}\\ {\text{37x}}_{\text{32}}{\text{+29x}}_{\text{33}}{\text{+28x}}_{44}+38{\text{x}}_{\text{35}}\\ {\text{20x}}_{\text{42}}{\text{+23x}}_{\text{43}}{\text{+22x}}_{\text{44}}+20{\text{x}}_{\text{45}}\ge \text{200}\end{array}$

Priority 4: Annual maintenance cost should not exceed $200,000. Thus the following constraints are added.

$\begin{array}{l}{\text{80x}}_{11}{\text{+80x}}_{\text{16}}\text{+}\\ {\text{36x}}_{\text{22}}{\text{+36x}}_{\text{23}}{\text{+36x}}_{\text{24}}+{\text{36x}}_{\text{25}}\\ {\text{50x}}_{\text{32}}{\text{+50x}}_{\text{33}}{\text{+50x}}_{35}\\ {\text{17x}}_{\text{42}}{\text{+17x}}_{\text{43}}{\text{+17x}}_{\text{44}}+17{\text{x}}_{\text{45}}\le \text{200}\end{array}$

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.

$S_{\text{i}}{}^{-}$= Amount by which numerically under the ${\text{i}}^{\text{th}}$ goal

$S_{\text{i}}{}^{+}$= Amount by which numerically exceed the ${\text{i}}^{\text{th}}$ goal

Therefore, the constraints become,

$\begin{array}{l}{\text{29x}}_{\text{22}}{\text{+38x}}_{\text{32}}{\text{+45x}}_{\text{42}}+S_1{}^{+}-S_1{}^{-}\text{=70}\\ {\text{29x}}_{\text{23}}{\text{+38x}}_{\text{33}}{\text{+45x}}_{\text{43}}+S_2{}^{+}-S_2{}^{-}\text{=80}\\ {\text{29x}}_{\text{24}}{\text{+38x}}_{\text{34}}{\text{+45x}}_{\text{44}}+S_3{}^{+}-S_3{}^{-}\text{=95}\\ {\text{29x}}_{\text{25}}{\text{+38x}}_{\text{35}}{\text{+45x}}_{\text{45}}+S_4{}^{+}-S_4{}^{-}\text{=120}\end{array}$

$\begin{array}{l}{\text{340x}}_{11}{\text{+340x}}_{\text{16}}\text{+}\\ {\text{300x}}_{\text{22}}{\text{+300x}}_{\text{23}}{\text{+300x}}_{\text{24}}+{\text{300x}}_{\text{25}}\\ {\text{840x}}_{\text{32}}{\text{+840x}}_{\text{33}}{\text{+840x}}_{\text{44}}+{\text{840x}}_{35}\\ {\text{85x}}_{\text{42}}{\text{+85x}}_{\text{43}}{\text{+85x}}_{\text{44}}+85{\text{x}}_{\text{45}}+S_5{}^{+}-S_5{}^{-}=1\text{200}\end{array}$

$\begin{array}{l}{\text{31x}}_{11}{\text{+27x}}_{\text{16}}\text{+}\\ {\text{25x}}_{\text{22}}{\text{+21x}}_{\text{23}}{\text{+32x}}_{\text{24}}+{\text{32x}}_{\text{25}}\\ {\text{37x}}_{\text{32}}{\text{+29x}}_{\text{33}}{\text{+28x}}_{44}+38{\text{x}}_{\text{35}}\\ {\text{20x}}_{\text{42}}{\text{+23x}}_{\text{43}}{\text{+22x}}_{\text{44}}+20{\text{x}}_{\text{45}}+S_6{}^{+}-S_6{}^{-}\text{=200}\end{array}$

$\begin{array}{l}{\text{80x}}_{11}{\text{+80x}}_{\text{16}}\text{+}\\ {\text{36x}}_{\text{22}}{\text{+36x}}_{\text{23}}{\text{+36x}}_{\text{24}}+{\text{36x}}_{\text{25}}\\ {\text{50x}}_{\text{32}}{\text{+50x}}_{\text{33}}{\text{+50x}}_{35}\\ {\text{17x}}_{\text{42}}{\text{+17x}}_{\text{43}}{\text{+17x}}_{\text{44}}+17{\text{x}}_{\text{45}}+S_7{}^{+}-S_7{}^{-}\text{=200}\end{array}$

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 $S_{\text{i}}{}^{-}$ is included in minimization equation. If left hand side is more than right hand side, then $S_{\text{i}}{}^{+}$ is included in minimizations equation.

Hence the minimization equation is,

Minimize, ${\text{z=P}}_{\text{1}}{S}_1{}^{-}{\text{+P}}_1{S}_2{}^{-}{\text{+P}}_1{S}_3{}^{-}{\text{+P}}_1{S}_4{}^{-}{\text{+P}}_2{S}_5{}^{-}{\text{+P}}_3{S}_6{}^{+}{\text{+P}}_4{S}_7{}^{-}$

For each Priority assign a value of ${\text{P}}_{\text{1}}=4,{\text{P}}_2=3,{\text{P}}_3=2,{\text{P}}_{\text{1}}=1$ for priority $1,2,3,4$ respectively