Chapter 4.16, Problem 12P

Explanation of Solution

Selecting the project using the preemptive goal programming:

Consider the following variables $x_1,x_2,\mathrm{........},x_8$,

${\text{x}}_{\text{i}}\text{=}\{\begin{array}{l}\text{1 If project i is build}\\ \text{0 otherwise}\end{array}$

For every project values related to each attribute are tabulated.

The table is given below,

	1	2	3	4	5	6	7	8
ROI	2070	456	670	350	495	380	1500	480
Cost	900	240	335	700	410	190	500	160
Productivity improvement	3	2	2	0	1	0	3	2
Manpower needed	18	18	27	36	42	6	48	24
Degree of risk	3	2	4	1	1	0	2	3

The company has the following goals.

Goal 1: Archive a return on investment of at least $\$3,250$. Thus the constraint formed is,

${\text{2070x}}_{\text{1}}{\text{+456x}}_{\text{2}}{\text{+670x}}_{\text{3}}{\text{+350x}}_{\text{4}}{\text{+495x}}_{\text{5}}{\text{+380x}}_{\text{6}}{\text{+1500x}}_{\text{7}}{\text{+480x}}_{\text{8}}\ge \text{3250}$

Goal 2: Limit cost to $\$1300$. Thus the constraint formed is,

${\text{900x}}_{\text{1}}{\text{+240x}}_{\text{2}}{\text{+335x}}_{\text{3}}{\text{+700x}}_{\text{4}}{\text{+410x}}_{\text{5}}{\text{+190x}}_{\text{6}}{\text{+500x}}_{\text{7}}{\text{+160x}}_{\text{8}}\le 1300$

Goal 3: Archive a productivity improvement of at least 6. Thus the constraint formed is,

${\text{3x}}_{\text{1}}{\text{+2x}}_{\text{2}}{\text{+2x}}_{\text{3}}{\text{+x}}_{\text{5}}{\text{+3x}}_{\text{7}}{\text{+2x}}_{\text{8}}\le 6$

Goal 4: Limit manpower use to 108. Thus the constraint formed is,

${\text{18x}}_{\text{1}}{\text{+18x}}_{\text{2}}{\text{+27x}}_{\text{3}}{\text{+36x}}_{\text{4}}+42{\text{x}}_{\text{5}}{\text{+6x}}_{\text{6}}+48{\text{x}}_{\text{7}}{\text{+24x}}_{\text{8}}\le 108$

Goal 5: Limit technological risk to a total of 4. Thus the constraint formed is

${\text{3x}}_{\text{1}}{\text{+2x}}_{\text{2}}{\text{+4x}}_{\text{3}}{\text{+x}}_{\text{4}}+{\text{x}}_{\text{5}}{\text{+6x}}_{\text{6}}+2{\text{x}}_{\text{7}}{\text{+3x}}_{\text{8}}\le 4$

Hence the linear programming can be formulated as given below,

Minimize $z={\text{x}}_{\text{1}}{\text{+x}}_{\text{2}}{\text{+x}}_{\text{3}}{\text{+x}}_{\text{4}}+{\text{x}}_{\text{5}}{\text{+x}}_{\text{6}}+{\text{x}}_{\text{7}}{\text{+x}}_{\text{8}}$

Subject to the constraints,

$\begin{array}{c}{\text{2070x}}_{\text{1}}{\text{+456x}}_{\text{2}}{\text{+670x}}_{\text{3}}{\text{+350x}}_{\text{4}}{\text{+495x}}_{\text{5}}{\text{+380x}}_{\text{6}}{\text{+1500x}}_{\text{7}}{\text{+480x}}_{\text{8}}\ge \text{3250}\\ {\text{900x}}_{\text{1}}{\text{+240x}}_{\text{2}}{\text{+335x}}_{\text{3}}{\text{+700x}}_{\text{4}}{\text{+410x}}_{\text{5}}{\text{+190x}}_{\text{6}}{\text{+500x}}_{\text{7}}{\text{+160x}}_{\text{8}}\le 1300\\ {\text{3x}}_{\text{1}}{\text{+2x}}_{\text{2}}{\text{+2x}}_{\text{3}}{\text{+x}}_{\text{5}}{\text{+3x}}_{\text{7}}{\text{+2x}}_{\text{8}}\le 6\\ {\text{18x}}_{\text{1}}{\text{+18x}}_{\text{2}}{\text{+27x}}_{\text{3}}{\text{+36x}}_{\text{4}}+42{\text{x}}_{\text{5}}{\text{+6x}}_{\text{6}}+48{\text{x}}_{\text{7}}{\text{+24x}}_{\text{8}}\le 108\end{array}$

$\begin{array}{c}{\text{3x}}_{\text{1}}{\text{+2x}}_{\text{2}}{\text{+4x}}_{\text{3}}{\text{+x}}_{\text{4}}+{\text{x}}_{\text{5}}{\text{+6x}}_{\text{6}}+2{\text{x}}_{\text{7}}{\text{+3x}}_{\text{8}}\le 4\\ {\text{x}}_{\text{i}}\ge \text{0}\end{array}$

Here, it is observed that for these set of constraints, there is no region which is feasible. That is all the constraints cannot be met. Hence e assign a cost value incurred if any of the goal is not meet. So introduce deviational variables as follows