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 $x_1,x_2,\mathrm{........},x_7$,

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

HAL has the following goals

Goal 1:

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

$40{\text{x}}_{\text{1}}{\text{+30x}}_{\text{2}}{\text{+60x}}_{\text{3}}{\text{+45x}}_{\text{4}}+55{\text{x}}_{\text{5}}{\text{+40x}}_{\text{6}}+90{\text{x}}_{\text{7}}\ge 200$

Goal 2:

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

$\text{0}{\text{.75x}}_{\text{1}}\text{+0}{\text{.70x}}_{\text{2}}\text{+0}{\text{.75x}}_{\text{3}}\text{+0}{\text{.90x}}_{\text{4}}+0.65{\text{x}}_{\text{5}}\text{+0}{\text{.60x}}_{\text{6}}+0.65{\text{x}}_{\text{7}}\ge 0.75$

Goal 3:

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

$20{\text{x}}_{\text{1}}{\text{+16x}}_{\text{2}}{\text{+12x}}_{\text{3}}{\text{+8x}}_{\text{4}}+18{\text{x}}_{\text{5}}{\text{+18x}}_{\text{6}}+19{\text{x}}_{\text{7}}\ge 15$

Goal 4:

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

$220{\text{x}}_{\text{1}}{\text{+140x}}_{\text{2}}{\text{+280x}}_{\text{3}}{\text{+240x}}_{\text{4}}+300{\text{x}}_{\text{5}}{\text{+200x}}_{\text{6}}+440{\text{x}}_{\text{7}}\le 1000$

Hence the LP is formulated as given below

Minimize, $0{\text{x}}_{\text{1}}{\text{+0x}}_{\text{2}}{\text{+0x}}_{\text{3}}{\text{+0x}}_{\text{4}}+0{\text{x}}_{\text{5}}{\text{+0x}}_{\text{6}}+0{\text{x}}_{\text{7}}$

Subjected to the constraints,

$\begin{array}{c}40{\text{x}}_{\text{1}}{\text{+30x}}_{\text{2}}{\text{+60x}}_{\text{3}}{\text{+45x}}_{\text{4}}+55{\text{x}}_{\text{5}}{\text{+40x}}_{\text{6}}+90{\text{x}}_{\text{7}}\ge 200\\ \text{0}{\text{.75x}}_{\text{1}}\text{+0}{\text{.70x}}_{\text{2}}\text{+0}{\text{.75x}}_{\text{3}}\text{+0}{\text{.90x}}_{\text{4}}+0.65{\text{x}}_{\text{5}}\text{+0}{\text{.60x}}_{\text{6}}+0.65{\text{x}}_{\text{7}}\ge 0.75\\ 20{\text{x}}_{\text{1}}{\text{+16x}}_{\text{2}}{\text{+12x}}_{\text{3}}{\text{+8x}}_{\text{4}}+18{\text{x}}_{\text{5}}{\text{+18x}}_{\text{6}}+19{\text{x}}_{\text{7}}\ge 15\\ 220{\text{x}}_{\text{1}}{\text{+140x}}_{\text{2}}{\text{+280x}}_{\text{3}}{\text{+240x}}_{\text{4}}+300{\text{x}}_{\text{5}}{\text{+200x}}_{\text{6}}+440{\text{x}}_{\text{7}}\le 1000\end{array}$

${\text{x}}_{\text{i}}\text{=0}$ or $\text{1}$

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.

$S_{\text{i}}{}^{-}$= 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

Hence the constraints become,

$\begin{array}{c}40{\text{x}}_{\text{1}}{\text{+30x}}_{\text{2}}{\text{+60x}}_{\text{3}}{\text{+45x}}_{\text{4}}+55{\text{x}}_{\text{5}}{\text{+40x}}_{\text{6}}+90{\text{x}}_{\text{7}}+S_1{}^{+}-S_1{}^{-}=200\\ \text{0}{\text{.75x}}_{\text{1}}\text{+0}{\text{.70x}}_{\text{2}}\text{+0}{\text{.75x}}_{\text{3}}\text{+0}{\text{.90x}}_{\text{4}}+0.65{\text{x}}_{\text{5}}\text{+0}{\text{.60x}}_{\text{6}}+0.65{\text{x}}_{\text{7}}+S_2{}^{+}-S_2{}^{-}=0.75\\ 20{\text{x}}_{\text{1}}{\text{+16x}}_{\text{2}}{\text{+12x}}_{\text{3}}{\text{+8x}}_{\text{4}}+18{\text{x}}_{\text{5}}{\text{+18x}}_{\text{6}}+19{\text{x}}_{\text{7}}+S_3{}^{+}-S_3{}^{-}=15\\ 220{\text{x}}_{\text{1}}{\text{+140x}}_{\text{2}}{\text{+280x}}_{\text{3}}{\text{+240x}}_{\text{4}}+300{\text{x}}_{\text{5}}{\text{+200x}}_{\text{6}}+440{\text{x}}_{\text{7}}+S_4{}^{+}-S_4{}^{-}=1000\end{array}$

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

${\text{P}}_{\text{1}}{\text{>>P}}_{\text{2}}{\text{>>P}}_{\text{3}}{\text{>>P}}_{\text{4}}$

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