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 $x1,x2,........,x7$ ,

$xi={1 If project i is undertaken0 otherwise$

HAL has the following goals

Goal 1:

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

$40x1+30x2+60x3+45x4+55x5+40x6+90x7≥200$

Goal 2:

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

$0.75x1+0.70x2+0.75x3+0.90x4+0.65x5+0.60x6+0.65x7≥0.75$

Goal 3:

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

$20x1+16x2+12x3+8x4+18x5+18x6+19x7≥15$

Goal 4:

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

$220x1+140x2+280x3+240x4+300x5+200x6+440x7≤1000$

Hence the LP is formulated as given below

Minimize, $0x1+0x2+0x3+0x4+0x5+0x6+0x7$

Subjected to the constraints,

$40x1+30x2+60x3+45x4+55x5+40x6+90x7≥2000.75x1+0.70x2+0.75x3+0.90x4+0.65x5+0.60x6+0.65x7≥0.7520x1+16x2+12x3+8x4+18x5+18x6+19x7≥15220x1+140x2+280x3+240x4+300x5+200x6+440x7≤1000$

$xi=0$ or $1$

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.

$Si−$ = Amount by which numerically under the $ith$ goal

$Si+$ = Amount by which numerically exceed the $ith$ goal

Hence the constraints become,

$40x1+30x2+60x3+45x4+55x5+40x6+90x7+S1+−S1−=2000.75x1+0.70x2+0.75x3+0.90x4+0.65x5+0.60x6+0.65x7+S2+−S2−=0.7520x1+16x2+12x3+8x4+18x5+18x6+19x7+S3+−S3−=15220x1+140x2+280x3+240x4+300x5+200x6+440x7+S4+−S4−=1000$

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

$P1>>P2>>P3>>P4$

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

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