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GOAL PROGRAMMING: FORMULATION AND GRAPHICAL SOLUTION 599

Figure 14.2 Portfolios that Satisfy the P 1 Goal

H
3000
Number of Shares of Hub Properties 2000 Feasible + – Available Funds: 25U + 50H = 80 000
Priority Level 1 Goal Equation
with d 1 = d 1 = 0; 0.5U + 0.25H = 700

+
d = 0
1
1000
Portfolios
+
That Will
d 1 > 0
Achieve Priority
Level 1 Goal
U
0 1000 2000 3000 4000
Number of Shares of UK Oil

þ
also result in the value of d ¼ 0. Thus, the shaded region contains all the feasible
1
solution points that achieve the priority level 1 goal.
At this point, we have solved the priority level 1 problem. Note that alternative
optimal solutions are possible; in fact, all solution points in the shaded region in
þ
Figure 14.2 maintain a portfolio risk index of 700 or less, and hence d ¼ 0.
1
The priority level 2 goal for the Nicolo Investment problem is to find a portfolio
that will provide an annual return of at least £9000. Is overachieving the target value
of £9000 a concern? Clearly, the answer is no because portfolios with an annual
return of more than £9000 correspond to higher returns. Is underachieving the
target value of £9000 a concern? The answer is yes because portfolios with an annual
return of less than £9000 are not acceptable to the client. Thus, the objective
function corresponding to the priority level 2 linear programme should minimize

the value of d . However, because goal 2 is a secondary goal, the solution to the
2
priority level 2 linear programme must not degrade the optimal solution to the
priority level 1 problem. Thus, the priority level 2 linear programme can now be
stated.

P 2 Problem
Min d
2
s:t:
25U þ 50H 80 000 Funds available
þ
0:50U þ 0:25H d þ d ¼ 700 P 1 goal
1
1
þ

3U þ 5H d þ d ¼ 9 000 P 2 goal
2 2
d þ ¼ 0 Maintain achievement of P 1 goal
1
þ

þ

U; H; d ; d ; d ; d 0
1 1 2 2
Note that the priority level 2 linear programme differs from the priority level 1 linear
programme in two ways. The objective function involves minimizing the amount by
which the portfolio annual return underachieves the level 2 goal, and another
constraint has been added to ensure that no amount of achievement of the priority
level 1 goal is sacrificed.
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