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GOAL PROGRAMMING: FORMULATION AND GRAPHICAL SOLUTION 595

To illustrate how to use the risk index per share to measure the total portfolio risk,
suppose that Nicolo chooses a portfolio that invests all £80 000 in UK Oil, the higher
risk, but higher return, investment. Nicolo could purchase £80 000/£25 ¼ 3200
shares of UK Oil, and the portfolio would have a risk index of 3200(0.50) ¼ 1600.
Conversely, if Nicolo purchases no shares of either stock, the portfolio will have no
risk, but also no return. Thus, the portfolio risk index will vary from 0 (least risk) to
1600 (most risk).
Nicolo’s client would like to avoid a high-risk portfolio; thus, investing all funds
in UK Oil would not be desirable. However, the client agreed that an acceptable
level of risk would correspond to portfolios with a maximum total risk index of 700.
Thus, considering only risk, one goal is to find a portfolio with a risk index of 700
or less.
Another goal of the client is to obtain an annual return of at least £9000. This goal
can be achieved with a portfolio consisting of 2000 shares of UK Oil [at a cost of
2000(£25) ¼ £50 000] and 600 shares of Hub Properties [at a cost of 600(£50) ¼
£30 000]; the annual return in this case would be 2000(£3) + 600(£5) ¼ £9000. Note,
however, that the portfolio risk index for this investment strategy would be
2000(0.50) + 600(0.25) ¼ 1150; thus, this portfolio achieves the annual return goal
but does not satisfy the portfolio risk index goal.
So, the portfolio selection problem is a multicriteria decision problem involving
two goals: one dealing with risk and one dealing with annual return. The goal
programming approach was developed precisely for this kind of problem. Goal
programming can be used to identify a solution that comes closest to achieving both
goals. Before applying the methodology, the client must determine which, if either,
goal is more important.
Suppose that the client’s top priority goal is to restrict the risk; that is, keeping the
portfolio risk index at 700 or less is so important that the client is not willing to trade
the achievement of this goal for any amount of an increase in annual return. As long
as the portfolio risk index does not exceed 700, the client seeks the best possible
return. Based on this statement of priorities, the goals for the problem are as
follows:

Primary Goal (Priority Level 1) Goal 1: Find a portfolio that has a risk index of
700 or less.

Secondary Goal (Priority Level 2) Goal 2: Find a portfolio that will provide an
annual return of at least £9000.

The primary goal is called a priority level 1 goal, and the secondary goal is called a
priority level 2 goal. In goal programming terminology, these are called preemptive
priorities because the decision maker is not willing to sacrifice any amount of
achievement of the priority level 1 goal for the lower priority goal. The portfolio
In goal programming with risk index of 700 is the target value for the priority level 1 (primary) goal, and the
preemptive priorities, we annual return of £9000 is the target value for the priority level 2 (secondary) goal.
never permit trade-offs
between higher and The difficulty in finding a solution that will achieve these goals is that only £80 000 is
lower level goals. available for investment.

Developing the Constraints and the Goal Equations
We begin by defining the decision variables:
U ¼ number of shares of UK Oil purchased
H ¼ number of shares of Hub Properties purchased

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