Example problem
An investor wishes to invest a certain amount of money. He is evaluating ten different securities (`shares') for his investment. He estimates the return on investment for a period of one year. The following table gives for each share its country of origin, the risk category (R: high risk, N: low risk) and the expected return on investment (ROI). The investor specifies certain constraints. To spread the risk he wishes to invest at most 30% of the capital into any share. He further wishes to invest at least half of his capital in North-American shares and at most a third in high-risk shares. How should the capital be divided among the shares to obtain the highest expected return on investment?
| Number | Description | Origin | Risk | ROI |
|---|---|---|---|---|
| 1 | treasury | Canada | N | 5 |
| 2 | hardware | USA | R | 17 |
| 3 | theater | USA | R | 26 |
| 4 | telecom | USA | R | 12 |
| 5 | brewery | UK | N | 8 |
| 6 | highways | France | N | 9 |
| 7 | cars | Germany | N | 7 |
| 8 | bank | Luxemburg | N | 6 |
| 9 | software | India | R | 31 |
| 10 | electronics | Japan | R | 21 |
To construct a mathematical model, we first identify the decisions that need to be taken to obtain a solution: in the present case we wish to know how much of every share to take into the portfolio. We therefore define decision variables fracs that denote the fraction of the capital invested in share s. That means, these variables will take fractional values between 0 and 1 (where 1 corresponds to 100% of the total capital). Indeed, every variable is bounded by the maximum amount the investor wishes to spend per share: at most 30% of the capital may be invested into every share. The following constraint establishes these bounds on the variables fracs (read: `for all s in SHARES ...').
In the mathematical formulation, we write SHARES for the set of shares that the investor may wish to invest in and RETs the expected ROI per share s. NA denotes the subset of the shares that are of North-American origin and RISK the set of high-risk values.
The investor wishes to spend all his capital, that is, the fractions spent on the different shares must add up to 100%. This fact is expressed by the following equality constraint:
We now also need to express the two constraints that the investor has specified: At most one third of the values may be high-risk values—i.e., the sum invested into this category of shares must not exceed 1/3 of the total capital:
The investor also insists on spending at least 50% on North-American shares:
These two constraints are inequality constraints.
The investor's objective is to maximize the return on investment of all shares, in other terms, to maximize the following sum:
This is the objective function of our mathematical model.
After collecting the different parts, we obtain the following complete mathematical model formulation:
∑s ∈ RISK fracs ≤ 1/3
∑s ∈ NA fracs ≥ 0.5
∑s ∈ SHARES fracs = 1
∀ s ∈ SHARES: 0 ≤ fracs ≤ 0.3
In the next chapter we shall see how this mathemetical model is transformed into a Mosel model that is then solved with Xpress Optimizer. In Chapter 10 we show how to use BCL for this purpose and Chapter 15 discusses how to input this model directly into the Optimizer without modeling support.
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