A client asks his stockbroker to invest $100,000 for maximum annual income, subject to the three conditions:

- 1. Spread the investment over no more than three different stocks.
- 2. Put no more than 40 percent of the money into any one stock.
- 3. Put a minimum of $10,000 into an oil stock.

The broker has identified three stocks for investment. Their estimated annual returns per share and price per share are shown in the following table:

Stock | Price per share | Estimate annual return per share |
---|---|---|

Oil | $120 | $11 |

Auto | $62 | $4 |

Pharmaceutical | $18 | $2 |

The client wishes to use LP to determine the optimal number of shares to buy for each stock, so as to maximize the total estimated return.

a) Write down the decision variables.

b) Write down the optimization statement for the objective function.

c) Write down the constraints.

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a) The decision variables are the number of shares to be bought for each of the three stocks.

Let x1 be the number of shares to be bought for the oil stock. Let x2 be the number of shares to be bought for the auto stock. Let x3 be the number of shares to be bought for the pharmaceutical stock.

b) The objective function to be maximized is the total estimated return of the investment. It can be written as follows:

maximize 11x1 + 4x2 + 2x3

This objective function is the sum of the estimated annual returns of each stock multiplied by the number of shares bought for that stock.

c) The constraints are as follows:

120x1 + 62x2 + 18x3 ≤ 100,000

This constraint ensures that the total investment does not exceed the available amount.

x1 + x2 + x3 ≤ 3

This constraint ensures that the investment is diversified across a maximum of three stocks.

x1 ≤ 0.4(120x1 + 62x2 + 18x3) x2 ≤ 0.4(120x1 + 62x2 + 18x3) x3 ≤ 0.4(120x1 + 62x2 + 18x3)

These constraints ensure that no more than 40% of the investment is put into any one stock.

x1 ≥ 10,000/120

This constraint ensures that the minimum investment required in the oil stock is met.

Therefore, the complete linear programming problem can be written as follows:

maximize 11x1 + 4x2 + 2x3 subject to: 120x1 + 62x2 + 18x3 ≤ 100,000 x1 + x2 + x3 ≤ 3 x1 ≤ 0.4(120x1 + 62x2 + 18x3) x2 ≤ 0.4(120x1 + 62x2 + 18x3) x3 ≤ 0.4(120x1 + 62x2 + 18x3) x1 ≥ 83.33