A problem frequently encountered by managers of banks, mutual funds, investment services, and insurance companies is the selection of specific investments from among a wide variety of alternatives. The manager’s overall objective is usually to maximize expected return on investment, given a set of legal, policy, or risk restraints.

For example, the International City Trust (ICT) invests in short-term trade credits, corporate bonds, gold stocks, and construction loans. To encourage a diversified portfolio, the board of directors has placed limits on the amount that can be committed to any one type of investment. ICT has $5 million available for immediate investment and wishes to do two things: (1) maximize the return on the investments made over the next 6 months and (2) satisfy the diversification requirements as set by the board of directors.

The specifics of the investment possibilities are as follows:

In addition, the board specifies that at least 55% of the funds invested must be in gold stocks and construction loans and that no less than 15% must be invested in trade credits.

In formulating this as an LP, the objective is to maximize the return. There are four separate constraints limiting the maximum amount in each investment option to the maximum given in the table. The fifth constraint specifies that the total amount in gold stocks and construction loans must be at least 55% of the total amount invested, and the next constraint specifies that the total amount in trade credit must be at least 15% of the total amount invested. The final constraint stipulates that the total amount invested cannot exceed $5 million (it could be less). Define the variables as

The total amount invested is $X_1+X_2+X_3+X_4,$ which may be less than $5 million. This is important when calculating 55% of the total amount invested and 15% of the total amount invested in two of the constraints.

Objective:

Program 8.6 shows the solution found using Solver in Excel. ICT maximizes its interest earned by making the following investment: $X_1=\$750,000, X_2=\$950,000, X_3=\$1,500,000,$ and $X_4=\$1,800,000.$ The total interest earned is $712,000.

The truck loading problem involves deciding which items to load on a truck so as to maximize the value of a load shipped. As an example, we consider Goodman Shipping, an Orlando firm owned by Steven Goodman. One of his trucks, with a capacity of 10,000 pounds, is about to be loaded.¹ Awaiting shipment are the following items:

Each of these six items, we see, has an associated dollar value and weight.

The objective is to maximize the total value of the items loaded onto the truck without exceeding the truck’s weight capacity. We let $X_i$ be the proportion of each item i loaded on the truck:

These final six constraints reflect the fact that at most one “unit” of an item can be loaded onto the truck. In effect, if Goodman can load a portion of an item (say, item 1 is a batch of 1,000 folding chairs, not all of which need be shipped together), the $X_i\text{s}$ will all be proportions ranging from 0 (nothing) to 1 (all of that item loaded).

To solve this LP problem, we turn to Excel’s Solver. Program 8.7 shows Goodman’s Excel formulation, input data, and the solution, which yields a total load value of $31,500.

The answer leads us to an interesting issue that we deal with in detail in Chapter 10. What does Goodman do if fractional values of items cannot be loaded? For example, if electric cars are the items being loaded, we clearly cannot ship one-third of a Tesla.

If the proportion of item 1 was rounded up to 1.00, the weight of the load would increase to 15,000 pounds. This would violate the maximum weight constraint of 10,000 pounds. Therefore, the fraction of item 1 must be rounded down to zero. This would drop the weight of the load to 7,500 pounds, leaving 2,500 pounds of the load capacity unused. Because no other item weighs less than 2,500 pounds, the truck cannot be filled up further.

Thus, we see that by using regular LP and rounding the fractional weights, the truck would carry only item 2, for a load weight of 7,500 pounds and a load value of $24,000.

QM for Windows and spreadsheet optimizers such as Excel’s Solver are capable of dealing with integer programming problems as well—that is, LP problems requiring integer solutions. Using Excel, the integer solution to Goodman’s problem is to load items 3, 4, and 6, for a total weight of 10,000 pounds and load value of $27,250.