The simplicity of the profitability-index method may sometimes outweigh its limitations. For example, it may not pay to worry about expenditures in subsequent years if you have only a hazy notion of future capital availability or investment opportunities. But there are also circumstances in which the limitations of the profitability-index method are intolerable. For such occasions we need a more general method for solving the capital rationing problem.
We begin by restating the problem just described. Suppose that we were to accept proportion xA of project A in our example. Then the net present value of our investment in the project would be 21xA. Similarly, the net present value of our investment in project B can be expressed as 16xB, and so on. Our objective is to select the set of projects with the highest total net present value. In other words we wish to find the values of x that maximize
Our choice of projects is subject to several constraints. First, total cash outflow in period 0 must not be greater than $10 million. In other words,
10xA + 5xB + 5xC + 0xD < 10 Similarly, total outflow in period 1 must not be greater than $10 million:
Finally, we cannot invest a negative amount in a project, and we cannot purchase more than one of each. Therefore we have
0 < xA < 1, 0 < xB < 1, ... Collecting all these conditions, we can summarize the problem as follows:
Maximize 21xA + 16xB + 12xC +13xD
One way to tackle such a problem is to keep selecting different values for the x's, noting which combination both satisfies the constraints and gives the highest net present value. But it's smarter to recognize that the equations above constitute a linear programming (LP) problem. It can be handed to a computer equipped to solve LPs.
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The answer given by the LP method is somewhat different from the one we obtained earlier. Instead of investing in one unit of project A and one of project D, we are told to take half of project A, all of project B, and three-quarters of D. The reason is simple. The computer is a dumb, but obedient, pet, and since we did not tell it that the x's had to be whole numbers, it saw no reason to make them so. By accepting "fractional" projects, it is possible to increase NPV by $2.25 million. For many purposes this is quite appropriate. If project A represents an investment in 1,000 square feet of warehouse space or in 1,000 tons of steel plate, it might be feasible to accept 500 square feet or 500 tons and quite reasonable to assume that cash flow would be reduced proportionately. If, however, project A is a single crane or oil well, such fractional investments make little sense.
When fractional projects are not feasible, we can use a form of linear programming known as integer (or zero-one) programming, which limits all the x's to integers.
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