The Standard Linear Programming Problem

Let us begin this chapter with a pair of examples.

Example 7.1a The following investment opportunity is available from Monday to Friday of the forthcoming week. For any amount x, if you invest x on a day and 2x on the next day then you will receive 4x at the beginning of the third day. The amount received at the beginning of a day can then be used that day for starting a new investment or for continuing an ongoing investment. There are no payments received after Friday (so one should not begin a new investment on Thursday or Friday). If you invest x but cannot put down 2x on the next day then you lose the initial investment x. For instance, starting with a fortune of 1, it does not pay to invest more than 1/3 on day 1. For if x > 1/3 is invested on day 1 then your remaining capital of 1 − x will not be sufficient to enable you to invest 2x on day 2, resulting in a forfeit of the initial investment x. On the other hand, you could invest 1/3 on day 1 and 2/3 on day 2, which yields the return 4/3 at the beginning of day 3; you could then use that amount to begin a second investment by investing 4/9 (one third of 4/3) on day 3 and 8/9 on day 4, thus earning a return of 16/9 on day 5. Can you do better?