This is the statement of my problem:
A new financial product is made so that if you invest X thousands of pounds a day, and a third more the next day, the third day we would recover triple of the investment from the first day. How should we invest 100,000 pounds during a full week for us to have the maximum money at the end of the week?
My attempt:
My decision variables are: $x_1$ = amount of pounds (in thousands of pounds) invested the first day, $x_2$ = amount of pounds invested the third day, $x_3$ = amount of pounds invested the fifth day.
It is supposed that you can reinvest the benefit you obtain during the week, and you can not invest more than the indicated (for instance, the second day you only invest $4/3x_1$).
Therefore, the objective would be: $3x_1 + 3x_2 + 3x_3 - \frac{7}3(x_1 + x_2 + x_3)$ discounting to profit what is spent.
And my restrictions: $0 \leq x_1 \leq 100$, $0 \leq x_2 \leq 100 - \frac{7}3x_1 + 3x_1$ and $0 \leq x_3 \leq 100 - \frac{7}3x_1+3x_1-\frac{7}3x_2+3x_2$
Could anyone check my solution and tell me if it is correct, or should I change anything?
Thank you!
Your formulation is over-predicting/assuming behaviors about the investments in that you’re only looking to invest on every third day. Since this is a seven day week in which our returns are delayed by two days, it’s possible to invest everyday like so:
$$\max z = 3(x_1 + x_2 + x_3 + x_4+x_5) - (1/3)(x_1 + x_2 + x_3 + x_4 + x_5)$$
Subject to:
$$\sum_{i=1}^5 x_i + (1/3)x_i \le 100$$
$$x_i \ge 0\quad\forall i\in[1,\ldots,5]$$
If we can reinvest our earnings like you mentioned, then it would go like the following:
$$\max z = 3(x_4+x_5) - (1/3)x_5$$
Subject to:
$$x_1 \le 100$$ $$x_2 \le 100 - (4/3)x_1$$ $$x_3 \le 100 + 3x_1 - (4/3)x_2$$ $$x_4 \le 100 + 3x_2 - (4/3)x_3$$ $$x_5 \le 100 + 3x_3 - (4/3)x_4$$
$$x_i \ge 0\quad\forall i\in[1,\ldots,5]$$