## Related questions with answers

The IRS has determined that during each of the next 12 months it will need the number of supercomputers given in Table 45. To meet these requirements, the IRS rents supercomputers for a period of one, two, or three months. It costs $100 to rent a supercomputer for one month,$180 for two months, and $250 for three months. At the beginning of month 1, the IRS has no supercomputers. Determine the rental plan that meets the next 12 months’ requirements at minimum cost. Note: You may assume that fractional rentals are okay, so if your solution says to rent 140.6 computers for one month we can round this up or down (to 141 or 140) without having much effect on the total cost. Table 45:

$\begin{matrix} \text{Month} & \text{Computer Requirements}\\ \text{1} & \text{800}\\ \text{2} & \text{1,000}\\ \text{3} & \text{600}\\ \text{4} & \text{500}\\ \text{5} & \text{1,200}\\ \text{6} & \text{400}\\ \text{7} & \text{800}\\ \text{8} & \text{600}\\ \text{9} & \text{400}\\ \text{10} & \text{500}\\ \text{11} & \text{800}\\ \text{12} & \text{600}\\ \end{matrix}$

Solution

VerifiedThe IRS rents supercomputers for periods of one,two or three months, at the different rates ofcourse. For one month the price is $\$100,$ for two months it is $\$180$ and for three months it is $\$250.$ Also let us assume that at the beggining of month $1$ there are no supercomputers. We are to construct rental plan that meets the next $12$ months requirements at minimum cost, ofcourse. Let us $x_t$ denote supercomputer rented at month $t$ for period of one month. Similarly, let $y_t$ and $z_t$ be decision variables that denote number of supercomputers rented at month $t$ for two or three months period respectively. Thus, observe the following minimization of the objective function $z:$

$\begin{align*} \min z&=100(x_1+x_2+x_3+x_4+x_5+x_6+x_7+x_8+x_9+x_{10}+x_{11}+x_{12})\\[10pt] &+180(y_1+y_2+y_3+y_4+y_5+y_6+y_7+y_8+y_9+y_{10}+y_{11}+\frac{1}{2}y_{12})\\[10pt] &+250(z_1+z_2+z_3+z_4+z_5+z_6+z_7+z_8+z_9+z_{10}+\frac{2}{3}z_{11}+\frac{1}{3}z_{12}). \end{align*}$

We are also given supercomputer requirements for each month. So let $i_t$ be the number of supercomputers in month $t.$ Obviously it must be:

$i_1=x_1+y_1+z_1,$

which gives us the general idea. Therefore it simply follows:

$i_2=i_1-x_1+x_2+y_2+z_2,i_3=i_2-x_2-y_1+x_3+y_3+z_3$

$i_t=i_{t-1}-x_{t-1}-y_{t-2}-z_{t-3}+x_t+y_t+z_t,$

ofcourse, we need to take care of indices. Since we established relationships between variables, now we see that:

$i_1\geq800,i_2\geq1000,i_3\geq600,i_4\geq500,i_5\geq1200,i_6\geq400,$

$i_7\geq800,i_8\geq600,i_9\geq400,i_{10}\geq500,i_{11}\geq800,i_{12}\geq600.$

Alos, note that all variables are nonnegative. In the assignment we assumed that renting supercomputer for three months, for example, in November means that IRS will pay only $\frac{2}{3}250.$ Ohterwise we would never use these variables since they clearly are not the optimal. With this, we are done.

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