## Related questions with answers

An insurance company believes that it will require the following numbers of personal computers during the next six months: January, 9; February, 5; March, 7; April, 9; May, 10; June, 5. Computers can be rented for a period of one, two, or three months at the following unit rates: one-month rate, $\$200$; two-month rate, $\$350$; three-month rate, $\$450$. Formulate an LP that can be used to minimize the cost of renting the required computers. You may assume that if a machine is rented for a period of time extending beyond June, the cost of the rental should be prorated. For example, if a computer is rented for three months at the beginning of May, then a rental fee of $\frac{2}{3}(450)=\$ 300$, not $\$450$, should be assessed in the objective function.

Solution

VerifiedThe following number of personal computers are required during the next six months:$9,5,7,9,10,5$ respectively. Price of one month rental of computer is $200.$ For two or three months the price is $350$ and $450$ respectively. We want to minimize the cost of renting the computers. Let $x_t$ be the number of rented computers in the month $t$ at one-month rate, while $y_t$ and $z_t$ are number of rented computers in month $t$ at two-month and three-month rates respectively. Since we want to minimize the cost of renting the required computers we have:

$\begin{align*} \min z &=200(x_1+x_2+x_3+x_4+x_5+x_6)+350(y_1+y_2+y_3+y_4+y_5+\frac{1}{2}y_6)\\[10pt] &=+450(z_1+z_2+z_3+z_4+\frac{2}{3}z_5+\frac{1}{3}z_6). \end{align*}$

However, we know precisely how much computers are required for each month. Therefore let us define: $i_1=x_1+y_1+z_1.$ This will imply:

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

$i_4=i_3-z_1-y_2-x_3+x_4+y_4+z_4,i_5=i_4-x_4-y_3-z_2+x_5+y_5+z_5$

$i_6=i_5-x_5-y_4-z_3+x_6+y_6+z_6.$

We established relationships between variables that count active computers in month. Now the other constraints are just:

$i_1\geq9,i_2\geq5,i_3\geq7,i_4\geq9,i_5\geq10,i_6\geq5.$

With this and the fact that all variables are nonnegative we specified all constraints. Hence, we are done.

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