Chapter 3.5, Problem 1P

Explanation of Solution

Formulation of LP:

Let,

$x_1$= Number of fulltime employees (FTE) who start work on Sunday

$x_2$= Number of FTE who start work on Monday

$x_3$= Number of FTE who start work on Tuesday

$x_4$= Number of FTE who start work on Wednesday

$x_5$= Number of FTE who start work on Thursday

$x_6$= Number of FTE who start work on Friday

$x_7$= Number of FTE who start work on Saturday

$x_8$= Number of part-time employees (PTE) who start work on Sunday,

$x_9$= Number of PTE who start work on Saturday.

$x_{10}$= Number of PTE who start work on Saturday.

$x_{11}$= Number of PTE who start work on Saturday.

$x_{12}$= Number of PTE who start work on Saturday.

$x_{13}$= Number of PTE who start work on Saturday.

$x_{14}$= Number of PTE who start work on Saturday.

The objective of the problem is to minimize the office’s weekly labour costs.

Objective function:

Minimize,

$z=15\left(8\right)\left(5\right)\left(x_1+x_2+x_3+x_4+x_5+x_6+x_7\right)+10\left(4\right)\left(5\right)\left(x_8+x_9+x_{10}+x_{11}\text{+}{x}_{12}+x_{13}+x_{14}\right)$

Explanation:

A full time employees work for $8\times 5$ hours a week and part time employee work for

$4\times 5$ hours in a week. The full time employees get $15 per hours and par time employees gets $10 per hour.

Constraints:

$\begin{array}{l}\text{8}\left({\text{x}}_{\text{1}}{\text{+x}}_{\text{4}}{\text{+x}}_{\text{5}}{\text{+x}}_{\text{6}}{\text{+x}}_{\text{7}}\right)\text{+4}\left({\text{x}}_{\text{8}}{\text{+x}}_{\text{11}}{\text{+x}}_{\text{12}}{\text{+x}}_{\text{13}}{\text{+x}}_{\text{14}}\right)\ge \text{88 }\left(\text{Sunday}\right)\\ \text{8}\left({\text{x}}_{\text{1}}{\text{+x}}_{\text{2}}{\text{+x}}_{\text{5}}{\text{+x}}_{\text{6}}{\text{+x}}_{\text{7}}\right)\text{+4}\left({\text{x}}_{\text{8}}{\text{+x}}_{\text{9}}{\text{+x}}_{\text{12}}{\text{+x}}_{\text{13}}{\text{+x}}_{\text{14}}\right)\ge \text{136 }\left(\text{Monday}\right)\\ \text{8}({\text{x}}_{\text{1}}{\text{+x}}_{\text{2}}{\text{+x}}_{}\end{array}$