Chapter 4.14, Problem 7P

Explanation of Solution

Maximizing profit of Clothco for the next 6 months:

Consider the following variables,

${\text{i}}_{\text{t}}$= Pair of pants in inventory at the end of month $\text{t}$.

$s_{\text{t}}$= Pair of pants sold in inventory at the end of month $\text{t}$.

$w_{\text{t}}$= Workers employed during moth $\text{t}$.

$z$= Profit generated.

Defining a new function for defining the workers working time:

${\text{x}}_{\text{t}}{\text{=x}}_{\text{t}}{\text{'-x}}_{\text{t}}\text{''}$

In the above equation,

${\text{x}}_{\text{t}}$= The change of number of workers from month $\text{t-1}$ to month $\text{t}$.

${\text{x}}_{\text{t}}\text{'}$= The number of workers hired at the start of month $\text{t}$.

${\text{x}}_{\text{t}}\text{''}$= The number of workers fired at the start of month $\text{t}$.

So, the objective function is given below,

Maximize, $\text{z=}\left(\begin{array}{c}{\text{40(s}}_{\text{1}}{\text{+s}}_{\text{2}}{\text{+s}}_{\text{3}}{\text{+s}}_{\text{4}}{\text{+s}}_{\text{5}}{\text{+s}}_{\text{6}}{\text{)-10(p}}_{\text{1}}{\text{+p}}_{\text{2}}{\text{+p}}_{\text{3}}{\text{+p}}_{\text{4}}{\text{+p}}_{\text{5}}{\text{+p}}_{\text{6}}\text{)}\\ {\text{-5(i}}_{\text{1}}{\text{+i}}_{\text{2}}{\text{+i}}_{\text{3}}{\text{+i}}_{\text{4}}{\text{+i}}_{\text{5}}{\text{+i}}_{\text{6}}{\text{)-2000(w}}_{\text{1}}{\text{+w}}_{\text{2}}{\text{+w}}_{\text{3}}{\text{+w}}_{\text{4}}{\text{+w}}_{\text{5}}{\text{+w}}_{\text{6}}\text{)}\\ {\text{-1000(x}}_{\text{1}}{\text{''+x}}_{\text{2}}{\text{''+x}}_{\text{3}}{\text{''+x}}_{\text{4}}{\text{''+x}}_{\text{5}}{\text{''+x}}_{\text{6}}\text{'')}\\ {\text{-1500(x}}_{\text{1}}{\text{'+x}}_{\text{2}}{\text{'+x}}_{\text{3}}{\text{'+x}}_{\text{4}}{\text{'+x}}_{\text{6}}{\text{'+x}}_{\text{7}}\text{')}\end{array}\right)$

Subject to the constraints,