Chapter 4.1, Problem 1P

Explanation of Solution

Giapetto problem

• Here ${\text{x}}_{\text{1}}$ is the number of soldiers produced each week and ${\text{x}}_{\text{2}}$ is the number of trains produced each week.
• As there are not more than 100 hours of finishing time to be used, hence the constraint is

  $\begin{array}{l}{\text{2x}}_{\text{1}}{\text{+ x}}_{\text{2}}\text{<= 100}\\ {\text{x}}_{\text{1}}{\text{+ x}}_{\text{2}}\text{<= 80}\\ {\text{x}}_{\text{1}}\text{<= 40}\end{array}$

• As there not more than 80 hours of carpentry time to be used, hence the constraint is ${\text{x}}_{\text{1}}{\text{+ x}}_{\text{2}}\text{<= 80}$
• As there are at most 40 soldiers produced per week, hence the constraint is ${\text{x}}_{\text{1}}\text{<= 40}$
• Hence the weekly profit is,

  $\text{Z = }\left({\text{27x}}_{\text{1}}{\text{+ 21x}}_{\text{2}}\right)\text{ – }\left({\text{10x}}_{\text{1}}{\text{+ 9x}}_{\text{2}}\right)\text{ – }\left({\text{14x}}_{\text{1}}{\text{+ 10x}}_{\text{2}}\right){\text{ = 3x}}_{\text{1}}{\text{+ 2x}}_{\text{2}}$

• Hence the objective function is,

  ${\text{Maximize Z = 3x}}_{\text{1}}\text{+ 2x2}$

  Subject to

  $\begin{array}{l}{\text{2x}}_{\text{1}}{\text{+ x}}_{\text{2}}\text{<= 100}\\ {\text{x}}_{\text{1}}\end{array}$