Chapter 4.16, Problem 10P

Explanation of Solution

Formulation of pre-emptive goal programming model:

As given in the problem, the store employs 5 full-time employees and 3 part-time employees.

From each sale, the Ricky record store gets a profit of $3

Let the variables ${\text{x}}_{\text{1}}$, ${\text{x}}_{\text{2}}$, ${\text{x}}_{\text{3}}$, ${\text{x}}_{\text{4}}$, ${\text{x}}_{\text{5}}$ be the number of hours worked by full-time employees and ${\text{x}}_{\text{6}}$, ${\text{x}}_{\text{7}}$, ${\text{x}}_{\text{8}}$ be the number of hours worked by part-time employees.

Per week, full time employees work for 40 hours and per week, part time employees work for 20 hours. If required full time employees have to work over time. From this, the user can obtain overtime hours worked,

$\begin{array}{l}{\text{x}}_{\text{1}}{\text{+y}}_{\text{1}}{}^{\text{+}}{\text{-y}}_{\text{1}}{}^{\text{-}}\text{=40}\\ {\text{x}}_2{\text{+y}}_2{}^{\text{+}}{\text{-y}}_2{}^{\text{-}}\text{=40}\\ {\text{x}}_3{\text{+y}}_3{}^{\text{+}}{\text{-y}}_3{}^{\text{-}}\text{=40}\\ {\text{x}}_4{\text{+y}}_4{}^{\text{+}}{\text{-y}}_4{}^{\text{-}}\text{=40}\end{array}$

From the above, ${\text{y}}_{\text{1}}{}^{\text{-}},{\text{y}}_2{}^{\text{-}},{\text{y}}_3{}^{\text{-}},{\text{y}}_4{}^{\text{-}}$ gives the number of overtime worked.

Per week, the part-time employees work for maximum of 20 hours. Thus, the constraints are,

$\begin{array}{l}{\text{x}}_{\text{6}}\le \text{20}\\ {\text{x}}_7\le \text{20}\\ {\text{x}}_8\le \text{20}\end{array}$

The full time employee is paid $6 in regular hours and $10 in overtime. Part-time employees are paid $3.

Ricky has a weekly budget of $500. Therefore, expenses should be less than 500. So,

${\text{6(x}}_{\text{1}}{\text{+x}}_{\text{2}}{\text{+x}}_{\text{3}}{\text{+x}}_{\text{4}}{\text{+x}}_{\text{5}}{\text{)+10(y}}_{\text{1}}{}^{\text{-}}{\text{+y}}_{\text{2}}{}^{\text{-}}{\text{+y}}_{\text{3}}{}^{\text{-}}{\text{+y}}_{\text{4}}{}^{\text{-}}{\text{)+3(x}}_{\text{6}}{\text{+x}}_{\text{7}}{\text{+x}}_{\text{8}}\text{)}\le \text{500}$

$\begin{array}{l}{\text{6x}}_{\text{1}}{\text{+6x}}_{\text{2}}{\text{+6x}}_{\text{3}}{\text{+6x}}_{\text{4}}{\text{+6x}}_{\text{5}}{\text{+10y}}_{\text{1}}{}^{\text{-}}{\text{+10y}}_{\text{2}}{}^{\text{-}}{\text{+10y}}_{\text{3}}{}^{\text{-}}{\text{+10y}}_{\text{4}}{}^{\text{-}}{\text{+3x}}_{\text{6}}{\text{+3x}}_{\text{7}}{\text{+3x}}_{\text{8}}\text{)}\le \text{500}\\ \end{array}$

Goal 1: Sell at least 1,600 records per week

The constraint formed is given below,

$\begin{array}{l}{\text{5(x}}_{\text{1}}{\text{+x}}_{\text{2}}{\text{+x}}_{\text{3}}{\text{+x}}_{\text{4}}{\text{+x}}_{\text{5}}{\text{)+3(x}}_{\text{6}}{\text{+x}}_{\text{7}}{\text{+x}}_{\text{8}}\text{)}\le \text{1600}\\ {\text{5x}}_{\text{1}}{\text{+5x}}_{\text{2}}{\text{+5x}}_{\text{3}}{\text{+5x}}_{\text{4}}{\text{+5x}}_{\text{5}}{\text{)+3x}}_{\text{6}}{\text{+3x}}_{\text{7}}{\text{+3x}}_{\text{8}}\le \text{1600}\end{array}$

Goal 2: Earn a profit of at least 2,200 per week.

The constraint formed is given below,

$\begin{array}{l}{\text{3(5(x}}_{\text{1}}{\text{+x}}_{\text{2}}{\text{+x}}_{\text{3}}{\text{+x}}_{\text{4}}{\text{+x}}_{\text{5}}{\text{)+3(x}}_{\text{6}}{\text{+x}}_{\text{7}}{\text{+x}}_{\text{8}}\text{))}\le 22\text{00}\\ {\text{15x}}_{\text{1}}{\text{+15x}}_{\text{2}}{\text{+15x}}_{\text{3}}{\text{+15x}}_{\text{4}}{\text{+15x}}_{\text{5}}{\text{)+9x}}_{\text{6}}{\text{+9x}}_{\text{7}}{\text{+9x}}_{\text{8}}\le 22\text{00}\end{array}$

Goal 3: full time employees should work at most 100 hours of overtime.

The constraint formed is given below,

${\text{y}}_{\text{1}}{}^{\text{-}}{\text{+y}}_{\text{2}}{}^{\text{-}}{\text{+y}}_{\text{3}}{}^{\text{-}}{\text{+y}}_{\text{4}}{}^{\text{-}}{\text{+y}}_{\text{5}}{}^{\text{-}}\le \text{100}$

From the above, the number of overtime hours worked is given by ${\text{y}}_{\text{1}}{}^{\text{-}}{\text{,y}}_{\text{2}}{}^{\text{-}}{\text{,y}}_{\text{3}}{}^{\text{-}}{\text{,y}}_{\text{4}}{}^{\text{-}}$

Goal 4: number of hours worked by full time employees should be close to 40.

The constraint formed is given below,

${\text{x}}_{\text{1}}{\text{+x}}_{\text{2}}{\text{+x}}_{\text{3}}{\text{+x}}_{\text{4}}{\text{+x}}_{\text{5}}\ge 200$

From the above, the user can observe that for all the above constraints, there is no feasible region means all the constraints cannot be met