Chapter 3, Problem 32RP

Explanation of Solution

 Formulation of Linear Programming (LP) to maximize Eastinghouse’s daily profit:

 The given table shows the company’s hourly salary paid to the employee, price charged for each capacitor made, and the number of defects in each capacitor produced during each shift.

 Let “x₁” be the number of workers assigned to shift 8AM to 4PM, “x₂” be the number of workers assigned to shift 4PM to midnight, and “x₃” be the number of workers assigned to shift midnight to 8AM.

 The total number of capacitors produced each day is given by,

 $10x_1+10x{2}_{}+10x{3}_{}$

 The revenue generated from these capacitors is,

 $18\left(10x_1\right)+22\left(10x{2}_{}\right)+24\left(10x{3}_{}\right)$

 The hourly labor cost is given by,

 $12x_1+16x{2}_{}+20x{3}_{}$

 Since, each worker works for 8 hours, so per day labor cost is,

 $8\left(12x_1+16x{2}_{}+20x{3}_{}\right)$

 The profit is given by,

 $\begin{array}{c}\text{Profit = Revenue - Cost}\\ =18\left(10x_1\right)+22\left(10x_2\right)+24\left(10x_3\right)-18\left(12x_1+16x_2+20x_3\right)\\ =84x_1+92x_2+80x_3\end{array}$

 The constraints of the problem are as follows:

 At most 10 workers can be assigned to any shift. Therefore,

 $x_1+x_2+x_3\le 25$

 At most 250 capacitors can be sold daily. Therefore,

 $\begin{array}{c}10\left(x_1+x_2+x_3\right)\le 250\\ x_1+x_2+x_3\le 25\end{array}$

 The average number of defects per capacitor cannot exceed 3