Chapter 5, Problem 5P

a)

Summary Introduction

To determine: Formulate integer linear programming for the given model.

Introduction: The linear programming model is used to minimize the cost and maximize the profit through algebraic and graphical presentation using different variables and satisfying them.

Explanation of Solution

$\begin{array}{c}{\text{x}}_1\text{is}\text{coat}\\ {\text{x}}_2\text{is}\text{slacks}\end{array}$

The Model

$\begin{array}{l}\text{Maximize Z}=50x_1+40x_2\\ \text{Subject to}\\ 3x_1+5x_2\le 150\text{sq}\text{.}\text{}\text{yards}\\ 10x_1+4x_2\le 200\text{hour}\\ x_1,x_2\ge 0,\text{integer}\end{array}$

b)

Summary Introduction

To determine: The integer solution and solution without integer constrain and compare the results.

Explanation of Solution

Profit with integer restriction:

Formula:

The profit with integer restriction is $1460 with $x_1=10,x_2=24$

Profit without integer restriction:

Formula:

The profit without integer restriction is $1473 with $x_1=10.5,x_2=23.7$. From the above calculations, it can be found that the profit is high when there are no integer restrictions. Therefore, the rounded solution is not optimal.