Chapter 2, Problem 1P

a)

Summary Introduction

To formulate: A linear programming model for the given problem.

Explanation of Solution

Formulation:

$\begin{array}{l}\text{Let }{x}_1\text{ be the number of cakes}\text{.}\\ \text{Let }{x}_2\text{ be the number of loaves of bread}\text{.}\end{array}$

$\begin{array}{c}\text{Maximize Z}=10x_1+6x_2\\ \text{Subject to:}\\ 3x_1+8x_2\le 20\left(\text{cups of flour}\right)\\ 45x_1+30x_2\le 180\left(\text{minutes}\right)\\ x_1,x_2\ge 0\end{array}$

b)

Summary Introduction

To solve: The problem graphically.

Explanation of Solution

Calculation of coordinates

Constraint 1:

$\begin{array}{c}3\left(0\right)+8x_2=20\\ x_2=\frac{20}{8}\\ =2.5\end{array}$

$\begin{array}{c}3x_1+8\left(0\right)=20\\ x_1=\frac{20}{3}\\ =6.67\end{array}$

The coordinates are (6.67, 2.5).

Constraint 2:

$\begin{array}{c}45\left(0\right)+30x_2=180\\ x_2=\frac{180}{30}\\ =6\end{array}$

$\begin{array}{c}45x_1+30\left(0\right)=180\\ x_1=\frac{180}{45}\\ =4\end{array}$

The coordinates are (4, 6).

Graph:

The three corner points encompassing the feasible region are A (0, 2.5), B (3.111, 1.333), and C (4, 0).

Calculation of object function value:

Formulas:

Answer:

The optimal solution is:

Number of cakes = 4

Number of loaves of bread = 0

Value of the objective function = 40