Chapter 5, Problem 24P

Summary Introduction

To identify: The design that will yield the highest profit.

Introduction:

Expected monetary value (EMV):

Expected monetary value (EMV) is a calculation system for expected returns for the certain decision made by a company.

Answer to Problem 24P

The design K1 can be used because of the highest profit.

Explanation of Solution

Given information:

Design	Cost	Probability	Number of non-defective (out of 100)
K1	$100,000.00	0.8	90
		0.2	70
K2	$130,000.00	0.85	90
		0.15	75
K3	$180,000.00	0.9	95
		0.1	80

$\begin{array}{c}\text{Number}\text{of}\text{sandwiches}\text{produced}\text{daily}=\text{500}\\ \text{Cost}\text{of}\text{each}\text{sandwich}=\text{\$1}\text{.30}\\ \text{Selling}\text{price}\text{of}\text{each}\text{sandwich}=\$\text{2}\text{.50}\\ \text{Number}\text{of}\text{days}\text{of operation}=\text{300}\end{array}$

Calculation of EMV:

Calculation of expected outcome and EMV:

Design K1:

Probability 0.80

$\begin{array}{l}=-\$100,000-\left(\frac{10}{100}\times 500\times 300\times \$1.30\right)+\left(\frac{90}{100}\times 500\times 300\times \$1.20\right)\\ =-\$100,000-\$19,500+\$162,000\\ =\$42,500\end{array}$

The expected outcome is calculated by adding the cost for the design K1, $100,000 and the cost of 10 unsold sandwiches ($1.20) and subtracting the resultant with the profit of 90 sandwiches sold ($2.50-$1.20=$1.30). The expected outcome is $42,500.

Probability 0.20

$\begin{array}{l}=-\$100,000-\left(\frac{30}{100}\times 500\times 300\times \$1.30\right)+\left(\frac{70}{100}\times 500\times 300\times \$1.20\right)\\ =-\$100,000-\$58,500+\$126,000\\ =-\$32,500\end{array}$

The expected outcome is calculated by adding the cost for the design K1, $100,000 and cost of 30 unsold sandwiches ($1.20) and subtracting the resultant with the profit of 70 sandwiches sold ($2.50-$1.20=$1.30). The expected outcome is -$32,500.

Calculation of EMV:

EMV is calculated by multiplying the payoff with the respective probabilities.

$\begin{array}{l}=\left(42,500\times 0.80\right)+\left(-32,500\times 0.20\right)\\ =\$27,500\end{array}$ (1)

The expected outcome for probability 0.80, $42,500 is multiplied and the expected outcome for probability 0.20, -$32,500 is multiplied and the resultant of both is added to give the EMV for design K1 is $27,500.

Design K2:

Probability 0.85

$\begin{array}{l}=-\$130,000-\left(\frac{10}{100}\times 500\times 300\times \$1.30\right)+\left(\frac{90}{100}\times 500\times 300\times \$1.20\right)\\ =-\$130,000-\$19,500+\$162,000\\ =\$12,500\end{array}$

The expected outcome is calculated by adding the cost for the design K2, $130,000 and the cost of 10 unsold sandwiches ($1.20) and subtracting the resultant with the profit of 90 sandwiches sold ($2.50-$1.20=$1.30). The expected outcome is $12,500.

Probability 0.15

$\begin{array}{l}=-\$130,000-\left(\frac{25}{100}\times 500\times 300\times \$1.30\right)+\left(\frac{75}{100}\times 500\times 300\times \$1.20\right)\\ =-\$130,000-\$48,500+\$135,000\\ =-\$43,750\end{array}$

The expected outcome is calculated by adding the cost for the design K2, $130,000  and the cost of 25 unsold  sandwiches ($1.20) and subtracting the resultant with the profit of 75 sandwiches sold ($2.50-$1.20=$1.30). The expected outcome is -$43,750.

Calculation of EMV:

EMV is calculated by multiplying the payoff with the respective probabilities.

$\begin{array}{l}=\left(12,500\times 0.85\right)+\left(-43,750\times 0.15\right)\\ =\$4,062.50\end{array}$ (2)

The expected outcome for probability 0.85, $12,500 is multiplied and the expected outcome for probability 0.15, -$43,750 is multiplied and the resultant of both is added to give the EMV for design K2 is $4,062.50.

Design K3:

Probability 0.90:

$\begin{array}{l}=-\$180,000-\left(\frac{5}{100}\times 500\times 300\times \$1.30\right)+\left(\frac{95}{100}\times 500\times 300\times \$1.20\right)\\ =-\$180,000-\$9,750+\$171,000\\ =-\$18,750\end{array}$

The expected outcome is calculated by adding the cost for the design K3, $180,000 and the cost of 5 unsold sandwiches ($1.20) and subtracting the resultant with the profit of 95 sandwiches sold ($2.50 - $1.20=$1.30). The expected outcome is - $18,750.

Probability 0.10:

$\begin{array}{l}=-\$180,000-\left(\frac{20}{100}\times 500\times 300\times \$1.30\right)+\left(\frac{80}{100}\times 500\times 300\times \$1.20\right)\\ =-\$180,000-\$39,000+\$144,000\\ =-\$75,000\end{array}$

The expected outcome is calculated by adding the cost for the design K3, $180,000  and cost of 20 unsold  sandwiches ($1.20) and subtracting the resultant with the profit of 80 sandwiches sold ($2.50-$1.20=$1.30). The expected outcome is - $75,000.

Calculation of EMV:

EMV is calculated by multiplying the payoff with the respective probabilities.

$\begin{array}{l}=\left(-18,750\times 0.90\right)+\left(-75,000\times 0.1\right)\\ =-\$24.375\end{array}$ (3)

The probability 0.90 and its expected outcome, -$18,750 is multiplied and the expected outcome for probability 0.10, -$75,000 is multiplied and the resultant of both is added to give the EMV for design K3 is - $24.375.

From the calculated EMV of three designs K1, K2 and K3, it can be found that EMV for design K1 is higher than other two (refer equation (1),(2) and (3)). Therefore, it is recommended to use design K1.

Hence, the design K1 can be used because of the highest profit.