Chapter 8, Problem 36E

1.

To determine

Compute cost equation for maintenance costs.

Explanation of Solution

The model to fit the cost is given below:

Maintenance costs: $\text{Machine costs}=\left(\text{a+b}\times \text{M}\right)$

The months 5 and 10 are the highest and lowest points.

High-low method:

Calculate the change in total maintenance costs.

$\begin{array}{c}\left(\begin{array}{l}\text{Change in total }\\ \text{maintenance costs}\end{array}\right)=\left(\begin{array}{l}\text{5th month maintenance cost}\\ -\text{10th month maintenance cost}\end{array}\right)\\ =\$3,100-\$2,220\\ =\$880\end{array}$

Calculate the change in total maintenance hours.

$\begin{array}{c}\left(\begin{array}{l}\text{Change in total }\\ \text{maintenance hours}\end{array}\right)=\left(\begin{array}{l}5\text{th month machine hours}\\ -10\text{th month machine hours}\end{array}\right)\\ =1,900-1,100\\ =800\end{array}$

Calculate slope (b).

$\begin{array}{c}\text{Slope}\left(\text{b}\right)=\frac{\left(\begin{array}{l}\text{Change in total }\\ \text{maintenance costs}\end{array}\right)}{\left(\begin{array}{l}\text{Change in total }\\ \text{machine hours}\end{array}\right)}\\ =\frac{\$880}{800}\\ =\$1.10\end{array}$

Calculate the constant (a).

$\begin{array}{c}\text{Constant }\left(\text{a}\right)=\left(\begin{array}{l}\left(\begin{array}{l}\text{5th Month }\\ \text{maintenance cost}\end{array}\right)\\ -\left(\text{b}\times \text{5th month machine hours}\right)\end{array}\right)\\ =\left(\$3,100-\left(\$1.10\times 1,900\right)\right)\\ =\$1,010\end{array}$

Calculate the maintenance costs.

$\begin{array}{c}\text{Maintenance cost}=\left(\text{a}+\left(\text{b}\times \text{M}\right)\right)\\ =\left(\$1,010+\left(\$1.10\times \text{M}\right)\right)\end{array}$

An alternative solution is preferred. A view of graph of machine hours versus maintenance expense, as chosen lowest data point is not characteristic of the relationships in the data for the 11^th month.

The 10^th month point is far (left) from the remaining data points whereas, 11^th month is slightly closer to the remaining data points.

Recalculation of high-low method with 11^th month:

Calculate the change in total maintenance costs.

$\begin{array}{c}\left(\begin{array}{l}\text{Change in total }\\ \text{maintenance costs}\end{array}\right)=\left(\begin{array}{l}\text{5th month maintenance cost}\\ -\text{11th month maintenance cost}\end{array}\right)\\ =\$3,100-\$2,230\\ =\$870\end{array}$

Calculate the change in total maintenance hours.

$\begin{array}{c}\left(\begin{array}{l}\text{Change in total }\\ \text{maintenance hours}\end{array}\right)=\left(\begin{array}{l}5\text{th month machine hours}\\ -11\text{th month machine hours}\end{array}\right)\\ =1,900-1,300\\ =600\end{array}$

Calculate slope (b).

$\begin{array}{c}\text{Slope}\left(\text{b}\right)=\frac{\left(\begin{array}{l}\text{Change in total }\\ \text{maintenance costs}\end{array}\right)}{\left(\begin{array}{l}\text{Change in total }\\ \text{machine hours}\end{array}\right)}\\ =\frac{\$870}{600}\\ =\$1.45\end{array}$

Calculate the constant (a).

$\begin{array}{c}\text{Constant }\left(\text{a}\right)=\left(\begin{array}{l}\left(\begin{array}{l}\text{5th Month }\\ \text{maintenance cost}\end{array}\right)\\ -\left(\text{b}\times \text{5th month machine hours}\right)\end{array}\right)\\ =\left(\$3,100-\left(\$1.45\times 1,900\right)\right)\\ =\$345\end{array}$

Calculate the maintenance costs.

$\begin{array}{c}\text{Maintenance cost}=\left(\text{a}+\left(\text{b}\times \text{M}\right)\right)\\ =\left(\$345+\left(\$1.45\times \text{M}\right)\right)\end{array}$

2.

To determine

Compute mean absolute percentage error for the cost equation developed in part 1.

Explanation of Solution

The graph of expense versus hours shows the point for 10^th month to possibly be an outlier. The point for 11^th month is also an outlier but the 12^th month data has to be used instead.

Compute expense.

$\text{Expense}=\left(-\$883+\left(\$2.097\times \text{Hours}\right)\right)$

The graph drawn below shows the relationship between machine-hours and maintenance expense and also the position of 10^th, 11^th and 12^th month’s points.

Figure (1)

Compute of the mean absolute percentage error for high low model.

S.No	Machine Hours	Maintenance Costs ($)	High-low Estimate ($)	Absolute Value of Difference ($)
1	1,690	2,600	2,869	269
2	1,770	2,760	2,957	197
3	1,850	2,910	3,045	135
4	1,870	3,020	3,067	47
5	1,900	3,100	3,100	0
6	1,880	3,070	3,078	8
7	1,860	3,010	3,056	46
8	1,840	2,850	3,034	184
9	1,700	2,620	2,880	260
10	1,100	2,220	2,220	0
11	1,300	2,230	2,440	210
12	1,590	2,450	2,759	309
	Total	32,840		1,665

Table (1)

$\begin{array}{c}\text{MAPE}=\frac{\text{Absolute value of difference}}{\text{Maintenance costs}}\\ =\frac{\$1,665}{\$32,840}\\ =5.1\%\end{array}$