Chapter 8, Problem 26P

a

Summary Introduction

Interpretation:

Whether the following statement is true or not.

Concept Introduction: The weighted average method is a method to calculate the mean in which weights or numerical value is given to qualitative data according to the importance.

Explanation of Solution

The following is the available information:

Factor	Weight	Location 1	Location 2	Location 3
Cost of available land parcel	0.3	4	2	1
Accessibility	0.5	5	2	4
Size of available land parcel	0.2	1	5	3

1. False
Based on the weights given to the factor, the highest importance is given to accessibility (0.5). The second highest importance is given to the cost of available land parcels. Therefore, the statement is false.
2. False
Based on the available information, location 3 is rated with 4 points and location 2 is rated with 2 points. It clearly tells that location 3 is better accessible than location 2. Therefore, the statement is false.
3. True
Based on the weight given to the factor, the highest importance is given to accessibility (0.5). The second highest importance is given to the cost of the available land parcel (0.3) and the least important factor is given to the size of the available land parcel (0.2). Therefore, the given statement is true.

b

Summary Introduction

Interpretation:

The optimal location for new fire station.

Concept Introduction: The weighted average method is a method to calculate the mean in which weights or numerical value is given to qualitative data according to the importance.

Explanation of Solution

The formula is given below to determine the optimal location.

Score = ${\displaystyle \sum {\mathrm{W}}_{\mathrm{i}}{\mathrm{S}}_{\mathrm{i}}}$

Here,

  ${\mathrm{W}}_{\mathrm{i}}$ = weight associated with factor i

  ${\mathrm{S}}_{\mathrm{i}}$ = Score that location earned on factor i

Now, calculate factor rating score for location 1:

The above formula denotes that the each factor should be multiplied by its weight, further, the answer of factor should be added.

Score = ${\displaystyle \sum {\mathrm{W}}_{\mathrm{i}}{\mathrm{S}}_{\mathrm{i}}}$

  $\begin{array}{l}=(0.3\times 4)+(0.5\times 5)+(0.2\times 1)\\ =1.2+2.5+0.2\\ =3.9\end{array}$

Hence, the rating score is 3.9

Calculation of factor rating score for location 2:

  $\mathrm{S}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}={\displaystyle \sum {\mathrm{W}}_{\mathrm{i}}{\mathrm{S}}_{\mathrm{i}}}$

  $\begin{array}{l}\mathrm{S}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}=(0.3\times 2)+(0.5\times 2)+(0.2\times 5)\\ =0.6+1+1\\ =2.6\end{array}$

Hence, the rating score is 2.6

Calculation of factor rating for location 3:

  $\mathrm{S}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}={\displaystyle \sum {\mathrm{W}}_{\mathrm{i}}{\mathrm{S}}_{\mathrm{i}}}$

  $\begin{array}{l}\mathrm{S}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}=(0.3\times 1)+(0.5\times 4)+(0.2\times 3)\\ =0.3+2+0.6\\ =2.9\end{array}$

Hence, the factor rating score is 2.9

Hence, the company will select Location 1 to build new fire station. It is because the score of new location A is higher than the other two location.

c

Summary Introduction

Interpretation:

The weight of each factor.

Concept Introduction: The weighted average method is a method to calculate the mean in which weights or numerical value is given to qualitative data according to the importance.

Explanation of Solution

Assume that the given weights were wrong, the company considered accessibility is three times important than the other two factors. Further, the other two factors are equally important.

The cost of available land parcel and the size of the available land parcels is equally important. Therefore, both are scored as one. Further, accessibility is three times important than other factors. Therefore, it should be scored as three. Finally, the total score is five, which is 1+1+3 = 5.

Hence, the weight of the cost of available land parcel is 0.20 (1/5), the weight of size of available land parcel is 0.20 (1/5), and accessibility is 0.60 (3/5).

d

Summary Introduction

Interpretation:

Location of new fire station.

Concept Introduction: The weighted average method is a method to calculate the mean in which weights or numerical value is given to qualitative data according to the importance.

Explanation of Solution

Calculation of factor rating score for location 1:

The above formula denotes that each factor should be multiplied by its weight. Further, the answer of factor should be added.

  $\mathrm{S}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}={\displaystyle \sum {\mathrm{W}}_{\mathrm{i}}{\mathrm{S}}_{\mathrm{i}}}$

  $\begin{array}{l}\mathrm{S}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}=(0.2\times 4)+(0.6\times 5)+(0.2\times 1)\\ =0.8+3+0.2\\ =4\end{array}$

Hence, the rating score is 4.

Calculation of factor rating score for location 2:

  $\mathrm{S}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}={\displaystyle \sum {\mathrm{W}}_{\mathrm{i}}{\mathrm{S}}_{\mathrm{i}}}$

  $\begin{array}{l}\mathrm{S}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}=(0.2\times 2)+(0.6\times 2)+(0.2\times 5)\\ =0.4+1.2+1\\ =2.6\end{array}$

Hence, the rating score is 2.6

Calculation of factor rating score for location 3:

  $\mathrm{S}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}={\displaystyle \sum {\mathrm{W}}_{\mathrm{i}}{\mathrm{S}}_{\mathrm{i}}}$

  $\begin{array}{l}\mathrm{S}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}=(0.2\times 1)+(0.6\times 4)+(0.2\times 3)\\ =0.2+2.4+0.6\\ =3.2\end{array}$

Hence, the rating score is 3.2

Hence, the company will select location A to build new fire station. It is because the score of location A is higher than the other two.