Chapter 8, Problem 23P

a)

Summary Introduction

To determine: The center-of-gravity coordinates the new store to be located.

Introduction: Location is one of the important element for a business that controls the cost and expenses. Location strategies support in framing other strategies for a firm where optimal location point will provide competitive advantage to a firm.

Answer to Problem 23P

The coordinates of the center of gravity are (66.69, 30.22).

Explanation of Solution

Given information:

Census Tract	Population in census tract	X,Y
101	2,000	(25, 45)
102	5,000	(25, 25)
103	10,000	(55, 45)
104	7,000	(50, 20)
105	10,000	(80, 50)
106	20,000	(70, 20)
107	14,000	(90, 25)

Center-of-gravity coordinates of the new store:

Derive the coordinates of the Center of gravity by proceeding sequentially as per the following steps given the population data and the coordinates of various census tracts.

The various Census tracts under consideration are shown below in the graph.

The x coordinate of the centre of gravity C_x is given by the formula

${\text{C}}_{\text{x}}=\frac{{\displaystyle \sum _{\text{i}}{\text{d}}_{\text{ix}}\times {\text{Q}}_{\text{i}}}}{{\displaystyle \sum _{\text{i}}{\text{Q}}_{\text{i}}}}$ (1)

where d_ix is the x coordinate of the location i and Q_i is the shipping load to and from the location i

The y coordinate of the centre of gravity C_y is given by the formula

${\text{C}}_{\text{y}}=\frac{{\displaystyle \sum _{\text{i}}{\text{d}}_{\text{iy}}\times {\text{Q}}_{\text{i}}}}{{\displaystyle \sum _{\text{i}}{\text{Q}}_{\text{i}}}}$ (2)

where d_iy is the y coordinate of the location i and Q_i is the shipping load to and from the location i

Use equation (1) and calculate C_x as follows,

$\begin{array}{c}{\text{C}}_{\text{x}}=\frac{\begin{array}{l}25\times 2,000+25\times 5,000+55\times 10,000+50\times 7,000\\ +80\times 10,000+70\times 20,000+90\times 14,000\end{array}}{2,000+5,000+10,000+7,000+10,000+20,000+14,000}\\ =\frac{4,535,000}{68,000}\\ =66.69\end{array}$

Use equation (2) and calculate C_y,

$\begin{array}{c}{\text{C}}_{\text{y}}=\frac{\begin{array}{l}45\times 2,000+25\times 5,000+45\times 10,000+20\times 7,000\\ +50\times 10,000+20\times 20,000+25\times 14,000\end{array}}{2,000+5,000+10,000+7,000+10,000+20,000+14,000}\\ =\frac{2,055,000}{68,000}\\ =30.22\end{array}$

The coordinates of the center of gravity are (66.69, 30.22), marked in red color in the graph below.

Hence, the coordinates of the center of gravity are (66.69, 30.22).

b)

Summary Introduction

To determine: The change in center-of-gravity coordinates of the new store when 103 and 105 census tracts grow by 20%.

Answer to Problem 23P

The new coordinates of the center of gravity marginally shifts to (66.74, 31.18).

Explanation of Solution

Given information:

Census tract 103 and 105 grow by 20%.

Census Tract	Population in census tract	X,Y
101	2,000	(25, 45)
102	5,000	(25, 25)
103	12,000	(55, 45)
104	7,000	(50, 20)
105	12,000	(80, 50)
106	20,000	(70, 20)
107	14,000	(90, 25)

Calculation of change in center-of-gravity coordinates:

Derive the center of gravity, given that the population growth in census tracts 103 and 105 has increased by 20% as shown in the table.

Use equation (1) and calculate C_x as follows

$\begin{array}{c}{\text{C}}_{\text{x}}=\frac{\begin{array}{l}25\times 2,000+25\times 5,000+55\times 12,000+50\times 7,000\\ +80\times 12,000+70\times 20,000+90\times 14,000\end{array}}{2,000+5,000+12,000+7,000+12,000+20,000+14,000}\\ =\frac{4,805,000}{72,000}\\ =66.74\end{array}$

Use equation (2) and calculate C_y

$\begin{array}{c}{\text{C}}_{\text{y}}=\frac{\begin{array}{l}45\times 2,000+25\times 5,000+45\times 12,000+20\times 7,000\\ +50\times 12,000+20\times 20,000+25\times 14,000\end{array}}{2,000+5,000+12,000+7,000+12,000+20,000+14,000}\\ =\frac{2,245,000}{72,000}\\ =31.18\end{array}$

The new coordinates of the center of gravity marginally shifts to (66.74, 31.18) as shown below.

Hence, the new coordinates of the center of gravity marginally shifts to (66.74, 31.18).