Chapter 8, Problem 20P

a)

Summary Introduction

To determine: The map coordinates for location of the hub.

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 20P

The map coordinates for the location of hub is (5.15, 7.30).

Explanation of Solution

Given information:

City	Map Coordinates (X,Y)	Shipping load
A	(5, 10)	5
B	(6, 8)	10
C	(4, 9)	15
D	(9, 5)	5
E	(7, 9)	15
F	(3, 2)	10
G	(2, 6)	5

Calculation of map coordinates for location of the hub:

Given the map coordinates and shipping loads for each city location as shown below, use the Centre of Gravity method as shown in the sequential steps described below.

Show the city locations in a graph as shown below.

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

$\begin{array}{c}{\text{C}}_{\text{x}}=\frac{5\times 5+6\times 10+4\times 15+9\times 5+7\times 15+3\times 10+2\times 5}{5+10+15+5+15+10+5}\\ =\frac{335}{65}\\ =5.15\end{array}$

Use equation (2) and calculate C_y as follows,

$\begin{array}{c}{\text{C}}_{\text{y}}=\frac{10\times 5+8\times 10+9\times 15+5\times 5+9\times 15+2\times 10+6\times 5}{5+10+15+5+15+10+5}\\ =\frac{475}{65}\\ =7.30\end{array}$

The coordinates of the Centre of Gravity are given as (5.15,7.30).

The proposed hub highlighted in red color in figure shown below should be close to the coordinates (5.15, 7.30) to minimize the transportation costs.

Hence, the map coordinates for the location of hub is (5.15, 7.30).

b)

Summary Introduction

To determine: The change in the map coordinates for location of the hub when shipment A is tripled.

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 20P

The hub location should be as close to the coordinates (5.13, 7.67) as possible to minimize transport costs.

Explanation of Solution

Given information:

Shipment A is tripled.

City	Map Coordinates (X,Y)	Shipping load
A	(5, 10)	15
B	(6, 8)	10
C	(4, 9)	15
D	(9, 5)	5
E	(7, 9)	15
F	(3, 2)	10
G	(2, 6)	5

Calculation of change in the map coordinates for location of the hub when shipment A is tripled:

If the shipments from City A triple, calculate the center of gravity as per calculations shown later.

Use equation (1) and calculate C_x as follows,

$\begin{array}{c}{\text{C}}_{\text{x}}=\frac{15\times 5+6\times 10+4\times 15+9\times 5+7\times 15+3\times 10+2\times 5}{15+10+15+5+15+10+5}\\ =\frac{385}{75}\\ =5.13\end{array}$

Use equation (2) and calculate C_y as follows,

$\begin{array}{c}{\text{C}}_{\text{y}}=\frac{10\times 15+8\times 10+9\times 15+5\times 5+9\times 15+2\times 10+6\times 5}{5+10+15+5+15+10+5}\\ =\frac{575}{75}\\ =7.67\end{array}$

Hence, the hub location should be as close to the coordinates (5.13, 7.67) as possible to minimize transport costs.