Question

Rauschenberg Manufacturing is investigating which locations would best position its new plant relative to three important customers​ (located in cities​ A, B, and​ C). As shown in the table​ below, all three customers require multiple daily deliveries. Management limited the search for this plant to those three locations and compiled the following​ information:    

Location   Deliveries per day   x-coordinate   y-coordinate
A   4 100 100
B   6   500 200
C   7   300 100

a. Which of these three locations yields the smallest total travel​ distance, based on Euclidean​ distances?

Location ____ yields the shortest travel​ (transportation) distance of ____miles. ​(Enter your response rounded to one decimal​ place.) b. Which of these locations is best, based on rectilinear distances?

c. What are the coordinates of the center of gravity?

Answer 1

a)

Euclidean distance is the normal straight line distance obtained using the Pythagoras formula.

Considering A as the plant location, total travel distance = 4*0+6*sqrt((500-100)^2+(200-100)^2)+7*sqrt((300-100)^2+(100-100)^2)

= 3,873.9 miles

Considering B as the plant location, total travel distance = 4*sqrt((500-100)^2+(200-100)^2)+6*0+7*sqrt((300-500)^2+(100-200)^2)

= 3,214.5 miles

Considering C as the plant location, total travel distance = 4*sqrt((300-100)^2+(100-100)^2)+6*sqrt((300-500)^2+(100-200)^2)+7*0

= 2,141.6 miles

Location __C__ yields the shortest travel​ (transportation) distance of 2,141.6  miles. ​

b)

Rectilinear distance is obtained by finding the absolute difference of X coordinates of the selected location and other location and Y coordinates of the selected location and other location and these adding these difference.

For example rectilinear distance between location A and B is : ABS(100-500)+ABS(100-200) = 400+100 = 500

Considering A as the plant location, total travel distance = 4*0+6*(ABS(500-100)+ABS(200-100))+7*(ABS(300-100)+ABS(100-100))

= 4,400 miles

Considering B as the plant location, total travel distance = 4*(ABS(500-100)+ABS(200-100))+6*0+7*(ABS(300-500)+ABS(100-200))

= 4,100 miles

Considering C as the plant location, total travel distance = 4*(ABS(300-100)+ABS(100-100))+6*(ABS(300-500)+ABS(100-200))+7*0

= 2,600 miles

Based on rectlinear distance,

Location __C__ yields the shortest travel​ (transportation) distance of 2,600  miles. ​

c)

Coordinates of Center of Gravity are determined as below:

X coordinate = (100*4+500*6+300*7)/(4+6+7) =  323.5 ~ 324

Y coordinate = (100*4+200*6+100*7)/(4+6+7) = 135.3 ~ 135