The following table gives the map coordinates and the shipping loads for a set o

Question

The following table gives the map coordinates and the shipping loads for a set of cities that we wish to connect through a central hub.

  

City

(x,y)

Shipping Load

Upper A

(6,11)

4

Upper B

(5,8)

9

Upper C

(5,9)

15

Upper D

(9,6)

4

Upper E

(6,9)

15

Upper F

(2,2)

9

Upper G

(1,5)

4

a) For the location of the proposed new central hub, the coordinates should be near:

x = _____ (round your response to two decimal places).

y = ______(round your response to two decimal places).

B.) If the shipments from city A double for the location of the proposed new hub, the coordinates should be near : (round 2 decimal places)

x = _____

y = ______

City

Map Coordinates

(x,y)

Shipping Load

Upper A

(6,11)

4

Upper B

(5,8)

9

Upper C

(5,9)

15

Upper D

(9,6)

4

Upper E

(6,9)

15

Upper F

(2,2)

9

Upper G

(1,5)

4

Explanation / Answer

The problem will be solved using Center of gravity method.

Accordingly,

Sum of all shipping loads x P = Sum of ( X coordinate x Shipping load ) for 7 cities

Or, ( 4 + 9 + 15 + 4 + 15 + 9 + 4).P = 6X4 + 5X9 + 5X15 + 9X4 + 6X15 +2X9 + 1X4

Or, 60.P = 24 + 45 + 75 + 36 + 90 + 18 + 4 = 292

Or, P = 292/60 = 4.87 ( rounded to 2 decimal places )

Sum of all shipping loads x Q = Sum of ( Y coordinate x Shipping load ) for 7 cities

Or, ( 4 +9 + 15 + 4+ 15 + 9+4 ).Q = 11X 4 + 8 X 9 + 9X 15 + 6 X 4 + 9 X 15 + 2X 9 + 5X 4

Or, 60 .Q = 44 + 72 + 135 + 24 + 135 + 18 + 20

Or, Q = 448/60 = 7.47 ( rounded to 2 decimal places )

X = 4.87

Y = 7.47

Let the coordinate of the center of gravity i.e. the central hub = P1, Q1

Accordingly,

Sum of all shipping loads x P = Sum of ( X coordinate x Shipping load ) for 7 cities

Or, ( 8 + 9 + 15 + 4 + 15 + 9+ 4 ).P1 = 6X8 + 5X9 + 5X15 + 9X4 + 6X15 +2X9 + 1X4

Or, 64.P1 = 48 + 72 + 135 + 24 + 135 + 18 + 20

Or, P1 = 316/64 = 4.94

Sum of all shipping loads x Q = Sum of ( Y coordinate x Shipping load ) for 7 cities

Or, ( 8 +9 + 15 + 4 +15+9+4 ).Q = 11X 8+ 8 X 9 + 9X 15 + 6 X 4 + 9 X 15 + 2X 9 + 5X 4

Or, 64 .Q1 = 88 + 72 + 135 + 24 + 135 + 18 + 20

Or, Q1 = 492/64 = 7.69 ( rounded to 2 decimal places )

X = 4.94

Y = 7.69

X = 4.87

Y = 7.47

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