Question

A larger and more modern main post office is to be constructed at a new location in​ Davis, California. Growing suburbs caused a shift in the population density from where it was 40 years​ ago, when the current facility was built. Annette​ Werk, the​ postmaster, asked her assistants to draw a grid map of the seven points where mail is picked up and delivered in bulk. The coordinates and trips per day to and from the seven mail source points and the current main post​ office, M, are shown in the following table. M will continue to act as a mail source point after relocation.  

Mail Source Point	Round trips per day	​x, y, Coordinates​ (miles)
1	4	​(7​, 2​)
2	2	​(3​, 8)
3	2	​(18​, 14)
4	13	​(9​, 8)
5	6	​(13​, 9​)
6	5	​(16​, 6​)
7	7	​(8, 11)
M	8	​(12​, 12)

a. Calculate the center of gravity as a possible location for the new facility. ​(Enter your responses rounded to the nearest whole​ number.)

The center of gravity​ "X" coordinate​ =

The center of gravity​ "Y" coordinate​ =

b. Compare the​ load-distance scores for the location in part​ (a) and the current​ location, using rectilinear distance. ​(Round all intermediate calculations to the nearest whole​ number.)

The​ load-distance score to​ "M" is

​ (Enter your response as an​ integer.)

The​ load-distance score to the center of gravity is

​(Enter your response rounded to the nearest whole​ number.)

Answer 1

Question – a:

As per center of gravity method, the location coordinates will be:

x-coordinate = sum total of (x-coordinate*roundtrip per day) for all mail sources divided by sum total of roundtrips of all mail sources.

y-coordinate = sum total of (y-coordinate*roundtrip per day) for all mail sources divided by sum total of roundtrips of all mail sources.

X = (4*7 + 2*3 + 2*18 + 13*9 + 6*13 + 5*16 + 7*8 + 8*12)/(4 + 2 + 2 + 13 + 6 + 5 + 7 + 8)

= 497/47

= 10.57

= 11 (rounded to nearest whole)

Y = (4*2 + 2*8 + 2*14 + 13*8 + 6*9 + 5*6 + 7*11 + 8*12)/(4 + 2 + 2 + 13 + 6 + 5 + 7 + 8)

= 413/47

= 8.79

= 9 (rounded to nearest whole)

Answer is:

The center of gravity​ "X" coordinate​ = 11

The center of gravity​ "Y" coordinate​ = 9

Question – b:

Rectilinear distance = |x1 – x2| + |y1 – y2|

The​ load-distance score to​ "M" is

Mail Source Point	Number of Trips	x,y coordinate	Rectilinear Distance
1	4	(7, 2)	15
2	2	(3, 8)	13
3	2	(18, 14)	8
4	13	(9, 8)	7
5	6	(13, 9)	4
6	5	(16, 6)	10
7	7	(8, 11)	5

The​ load-distance score to​ "M" is: (4*15 + 2*13 + 2*8 + 13*7 + 6*4 + 5*10 + 7*5) = 302

The​ load-distance score to the center of gravity is

Mail Source Point	Number of Trips	x,y coordinate	Rectilinear Distance
1	4	(7, 2)	11
2	2	(3, 8)	9
3	2	(18, 14)	12
4	13	(9, 8)	3
5	6	(13, 9)	2
6	5	(16, 6)	8
7	7	(8, 11)	5
M	8	(12, 12)	4

The load-distance score to the center of gravity is: (4*11 + 2*9 + 2*12 + 13*3 + 6*2 + 5*8 + 7*5 + 8*4) = 244

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