Question

Mega-Mart, a discount store chain, wants to build a new superstore in an area in southwest Virginia near four small towns with expected daily number of customers between 970 and 2600. The coordinates (in miles) of these four towns and the expected daily number of customers in each are as follows: Coordinates Whitesburg Alton Ville Camburg Milligan 12 18 32 y 20 15 7 25 Expected daily number of customers 2600 1200 1830 970 30 a) (10 pts) Determine the location of new superstore using the center-of-gravity technique. b) (10 pts) Suppose that the coordinates you found by center of gravity in (a) is not technically feasible and you need to move your location to one of two possible sites. The coordinates (in miles) of these two potential sites are site 1 (25, 25) and site 2 (35, 5). Use the load-distance technique to select the best feasible location for the new superstore. c) (5 pts) Plot the four small towns and the location according to center of gravity method and load- distance method on a grid map. If both location alternatives (found by center-of-gravity and load- distance method) are feasible which one would you prefer and why? Write your own comment and justify numerically. d) (10 pts) Suppose superstore has to arrange a shuttle to bring customers to the store for the first year of the operation. Daily operational cost for this shuttle is 1 cent per customer per mile. How much would the superstore afford to spend to make the location by center of gravity feasible? (Assume 300 operational days in one year).

Answer 1

Location	x	y	volume v
w	12	20	2600
a	18	15	1200
c	30	7	1830
m	32	25	970

a. Location of the new substore by CG method.

Location	x	y	volume v	vixi	viyi
w	12	20	2600	31200	52000
a	18	15	1200	21600	18000
c	30	7	1830	54900	12810
m	32	25	970	31040	24250
			6600	138740	107060
?vi=6600
?vixi=138740
?viyi=107060
Cooordinates of the centre of gravity xc and yc can now be determined:
xc=?vixi/?vi=138740/6600	21.02
yc=?viyi/?vi=107060/6600	16.22
Thus the coordinates of the location by CG method=21,16

b. Possible two sites S1 (25,25 ) and S2 (35,5)

Let us calculate the load distances by the load distance method:

b1. For Site S1.

S1=25,25
Location	x	y	volume v	Distance D	v*D
w	12	20	2600	13.93	36218
a	18	15	1200	12.21	14652
c	30	7	1830	18.68	34184.4
m	32	25	970	7.00	6790
			6600		91844.4
Distance= √ [(25-12)2 + (25-20)2 ]=√194=13.93			194
(Only one calculation has been verified and the rest taken as shown in the problem. b2. For site 2			13.92839
Similarly for S2 (35,5)
Location	x	y	volume v	Distance D	v*D
w	12	20	2600	27.46	71396
a	18	15	1200	19.72	23664
c	30	7	1830	5.39	9863.7
m	32	25	970	20.22	19613.4
			6600		124537.1
Distance= √ [(35-12)2 + (05-20)2 ]=√				754
(Only one calculation has been verified and the rest taken as shown in the problem.			27.45906
vD1	91844.4
vD2	124537.1
∴ Site 1 having a total load distance of 91,844 is better than site 2 having a load distance of 124,537

c.

Grid showing 1. all four towns, 2. location as per CG method and 3. two locations proposed (for Load distance method). 30 25 LD1 25,25 m 20 WE 15 CG 21,16 0 a 10 5 LD2 35,5 O o 5 10 15 20 25 30 35 40

The centre of gravity location has been determined while th etwo locations for load distance

have ben specified in the problme.

Now one of the locations S2 proposed is definitely not feasible because the site S1 has a smaller load distance.

Now we ned to compare the S1 and the CG (Location by CG method)

We have not calculated the load distance for the CG, while the load distance for S1 has been calclulated. Si being better than S2 is no reason why it will be better than the CG.

Similarly for CG (21,16)	(ignoring the fractions)
Location	x	y	volume v	Distance D	v*D		Distance= √ [(21-12)2 + (16-20)2 ]=√97 =	97	9.848858
w	12	20	2600	9.85	25610
a	18	15	1200	3.16	3792		Distance= √ [(21-18)2 + (16-15)2 ]=√10 =3.16		3.162278
c	30	7	1830	12.73	23296		Distance= √ [(21-30)2 + (16-07)2 ]=√162 =12.73		12.72792
m	32	25	970	14.21	13784		Distance= √ [(21-32)2 + (16-25)2 ]=√ 202 =14.21	202	14.21267
			6600		66482
Total load distance =66,482

Load distance from the CG is much less than the S1.

The CG location is definitely the optimum of the two.

Therefore CG is the best location.

SHUTTLE

There is a slight difference in the load distance calculated above from those given in the problem (66482 v/s 66453.2- 0.04 % only) This is because, here the fractions in the grid values of CG have been ignored.

Total cost =66482*300*0.01= $199,446. This amount has to be borne by the superstore if CG is chosen as the location.

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