Question

Question General Ford produces cars at Malatya and Antalya and has a warehouse in Eskisehir; the company supplies cars to customers in Ankara and Istanbul. The cost of shipping a car between points is given in the table below. (6-" means that a shipment is not allowed). Malatya can produce as many as 1,300 cars, and Antalya can produce as many as 1,900 cars. Ankara must receive 1,400 cars, and Istanbul must receive 1,700 cars. Malatya 0 Antalya Istanbul Ankara 90 To ($) Eskisehir 100 121 0 0 From Malatya Antalya Eskisehir Ankara Istanbul 110 139 78 113 0 0 The company wishes to minimize the shipping costs incurred in meeting demands at Houston and Tampa a) Formulate the problem as an LP.

Answer 1

a)

LP is following:

Let Xij = number of cars to be shipped from location i to j, where i,j = {1,2,3,4,5} for {Malatya,Antalya,Eskisehir,Ankara,Istanbul}

Min 100X13+90X14+121X23+110X24+139X25+113X34+78X35

s.t.

X13+X14 <= 1300

X23+X24+X25 <= 1900

X13+X23-X34-X35 = 0

X14+X24+X34 = 1400

X25+X35 = 1700

Xij >= 0

--------------------------------------------------

b)

Balanced transportation problem is formulated as below:

Cost from/to	Eskisehir	Ankara	Istanbul	Dummy	Capacity
Malatya	100	90	1000	0	1300
Antalya	121	110	139	0	1900
Eskisehir	0	113	78	0	3100
Demand	3100	1400	1700	100

Shipping cost of Malatya to Istanbul is written as 1000, which is hypothetically large to prohibit allocation to this cell, as shipment from Malatya to Istanbul is not allowed.

Total capacity (1300+1900=3200) is more than total demand (1400+1700=3100) . Therefore, a dummy demand node is created for balance 100 cars with shipping cost of 0.

Total demand and capacity of warehouse in Eskisehir is equal to total demand, i.e. 3100 cars

Basic Feasible Solution (bfs) is determined by northwest corner (NWC) method as follows:

In NWC method, we start with the Northwest Corner cell and allocate the maximum quantity possible and then move on to either the next cell to the right or below, depending upon the remaining demand or supply in that particular row or column. The process is repeated until all the demand is satisfied.

Resulting tableau showing the shipping quantity from each plant to warehouse to customers is following:

	Eskisehir	Ankara	Istanbul	Dummy	Capacity
Malatya	1300	0	0	0	1300
Antalya	1800	100	0	0	1900
Eskisehir	0	1300	1700	100	3100
Demand	3100	1400	1700	100

Total cost = 1300*100+1800*121+100*110+1300*113+1700*78+100*0

= $ 638,300