Mercedes produces cars at Aksaray and Istanbul and has a warehouse in Gebze; the
ID: 450620 • Letter: M
Question
Mercedes produces cars at Aksaray and Istanbul and has a warehouse in Gebze; the company supplies cars to customers in Izmir and Antalya. The cost of shipping a car between two points is given in the table below: ("-" means that shipment is not allowed between these two points). Aksaray can produce as many as 1100 cars and Istanbul can produce as many as 2900 cars. Izmir must receive 2400 cars and Antalya must receive 1500 cars. Construct a balanced transportation tableau for the problem. Explain how you have constructed the balanced tableau. Also, show the cost calculations on the transportation tableau. Formulate a balanced transportation problem that could be used to determine how to minimize the total distribution cost of meeting requirements. Solve the problem by using IBM ILOG CPLEX Optimization Studio. Write down the optimal solution and make comments. Submit also your 0PL output with your homework.Explanation / Answer
Formulation as a balanced Transportation problem
As per the information, producing units are at Aksaray (capacity 1100 units) and at Istanbul (capacity 2900 units) . Therefore total supply is 4000 units available at the places. Gebze is the place for warehouse.
The demand centers are at Izmir (demand 2400 units) and at Antalaya (1500 units). Therefore Total demand is 3900 units. Supply is more than the demand, therefore we require a dummy demand center with demand of 100 units.
In the above formulation it is presumed Gebze is not playing any role, as a supplier and or as a customer but it is supposed to act both as supplier and as well as customers, supposed take units from the producers and supply to the customers. Nothing is mentioned about its capacity so let us assume that it has the capacity to handle the maximum total production. With the constraint that its supply is equal to its demand, so reformulation is as follows:
Another formulation:
Let A1, A2, A3 and A4 are the number of units transported from Aksaray to Gebze, Izmir, Antalay and Dummy respectively. Similarly B1, B2, B3 and B4 are the number of units transported from Istanbul to Gebze, Izmir, Antalya and Dummy respectively. Finally C1, C2 and C3 be the number of units transported from Gebze to Izmir, Antalya and Dummy respectively.
Objective function is to minimize the total cost of transportation as follows:
Minimize100A1+20A2+225A3+0A4+111B1+110B2+119B3+0B4+113C1+78C2+0C3
Subject to constraint:
Supply A1+A2+A3+A4 = 1100 Production at Aksaray
B1+B2+B3+B4 = 2900 Production at Istanbul
Demand A2+B2+C1 = 2400 Demand at Izmir
A3+B3+C2 = 1500 Demand at Antalya
A4+B4+C3 = 100 Demand at Dummy
Constraint of warehouse at at Gebze: Balancing supply = demand A1+B1 = C1+C2+C3
Non-negativity constraints A1,A2,A3,A4,B1,B2,B3,B4,C1,C2,C3 >=0 being quantities can not be negative.
Producer Customers Aksaray Istanbul Gebze Izmir Antalya Dummy Production/Supply Aksaray 0 140 100 90 225 0 1100 Istanbul 145 0 111 110 119 0 2900 Gebze 105 115 0 113 78 0 0 Izmir 89 109 121 0 --- 0 0 Antalya 210 117 82 -- 0 0 0 Demand 0 0 0 2400 1500 100 4000