A company supplies goods to three customers, who each require 30 units. The comp

Question

A company supplies goods to three customers, who each require 30 units. The company has two warehouses. Warehouse 1 has 40 units available, and warehouse 2 has 30 units available. The costs of shipping 1 unit from warehouse to customer are shown in the Table. There is a penalty for each unmet customer unit of demand: With customer 1, a penalty cost of $90 is incurred; with customer 2, $80; and with customer 3, $110. Formulate a balanced transportation problem (which means write down the LP and then develop the Transportation Tableau) to minimize the sum of shortage and shipping costs. (you don't need to solve it)

Explanation / Answer

Let the units sold to the customers1,2 and 3 be x,y and z respectively.

Then,

x+y+z <= 70     - (1)

Let a1,b1 be the units transported from W1 and W2 resp. for customer 1. Similarily, a2,b2 for C2 and a3,b3 for C3

Then,

a1+a2+a3 <= 40     - (2)

b1+b2+b3 <= 30     - (3)

Also,

a1+b1 = x     - (4)

a2+b2 = y     - (5)

a3+b3 = z     - (6)

The total expenditure would then be,

T = [90*(30-x)+15a1+10b1]+[80*(30-y)+35a2+50b2]+[110(30-z)+25a3+40b3]     - (7)

Using equations (1) to (7)  

T = 3600 + 30y - 10b1

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