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Example 11- (Transportation problem) There are two factories - Transport problems

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Example 11 (Transportation problem) There are two factories located one at place P and the other at place Q. From these locations, a certain commodity is to be delivered to each of the three depots situated at A, B and C. The weekly requirements of the depots are respectively 5, 5 and 4 units of the commodity while the production capacity of the factories at P and Q are respectively 8 and 6 units. The cost of transportation per unit is given below: How many units should be transported from each factory to each depot in order that the transportation cost is minimum. What will be the minimum transportation cost? Since number of units transported to each depot must be greater than or equal to zero -: ∴ x ≥ 0, y ≥ 0 8 − (x + y) ≥ 0 ⇒ x + y ≤ 8 5 − x ≥ 0 ⇒ x ≤ 5 5 − y ≥ 0 ⇒ y ≤ 5 x + y − 4 ≥ 0 ⇒ x + y ≥ 4 As we need to minimize the cost of transportation, Hence the function used is minimize Z. Total transportation cost will be Z = 160 x + 100y + 150 [8 − x + y] + 100 (5 – x) + 120 (5 − y) + 100 (x + y − 4) Z = 10 x − 70y + 1900 Combining all constraints : Min, Z = 10x − 70y + 1900 Subject to constraints : x + y ≤ 8 x ≤ 5 y ≤ 5 x + y ≥ 4 x ≥ 0 , y ≥ 0

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