Example 1: The Transportation Problem

There are three plants making widgets, and three widget-buying customers. Each plant has a fixed supply, and each customer has a fixed demand. The table below shows the supply and demand information, along with the cost of shipping from a given plant to a given customer.

	Customer 1	Customer 2	Customer 3	Total Supply
Plant 1	6	1	4	100
Plant 2	14	5	8	300
Plant 3	20	14	15	300
Total Demand	300	200	200	700

Our objective is to minimize the shipping cost.

Formulate the Problem:

Define 9 variables, each with a double subscript that refers to the given Plant and Customer.

Objective Function :

Min Z = 6x₁₁ + 1x₁₂ + 4x₁₃ + 14x₂₁ + 5x₂₂ + 8x₂₃ + 20x₃₁ + 14x₃₂ + 15x₃₃

ST : x₁₁ + x₁₂ + x₁₃ <= 100

x₂₁ + x₂₂ + x₂₃ <= 300

x₃₁ + x₃₂ + x₃₃ <= 300

x₁₁ + x₂₁ + x₃₁ = 300

x₁₂ + x₂₂ + x₃₂ = 200

x₁₃ + x₂₃ + x₃₃ = 200

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