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2_Transportation,_Transshipment_and_Assignment

# 2_Transportation,_Transshipment_and_Assignment -...

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1 Transportation, Transshipment, and Assignment (TTA) Chapter 6 2 Chapter Outline To examine special types of LP problems: Transportation Problem Transshipment Problem Assignment Problem 3 Transportation Model: Characteristics (1 of 2) Consider the transportation of a product from supply source S i to demand destination D j 4 Transportation Model: Characteristics (2 of 2) A product is transported from a number of sources to a number of destinations with the possible objective : Minimization of total transportation cost , or Maximization of total profit , or Minimization of total distance traveled Each source is able to supply a fixed number of units of the product, and each destination has a fixed demand for the product with constraints : Satisfy the demand requirement at each destination Satisfy the supply capacity available at each source All constraints are equalities in a balanced transportation model where supply equals demand Constraints contain inequalities in unbalanced models where supply does not equal demand Assumption : Transportation cost is proportional to number of units transported

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5 Transport Cost from Grain Elevator to Mill (\$/ton) Grain Elevator A. Chicago B. St. Louis C. Cincinnati 1. Kansas City 2. Omaha 3. Des Moines \$ 6 7 4 \$ 8 11 5 \$ 10 11 12 Transportation Model Example Problem Definition and Data [Supply=Demand] (1 of 9) How many tons of wheat to transport from each grain elevator to each mill on a monthly basis in order to minimize the total cost of transportation? Grain Elevator Supply Mill Demand 1. Kansas City 150 A. Chicago 220 2. Omaha 175 B. St. Louis 100 3. Des Moines 275 C. Cincinnati 300 Total 600 tons Total 600 tons 6 Transportation Model Example Transportation Network Routes (2 of 9) Figure 6.1 Network of Transportation Routes for Wheat Shipments 7 The 3 mills demand 600 total tons, which is the exact amount that can be supplied by the 3 grain elevators (300 total tons) => balanced transportation model (supply exactly equals demand) Let x ij = tons of wheat from each grain elevator i (i = 1, 2, 3) to each mill j (j = A,B,C) Minimize total transportation cost, Z = 6x 1A + 8x 1B + 10x 1C + 7x 2A + 11x 2B + 11x 2C + 4x 3A + 5x 3B + 12x 3C subject to: x 1A + x 1B + x 1C = 150 x 2A + x 2B + x 2C = 175 x 3A + x 3B + x 3C = 275 x 1A + x 2A + x 3A = 200 x 1B + x 2B + x 3B = 100 x 1C + x 2C + x 3C = 300 x ij 0 Transportation Model Example Model Formulation [Supply=Demand] (3 of 9) Supply Capacity Demand Requirement 8 Transport Cost from Grain Elevator to Mill (\$/ton) Grain Elevator A. Chicago B. St. Louis C. Cincinnati 1. Kansas City 2. Omaha 3. Des Moines \$ 6 7 4 \$ 8 11 5 \$ 10 11 12 Transportation Model Example Problem Definition and Data [Supply<Demand] (4 of 9)
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2_Transportation,_Transshipment_and_Assignment -...

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