duality - Chapter 7 Transportation, Assignment and...

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. 1 Chapter 7 Transportation, Assignment and Transshipment Problems
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. 2 Applications Physical analog of nodes Physical analog of arcs Flow Communication systems phone exchanges, computers, transmission facilities, satellites Cables, fiber optic links, microwave relay links Voice messages, Data, Video transmissions Hydraulic systems Pumping stations Reservoirs, Lakes Pipelines Water, Gas, Oil, Hydraulic fluids Integrated computer circuits Gates, registers, processors Wires Electrical current Mechanical systems Joints Rods, Beams, Springs Heat, Energy Transportation systems Intersections, Airports, Rail yards Highways, Airline routes Railbeds Passengers, freight, vehicles, operators Applications of Network Optimization
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. 3 Description A transportation problem basically deals with the problem, which aims to find the best way to fulfill the demand of n demand points using the capacities of m supply points. While trying to find the best way, generally a variable cost of shipping the product from one supply point to a demand point or a similar constraint should be taken into consideration.
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. 4 7.1 Formulating Transportation Problems Example 1: Powerco has three electric power plants that supply the electric needs of four cities. The associated supply of each plant and demand of each city is given in the table 1. The cost of sending 1 million kwh of electricity from a plant to a city depends on the distance the electricity must travel.
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. 5 Transportation tableau A transportation problem is specified by the supply, the demand, and the shipping costs. So the relevant data can be summarized in a transportation tableau. The transportation tableau implicitly expresses the supply and demand constraints and the shipping cost between each demand and supply point.
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. 6 Table 1. Shipping costs, Supply, and Demand for Powerco Example From To City 1 City 2 City 3 City 4 Supply (Million kwh) Plant 1 $8 $6 $10 $9 35 Plant 2 $9 $12 $13 $7 50 Plant 3 $14 $9 $16 $5 40 Demand (Million kwh) 45 20 30 30 Transportation Tableau
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. 7 Solution 1. Decision Variable: Since we have to determine how much electricity is sent from each plant to each city; X ij = Amount of electricity produced at plant i and sent to city j X 14 = Amount of electricity produced at plant 1 and sent to city 4
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. 8 2. Objective function Since we want to minimize the total cost of shipping from plants to cities; Minimize Z = 8X 11 +6X 12 +10X 13 +9X 14 +9X 21 +12X 22 +13X 23 +7X 24 +14X 31 +9X 32 +16X 33 +5X 34
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. 9 3. Supply Constraints Since each supply point has a limited production capacity; X 11 +X 12 +X 13 +X 14 <= 35 X 21 +X 22 +X 23 +X 24 <= 50 X 31 +X 32 +X 33 +X 34 <= 40
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. 10 4. Demand Constraints Since each supply point has a limited production capacity; X 11 +X 21 +X 31 >= 45 X 12 +X 22 +X 32 >= 20 X 13 +X 23 +X 33 >= 30 X 14 +X 24 +X 34 >= 30
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. 11 5. Sign Constraints Since a negative amount of electricity can not be shipped all Xij’s must be non negative; Xij >= 0 (i= 1,2,3; j= 1,2,3,4)
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This note was uploaded on 05/07/2011 for the course INDUSTRIAL IE 208 taught by Professor Serolbulkan during the Fall '11 term at Marmara Üniversitesi.

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duality - Chapter 7 Transportation, Assignment and...

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