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Chapter 6 Transportation and Assignment Problems The Transportation Problem • Given: – Capacity of each source; – Demand of each destination; – Transportation cost to ship one unit from a source to a destination. • To find the most economical way of satisfying the demands of the destinations by using the resources. A Transportation Example, p.227-228 (234-235) Suppliers (Sources) Demanders (Destinations) Chicago St. Louis Cincinnati Capacities of Suppliers Kansas City $6 /ton $8 /ton $10 /ton 150 tons Omaha $7 /ton $11 /ton $11 /ton 175 tons Des Moines $4 /ton $5 /ton $12 /ton 275 tons 200 tons 100 tons 300 tons Demands of Demanders How to satisfy the demands by using the sources with lowest total cost? (That is: How many tons should be shipped from each source to each destination?) Solving Transportation Problem • The solution method (algorithm) is elegant. But, as business people, we do not need to know the details since computers can help us solve it. • Use the ‘transportation module’ in QM. Total Supply and Total Demand • Total supply is not necessary equal to total demand. • A dummy source or a dummy destination appears in the QM result if total supply is not equal to total demand. Dummy Source or Destination • A dummy source in the result of QM indicates an overall shortage, and at which destinations shortages will occur. • A dummy destination in the result of QM indicates an overall surplus, and which sources will have surpluses. Prohibited Route • If a route is prohibited to use, just set the unit transportation cost of that route to a very large number. The Assignment Problem • Given – The cost (or efficiency index) for a person to a job. • To assign Y persons to Y jobs so that the total cost is minimized or total efficiency is maximized. An Assignment example, p.240 (247) Distances to drive for each official. Official (persons) Raleigh Game Sites (jobs) Atlanta Durham Clemson A 210 90 180 160 B 100 70 130 200 C 175 105 140 170 D 80 65 105 120 Solving Assignment Problem • It is a special transportation problem (why?), so it can be solved by using ‘Transportation Module’ in QM. • More conveniently, we use the ‘Assignment Module’ in QM. Assignment Problem vs. Transportation Problem • The assignment problem is a special case of the transportation problem in which demands of all destinations are 1, and capacities of all sources are 1.