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Linear Programming

Linear Programming. Example 5 Transportation Problem. Transportation Problem.

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Linear Programming

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  1. Linear Programming Example 5 Transportation Problem

  2. Transportation Problem • The transportation problem seeks to minimize the total shipping costs of transporting goods from m origins (each with a supply si) to n destinations (each with a demand dj), when the unit shipping cost from an origin, i, to a destination, j, is cij. • The network representation for a transportation problem with two sources and three destinations is given on the next slide.

  3. Network Representation 1 d1 c11 1 c12 s1 c13 2 d2 c21 c22 2 s2 c23 3 d3 Sources Destinations

  4. LP Formulation The LP formulation in terms of the amounts shipped from the origins to the destinations, xij , can be written as: Min cijxij i j s.t. xij<si for each origin i j xij = dj for each destination j i xij> 0 for all i and j

  5. Example: Acme Block Co. • Acme Block Company has orders for 80 tons ofconcrete blocks at three suburban locationsas follows: Northwood -- 25 tons, Westwood -- 45 tons, and Eastwood -- 10 tons. • Acmehas two plants, each of whichcan produce 50 tons per week. • Delivery cost per ton from each plant to each suburban location is shown on the next slide. How should end of week shipments be made to filltheabove orders?

  6. Delivery Cost • Delivery Cost Per Ton NorthwoodWestwoodEastwood Plant 1 24 30 40 Plant 2 30 40 42

  7. LP Model Decision Variable X11: Tons of Concrete shipped from Plan 1 to Northwood X12: Tons of Concrete shipped from Plan 1 to Westwood X13: Tons of Concrete shipped from Plan 1 to Eastwood X21: Tons of Concrete shipped from Plan 2 to Northwood X22: Tons of Concrete shipped from Plan 2 to Westwood X13: Tons of Concrete shipped from Plan 2 to Eastwood Objective Function Min. 24X11+30X12+40X13+30X21+40X22+42X23

  8. LP Model Constraints X11 + X12 +X13 <= 50 Plant 1 X21 + X22 + X23 <= 50 Plant 2 X11+X21 = 25 Northwood X12+X22 = 45 Westwood X13+X23 = 10 Eastwood Non-Negativity X11, X12, X13, X21, X22, X23 >=0

  9. Excel Input

  10. Solver Solution

  11. Solution • Optimal Solution FromToAmountCost Plant 1 Northwood 5 120 Plant 1 Westwood 45 1,350 Plant 2 Northwood 20 600 Plant 2 Eastwood 10 420 Total Cost = $2,490

  12. Sensitivity Report • Partial Sensitivity Report (first half)

  13. Sensitivity Report • Partial Sensitivity Report (second half)

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