Linear Programming Example 5 Transportation Problem

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

Network Representation 2 2 c 11 c 12 c 13 c 21 c 22 c 23 d1d1d1d1 d2d2d2d2 d3d3d3d3 s1s1s1s1 s2s2 SourcesDestinations

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

Example: Acme Block Co. n Acme Block Company has orders for 80 tons of concrete blocks at three suburban locations as follows: Northwood tons, Westwood tons, and Westwood tons, and Eastwood tons. Eastwood tons. n Acme has two plants, each of which can produce 50 tons per week. n 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 fill the above orders?

n Delivery Cost Per Ton Northwood Westwood Eastwood Northwood Westwood Eastwood Plant Plant Plant Plant Delivery Cost

Decision Variable X 11 : Tons of Concrete shipped from Plan 1 to Northwood X 12 : Tons of Concrete shipped from Plan 1 to Westwood X 13 : Tons of Concrete shipped from Plan 1 to Eastwood X 21 : Tons of Concrete shipped from Plan 2 to Northwood X 22 : Tons of Concrete shipped from Plan 2 to Westwood X 13 : Tons of Concrete shipped from Plan 2 to Eastwood Objective Function Min. 24X X X X X X 23 LP Model

Constraints X 11 + X 12 + X 13 <= 50Plant 1 X 21 + X 22 + X 23 <= 50Plant 2 X 11 +X 21 = 25Northwood X 12 +X 22 = 45Westwood X 13 +X 23 = 10Eastwood Non-Negativity X 11, X 12, X 13, X 21, X 22, X 23 >=0 LP Model

Excel Input

Solver Solution

n Optimal Solution From To Amount Cost From To Amount Cost Plant 1 Northwood Plant 1 Westwood 45 1,350 Plant 1 Westwood 45 1,350 Plant 2 Northwood Plant 2 Northwood Plant 2 Eastwood Plant 2 Eastwood Total Cost = $2,490 Total Cost = $2,490 Solution

n Partial Sensitivity Report (first half) Sensitivity Report

n Partial Sensitivity Report (second half) Sensitivity Report