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P&T Company Distribution Problem. Shipping Data. Current Shipping Plan. Shipping Cost per Truckload. Total shipping cost = 75($464) + 5($352) + 65($416) + 55($690) + 15($388) + 85($685) = $165,595. Terminology for a Transportation Problem. Characteristics of Transportation Problems.
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Shipping Cost per Truckload Total shipping cost = 75($464) + 5($352) + 65($416) + 55($690) + 15($388) + 85($685) = $165,595
Characteristics of Transportation Problems • The Requirements Assumption • Each source has a fixed supply of units, where this entire supply must be distributed to the destinations. • Each destination has a fixed demand for units, where this entire demand must be received from the sources. • The Feasible Solutions Property • A transportation problem will have feasible solutions if and only if the sum of its supplies equals the sum of its demands. • The Cost Assumption • The cost of distributing units from any particular source to any particular destination is directly proportional to the number of units distributed. • This cost is just the unit cost of distribution times the number of units distributed.
The Transportation Model Any problem (whether involving transportation or not) fits the model for a transportation problem if • It can be described completely in terms of a table like Table 6.5 that identifies all the sources, destinations, supplies, demands, and unit costs, and • satisfies both the requirements assumption and the cost assumption. The objective is to minimize the total cost of distributing the units.
The Transportation Problem is an LP Let xij = the number of truckloads to ship from cannery i to warehouse j (i = 1, 2, 3; j = 1, 2, 3, 4)Minimize Cost = $464x11 + $513x12 + $654x13 + $867x14 + $352x21 + $416x22 + $690x23 + $791x24 + $995x31 + $682x32 + $388x33 + $685x34subject to Cannery 1: x11 + x12 + x13 + x14 = 75 Cannery 2: x21 + x22 + x23 + x24 = 125 Cannery 3: x31 + x32 + x33 + x34 = 100 Warehouse 1: x11 + x21 + x31 = 80 Warehouse 2: x12 + x22 + x32 = 65 Warehouse 3: x13 + x23 + x33 = 70 Warehouse 4: x14 + x24 + x34 = 85andxij ≥ 0 (i = 1, 2, 3; j = 1, 2, 3, 4)
Integer Solutions Property As long as all its supplies and demands have integer values, any transportation problem with feasible solutions is guaranteed to have an optimal solution with integer values for all its decision variables. Therefore, it is not necessary to add constraints to the model that restrict these variables to only have integer values.
