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Bus 480 – Lecture 2 Transportation and Assignment models

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Transportation Model Product is transported from location to location minimum cost Each source able to supply a fixed number of units Each destination has a demand for a fixed number of units Can have multiple solutions, but the same minimum cost

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Transportation Model Let X ij = amount of product from point i to point j We will minimize cost The constraints will be the amount that is shipped out of point i to point j –Supply Constraints –Demand Constraints Each row will be its own constraint Each column will be its own constraint

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Example #4 From/ToSite 1Site 2Site 3Supply A$6$9$11130 B$12$3$570 C$4$8$11100 Demand Each cell represents the cost of shipping one unit from site i to site j The cost of sending one unit of product from site A to site 1 is $6 Site 1 has a demand of 80 units Site A can supply 130 units

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Example #4 Let X A1 =amount shipped from site A to site 1 X A2 =amount shipped from site A to site 2 X A3 =amount shipped from site A to site 3 X B1 =amount shipped from site B to site 1 X B2 =amount shipped from site B to site 2 X B3 =amount shipped from site B to site 3 X C1 =amount shipped from site C to site 1 X C2 =amount shipped from site C to site 2 X C3 =amount shipped from site C to site 3

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Demand/Supply Check Check to see if total supply equals total demand. –If yes, then constraints are equality –If no, then if supply > demand then supply constraints are <= and demand is = –if supply < demand then supply constraints are = and demand is <=

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Example 4 Total demand : = 250 Total supply : = 300 Supply is greater than demand. –Cannot ship more product than is demanded Supply constraints will be <= –Can meet all the demand Demand constraints will be =

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Example 4 Objective Function Min 6X A1 + 9X A2 + 11X A3 + 12X B1 + 3X B2 + 5X B3 + 4X C1 + 8X C2 + 11X C3 Note the structure. Its the same as the table

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Example 4 Supply Constraints X A1 + X A2 + X A3 130 X B1 + X B2 + X B3 70 X C1 + X C2 + X C3 100 Why are the signs ? Demand Constraints X A1 + X B1 + X C1 = 80 X A2 + X B2 + X C2 = 110 X A3 + X B3 + X C3 = 60 Why are the signs =?

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Example 4 Hint : Copy the table twice. Once for the decision variables and one for costs Each row will be a sum. Each column will be a sum. Objective function will be sumproduct

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Example 4

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Example 4 - Solution Site A : Ship 80 units to site 2 Site B : Ship 10 units to site 2 and 60 units to site 3 Site C : Ship 80 units to site 1 and 20 units to site 2 Minimum cost = $1530 Check All constraints to verify the results

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Transshipment Problems Have intermediate sites If an intermediate site cannot hold product then everything going into the site must equal that going out of the site Still have supply and demand constraints One extra constraint for each site

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Transshipment Diagram Site D and E are intermediary Sites Sites A, B, C are supply sites Sites F, G, H are demand sites A B C E D H G F

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Transshipment Constraints Everything going into site d must equal everything leaving site d X ad + X bd + X cd = X df + X dg + X dh In Excel, move all variables to the left side of the equation X ad + X bd + X cd - X df - X dg - X dh = 0

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Example 41 Ship from European port to US port Once product gets to US port, it is immediately shipped to the inland port –This makes the transshipment constraints=0 –If the port can hold product, then 0 will change to the amount of product the port can hold, typically. Why?

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Example 41- Solution

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Cost function is the addition of multiple sumproducts

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Assignment Problem Set up identical the transportation problem All constraints are equal to 1 X ij will be binary = 1 if selected, 0 otherwise

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Assignment Example

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