Presentation on theme: "Bus 480 – Lecture 2 Transportation and Assignment models"— Presentation transcript:
1 Bus 480 – Lecture 2 Transportation and Assignment models
2 Transportation ModelProduct is transported from location to location minimum costEach source able to supply a fixed number of unitsEach destination has a demand for a fixed number of unitsCan have multiple solutions, but the same minimum cost
3 Transportation ModelLet Xij = amount of product from point i to point jWe will minimize costThe constraints will be the amount that is shipped out of point i to point jSupply ConstraintsDemand ConstraintsEach row will be its own constraintEach column will be its own constraint
4 Example #4 From/To Site 1 Site 2 Site 3 Supply A $6 $9 $11 130 B $12 $3$570C$4$8100Demand8011060Each cell represents the cost of shipping one unit from site i to site jThe cost of sending one unit of product from site A to site 1 is $6Site 1 has a demand of 80 unitsSite A can supply 130 units
5 Example #4 Let XA1=amount shipped from site A to site 1 XB1=amount shipped from site B to site 1XB2=amount shipped from site B to site 2XB3=amount shipped from site B to site 3XC1=amount shipped from site C to site 1XC2=amount shipped from site C to site 2XC3=amount shipped from site C to site 3
6 Demand/Supply Check Check to see if total supply equals total demand. If yes, then constraints are equalityIf no, then if supply > demand then supply constraints are <= and demand is =if supply < demand then supply constraints are = and demand is <=
7 Example 4 Total demand : 80 + 110 + 60 = 250 Total supply : = 300Supply is greater than demand.Cannot ship more product than is demandedSupply constraints will be <=Can meet all the demandDemand constraints will be =
8 Example 4 Objective Function Min 6XA1 + 9XA2 + 11XA3 + 12XB1 + 3XB XB3 +4XC1 + 8XC2 + 11XC3Note the structure. It’s the same as the table
12 Example 4 - Solution Site A : Ship 80 units to site 2 Site B : Ship 10 units to site 2 and 60 units to site 3Site C : Ship 80 units to site 1 and 20 units to site 2Minimum cost = $1530Check All constraints to verify the results
13 Transshipment Problems Have intermediate sitesIf an intermediate site cannot hold product then everything going into the site must equal that going out of the siteStill have supply and demand constraintsOne extra constraint for each site
14 Transshipment Diagram Site D and E are intermediary SitesSites A, B, C are supply sitesSites F, G, H are demand sitesABCEDHGF
15 Transshipment Constraints Everything going into site d must equal everything leaving site dXad + Xbd + Xcd = Xdf + Xdg + XdhIn Excel, move all variables to the left side of the equationXad + Xbd + Xcd - Xdf - Xdg - Xdh = 0
16 Example 41 Ship from European port to US port Once product gets to US port, it is immediately shipped to the inland portThis makes the transshipment constraints=0If the port can hold product, then 0 will change to the amount of product the port can hold, typically ≤ . Why?
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