Bus 480 – Lecture 2 Transportation and Assignment models

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

Transportation Model Let Xij = 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

Example #4 From/To Site 1 Site 2 Site 3 Supply A $6 $9 $11 130 B $12 $3 $5 70 C $4 $8 100 Demand 80 110 60 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

Example #4 Let XA1=amount shipped from site A to site 1 XB1=amount shipped from site B to site 1 XB2=amount shipped from site B to site 2 XB3=amount shipped from site B to site 3 XC1=amount shipped from site C to site 1 XC2=amount shipped from site C to site 2 XC3=amount shipped from site C to site 3

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 <=

Example 4 Total demand : 80 + 110 + 60 = 250 Total supply : 130 + 70 + 100 = 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 =

Example 4 Objective Function Min 6XA1 + 9XA2 + 11XA3 + 12XB1 + 3XB2 + 5XB3 + 4XC1 + 8XC2 + 11XC3 Note the structure. It’s the same as the table

Example 4 Supply Constraints XA1 + XA2 + XA3 ≤ 130 XB1 + XB2 + XB3 ≤ 70 XC1 + XC2 + XC3 ≤ 100 Why are the signs ≤ ? Demand Constraints XA1 + XB1 + XC1 = 80 XA2 + XB2 + XC2 = 110 XA3 + XB3 + XC3 = 60 Why are the signs =?

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

Example 4

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

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

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

Transshipment Constraints Everything going into site d must equal everything leaving site d Xad + Xbd + Xcd = Xdf + Xdg + Xdh In Excel, move all variables to the left side of the equation Xad + Xbd + Xcd - Xdf - Xdg - Xdh = 0

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?

Example 41- Solution

Example 41- Solution

Example 41- Solution

Example 41- Solution Cost function is the addition of multiple sumproducts

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

Assignment Example