1. Bus 480 – Lecture 2 Transportation and Assignment models

2. 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

3. 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

4. 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

5. 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

6. 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 <=

7. Example 4 Total demand : 80 + 110 + 60 = 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 =

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

9. 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 =?

10. 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

11. Example 4

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 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

13. 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

14. 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

15. 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

16. 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?

17. Example 41- Solution

18. Example 41- Solution

19. Example 41- Solution

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

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

22. Assignment Example