1. . 1 Transportation and Assignment Problems

2. . 2 Applications Physical analog of nodes Physical analog of arcs Flow Communication systems phone exchanges, computers, transmission facilities, satellites Cables, fiber optic links, microwave relay links Voice messages, Data, Video transmissions Hydraulic systems Pumping stations Reservoirs, Lakes Pipelines Water, Gas, Oil, Hydraulic fluids Integrated computer circuits Gates, registers, processors WiresElectrical current Mechanical systemsJoints Rods, Beams, Springs Heat, Energy Transportation systems Intersections, Airports, Rail yards Highways, Airline routes Railbeds Passengers, freight, vehicles, operators Applications of Network Optimization

3. . 3 Description A transportation problem basically deals with the problem, which aims to find the best way to fulfill the demand of n demand points using the capacities of m supply points. While trying to find the best way, generally a variable cost of shipping the product from one supply point to a demand point or a similar constraint should be taken into consideration.

4. . 4 Formulating Transportation Problems Example 1: Powerco has three electric power plants that supply the electric needs of four cities. The associated supply of each plant and demand of each city is given in the table 1. The cost of sending 1 million kwh of electricity from a plant to a city depends on the distance the electricity must travel.

5. . 5 Transportation tableau A transportation problem is specified by the supply, the demand, and the shipping costs. So the relevant data can be summarized in a transportation tableau. The transportation tableau implicitly expresses the supply and demand constraints and the shipping cost between each demand and supply point.

6. . 6 Table 1. Shipping costs, Supply, and Demand for Powerco Example FromTo City 1City 2City 3City 4Supply (Million kwh) Plant 1$8$6$10$935 Plant 2$9$12$13$750 Plant 3$14$9$16$540 Demand (Million kwh) 452030 Transportation Tableau

7. . 7 Solution 1.Decision Variable: Since we have to determine how much electricity is sent from each plant to each city; X ij = Amount of electricity produced at plant i and sent to city j X 14 = Amount of electricity produced at plant 1 and sent to city 4

8. . 8 2. Objective function Since we want to minimize the total cost of shipping from plants to cities; Minimize Z = 8X 11 +6X 12 +10X 13 +9X 14 +9X 21 +12X 22 +13X 23 +7X 24 +14X 31 +9X 32 +16X 33 +5X 34

9. . 9 3. Supply Constraints Since each supply point has a limited production capacity; X 11 +X 12 +X 13 +X 14 <= 35 X 21 +X 22 +X 23 +X 24 <= 50 X 31 +X 32 +X 33 +X 34 <= 40

10. . 10 4. Demand Constraints Since each supply point has a limited production capacity; X 11 +X 21 +X 31 >= 45 X 12 +X 22 +X 32 >= 20 X 13 +X 23 +X 33 >= 30 X 14 +X 24 +X 34 >= 30

11. . 11 5. Sign Constraints Since a negative amount of electricity can not be shipped all Xij’s must be non negative; Xij >= 0 (i= 1,2,3; j= 1,2,3,4)

12. . 12 LP Formulation of Powerco’s Problem Min Z = 8X 11 +6X 12 +10X 13 +9X 14 +9X 21 +12X 22 +13X 23 +7X 24 +14X 31 +9X 32 +16X 33 +5X 34 S.T.:X 11 +X 12 +X 13 +X 14 <= 35 (Supply Constraints) X 21 +X 22 +X 23 +X 24 <= 50 X 31 +X 32 +X 33 +X 34 <= 40 X 11 +X 21 +X 31 >= 45 (Demand Constraints) X 12 +X 22 +X 32 >= 20 X 13 +X 23 +X 33 >= 30 X 14 +X 24 +X 34 >= 30 Xij >= 0 (i= 1,2,3; j= 1,2,3,4)

13. . 13 General Description of a Transportation Problem 1.A set of m supply points from which a good is shipped. Supply point i can supply at most s i units. 2.A set of n demand points to which the good is shipped. Demand point j must receive at least d i units of the shipped good. 3.Each unit produced at supply point i and shipped to demand point j incurs a variable cost of c ij.

14. . 14 X ij = number of units shipped from supply point i to demand point j

15. . 15 Balanced Transportation Problem If Total supply equals to total demand, the problem is said to be a balanced transportation problem:

16. . 16 Methods to find the bfs for a balanced TP There are two basic methods: 1.Northwest Corner Method 2.Vogel’s Method

17. . 17 1. Northwest Corner Method To find the bfs by the NWC method: Begin in the upper left (northwest) corner of the transportation tableau and set x 11 as large as possible (here the limitations for setting x 11 to a larger number, will be the demand of demand point 1 and the supply of supply point 1. Your x 11 value can not be greater than minimum of this 2 values).

18. . 18 According to the explanations in the previous slide we can set x 11 =3 (meaning demand of demand point 1 is satisfied by supply point 1).

19. . 19 After we check the east and south cells, we saw that we can go east (meaning supply point 1 still has capacity to fulfill some demand).

20. . 20 After applying the same procedure, we saw that we can go south this time (meaning demand point 2 needs more supply by supply point 2).

21. . 21 Finally, we will have the following bfs, which is: x 11 =3, x 12 =2, x 22 =3, x 23 =2, x 24 =1, x 34 =2

22. . 22 3. Vogel’s Method Begin with computing each row and column a penalty. The penalty will be equal to the difference between the two smallest shipping costs in the row or column. Identify the row or column with the largest penalty. Find the first basic variable which has the smallest shipping cost in that row or column. Then assign the highest possible value to that variable, and cross-out the row or column as in the previous methods. Compute new penalties and use the same procedure.

23. . 23 An example for Vogel’s Method Step 1: Compute the penalties.

24. . 24 Step 2: Identify the largest penalty and assign the highest possible value to the variable.

25. . 25 Step 3: Identify the largest penalty and assign the highest possible value to the variable.

26. . 26 Step 4: Identify the largest penalty and assign the highest possible value to the variable.

27. . 27 Step 5: Finally the bfs is found as X 11 =0, X 12 =5, X 13 =5, and X 21 =15

28. . 28 The Transportation Simplex Method In this section we will explain how the simplex algorithm is used to solve a transportation problem.

29. . 29 How to Pivot a Transportation Problem Based on the transportation tableau, the following steps should be performed. Step 1. Determine (by a criterion to be developed shortly, for example northwest corner method) the variable that should enter the basis. Step 2. Find the loop (it can be shown that there is only one loop) involving the entering variable and some of the basic variables. Step 3. Counting the cells in the loop, label them as even cells or odd cells.

30. . 30 Step 4. Find the odd cells whose variable assumes the smallest value. Call this value θ. The variable corresponding to this odd cell will leave the basis. To perform the pivot, decrease the value of each odd cell by θ and increase the value of each even cell by θ. The variables that are not in the loop remain unchanged. The pivot is now complete. If θ=0, the entering variable will equal 0, and an odd variable that has a current value of 0 will leave the basis. In this case a degenerate bfs existed before and will result after the pivot. If more than one odd cell in the loop equals θ, you may arbitrarily choose one of these odd cells to leave the basis; again a degenerate bfs will result

31. . 31 Assignment Problems Example: Machineco has four jobs to be completed. Each machine must be assigned to complete one job. The time required to setup each machine for completing each job is shown in the table below. Machinco wants to minimize the total setup time needed to complete the four jobs.

32. . 32 Setup times (Also called the cost matrix) Time (Hours) Job1Job2Job3Job4 Machine 114587 Machine 221265 Machine 37839 Machine 424610

33. . 33 The Model According to the setup table Machinco’s problem can be formulated as follows (for i,j=1,2,3,4):

34. . 34 For the model on the previous page note that: X ij =1 if machine i is assigned to meet the demands of job j X ij =0 if machine i is not assigned to meet the demands of job j In general an assignment problem is balanced transportation problem in which all supplies and demands are equal to 1.

35. . 35 The Assignment Problem In general the LP formulation is given as Minimize Each supply is 1 Each demand is 1