T ransportation M odel

1. Transportation Model Winston S. Sirug, PhD Full Professor

2. Topics  Basic Concepts  Initial Feasible Solution Optimal Feasible Solution  Minimization: Optimal Feasible Solution  Maximization: Optimal Solution  Alternative Optimal Solution  Balanced and Unbalanced Transportation Model  Degenerate and Non-degenerate Transportation Model

3. Historical Facts The transportation model was first proposed in 1941, the study is entitled: The Distribution of a Product from Several Sources to Numerous Localities, which was written by Frank Lauren Hitchcock. Frank Lauren Hitchcock (1875–1957) was an American mathematician and physicist notable for vector analysis. He was also an expert in mathematical chemistry and quaternions (number system that extends the complex numbers. He received his AB from Harvard in 1896. and in 1910 he completed his PhD at Harvard with a thesis entitled, Vector Functions of a Point.

4. Historical Facts In 1947, a second major contribution was written independently by Tjalling Charles Koopmans entitled: Optimum Utilization of the Transportation System. T. C. Koopmans (Aug 28, 1910-Feb 26, 1985) was the joint winner, with Leonid Kantorovich, of the 1975 Nobel Memorial Prize in Economic Sciences.

5. Historical Facts In 1953, the Stepping Stone Method was developed by Abraham Charnesand William Wager Cooper. Abraham Charnes (Sept. 4, 1917-Dec. 19, 1992), professor emeritus of management science and information systems. He received bachelor's, master's, and PhD degrees from the University of Illinois. W.W. Cooper (Jul. 19, 1914-June 20, 2012) has an A.B. degree, majoring in economics and Ph.D. in business at Columbia University.A high school dropout and former boxing champ, he went on to revolutionize business education and research.

6. Historical Facts In 1955, the Modified Distribution Method was developed.

7. Basic Concepts The transportation model is a special procedure of linear programming for finding the minimum cost for distributing homogeneous units of a product from several points of supply (origin) to a number of points of demand (destinations).  the number of sources and their capacity to supply the products to distribute.  the number of destinations and their demand.  the per-unit cost of shipping the products from a certain source to a certain destination.

8. Source and Destination Source Destination 1 A In units In units 2 B In units In units 3 C In units In units Shipping Routes Requirements Capacities

9. Transportation Modeling  The goal is to select the shipping routes and units to be shipped to minimize total transportation cost.  The relationships among variables is linear: transportation costs are treated as a direct linear function of the number of units shipped  The more units we send somewhere, the more it’s going to cost us.

10. Assumptions on Transportation Model  The items to be shipped are homogeneous (or the same/similar).  Shipping cost per unit is the same regardless of the number of units shipped.  There is only one route or mode of transportation being used between each origin and each destination.

11. Terms A Transportation Modelis a LP problem in which a product is to be transported from a no. of sources to a no. of destinations at the minimum cost (for minimization problem) or maximum profit (maximum problem). A destination is a point of demand in a transportation problem Originis the source or supply location in a transportation problem. Unused squares(or unused cells) are squares, which represent routes where no quantity is shipped between a source and a destination Stone squares(or used cells) are used squares in the transportation problem are called

12. Transportation Model: Minimization Problem Example: The WSS Company sells desktop computers to IT companies in Metro Manila, and ships them from three distribution warehouses located in three (3) different areas. The company is able to supply the following numbers of desktop computers to IT companies by the beginning of the year: IT companies have ordered desktop computers that must be delivered and installed by the beginning of the year

13. Transportation Model: Minimization Problem The shipping costs per desktop computer from each distributor to each company are as follows: With cost minimization as a criterion, WSS Company wants to determine how many desktop computers should be shipped from each warehouse to each IT company

14. Solution Let 1 = Warehouse 1 2 = Warehouse 2 3 = Warehouse 3 A = AUS Link B = SJS Networking Inc. C = RFS Data Limited

15. Source and Destination Warehouse IT Company 1 A 150 units 100 units 2 B 200 units 80 units 3 C 50 units 220 units Requirements Capacities

16. Source and Destination Warehouse IT Company 7 1 A 150 units 100 units 5 9 10 12 2 B 200 units 80 units 10 6 3 3 C 50 units 220 units 14 Shipping Routes Requirements Capacities

17. Transportation Table Destinations Distribution x1A Warehouse 1 to IT Company A x1B Warehouse 1 to IT Company B x1C Warehouse 1 to IT Company C x2A Warehouse 2 to IT Company A x2B Warehouse 2 to IT Company B x2C Warehouse 2 to IT Company C x3A Warehouse 3 to IT Company A x3B Warehouse 3 to IT Company B x3C Warehouse 3 to IT Company C Origin

18. Transportation Problem Objective Function: Minimize: C = 7x1A+ 5x1B+ 9x1C+ 10x2A+ 12x2B+ 10x2C+ 6x3A+ 3x3B+ 14x3C Constraints: Subject to: x1A+ x1B+ x1C= 150 x2A+ x2B+ x2C= 200 x3A+ x3B+ x3C= 50 x1A+ x2A+ x3A= 100 x1B+ x2B + x3B= 80 x1C+ x2C+ x3C= 220 xij 0

19. Methods in Establishing Initial Solution • Northwest Corner Rule (NCR). It starts with allocating units to the upper left-hand corner and ends in the lower right corner of any transportation problem. Greedy Method or Minimum Cost Method (MCM). This method allocates the least-cost/highest-profit cell. Ties may be broken arbitrarily. The procedure is completed when all row and column requirements are addressed. • Vogel’s Approximation Method (VAM).It considers the “penalty cost” of not using the cheapest available route. One study found that VAM yields an optimum solution in 80% of the sample problems tested.

20. Methods in Establishing Optimal Solution • Stepping Stone Method (SSM). It involves tracing closed paths from each unused square through stone squares. It evaluate each unused square to determine whether a shift into it is advantageous from a total-cost/total-profit stand point. Unused squares were referred to as "water" and used cells as "stones.”. • Modified Distribution Method (MODI). It determines the per-unit cost/profit change associated with assigning flow to an unused square in the transportation problem. The tableau is modified with U (row) and V (column) variables. Allocated cell costs Cij= Ui+ Vj.

21. Northwest Corner Rule Northwest Choose the row/column with lower amount. 100

22. Northwest Corner Rule 50

23. Northwest Corner Rule 30

24. Northwest Corner Rule 170

25. Northwest Corner Rule 50

26. Northwest Corner Rule Solution

27. Minimum Cost Method (MCM) Determine the lowest transportation cost Choose the row/column with lower amount. 50

28. Minimum Cost Method (MCM) 30

29. Minimum Cost Method (MCM) 100

30. Minimum Cost Method (MCM) 20

31. Minimum Cost Method (MCM) 200

32. Minimum Cost Method (MCM)

33. Breaking of a Tie in a Minimum Cost Method (MCM) 80 Choose the higher allocation in the target squares. 80 50

34. Vogel’s Approximation Method (VAM) Choose the row/column with lower amount. 50 Largest Difference

35. Vogel’s Approximation Method (VAM) 30 Choose the row/column with lower amount. Largest Difference

36. Vogel’s Approximation Method (VAM) 100 Choose the row/column with lower amount. Largest Difference

37. Vogel’s Approximation Method (VAM) 20

38. Vogel’s Approximation Method (VAM) 200

39. Vogel’s Approximation Method (VAM)

40. Stepping Stone Method (SSM): Minimization Objective Function: Minimize: C = 7x1A+ 5x1B+ 9x1C+ 10x2A+ 12x2B+ 10x2C+ 6x3A+ 3x3B+ 14x3C Constraints: Subject to: x1A+ x1B+ x1C= 150 x2A+ x2B+ x2C= 200 x3A+ x3B+ x3C= 50 x1A+ x2A+ x3A= 100 x1B+ x2B + x3B= 80 x1C+ x2C+ x3C= 220 xij 0

41. Northwest Corner Rule Solution

42. Tracing of Closed Path Collect all unused squares: x1C, x2A, x3A, x3B Start with unused square x1C Tableau 1 Closed path of x1Cis +x1C – x2C + x2B – x1B With improvement index of +9 – 10 + 12 – 5 = 6 Improvement Indexis the increase/decrease in a total cost (for minimization problem) and total profit (for maximization problem) that would result from reallocating one unit to an unused square. Closed Path in unused squares are unique.

43. Tracing of Closed Path Tableau 1 Next unused square is x2A Closed path of x2A is +x2A – x1A + x1B – x2B With improvement index of +10 – 7 + 5 – 12 = –4

44. Tracing of Closed Path Tableau 1 Third unused square is x3A Closed path of x3A is +x3A – x1A + x1B – x2B + x2C – x3C With improvement index of +6 – 7 + 5 – 12 + 10 – 14 = –12

45. Tracing of Closed Path Tableau 1 Last unused square is x3B Closed path of x3C is +x3B – x2B + x2C – x3C With improvement index of +3 – 12 + 10 – 14 = –13

46. Improvement Index Computation Lowest Negative Negative improvement index indicates a deduction to TTC. Choose the lowest negative. The lowest negative improvement index indicates that amount deducted in every unit transported to x3B.

47. Change in Assignment Distribution Tableau 1 We need to allocate an amount to x3B Recall that the closed path of x3B is +x3B – x2B + x2C – x3C Select the lower amount w/c is x2B. Move 30 units to x3B.

48. Balance the Stone Squares Tableau 2 Recall the other unused squares x1C, x2A, x3A. 50 100 200 20

49. Transportation Cost Computation Previous TTC is 3,710 (30)(–13) = –390 3,320

50. Tracing of Closed Path Tableau 2 Collect all unused squares: x1C, x2A, x2B, x3A Start with unused square x1C Closed path of x1Cis +x1C – x3C+ x3B– x1B With improvement index of +9 – 14 + 3 – 5 = –7