1. Copyright 2006 John Wiley & Sons, Inc. Beni Asllani University of Tennessee at Chattanooga Operations Management - 6 hh Edition Chapter 11 Supplement Roberta Russell & Bernard W. Taylor, III Operational Decision-Making Tools: Transportation and Transshipment Models

2. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-2 Just how do you make decisions?  Emotional direction  Intuition  Analytic thinking  Are you an intuit, an analytic, what???  How many of you use models to make decisions??

3. 42  Arise whenever there is a perceived difference between what is desired and what is in actuality.  Problems serve as motivators for doing something  Problems lead to decisions Problems

4. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-4

5. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-5 Model Classification Criteria  Purpose  Perspective Use the perspective of the targeted decision-maker Use the perspective of the targeted decision-maker  Degree of Abstraction  Content and Form  Decision Environment  {This is what you should start any modeling facilitation meeting with}

6. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-6 Purpose  Planning  Forecasting  Training  Behavioral research

7. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-7 Perspective  Descriptive “Telling it like it is” “Telling it like it is” Most simulation models are of this type Most simulation models are of this type  Prescriptive “Telling it like it should be” “Telling it like it should be” Most optimization models are of this type Most optimization models are of this type

8. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-8 Degree of Abstraction  Isomorphic One-to-one One-to-one  Homomorphic One-to-many One-to-many

9. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-9 Content and Form  verbal descriptions  mathematical constructs  simulations  mental models  physical prototypes

10. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-10 Decision Environment  Decision Making Under Certainty TOOL: all of mathematical programming TOOL: all of mathematical programming  Decision Making under Risk and Uncertainty TOOL: Decision analysis--tables, trees, Bayesian revision TOOL: Decision analysis--tables, trees, Bayesian revision  Decision Making Under Change and Complexity TOOL: Structural models, simulation models TOOL: Structural models, simulation models

11. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-11 Mathematical Programming  Linear programming  Integer linear programming some or all of the variables are integer variables some or all of the variables are integer variables  Network programming (produces all integer solutions)  Nonlinear programming  Dynamic programming  Goal programming  The list goes on and on Geometric Programming Geometric Programming

12. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-12 A Model of this class  What would we include in it?

13. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-13 Management Science Models  A QUANTITATIVE REPRESENTATION OF A PROCESS THAT CONSISTS OF THOSE COMPONENTS THAT ARE SIGNIFICANT FOR THE PURPOSE BEING CONSIDERED

14. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-14 Mathematical programming models covered in Ch 11, Supplement  Transportation Model  Transshipment Model Not included are: Shortest Route Minimal Spanning Tree Maximal flow Assignment problem many others

15. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-15 Transportation Model  A transportation model is formulated for a class of problems with the following characteristics a product is transported from a number of sources to a number of destinations at the minimum possible cost a product is transported from a number of sources to a number of destinations at the minimum possible cost each source is able to supply a fixed number of units of product each source is able to supply a fixed number of units of product each destination has a fixed demand for the product each destination has a fixed demand for the product  Solution (optimization) Algorithms stepping-stone stepping-stone modified distribution modified distribution Excel’s Solver Excel’s Solver

16. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-16 Transportation Method: Example

17. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-17 Transportation Method: Example

18. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-18 Problem Formulation Using Excel Total Cost Formula

19. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-19 Using Solver from Tools Menu

20. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-20 Solution

21. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-21 Modified Problem Solution

22. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-22 The Underlying Network

23. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-23 For problems in which there is an underlying network:  There are easy (fast) solutions An exception is the traveling salesman problem An exception is the traveling salesman problem  The solutions are always integer ones  {How about solving a 50,000 node problem in less than a minute on a laptop??}

24. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-24 CARLTON PHARMACEUTICALS  Carlton Pharmaceuticals supplies drugs and other medical supplies.  It has three plants in: Cleveland, Detroit, Greensboro.  It has four distribution centers in: Boston, Richmond, Atlanta, St. Louis.  Management at Carlton would like to ship cases of a certain vaccine as economically as possible.

25. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-25  Data Unit shipping cost, supply, and demand Unit shipping cost, supply, and demand  Assumptions Unit shipping cost is constant. Unit shipping cost is constant. All the shipping occurs simultaneously. All the shipping occurs simultaneously. The only transportation considered is between sources and destinations. The only transportation considered is between sources and destinations. Total supply equals total demand. Total supply equals total demand.

26. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-26 NETWORK REPRESENTATI ON Boston Richmond Atlanta St.Louis DestinationsSources Cleveland Detroit Greensboro S 1 =1200 S 2 =1000 S 3 = 800 D 1 =1100 D 2 =400 D 3 =750 D 4 =750 37 40 42 32 35 40 30 25 35 15 20 28

27. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-27 The Associated Linear Programming Model The Associated Linear Programming Model The structure of the model is: The structure of the model is: Minimize Minimize ST [Amount shipped from a source] = [Supply at that source] [Amount received at a destination] = [Demand at that destination] Decision variables Decision variables X ij = amount shipped from source i to destination j. where: i=1 (Cleveland), 2 (Detroit), 3 (Greensboro) j=1 (Boston), 2 (Richmond), 3 (Atlanta), 4(St.Louis)

28. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-28 Boston Richmond Atlanta St.Louis D 1 =1100 D 2 =400 D 3 =750 D 4 =750 The supply constraints Cleveland S 1 =1200 X11 X12 X13 X14 Supply from Cleveland X11+X12+X13+X14 = 1200 Detroit S 2 =1000 X21 X22 X23 X24 Supply from Detroit X21+X22+X23+X24 = 1000 Greensboro S 3 = 800 X31 X32 X33 X34 Supply from Greensboro X31+X32+X33+X34 = 800

29. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-29 The complete mathematical programming model The complete mathematical programming model ==============

30. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-30 Excel Optimal Solution

31. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-31 Range of optimality WINQSB Sensitivity Analysis If this path is used, the total cost will increase by $5 per unit shipped along it

32. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-32 Range of feasibility Shadow prices for warehouses - the cost resulting from 1 extra case of vaccine demanded at the warehouse Shadow prices for plants - the savings incurred for each extra case of vaccine available at the plant

33. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-33 Transshipment Model

34. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-34 Transshipment Model: Solution

35. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-35 Copyright 2006 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in section 117 of the 1976 United States Copyright Act without express permission of the copyright owner is unlawful. Request for further information should be addressed to the Permission Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages caused by the use of these programs or from the use of the information herein.

36. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-36 DEPOT MAX A General Network Problem  Depot Max has six stores. Stores 5 and 6 are running low on the model 65A Arcadia workstation, and need a total of 25 additional units. Stores 5 and 6 are running low on the model 65A Arcadia workstation, and need a total of 25 additional units. Stores 1 and 2 are ordered to ship a total of 25 units to stores 5 and 6. Stores 1 and 2 are ordered to ship a total of 25 units to stores 5 and 6. Stores 3 and 4 are transshipment nodes with no demand or supply of their own. Stores 3 and 4 are transshipment nodes with no demand or supply of their own.

37. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-37  Other restrictions There is a maximum limit for quantities shipped on various routes. There is a maximum limit for quantities shipped on various routes. There are different unit transportation costs for different routes. There are different unit transportation costs for different routes.  Depot Max wishes to transport the available workstations at minimum total cost.

38. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-38 1 24 35 6 5 10 20 6 15 12 7 15 11 7 Transportation unit cost DATA: Network presentation –Supply nodes: Net flow out of the node] = [Supply at the node] X12 + X13 + X15 - X21 = 10 (Node 1) X21 + X24 - X12 = 15(Node 2) –Intermediate transshipment nodes: [Total flow out of the node] = [Total flow into the node] X34+X35 = X13 (Node 3) X46 = X24 + X34 (Node 4) –Demand nodes: [Net flow into the node] = [Demand for the node] X15 + X35 +X65 - X56 = 12 (Node 5) X46 +X56 - X65 = 13 (Node 6) Arcs: Upper bound and lower bound constraints:

39. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-39  The Complete mathematical model

40. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-40 WINQSB Input Data

41. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-41 WINQSB Optimal Solution

42. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-42 MONTPELIER SKI COMPANY Using a Transportation model for production scheduling MONTPELIER SKI COMPANY Using a Transportation model for production scheduling Montpelier is planning its production of skis for the months of July, August, and September. Montpelier is planning its production of skis for the months of July, August, and September. Production capacity and unit production cost will change from month to month. Production capacity and unit production cost will change from month to month. The company can use both regular time and overtime to produce skis. The company can use both regular time and overtime to produce skis. Production levels should meet both demand forecasts and end-of-quarter inventory requirement. Production levels should meet both demand forecasts and end-of-quarter inventory requirement. Management would like to schedule production to minimize its costs for the quarter. Management would like to schedule production to minimize its costs for the quarter.

43. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-43  Data: Initial inventory = 200 pairs Initial inventory = 200 pairs Ending inventory required =1200 pairs Ending inventory required =1200 pairs Production capacity for the next quarter = 400 pairs in regular time. Production capacity for the next quarter = 400 pairs in regular time. = 200 pairs in overtime. = 200 pairs in overtime. Holding cost rate is 3% per month per ski. Holding cost rate is 3% per month per ski. Production capacity, and forecasted demand for this quarter (in pairs of skis), and production cost per unit (by months) Production capacity, and forecasted demand for this quarter (in pairs of skis), and production cost per unit (by months)

44. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-44  Analysis of demand: Net demand to satisfy in July = 400 - 200 = 200 pairs Net demand to satisfy in July = 400 - 200 = 200 pairs Net demand in August = 600 Net demand in August = 600 Net demand in September = 1000 + 1200 = 2200 pairs Net demand in September = 1000 + 1200 = 2200 pairs  Analysis of Supplies: Production capacities are thought of as supplies. Production capacities are thought of as supplies. There are two sets of “supplies”: There are two sets of “supplies”: Set 1- Regular time supply (production capacity) Set 1- Regular time supply (production capacity) Set 2 - Overtime supply Set 2 - Overtime supply Initial inventory Forecasted demandIn house inventory Analysis of Unit costs Unit cost = [Unit production cost] + [Unit holding cost per month][the number of months stays in inventory] Example: A unit produced in July in Regular time and sold in September costs 25+ (3%)(25)(2 months) = $26.50

45. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-45 Network representation 2525.7526.50 0 3030.9031.80 0 +M 26 2626.78 0 +M 32 3232.96 0 +M+M 29 29 0 +M+M 37 37 0 Production Month/period Month sold July R/T July O/T Aug. R/T Aug. O/T Sept. R/T Sept. O/T July Aug. Sept. Dummy 1000 500 800 400 200 600 300 2200 Demand Production Capacity July R/T

46. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-46 Source: July production in R/T Destination: July‘s demand. Source: Aug. production in O/T Destination: Sept.’s demand 32+(.03)(32)=$32.96 Unit cost= $25 (production) Unit cost =Production+one month holding cost

47. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-47

48. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-48  Summary of the optimal solution In July produce at capacity (1000 pairs in R/T, and 500 pairs in O/T). Store 1500-200 = 1300 at the end of July. In July produce at capacity (1000 pairs in R/T, and 500 pairs in O/T). Store 1500-200 = 1300 at the end of July. In August, produce 800 pairs in R/T, and 300 in O/T. Store additional 800 + 300 - 600 = 500 pairs. In August, produce 800 pairs in R/T, and 300 in O/T. Store additional 800 + 300 - 600 = 500 pairs. In September, produce 400 pairs (clearly in R/T). With 1000 pairs retail demand, there will be In September, produce 400 pairs (clearly in R/T). With 1000 pairs retail demand, there will be (1300 + 500) + 400 - 1000 = 1200 pairs available for shipment to Ski Chalet. (1300 + 500) + 400 - 1000 = 1200 pairs available for shipment to Ski Chalet. Inventory +Production -Demand

49. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-49 Problem 4-25

50. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-50

51. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-51

52. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-52

53. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-53

54. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-54 6.3 The Assignment Problem  Problem definition m workers are to be assigned to m jobs m workers are to be assigned to m jobs A unit cost (or profit) C ij is associated with worker i performing job j. A unit cost (or profit) C ij is associated with worker i performing job j. Minimize the total cost (or maximize the total profit) of assigning workers to job so that each worker is assigned a job, and each job is performed. Minimize the total cost (or maximize the total profit) of assigning workers to job so that each worker is assigned a job, and each job is performed.

55. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-55 BALLSTON ELECTRONICS  Five different electrical devices produced on five production lines, are needed to be inspected.  The travel time of finished goods to inspection areas depends on both the production line and the inspection area.  Management wishes to designate a separate inspection area to inspect the products such that the total travel time is minimized.

56. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-56  Data: Travel time in minutes from assembly lines to inspection areas.

57. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-57 NETWORK REPRESENTATION 1 2 3 4 5 Assembly LineInspection Areas A B C D E S 1 =1 S 2 =1 S 3 =1 S 4 =1 S 5 =1 D 1= 1 D 2 =1 D 3 =1 D 4 =1 D 5 =1

58. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-58  Assumptions and restrictions The number of workers equals the number of jobs. The number of workers equals the number of jobs. Given a balanced problem, each worker is assigned exactly once, and each job is performed by exactly one worker. Given a balanced problem, each worker is assigned exactly once, and each job is performed by exactly one worker. For an unbalanced problem “dummy” workers (in case there are more jobs than workers), or “dummy” jobs (in case there are more workers than jobs) are added to balance the problem. For an unbalanced problem “dummy” workers (in case there are more jobs than workers), or “dummy” jobs (in case there are more workers than jobs) are added to balance the problem.

59. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-59  Computer solutions A complete enumeration is not efficient even for moderately large problems (with m=8, m! > 40,000 is the number of assignments to enumerate). A complete enumeration is not efficient even for moderately large problems (with m=8, m! > 40,000 is the number of assignments to enumerate). The Hungarian method provides an efficient solution procedure. The Hungarian method provides an efficient solution procedure.  Special cases A worker is unable to perform a particular job. A worker is unable to perform a particular job. A worker can be assigned to more than one job. A worker can be assigned to more than one job. A maximization assignment problem. A maximization assignment problem.

60. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-60 6.5 The Shortest Path Problem  For a given network find the path of minimum distance, time, or cost from a starting point, the start node, to a destination, the terminal node.  Problem definition There are n nodes, beginning with start node 1 and ending with terminal node n. There are n nodes, beginning with start node 1 and ending with terminal node n. Bi-directional arcs connect connected nodes i and j with nonnegative distances, d i j. Bi-directional arcs connect connected nodes i and j with nonnegative distances, d i j. Find the path of minimum total distance that connects node 1 to node n. Find the path of minimum total distance that connects node 1 to node n.

61. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-61 Fairway Van Lines Determine the shortest route from Seattle to El Paso over the following network highways. Determine the shortest route from Seattle to El Paso over the following network highways.

62. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-62 Salt Lake City 1 2 34 5 6 78 9 10 11 12 13 14 15 16 17 18 19 El Paso Seattle Boise Portland Butte Cheyenne Reno Sac. Bakersfield Las Vegas Denver Albuque. Kingman Barstow Los Angeles San Diego Tucson Phoenix 599 691 497 180 432 345 440 102 452 621 420 526 138 291 280 432 108 469 207 155 114 386 403 118 425314

63. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-63  Solution - a linear programming approach Decision variables Objective = Minimize  d ij X ij

64. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-64 7 2 Salt Lake City 1 34 Seattle Boise Portland 599 497 180 432 345 Butte [The number of highways traveled out of Seattle (the start node)] = 1 X12 + X13 + X14 = 1 In a similar manner: [The number of highways traveled into El Paso (terminal node)] = 1 X12,19 + X16,19 + X18,19 = 1 [The number of highways used to travel into a city] = [The number of highways traveled leaving the city]. For example, in Boise (City 4): X14 + X34 +X74 = X41 + X43 + X47. Subject to the following constraints: Nonnegativity constraints

65. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-65 WINQSB Optimal Solution

66. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-66  Solution - a network approach The Dijkstra’s algorithm: The Dijkstra’s algorithm: Find the shortest distance from the “START” node to every other node in the network, in the order of the closet nodes to the “START”. Find the shortest distance from the “START” node to every other node in the network, in the order of the closet nodes to the “START”. Once the shortest route to the m closest node is determined, the shortest route to the (m+1) closest node can be easily determined. Once the shortest route to the m closest node is determined, the shortest route to the (m+1) closest node can be easily determined. This algorithm finds the shortest route from the start to all the nodes in the network. This algorithm finds the shortest route from the start to all the nodes in the network.

67. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-67 SEA. BUT 599 POR 180 497 BOI 599 180 497 POR. BOI 432 SAC 602 + + = = 612 782 BOI BOI. 345 SLC + = 842 BUT. SLC 420 CHY. 691 + + = = 1119 1290 SLC. SLC SLC. SAC. An illustration of the Dijkstra’s algorithm … and so on until the whole network is covered.

68. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-68 6.6 The Minimal Spanning Tree  This problem arises when all the nodes of a given network must be connected to one another, without any loop.  The minimal spanning tree approach is appropriate for problems for which redundancy is expensive, or the flow along the arcs is considered instantaneous.

69. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-69 THE METROPOLITAN TRANSIT DISTRICT  The City of Vancouver is planning the development of a new light rail transportation system.  The system should link 8 residential and commercial centers.  The Metropolitan transit district needs to select the set of lines that will connect all the centers at a minimum total cost.  The network describes: feasible lines that have been drafted, feasible lines that have been drafted, minimum possible cost for taxpayers per line. minimum possible cost for taxpayers per line.

70. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-70 5 26 4 7 8 1 3 West Side North Side University Business District East Side Shopping Center South Side City Center 33 50 30 55 34 28 32 35 39 45 38 43 44 41 37 36 40 SPANNING TREE NETWORK PRESENTATION

71. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-71  Solution - a network approach The algorithm that solves this problem is a very easy (“trivial”) procedure. The algorithm that solves this problem is a very easy (“trivial”) procedure. It belongs to a class of “greedy” algorithms. It belongs to a class of “greedy” algorithms. The algorithm: The algorithm: Start by selecting the arc with the smallest arc length. Start by selecting the arc with the smallest arc length. At each iteration, add the next smallest arc length to the set of arcs already selected (provided no loop is constructed). At each iteration, add the next smallest arc length to the set of arcs already selected (provided no loop is constructed). Finish when all nodes are connected. Finish when all nodes are connected.  Computer solution Input consists of the number of nodes, the arc length, and the network description. Input consists of the number of nodes, the arc length, and the network description.

72. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-72 WINQSB Optimal Solution

73. Copyright 2006 John Wiley & Sons, Inc.Supplement 10-73 Shopping Center Loop 5 26 4 7 8 1 3 West Side North Side University Business District East Side South Side City Center 33 50 30 55 34 28 32 35 39 45 38 43 44 41 37 36 40 Total Cost = $236 million OPTIMAL SOLUTION NETWORK REPRESENTATION