1. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.1 Network Models Chapters 6 and 7

2. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.2 Where network flows arise Transportation –Transportation of goods over transportation networks –Scheduling of fleets of airplanes: time/space networks Manufacturing –Scheduling of goods for manufacturing –Flow of manufactured items within inventory systems Communications –Design and expansion of communication systems –Flow of information across networks Personnel Assignment –Assignment of crews to airline schedules –Assignment of drivers to vehicles

3. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.3 Network Optimization Problem Types Many optimization problems can be represented by a graphical network representation. Examples: –Distribution problems –Routing problems –Maximum flow problems –Designing computer / phone / road networks –Equipment replacement

4. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.4 Network examples Shortest path Maximum flow Transportation problem (Chapter 6) Assignment problem (Chapter 6) All are examples of a more general model type: –The Minimum-Cost-Network Flow Model

5. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.5 Advantages of Network Models They can be solved very quickly with specialized algorithms. They have naturally integer solutions. –By recognizing that a problem can be formulated as a network program, it is possible to solve special types of integer programs without resorting to the ineffective and time consuming integer programming algorithms. They are intuitive. –Network models provide a language for talking about problems that is much more intuitive than the “variables, objective, and constraints” language of linear and integer programming. These advantages come with a drawback (of course): –Network models cannot formulate the wide range of models that linear and integer programs can. –However, they occur often enough that they form an important tool for real decision making.

6. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.6 Table of Contents Chapter 7 (Network Optimization Problems) Minimum-Cost Flow Problems (Section 7.1) A Case Study: The BMZ Maximum Flow Problem (Section 7.2) Maximum Flow Problems (Section 7.3) Shortest Path Problems: Littletown Fire Department (Section 7.4) Shortest Path Problems: General Characteristics (Section 7.4) Shortest Path Problems: Minimizing Sarah’s Total Cost (Section 7.4) Shortest Path Problems: Minimizing Quick’s Total Time (Section 7.4) Minimum Spanning Trees: The Modern Corp. Problem (Section 7.5)

7. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.7 Distribution Unlimited Co. Problem The Distribution Unlimited Co. has two factories producing a product that needs to be shipped to two warehouses –Factory 1 produces 80 units. –Factory 2 produces 70 units. –Warehouse 1 needs 60 units. –Warehouse 2 needs 90 units. There are rail links directly from Factory 1 to Warehouse 1 and Factory 2 to Warehouse 2. Independent truckers are available to ship up to 50 units from each factory to the distribution center, and then 50 units from the distribution center to each warehouse. Question: How many units (truckloads) should be shipped along each shipping lane?

8. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.8 The Distribution Network

9. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.9 Data for Distribution Network Both transportation cost and arc capacity are considered.

10. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.10 A Network Model

11. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.11 The Optimal Solution

12. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.12 Terminology for Minimum-Cost Flow Problems 1.The model for any minimum-cost flow problem is represented by a network with flow passing through it. 2.The circles in the network are called nodes. 3.Each node where the net amount of flow generated (outflow minus inflow) is a fixed positive number is a supply node. 4.Each node where the net amount of flow generated is a fixed negative number is a demand node. 5.Any node where the net amount of flow generated is fixed at zero is a transshipment node. Having the amount of flow out of the node equal the amount of flow into the node is referred to as conservation of flow. 6.The arrows in the network are called arcs. 7.The maximum amount of flow allowed through an arc is referred to as the capacity of that arc.

13. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.13 Assumptions of a Minimum-Cost Flow Problem 1.At least one of the nodes is a supply node. 2.At least one of the other nodes is a demand node. 3.All the remaining nodes are transshipment nodes. 4.Flow through an arc is only allowed in the direction indicated by the arrowhead, where the maximum amount of flow is given by the capacity of that arc. (If flow can occur in both directions, this would be represented by a pair of arcs pointing in opposite directions.) 5.The network has enough arcs with sufficient capacity to enable all the flow generated at the supply nodes to reach all the demand nodes. 6.The cost of the flow through each arc is proportional to the amount of that flow, where the cost per unit flow is known. 7.The objective is to minimize the total cost of sending the available supply through the network to satisfy the given demand. (An alternative objective is to maximize the total profit from doing this.)

14. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.14 Properties of Minimum-Cost Flow Problems The Feasible Solutions Property: Under the previous assumptions, a minimum-cost flow problem will have feasible solutions if and only if the sum of the supplies from its supply nodes equals the sum of the demands at its demand nodes. The Integer Solutions Property: As long as all the supplies, demands, and arc capacities have integer values, any minimum-cost flow problem with feasible solutions is guaranteed to have an optimal solution with integer values for all its flow quantities.

15. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.15 Spreadsheet Model

16. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.16 The SUMIF Function The SUMIF formula can be used to simplify the node flow constraints. =SUMIF(Range A, x, Range B) For each quantity in (Range A) that equals x, SUMIF sums the corresponding entries in (Range B). The net outflow (flow out – flow in) from node x is then =SUMIF(“From labels”, x, “Flow”) – SUMIF(“To labels”, x, “Flow”)

17. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.17 Typical Applications of Minimum-Cost Flow Problems Kind of Application Supply Nodes Transshipment Nodes Demand Nodes Operation of a distribution network Sources of goods Intermediate storage facilities Customers Solid waste management Sources of solid waste Processing facilitiesLandfill locations Operation of a supply network Vendors Intermediate warehouses Processing facilities Coordinating product mixes at plants Plants Production of a specific product Market for a specific product Cash flow management Sources of cash at a specific time Short-term investment options Needs for cash at a specific time

18. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.18 The BMZ Maximum Flow Problem The BMZ Company is a European manufacturer of luxury automobiles. Its exports to the United States are particularly important. BMZ cars are becoming especially popular in California, so it is particularly important to keep the Los Angeles center well supplied with replacement parts for repairing these cars. BMZ needs to execute a plan quickly for shipping as much as possible from the main factory in Stuttgart, Germany to the distribution center in Los Angeles over the next month. The limiting factor on how much can be shipped is the limited capacity of the company’s distribution network. Question: How many units should be sent through each shipping lane to maximize the total units flowing from Stuttgart to Los Angeles?

19. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.19 The BMZ Distribution Network

20. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.20 A Network Model for BMZ

21. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.21 Spreadsheet Model for BMZ

22. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.22 Assumptions of Maximum Flow Problems 1.All flow through the network originates at one node, called the source, and terminates at one other node, called the sink. (The source and sink in the BMZ problem are the factory and the distribution center, respectively.) 2.All the remaining nodes are transshipment nodes. 3.Flow through an arc is only allowed in the direction indicated by the arrowhead, where the maximum amount of flow is given by the capacity of that arc. At the source, all arcs point away from the node. At the sink, all arcs point into the node. 4.The objective is to maximize the total amount of flow from the source to the sink. This amount is measured in either of two equivalent ways, namely, either the amount leaving the source or the amount entering the sink.

23. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.23 BMZ with Multiple Supply and Demand Points BMZ has a second, smaller factory in Berlin. The distribution center in Seattle has the capability of supplying parts to the customers of the distribution center in Los Angeles when shortages occur at the latter center. Question: How many units should be sent through each shipping lane to maximize the total units flowing from Stuttgart and Berlin to Los Angeles and Seattle?

24. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.24 Network Model for the expanded BMZ Problem

25. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.25 Spreadsheet Model

26. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.26 Some Applications of Maximum Flow Problems 1.Maximize the flow through a distribution network, as for BMZ. 2.Maximize the flow through a company’s supply network from its vendors to its processing facilities. 3.Maximize the flow of oil through a system of pipelines. 4.Maximize the flow of water through a system of aqueducts. 5.Maximize the flow of vehicles through a transportation network.

27. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.27 Littletown Fire Department Littletown is a small town in a rural area. Its fire department serves a relatively large geographical area that includes many farming communities. Since there are numerous roads throughout the area, many possible routes may be available for traveling to any given farming community. Question: Which route from the fire station to a certain farming community minimizes the total number of miles?

28. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.28 The Littletown Road System

29. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.29 The Network Representation

30. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.30 Spreadsheet Model

31. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.31 Assumptions of a Shortest Path Problem 1.You need to choose a path through the network that starts at a certain node, called the origin, and ends at another certain node, called the destination. 2.The lines connecting certain pairs of nodes commonly are links (which allow travel in either direction), although arcs (which only permit travel in one direction) also are allowed. 3.Associated with each link (or arc) is a nonnegative number called its length. (Be aware that the drawing of each link in the network typically makes no effort to show its true length other than giving the correct number next to the link.) 4.The objective is to find the shortest path (the path with the minimum total length) from the origin to the destination.

32. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.32 Applications of Shortest Path Problems 1.Minimize the total distance traveled. 2.Minimize the total cost of a sequence of activities. 3.Minimize the total time of a sequence of activities.

33. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.33 Minimizing Total Cost: Sarah’s Car Fund Sarah has just graduated from high school. As a graduation present, her parents have given her a car fund of $21,000 to help purchase and maintain a three-year-old used car for college. Since operating and maintenance costs go up rapidly as the car ages, Sarah may trade in her car on another three-year-old car one or more times during the next three summers if it will minimize her total net cost. (At the end of the four years of college, her parents will trade in the current used car on a new car for Sarah.) Question: When should Sarah trade in her car (if at all) during the next three summers?

34. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.34 Sarah’s Cost Data Operating and Maintenance Costs for Ownership Year Trade-in Value at End of Ownership Year Purchase Price12341234 $12,000$2,000$3,000$4,500$6,500$8,500$6,500$4,500$3,000

35. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.35 Shortest Path Formulation

36. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.36 Spreadsheet Model

37. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.37 Minimizing Total Time: Quick Company The Quick Company has learned that a competitor is planning to come out with a new kind of product with great sales potential. Quick has been working on a similar product that had been scheduled to come to market in 20 months. Quick’s management wishes to rush the product out to meet the competition. Each of four remaining phases can be conducted at a normal pace, at a priority pace, or at crash level to expedite completion. However, the normal pace has been ruled out as too slow for the last three phases. $30 million is available for all four phases. Question: At what pace should each of the four phases be conducted?

38. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.38 Time and Cost of the Four Phases Level Remaining ResearchDevelopment Design of Mfg. System Initiate Production and Distribution Normal5 months——— Priority4 months3 months5 months2 months Crash2 months 3 months1 month Level Remaining ResearchDevelopment Design of Mfg. System Initiate Production and Distribution Normal$3 million——— Priority6 million$6 million$9 million$3 million Crash9 million 12 million6 million

39. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.39 Shortest Path Formulation

40. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.40 Spreadsheet Model

41. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.41 The Optimal Solution PhaseLevelTimeCost Remaining researchCrash2 months$9 million DevelopmentPriority3 months6 million Design of manufacturing systemCrash3 months12 million Initiate production and distributionPriority2 months3 million Total10 months$30 million

42. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.42 Minimum Spanning Trees: The Modern Corp. Problem Modern Corporation has decided to have a state-of-the-art fiber- optic network installed to provide high-speed communication (data, voice, and video) between its major centers. Any pair of centers do not need to have a cable directly connecting them in order to take advantage of the technology. All that is necessary is to have a series of cables that connect the centers. Question: Which cables should be installed to provide high- speed communications between every pair of centers.

43. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.43 Modern Corporation’s Major Centers

44. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.44 The Optimal Solution

45. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.45 Assumptions of a Minimum-Spanning Tree Problem 1.You are given the nodes of a network but not the links. Instead, you are given the potential links and the positive cost (or a similar measure) for each if it is inserted into the network. 2.You wish to design the network by inserting enough links to satisfy the requirement that there be a path between every pair of nodes. 3.The objective is to satisfy this requirement in a way that minimizes the total cost of doing so.

46. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.46 Algorithm for a Minimum-Spanning-Tree Problem 1.Choice of the first link: Select the cheapest potential link. 2.Choice of the next link: Select the cheapest potential link between a node that already is touched by a link and a node that does not yet have such a link. 3.Repeat step 2 over and over until every node is touched by a link (perhaps more than one). At that point, an optimal solution (a minimum spanning tree) has been obtained. (Ties for the cheapest potential link at each step may be broken arbitrarily.)

47. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.47 Application of Algorithm to Modern Corp.: First Link

48. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.48 Application of Algorithm to Modern Corp.: Second Link

49. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.49 Application of Algorithm to Modern Corp.: Third Link

50. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.50 Application of Algorithm to Modern Corp.: Fourth Link

51. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.51 Application of Algorithm to Modern Corp.: Fifth Link

52. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.52 Application of Algorithm to Modern Corp.: Final Link

53. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.53 Applications of Minimum-Spanning-Tree Problems 1.Design of telecommunication networks (computer networks, lease- line telephone networks, cable television networks, etc.) 2.Design of a lightly-used transportation network to minimize the total cost of providing the links (rail lines, roads, etc.) 3.Design of a network of high-voltage electrical power transmission lines. 4.Design of a network of wiring on electrical equipment (e.g., a digital computer system) to minimize the total length of the wire. 5.Design of a network of pipelines to connect a number of locations.

54. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.54 Network Optimization Problems Many optimization problems can be represented by a graphical network representation. Examples: –Distribution problems –Routing problems –Maximum flow problems –Designing computer / phone / road networks –Equipment replacement

55. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.55 Components of a Minimum-Cost-Flow Model Nodes –can represent a location, point in time, or state –supply node (flow is generated) –demand node (flow is consumed) –transshipment node (flow in = flow out) Arcs –can represent potential flow (e.g., a shipping lane) or a transition from state to state. –directional (one-way) if both ways, use two arcs –cost (assumed proportional to flow) –may have capacity limitations

56. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.56 Minimum-Cost-Flow Model Objective: Minimize the total cost of all flow, while sending supply, subject to constraints, through the network to satisfy demand. Integer Solutions Property: If supplies, demands, and arc capacities are integer, then the optimal flow will also be integer. Network Simplex Method: A streamlined version of the simplex method. –extremely efficient –computer software may have graphical interface (with nodes and arcs) Excel uses the standard simplex method. However, the minimum- cost-flow model is a useful tool for modeling a problem: –visual –intuitive –easy to set up –transforms easily to a spreadsheet model

57. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.57 Minimum-Cost-Flow Model Consider a directed network with n nodes. The decision variables are x ij, the flow through arc (i, j). The given information includes: –c ij : cost per unit of flow from i to j (may be negative), –u ij : capacity (or upper bound) on flow from i to j, –b i : net flow generated at i. This last value has a sign convention: –b i > 0 if i is a supply node, –b i < 0 if i is a demand node, –b i = 0 if i is a transshipment node. The objective is to minimize the total cost of sending the supply through the network to satisfy the demand.

58. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.58 Minimum-Cost-Flow Model Linear programming formulation for this model is…

59. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.59 Minimum-Cost-Flow Model Things you can do with this model… –Lower bounds on arcs. If a variable x ij has a lower bound of l ij, upper bound of u ij, and cost of c ij, change the problem as follows: Replace the upper bound with u ij - l ij, Replace the supply at i with b i - l ij, Replace the supply at j with b i + l ij, –Now this is a minimum cost flow problem. Add c ij l ij to the objective after solving and l ij to the flow on arc (i, j) to obtain a solution of the original problem. –Upper bounds on flow through a node. Replace the node i with nodes i' and i''. Create an arc from i' to i'' with the appropriate capacity, and cost 0. Replace every arc (j, i) with one from j to i' and every arc (i, j) with one from i'' to j. Lower bounds can also be handled this way. –Convex, piecewise linear costs on arc flows (for minimization). This is handled by introducing multiple arcs between the nodes, one for each portion of the piecewise linear function. The convexity will assure that costs are handled correctly in an optimal solution.

60. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.60 Multi-Echelon Distribution Consider a multi-echelon distribution problem. Product must be distributed from a pair of factories to three warehouses. Product is then shipped to five distribution centers. A private trucking fleet is used for all shipping. Some shipping lanes are currently capacitated due to a limited number of trucks. Question: How many units should be shipped along each shipping lane?

61. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.61 Spreadsheet Model

62. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.62 The SUMIF Function The SUMIF formula can be used to simplify the node flow constraints. =SUMIF(Range A, x, Range B) For each quantity in (Range A) that equals x, SUMIF sums the corresponding entries in (Range B). The net outflow (flow out – flow in) from node x is then =SUMIF(“From labels”, x, “Flow”) – SUMIF(“To labels”, x, “Flow”)

63. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.63 Spreadsheet Model using SUMIF

64. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.64 The Minimum-Cost-Flow Model is an LP Any minimum cost flow model consists of a set of nodes: –Supply node(s), with supply s i –Demand node(s), with demand d i –Transshipment nodes A set of arcs from node i to node j –with cost c ij –some with limited capacity k ij LP Formulation: Let x ij = flow from i to j Minimize Cost = ∑ ij c ij x ij subject to Flow: ∑ all j flowing out of i x ij – ∑ all j flowing into i x ji = (s i, d i, or 0) Capacity: x ij ≤ k ij and x ij ≥ 0.

65. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.65 Maximum Flow Problem An oil company has the following pipeline network, where each pipeline is labeled with its maximum flow rate (in thousands of gallons per hour). Question: What is the maximum possible flow rate from A to G?

66. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.66 Spreadsheet Model

67. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.67 Shortest Path Problem The travel times along various routes in the Pacific Northwest is shown below. Question: What is the quickest route from Seattle to Denver?

68. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.68 Spreadsheet Model

69. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.69 Equipment Replacement A production department needs to purchase a new machine. As the machine ages, it requires additional maintenance and also has a higher defect rate. The production department plans to replace the machine every few years. The purchase price of a new machine is $10,000. The maintenance cost and cost of defective product is given below. A used machine has no resale value. Age of machineMaintenance CostCost of Defects First Year$3,000$2,000 Second Year$4,000 Third Year$6,000$7,000 Fourth Year$10,000$11,000 Fifth Year$20,000$24,000 Question: What is the best replacement policy over the next five years?

70. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.70 Spreadsheet Solution

71. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 7.71 Planning Vehicle Replacement at Phillips Petroleum Phillips Petroleum had a fleet of 1,500 cars and 3,800 trucks. Modeled replacement strategy as shortest path model (20-year time horizon)—solved model once for each class of vehicle. Could keep, purchase (replace), or lease, at 3-month intervals. Costs considered included: –Maintenance and operating costs (fuel, oil, repair), –Leasing cost for leased vehicles, –Purchasing cost for purchased vehicles, –State license fees and road taxes, –Tax effects (investment tax credits, depreciation) First used to make lease-or-buy decision, then vehicle-replacement strategy, and more recently for other equipment (non-vehicle). For more details, see Waddell (1983) Jul-Aug Interfaces article, “A Model for Equipment Replacement Decisions and Policies”, downloadable at www.mhhe.com/hillier2e/articles.