1. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.1 Chapter 7 (Binary Integer Programming) Including two CD-rom supplements

2. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.2 Assumptions of linear programming Proportionality: the gross margin and resource requirements per unit of activity are assumed to be constant regardless of the level of the activity use Additivity: no interaction effects between activities Homogeneity: all units of the same resource or activity are identical Continuity: resources can be used and activities produced in any fractional units Deterministic coefficients: all coefficients in the model are known with certainty Optimization: One proper objective function to be maximized or minimized Finiteness: only a finite number of activities and constraints is considered

3. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.3 Types of Integer Programming Problems Pure integer programming problems are those where all the decision variables must be integers. Mixed integer programming problems only require some of the variables (the “integer variables”) to have integer values so the divisibility assumption holds for the rest (the “continuous variables”). Binary variables are variables whose only possible values are 0 and 1. Binary integer programming (BIP) problems are those where all the decision variables restricted to integer values are further restricted to be binary variables. –Such problems can be further characterized as either pure BIP problems or mixed BIP problems, depending on whether all the decision variables or only some of them are binary variables.

4. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.4 The TBA Airlines Problem TBA Airlines is a small regional company that specializes in short flights in small airplanes. The company has been doing well and has decided to expand its operations. The basic issue facing management is whether to purchase more small airplanes to add some new short flights, or start moving into the national market by purchasing some large airplanes, or both. Question: How many airplanes of each type should be purchased to maximize their total net annual profit?

5. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.5 Data for the TBA Airlines Problem Small Airplane Large Airplane Capital Available Net annual profit per airplane$1 million$5 million Purchase cost per airplane5 million50 million$100 million Maximum purchase quantity2—

6. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.6 Linear Programming Formulation LetS = Number of small airplanes to purchase L = Number of large airplanes to purchase Maximize Profit = S + 5L ($millions) subject to Capital Available:5S + 50L ≤ 100 ($millions) Max Small Planes:S ≤ 2 and S ≥ 0, L ≥ 0.

7. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.7 Graphical Method for Linear Programming

8. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.8 Violates Divisibility Assumption of LP Divisibility Assumption of Linear Programming: Decision variables in a linear programming model are allowed to have any values, including fractional values, that satisfy the functional and nonnegativity constraints. Thus, these variables are not restricted to just integer values. Since the number of airplanes purchased by TBA must have an integer value, the divisibility assumption is violated.

9. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.9 Integer Programming Formulation LetS = Number of small airplanes to purchase L = Number of large airplanes to purchase Maximize Profit = S + 5L ($millions) subject to Capital Available:5S + 50L ≤ 100 ($millions) Max Small Planes:S ≤ 2 and S ≥ 0, L ≥ 0 S, L are integers.

10. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.10 Graphical Method for Integer Programming When an integer programming problem has just two decision variables, its optimal solution can be found by applying the graphical method for linear programming with just one change at the end. We begin as usual by graphing the feasible region for the LP relaxation, determining the slope of the objective function lines, and moving a straight edge with this slope through this feasible region in the direction of improving values of the objective function. However, rather than stopping at the last instant the straight edge passes through this feasible region, we now stop at the last instant the straight edge passes through an integer point that lies within this feasible region. This integer point is the optimal solution.

11. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.11 Graphical Method for Integer Programming

12. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.12 Spreadsheet Model

13. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.13 The default Tolerance field on the Solver Options dialog (relevant only for ILP models) is 5%. This means that the Solver ILP optimization procedure is continued only until the ILP solution OV is within 5% of the ILP’s optimum OV. A higher Tolerance speeds up Solver at the risk of a reported solution further from the true ILP optimum. Setting Tolerance to 0% forces Solver to find the ILP optimum but with much longer solution times.

14. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.14 Applications of Binary Variables Since binary variables only provide two choices, they are ideally suited to be the decision variables when dealing with yes-or-no decisions. Examples: –Should we undertake a particular fixed project? –Should we make a particular fixed investment? –Should we locate a facility in a particular site?

15. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.15 California Manufacturing Company The California Manufacturing Company is a diversified company with several factories and warehouses throughout California, but none yet in Los Angeles or San Francisco. A basic issue is whether to build a new factory in Los Angeles or San Francisco, or perhaps even both. Management is also considering building at most one new warehouse, but will restrict the choice to a city where a new factory is being built. Question: Should the California Manufacturing Company expand with factories and/or warehouses in Los Angeles and/or San Francisco?

16. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.16 Data for California Manufacturing Decision Number Yes-or-No Question Decision Variable Net Present Value (Millions) Capital Required (Millions) 1Build a factory in Los Angeles?x1x1 $8$6 2Build a factory in San Francisco?x2x2 53 3Build a warehouse in Los Angeles?x3x3 65 4Build a warehouse in San Francisco?x4x4 42 Capital Available: $10 million

17. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.17 Binary Decision Variables Decision Number Decision Variable Possible Value Interpretation of a Value of 1 Interpretation of a Value of 0 1x1x1 0 or 1 Build a factory in Los Angeles Do not build this factory 2x2x2 0 or 1 Build a factory in San Francisco Do not build this factory 3x3x3 0 or 1 Build a warehouse in Los Angeles Do not build this warehouse 4x4x4 0 or 1 Build a warehouse in San Francisco Do not build this warehouse

18. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.18 Algebraic Formulation Letx 1 = 1 if build a factory in L.A.; 0 otherwise x 2 = 1 if build a factory in S.F.; 0 otherwise x 3 = 1 if build a warehouse in Los Angeles; 0 otherwise x 4 = 1 if build a warehouse in San Francisco; 0 otherwise Maximize NPV = 8x 1 + 5x 2 + 6x 3 + 4x 4 ($millions) subject to Capital Spent:6x 1 + 3x 2 + 5x 3 + 2x 4 ≤ 10 ($millions) Max 1 Warehouse:x 3 + x 4 ≤ 1 Warehouse only if Factory:x 3 ≤ x 1 x 4 ≤ x 2 and x 1, x 2, x 3, x 4 are binary variables.

19. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.19 Spreadsheet Model

20. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.20 Sensitivity Analysis with Solver Table

21. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.21 Management’s Conclusion Management’s initial tentative decision had been to make $10 million of capital available. With this much capital, the best plan would be to build a factory in both Los Angeles and San Francisco, but no warehouses. An advantage of this plan is that it only uses $9 million of this capital, which frees up $1 million for other projects. A heavy penalty (a reduction of $4 million in total net present value) would be paid if the capital made available were to be reduced below $9 million. Increasing the capital made available by $1 million (to $11 million) would enable a substantial ($4 million) increase in the total net present value. Management decides to do this. With this much capital available, the best plan is to build a factory in both cities and a warehouse in San Francisco.

22. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.22 Some Other Applications Investment Analysis –Should we make a certain fixed investment? –Examples: Turkish Petroleum Refineries (1990), South African National Defense Force (1997), Grantham, Mayo, Van Otterloo and Company (1999) Site Selection –Should a certain site be selected for the location of a new facility? –Example: AT&T (1990) Designing a Production and Distribution Network –Should a certain plant remain open? Should a certain site be selected for a new plant? Should a distribution center remain open? Should a certain site be selected for a new distribution center? Should a certain distribution center be assigned to serve a certain market area? –Examples: Ault Foods (1994), Digital Equipment Corporation (1995) All references available for download at www.mhhe.com/hillier2e/articles

23. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.23 Some Other Applications Dispatching Shipments –Should a certain route be selected for a truck? Should a certain size truck be used? Should a certain time period for departure be used? –Examples: Quality Stores (1987), Air Products and Chemicals, Inc. (1983), Reynolds Metals Co. (1991), Sears, Roebuck and Company (1999) Scheduling Interrelated Activities –Should a certain activity begin in a certain time period? –Examples: Texas Stadium (1983), China (1995) Scheduling Asset Divestitures –Should a certain asset be sold in a certain time period? –Example: Homart Development (1987) Airline Applications: –Should a certain type of airplane be assigned to a certain flight leg? Should a certain sequence of flight legs be assigned to a crew? –Examples: American Airlines (1989, 1991), Air New Zealand (2001) All references available for download at www.mhhe.com/hillier2e/articles

24. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.24 Wyndor with Setup Costs (Variation 1) Suppose that two changes are made to the original Wyndor problem: 1.For each product, producing any units requires a substantial one-time setup cost for setting up the production facilities. 2.The production runs for these products will be ended after one week, so D and W in the original model now represent the total number of doors and windows produced, respectively, rather than production rates. Therefore, these two variables need to be restricted to integer values.

25. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.25 Graphical Solution to Original Wyndor Problem

26. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.26 Net Profit for Wyndor Problem with Setup Costs Net Profit ($) Number of Units ProducedDoorsWindows 00(300) – 0 = 00 (500) – 0 = 0 11(300) – 700 = –4001(500) – 1,300 = –800 22(300) – 700 = –1002(500) – 1,300 = –300 33(300) – 700 = 2003(500) – 1,300 = 200 44(300) – 700 = 5004(500) – 1,300 = 700 5Not feasible5(500) – 1,300 = 1,200 6Not feasible6(500) – 1,300 = 1,700

27. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.27 Feasible Solutions for Wyndor with Setup Costs

28. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.28 Algebraic Formulation LetD = Number of doors to produce, W = Number of windows to produce, y 1 = 1 if perform setup to produce doors; 0 otherwise, y 2 = 1 if perform setup to produce windows; 0 otherwise. Maximize P = 300D + 500W – 700y 1 – 1,300y 2 subject to Original Constraints: Plant 1:D ≤ 4 Plant 2:2W ≤ 12 Plant 3:3D + 2W ≤ 18 Produce only if Setup: Doors:D ≤ 99y 1 Windows:W ≤ 99y 2 and D ≥ 0, W ≥ 0, y 1 and y 2 are binary.

29. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.29 Spreadsheet Model

30. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.30 Wyndor with Mutually Exclusive Products (Variation 2) Suppose that now the only change from the original Wyndor problem is: The two potential new products (doors and windows) would compete for the same customers. Therefore, management has decided not to produce both of them together. –At most one can be chosen for production, so either D = 0 or W = 0, or both.

31. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.31 Feasible Solution for Wyndor with Mutually Exclusive Products

32. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.32 Algebraic Formulation LetD = Number of doors to produce, W = Number of windows to produce, y 1 = 1 if produce doors; 0 otherwise, y 2 = 1 if produce windows; 0 otherwise. Maximize P = 300D + 500W subject to Original Constraints: Plant 1:D ≤ 4 Plant 2:2W ≤ 12 Plant 3:3D + 2W ≤ 18 Auxiliary variables must =1 if produce any: Doors:D ≤ 99y 1 Windows:W ≤ 99y 2 Mutually Exclusive:y 1 + y 2 ≤ 1 and D ≥ 0, W ≥ 0, y 1 and y 2 are binary.

33. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.33 Spreadsheet Model

34. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.34 Wyndor with Either-Or Constraints (Variation 3) Suppose that now the only change from the original Wyndor problem is: The company has just opened a new plant (plant 4) that is similar to plant 3, so the new plant can perform the same operations as plant 3 to help produce the two new products (doors and windows). However, management wants just one of the plants to be chosen to work on these new products. The plant chosen should be the one that provides the most profitable product mix.

35. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.35 Data for Wyndor with Either-Or Constraints (Variation 3) Production Time Used for Each Unit Produced (Hours) Production Time Available per Week (Hours) PlantDoorsWindows 1104 20212 33218 42428 Unit Profit$300$500

36. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.36 Graphical Solution with Plant 3 or Plant 4

37. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.37 Algebraic Formulation LetD = Number of doors to produce, W = Number of windows to produce, y = 1 if plant 4 is used; 0 if plant 3 is used Maximize P = 300D + 500W subject to Plant 1:D ≤ 4 Plant 2:2W ≤ 12 Plant 3:3D + 2W ≤ 18 + 99y Plant 4:2D + 4W ≤ 28 + 99(1 – y) and D ≥ 0, W ≥ 0, y 1 and y 2 are binary.

38. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.38 Spreadsheet Model

39. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.39 Good Products Company Production Planning The Research and Development Division of the Good Products Company has developed three possible new products. To avoid undue diversification of the company’s product line, management has imposed the following restriction: –From the three possible new products, at most two should be chosen to be produced. Each of these products can be produced in either of two plants. For administrative reasons, management has imposed the following restriction: –Just one of the two plants should be chosen to be the sole producer of the two new products.

40. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.40 Data for the Good Products Company Production Time Used for Each Unit Produced (Hours) Production Time Available per Week (Hours) PlantProduct 1Product 2Product 3 134230 246240 Unit Profit573($thousands) Sales potential759(units per week)

41. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.41 Algebraic Formulation Letx i = Number of units of product i to produce per week (i = 1, 2, 3), y i = 1 if product i is produced; 0 otherwise (i = 1, 2, 3), y 4 = 1 if plant 2 is used; 0 if plant 1 is used Maximize Profit = 5x 1 + 7x 2 + 3x 3 ($thousands) subject to Auxiliary variables must =1 if produce any & Max Sales: Product 1:x 1 ≤ 7y 1 Product 2:x 2 ≤ 5y 2 Product 3:x 3 ≤ 9y 3 Either plant 1 (y 4 = 0) or plant 2 (y 4 = 1): Plant 1:3x 1 + 4x 2 + 2x 3 + 99y 4 ≤ 30 Plant 2:4x 1 + 6x 2 + 2x 3 + 99(1 – y 4 ) ≤ 40 and x i ≥ 0 (i = 1, 2, 3), y i are binary (i = 1, 2, 3, 4).

42. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.42 Spreadsheet Model

43. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.43 Supersuds Corporation Marketing Plan The Supersuds Corporation is developing its marketing plan for next year’s new products. For three of these products, the decision has been made to purchase a total of five TV spots for commercials on national television networks. Each spot will feature a single product. Question: How should the five spots be allocated to these three products?

44. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.44 Data for the Supersuds Corp. Problem Profit (Millions) Number of TV SpotsProduct 1Product 2Product 3 0$0 110–1 2322 3334

45. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.45 Algebraic Formulation Lety ij = 1 if there are j TV spots for product i; 0 otherwise (i = 1, 2, 3; j = 1, 2, 3) Maximize Profit = y 11 + 3y 12 + 3y 13 + 2y 22 + 3y 23 – y 31 + 2y 32 + 4y 33 ($millions) subject to Mutually Exclusive: Product 1:y 11 + y 12 + y 13 ≤ 1 Product 2:y 21 + y 22 + y 23 ≤ 1 Product 3:y 31 + y 32 + y 33 ≤ 1 Total available spots:y 11 + 2y 12 + 3y 13 + y 21 + 2y 22 + 3y 23 + y 31 + 2y 32 + 3y 33 ≤ 5 and y ij are binary (i = 1, 2, 3; j = 1, 2, 3).

46. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.46 Spreadsheet Model

47. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.47 Southwestern Airways Crew Scheduling Southwestern Airways needs to assign crews to cover all its upcoming flights. We will focus on assigning 3 crews based in San Francisco (SFO) to 11 flights. Question: How should the 3 crews be assigned 3 sequences of flights so that every one of the 11 flights is covered?

48. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.48 Southwestern Airways Flights

49. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.49 Data for the Southwestern Airways Problem Feasible Sequence of Flights Flights123456789101112 1. SFO–LAX1111 2. SFO–DEN1111 3. SFO–SEA1111 4. LAX–ORD22323 5. LAX–SFO2355 6. ORD–DEN334 7. ORD–SEA33334 8. DEN–SFO2445 9. DEN–ORD222 10. SEA–SFO2445 11. SEA–LAX22442 Cost, $1,000s234675789989

50. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.50 Algebraic Formulation Letx j = 1 if flight sequence j is assigned to a crew; 0 otherwise. (j = 1, 2, …, 12). Minimize Cost = 2x 1 + 3x 2 + 4x 3 + 6x 4 + 7x 5 + 5x 6 + 7x 7 + 8x 8 + 9x 9 + 9x 10 + 8x 11 + 9x 12 (in $thousands) subject to Flight 1 covered:x 1 + x 4 + x 7 + x 10 ≥ 1 Flight 2 covered:x 2 + x 5 + x 8 + x 11 ≥ 1 :: Flight 11 covered:x 6 + x 9 + x 10 + x 11 + x 12 ≥ 1 Three Crews:x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 + x 9 + x 10 + x 11 + x 12 ≤ 3 and x j are binary (j = 1, 2, …, 12).

51. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.51 Spreadsheet Model

52. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.52 Integer Programming When are “non-integer” solutions okay? –Solution is naturally divisible e.g., $, pounds, hours –Solution represents a rate e.g., units per week –Solution only for planning purposes When is rounding okay? –When numbers are large e.g., rounding 114.286 to 114 is probably okay. When is rounding not okay? –When numbers are small e.g., rounding 2.6 to 2 or 3 may be a problem. –Binary variables yes-or-no decisions

53. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.53 The Challenges of Rounding Rounded Solution may not be feasible. Rounded solution may not be close to optimal. There can be many rounded solutions. –Example: Consider a problem with 30 variables that are non- integer in the LP-solution. How many possible rounded solutions are there?

54. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.54 How Integer Programs are Solved

55. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.55 How Integer Programs are Solved

56. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.56 Applications of Binary Variables Making “yes-or-no” type decisions –Build a factory? –Manufacture a product? –Do a project? –Assign a person to a task? Set-covering problems –Make a set of assignments that “cover” a set of requirements. Fixed costs –If a product is produced, must incur a fixed setup cost. –If a warehouse is operated, must incur a fixed cost.

57. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.57 Example #1 (Capital Budgeting) Norwood Development is considering the potential of four different development projects. Each project would be completed in at most three years. The required cash outflow for each project is given in the table below, along with the net present value of each project to Norwood, and the cash that is available each year. Cash Outflow Required ($million) Cash Available ($million) Project 1Project 2Project 3Project 4 Year 19761128 Year 2643013 Year 3604010 NPV30162214 Question: Which projects should be undertaken?

58. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.58 Algebraic Formulation Let y i = 1 if project i is undertaken; 0 otherwise (i = 1, 2, 3, 4). Maximize NPV = 30y 1 + 16y 2 + 22y 3 + 14y 4 subject to Year 1:9y 1 + 7y 2 + 6y 3 + 11y 4 ≤ 28 ($million) Year 2 (cumulative):15y 1 + 11y 2 + 9y 3 + 11y 4 ≤ 41 ($million) Year 3 (cumulative):21y 1 + 11y 2 + 13y 3 + 11y 4 ≤ 51 ($million) and y i are binary (i = 1, 2, 3, 4).

59. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.59 Spreadsheet Solution

60. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.60 Additional Considerations (Logic and Dependency Constraints) At least one of projects 1, 2, or 3 Project 2 can’t be done unless project 3 is done Either project 3 or project 4, but not both No more than two projects total Question: What constraints would need to be added for each of these additional considerations?

61. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.61 Example #2 (Set Covering Problem) The Washington State legislature is trying to decide on locations at which to base search-and-rescue teams. The teams are expensive, so they would like as few as possible. Response time is critical, so they would like every county to either have a team located in that county or in an adjacent county. Question: Where should search-and-rescue teams be located?

62. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.62 The Counties of Washington State

63. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.63 Algebraic Formulation Let y i = 1 if a team is located in county i; 0 otherwise (i = 1, 2, …, 37). Minimize Number of Teams = y 1 + y 2 + … + y 37 subject to County 1 covered:y 1 + y 2 ≥ 1 County 2 covered:y 1 + y 2 + y 3 + y 6 + y 7 ≥ 1 County 3 covered:y 2 + y 3 + y 4 + y 7 + y 8 + y 14 ≥ 1 : County 37 covered:y 32 + y 36 + y 37 ≥ 1 and y i are binary (i = 1, 2, …, 37).

64. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.64 Spreadsheet Solution

65. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.65 Example #3 (Fixed Costs) Woodridge Pewter Company is a manufacturer of three pewter products: platters, bowls, and pitchers. The manufacture of each product requires Woodridge to have the appropriate machinery and molds available. The machinery and molds for each product can be rented at the following rates: for the platters, $400/week; for the bowls, $250/week; for the pitcher, $300/week. Each product requires the amounts of labor and pewter given in the table below. The sales price and variable cost are also given in the table. Labor Hours Pewter (pounds) Sales Price Variable Cost Platter35$100$60 Bowl148550 Pitcher437540 Available130240 Question: Which products should be produced, and in what quantity?

66. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.66 Algebraic Formulation Let x 1 = Number of platters produced, x 2 = Number of bowls produced, x 3 = Number of pitchers produced, y i = 1 if lease machine and mold for product i; 0 otherwise (i = 1, 2, 3). Maximize Profit = ($100–$60)x 1 + ($85–$50)x 2 + ($75–$40)x 3 – $400y 1 – $250y 2 – $300y 3 subject to Labor:3x 1 + x 2 + 4x 3 ≤ 130 hours Pewter:5x 1 + 4x 2 + 3x 3 ≤ 240 pounds Allow production only if machines and molds are purchased: x 1 ≤ 99y 1 x 2 ≤ 99y 2 x 3 ≤ 99y 3 and x i ≥ 0, and y i are binary (i = 1, 2, 3).

67. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.67 Spreadsheet Solution

68. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.68 Applications of Binary Variables Making “yes-or-no” type decisions –Build a factory? –Manufacture a product? –Do a project? –Assign a person to a task? Fixed costs –If a product is produced, must incur a fixed setup cost. –If a warehouse is operated, must incur a fixed cost. Either-or constraints –Production must either be 0 or ≥ 100. Subset of constraints –meet 3 out of 4 constraints.

69. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.69 Capital Budgeting with Contingency Constraints (Yes-or-No Decisions) A company is planning their capital budget over the next several years. There are 10 potential projects they are considering pursuing. They have calculated the expected net present value of each project, along with the cash outflow that would be required over the next five years. Also, suppose there are the following contingency constraints: –at least one of project 1, 2 or 3 must be done, –project 4 and project 5 cannot both be done, –project 7 can only be done if project 6 is done. Question: Which projects should they pursue?

70. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.70 Data for Capital Budgeting Problem Cash Outflow Required ($million) Cash Available ($million) Project 12345678910 Year 1140443282625 Year 2222224233625 Year 3325242348225 Year 4445453121125 Year 5110655511225 NPV20252230422518352833($million)

71. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.71 Spreadsheet Solution

72. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.72 Electrical Generator Startup Planning (Fixed Costs) An electrical utility company owns five generators. To generate electricity, a generator must be started up, and associated with this is a fixed startup cost. All of the generators are shut off at the end of each day. Generator ABCDE Fixed Startup Cost$2,450$1,600$1,000$1,250$2,200 Variable Cost (per MW)$3$4$6$5$4 Capacity (MW)2,0002,8004,3002,1002,000 Question: Which generators should be started up to meet the total capacity needed for the day (6000 MW)?

73. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.73 Spreadsheet Solution

74. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.74 Quality Furniture (Either-Or Constraints) Reconsider the Quality Furniture Problem: –The Quality Furniture Corporation produces benches and picnic tables. The firm has a limited supply of two resources: labor and wood. 1,600 labor hours are available during the next production period. The firm also has a stock of 9,000 pounds of wood available. Each bench requires 3 labor hours and 12 pounds of wood. Each table requires 6 labor hours and 38 pounds of wood. The profit margin on each bench is $8 and on each table is $18. Now suppose that they would not produce any fewer than 200 units of either product (i.e., either produce 0 or at least 200). Question: What product mix will maximize their total profit?

75. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.75 Spreadsheet Solution

76. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.76 Meeting a Subset of Constraints Consider a linear programming model with the following constraints, and suppose that meeting 3 out of 4 of these is good enough –12x 1 + 24x 2 + 18x 3 ≥ 2,400 –15x 1 + 32x 2 + 12x 3 ≥ 1,800 –20x 1 + 15x 2 + 20x 3 ≤ 2,000 –18x 1 + 21x 2 + 15x 3 ≤ 1,600

77. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.77 Meeting a Subset of Constraints Let y i = 1 if constraint i is enforced; 0 otherwise. Constraints: y 1 + y 2 + y 3 + y 4 ≥ 3 12x 1 + 24x 2 + 18x 3 ≥ 2,400y 1 15x 1 + 32x 2 + 12x 3 ≥ 1,800y 2 20x 1 + 15x 2 + 20x 3 ≤ 2,000 + M (1 – y 3 ) 18x 1 + 21x 2 + 15x 3 ≤ 1,600 + M (1 – y 4 ) where M is a large number.

78. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.78 Facility Location Consider a company that operates 5 plants and 3 warehouses that serve customers in 4 different regions. To lower costs, they are considering streamlining by closing one or more plants and warehouses. Associated with each plant are fixed costs, shipping costs, and production costs. Each plant has a limited capacity. Associated with each warehouse are fixed costs and shipping costs. Each warehouse has a limited capacity. Questions: Which plants should they keep open? Which warehouses should they keep open? How should they divide production among the open plants? How much should be shipped from each plant to each warehouse, and from each warehouse to each customer?

79. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.79 Data for Facility Location Problem Fixed Cost (per month) (Shipping + Production) Cost (per unit) Capacity (units per month) WH #1WH #2WH #3 Plant 1$42,000$650$750$850400 Plant 250,000500350550300 Plant 345,000450 350300 Plant 450,000400500600350 Plant 547,000550450350375 Fixed Cost (per month) Shipping Cost (per unit) Capacity (per month) Cust. 1Cust. 2Cust. 3Cust. 4 WH #1$45,000$25$65$70$35600 WH #225,00050254060400 WH #365,00060204045900 Demand:250225200275

80. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 9.80 Spreadsheet Solution