1. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.1 Table of Contents Chapter 4 (Linear Programming: Formulation and Applications) Taking stock – Where are we going with all this? Super Grain Corp. Advertising-Mix Problem (Section 4.1) Resource Allocation Problems and Think-Big Capital Budgeting (Section 4.2 Cost-Benefit-Trade-Off Problems and Union Airways (Section 4.3) Distribution-Network Problems and Big M Co. (Section 4.4) Continuing the Super Grain Corp. Case Study (Section 4.5) Mixed Formulations and Save-It Solid Waste Reclamation (Section 4.6)

2. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.2 Supply Chains – the generic model Raw materials supplier Manufacturing plant Distribution center Customers/ Retailers upstream downstream

3. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.3 Integrated Oil Company Oil Field: Exploration and Porduction Other Oil Companies Oil Field: Exploration and Porduction Refinery: Cracking and Blending Marketing Area Refinery: Cracking and Blending Marketing Area Products Crude Products Stocks Crude Crude Exchange Crude Crude Exchange

4. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.4 Forest industry supply chain Wagner, H.M. (1975). Principles of Operations Research 2 nd ed. Englewood Cliffs NJ: Prentice-Hall

5. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.5 Demand Allocation Model Ware- house 1 Ware- house 2 Ware- house 3 Plant 1 Plant 2 Plant 3 Variable costs Quantities Demands Capacities

6. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.6 Demand Allocation Model Parameters C ij variable costs per unit transported from Plant i to Region j K i capacity for Plant i D j demand for Region j mnumber of regions nnumber of plants Decision variables x ij quantity transported from Plant i to Region j

7. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.7 The Capacitated Plant Location Model (CPLM) Open/Closed Fixed costs Capacities Variable costs Quantities Demands Ware- house 1 Ware- house 2 Ware- house 3 Plant 1 Plant 2 Plant 3

8. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.8 The Capacitated Plant Location Model (CPLM) Decision variables y i binary variable indicating whether Plant i should be open (1) or closed (0) x ij quantity transported from Plant i to Region j Parameters F i fixed costs for Plant i C ij variable costs per unit transported from Plant i to Region j K i capacity for Plant i D j demand for Region j mnumber of regions nnumber of potential plants

9. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.9 The CPLM with single sourcing Ware- house 1 Ware- house 2 Ware- house 3 Plant 1 Plant 2 Plant 3 Variable costs Open/Closed Fixed costs Capacities Assigning plants to warehouses Demands

10. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.10 The CPLM with single sourcing Parameters F i fixed costs for Plant i C ij variable costs per unit transported from Plant i to Region j K i capacity for Plant i D j demand for Region j mnumber of regions nnumber of potential plants Decision variables y i binary variable indicating whether Plant i should be open (1) or closed (0) x ij binary variable indicating whether Plant i should supply market in Region j

11. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.11 Transshipment model with sourcing Market 1 Market 1 Market 2 Market 2 Market 3 Market 3 Plant 1 Supplier 1 Supplier 1 Supplier 2 Supplier 2 Plant 2 Plant 3 Ware- house 1 Ware- house 1 Ware- house 2 Ware- house 2 Variable costs Open/ Closed Quantities Capacities Fixed costs Demands Open/ Closed Quantities Variable costs Fixed costs Capacities (Combines plant location, warehouse location and sourcing)

12. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.12 Transshipment model with sourcing subject to for i = 1, 2, 3, …, n for e = 1, 2, 3, …, t for j = 1, 2, 3, …, m for h = 1, 2, 3, …, l

13. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.13 Transshipment model with sourcing Market 1 D 1 Market 1 D 1 Market 2 D 2 Market 2 D 2 Market m D m Market m D m Plant 1 K 1, F 1, y i Plant 1 K 1, F 1, y i Supplier 1 S 1 Supplier 1 S 1 Supplier l S l Supplier l S l Plant 2 K 2, F 2, y i Plant 2 K 2, F 2, y i Plant n K n, F n, y i Plant n K n, F n, y i Ware- house 1 W 1, f 1, y e Ware- house 1 W 1, f 1, y e Ware- house t W t, f t, y e Ware- house t W t, f t, y e X 12, c 12 X tm, c tm X 21, c 21 X t2, c t2 X 11, c 11 X l2, c l2 X ln, c ln X 11, c 11 X 21, c 21 X 2t, c 2t X nt, c nt x ie, c ie x ej, c ej x hi, c hi

14. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.14 Transshipment model with sourcing Parameters mnumber of markets or demand points nnumber of potential factory locations lnumber of suppliers tnumber of potential warehouse locations D j annual demand from customer j K i potential capacity of factory at site i S h supply capacity at supplier h W e potential warehouse capacity at site e F i fixed cost of locating a plant at site i f e fixed cost of locating a warehouse at site e c hi cost of shipping one unit from supply source h to factory i c ie cost of producing and shipping one unit from factory i to warehouse e c ej cost of shipping one unit from warehouse e to customer j Decision variables y i 1 if factory is located at site i, 0 otherwise y e 1 if warehouse is located at site e, 0 otherwise x ei quantity shipped from warehouse e to market j x ie quantity shipped from factory at site i to warehouse e x hi quantity shipped from supplier h to factory at site i

15. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.15 Integer programming With p = n, every variable must be integer-valued (Knapsack problem) With p < n, the problem is a Mixed Integer Programming case x j integer-valued x j ≥ 0 subject to for j = 1, 2, 3, …, n for i = 1, 2, 3, …, p (≤ n) for i = 1, 2, 3, …, m Drops the Continuity assumption of linear programming

16. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.16 Importance of integer programming problems Equipment utilisation –Define variable x j to be pieces of equipment that are to operate during the model’s planning horizon. If each piece is expensive and has large capacity, then fractional values may be meaningless in the decision context. Setup costs –Consider an activity that incurs a fixed cost C j whenever the corresponding level x j > 0, where C j is independent of the actual level of x j. Batch sizes –In production planning situations, one may want to restrict the level of x j to be either x j = 0 or x j ≥ L j. This is an either-or restriction. ”Go-No-Go” decisions –Other forms of either-or restrictions, common in capital budgeting decisions.

17. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.17 Facility Location models – more examples Source : Klose, A.and A. Drexl (2005). “Facility location models for distribution system design,” European Journal of Operational Research 162, pp.4-29. Applications of facility location models are not restricted to the primary application area of this article, that is, the design of distribution systems (for applications of facility location models to practical problem solving in the area of logistics network design see, e.g., Geoffrion and Graves, 1974; Geoffrion et al., 1982; Gelders et al., 1987; Robinson Jr. et al., 1993; Fleischmann, 1993; Geoffrion and Powers, 1995; Küksalan et al., 1995; Tüshaus and Wittmann, 1998; Engeler et al., 1999; Bruns et al., 2000; Galvão et al., 2002; Boffey et al., 2003; Vasko et al., 2003). By contrast many other problems where location and allocation decisions are interdependent are covered also. For the sake of brevity some of them shall be sketched out as follows: Cluster analysis : The topic of cluster analysis is to group items in such a way that items belonging to one group are homogeneous and items belonging to different groups are heterogeneous. Location then means to select representative items from the overall set of items while allocation corresponds to the assignment of the remaining items to the chosen clusters. Cclustering is important in the problem setting of vehicle routing and scheduling (see Fisher and Jaikumar, 1981; Bramel and Simchi-Levi, 1995), and in the area of combined routing location (see Klose and Wittmann, 1995; Klose, 1996). Location of bank accounts : A company which has to pay suppliers has to decide which bank accounts to use for this purpose. Depending on the location of the used accounts float can be optimized. Cornuejols et al. (1977) model this problem, the so-called account location problem, as an UFLP with the additional constraint (5). Nauss and Markland (1981) study the reverse problem of locating bank accounts in order to receive customer payments, the so-called lock box location problem. Vendor selection : Each company must choose vendors for the supply of products. Vendor selection is based on multiple criteria such as price, quality, know-how, etc. Location in this setting means selecting some vendors from a given set of vendors. Allocation relates to the decision which product to buy from which vendor. Current and Weber (1994) discuss, among other topics, that this problem can be tackled using well-known location models such as the UFLP and the CFLP. Location and sizing of offshore platforms for oil exploration : Hansen et al. (1992, 1994) use a capacitated multi-type location model in order to locate offshore platforms for oil exploration. Different platform types relate to potential platform capacities. Database location in computer networks : Within a computer network databases can be installed on certain nodes. Installation and maintenance of databases gives raise to fixed cost while transmission times or cost may decrease, hence, once more, a certain location- allocation problem arises. Fisher and Hochbaum (1980) model this problem as an extended UFLP. Concentrator location : The design of efficient telecommunication and computer networks poses several complex, interdependent problems. Star-like networks comprise a simple topology, connecting terminals with a central machine. Such a topology is inefficient in the case of many terminals and large distances. Probably, the installation of concentrators having powerful links to the central machine or another (backbone) network then is necessary. To determine the layout of a concentrator-based network results in a typical location- allocation problem, also called concentrator location problem by Mirzaian (1985) and Pirkul (1987). Chardaire (1999) and Chardaire et al. (1999) study the case where concentrators can be located on two different layers of the network. Index selection for database design : Databases comprise a set of tables, each of which consists of several arrays. Relating indices to arrays allows storing entries in a sorted manner yielding fast queries. Caprara and Salazar (1995, 1999) and Caprara et al. (1995) study the index selection problem as an important optimization problem in the physical design of relational databases. Moreover, it is shown that this problem can be formulated as an UFLP.

18. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.18

19. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.19 Table of Contents Chapter 4 (Linear Programming: Formulation and Applications) Super Grain Corp. Advertising-Mix Problem (Section 4.1)4.2–4.5 Resource Allocation Problems & Think-Big Capital Budgeting (Section 4.2)4.6–4.10 Cost-Benefit-Trade-Off Problems & Union Airways (Section 4.3)4.11–4.15 Distribution-Network Problems & Big M Co. (Section 4.4)4.16–4.20 Continuing the Super Grain Corp. Case Study (Section 4.5)4.21–4.24 Mixed Formulations & Save-It Solid Waste Reclamation (Section 4.6)4.25–4.30 Applications of Linear Programming with Spreadsheets (UW Lecture)4.31–4.50 These slides are based upon lectures to first-year MBA students at the University of Washington that discuss the application and formulation of linear programming models (as taught by one of the authors).

20. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.20 Super Grain Corp. Advertising-Mix Problem Goal: Design the promotional campaign for Crunchy Start. The three most effective advertising media for this product are –Television commercials on Saturday morning programs for children. –Advertisements in food and family-oriented magazines. –Advertisements in Sunday supplements of major newspapers. The limited resources in the problem are –Advertising budget ($4 million). –Planning budget ($1 million). –TV commercial spots available (5). The objective will be measured in terms of the expected number of exposures. Question: At what level should they advertise Crunchy Start in each of the three media?

21. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.21 Cost and Exposure Data Costs Cost Category Each TV Commercial Each Magazine Ad Each Sunday Ad Ad Budget$300,000$150,000$100,000 Planning budget90,00030,00040,000 Expected number of exposures 1,300,000600,000500,000

22. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.22 Spreadsheet Formulation

23. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.23 Algebraic Formulation LetTV = Number of commercials for separate spots on television M = Number of advertisements in magazines. SS = Number of advertisements in Sunday supplements. Maximize Exposure = 1,300TV + 600M + 500SS subject to Ad Spending:300TV + 150M + 100SS ≤ 4,000 ($thousand) Planning Cost:90TV + 30M + 30SS ≤ 1,000 ($thousand) Number of TV Spots:TV ≤ 5 and TV ≥ 0, M ≥ 0, SS ≥ 0.

24. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.24 Think-Big Capital Budgeting Problem Think-Big Development Co. is a major investor in commercial real-estate development projects. They are considering three large construction projects –Construct a high-rise office building. –Construct a hotel. –Construct a shopping center. Each project requires each partner to make four investments: a down payment now, and additional capital after one, two, and three years. Question: At what fraction should Think-Big invest in each of the three projects?

25. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.25 Financial Data for the Projects Investment Capital Requirements YearOffice BuildingHotelShopping Center 0$40 million$80 million$90 million 160 million80 million50 million 290 million80 million20 million 310 million70 million60 million Net present value$45 million$70 million$50 million

26. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.26 Spreadsheet Formulation

27. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.27 Algebraic Formulation LetOB = Participation share in the office building, H = Participation share in the hotel, SC = Participation share in the shopping center. Maximize NPV = 45OB + 70H + 50SC subject to Total invested now:40OB + 80H + 90SC ≤ 25 ($million) Total invested within 1 year:100OB + 160H + 140SC ≤ 45 ($million) Total invested within 2 years:190OB + 240H + 160SC ≤ 65 ($million) Total invested within 3 years:200OB + 310H + 220SC ≤ 80 ($million) and OB ≥ 0, H ≥ 0, SC ≥ 0.

28. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.28 A side note: Creative use of constraints – modeling logical conditions with binary variables Note: X i is a proposition (perhaps one of the alternative actions in the model) and u i is the associated binary variable (u i = 1 if X i is true, and 0 otherwise) 1.X 1 or X 2 is equivalent to u 1 + u 2 ≥ 1 2.X 1 and X 2 is equivalent to u 1 + u 2 = 2 3.not X 1 is equivalent to u 1 = 0 4.X 1 → X 2 is equivalent to u 1 - u 2 ≤ 0 5.X 1 ↔ X 2 is equivalent to u 1 - u 2 = 0 For example, –If a requirement was to have both a Hotel and a Shopping Center, then use constraint combination 2 in addition to the existing constraints. –If a requirement was to have a Shopping Center if a Hotel is selected, then use constraint combination 4 in addition to the existing constraints.

29. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.29 Summary of Formulation Procedure for Resource- Allocation Problems 1.Identify the activities for the problem at hand. 2.Identify an appropriate overall measure of performance (commonly profit). 3.For each activity, estimate the contribution per unit of the activity to the overall measure of performance. 4.Identify the resources that must be allocated. 5.For each resource, identify the amount available and then the amount used per unit of each activity. 6.Enter the data in steps 3 and 5 into data cells. 7.Designate changing cells for displaying the decisions. 8.In the row for each resource, use SUMPRODUCT to calculate the total amount used. Enter ≤ and the amount available in two adjacent cells. 9.Designate a target cell. Use SUMPRODUCT to calculate this measure of performance.

30. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.30 Union Airways Personnel Scheduling Union Airways is adding more flights to and from its hub airport and so needs to hire additional customer service agents. The five authorized eight-hour shifts are –Shift 1:6:00 AM to 2:00 PM –Shift 2:8:00 AM to 4:00 PM –Shift 3:Noon to 8:00 PM –Shift 4:4:00 PM to midnight –Shift 5:10:00 PM to 6:00 AM Question: How many agents should be assigned to each shift?

31. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.31 Schedule Data Time Periods Covered by Shift Time Period12345 Minimum Number of Agents Needed 6 AM to 8 AM√48 8 AM to 10 AM√√79 10 AM to noon√√65 Noon to 2 PM√√√87 2 PM to 4 PM√√64 4 PM to 6 PM√√73 6 PM to 8 PM√√82 8 PM to 10 PM√43 10 PM to midnight√√52 Midnight to 6 AM√15 Daily cost per agent$170$160$175$180$195

32. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.32 Spreadsheet Formulation

33. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.33 Algebraic Formulation LetS i = Number working shift i (for i = 1 to 5), Minimize Cost = $170S 1 + $160S 2 + $175S 3 + $180S 4 + $195S 5 subject to Total agents 6AM–8AM:S 1 ≥ 48 Total agents 8AM–10AM:S 1 + S 2 ≥ 79 Total agents 10AM–12PM:S 1 + S 2 ≥ 65 Total agents 12PM–2PM:S 1 + S 2 + S 3 ≥ 87 Total agents 2PM–4PM:S 2 + S 3 ≥ 64 Total agents 4PM–6PM:S 3 + S 4 ≥ 73 Total agents 6PM–8PM:S 3 + S 4 ≥ 82 Total agents 8PM–10PM:S 4 ≥ 43 Total agents 10PM–12AM:S 4 + S 5 ≥ 52 Total agents 12AM–6AM:S 5 ≥ 15 and S i ≥ 0 (for i = 1 to 5)

34. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.34 Summary of Formulation Procedure for Cost-Benefit-Tradeoff Problems 1.Identify the activities for the problem at hand. 2.Identify an appropriate overall measure of performance (commonly cost). 3.For each activity, estimate the contribution per unit of the activity to the overall measure of performance. 4.Identify the benefits that must be achieved. 5.For each benefit, identify the minimum acceptable level and then the contribution of each activity to that benefit. 6.Enter the data in steps 3 and 5 into data cells. 7.Designate changing cells for displaying the decisions. 8.In the row for each benefit, use SUMPRODUCT to calculate the level achieved. Enter ≤ and the minimum acceptable level in two adjacent cells. 9.Designate a target cell. Use SUMPRODUCT to calculate this measure of performance.

35. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.35 The Big M Distribution-Network Problem The Big M Company produces a variety of heavy duty machinery at two factories. One of its products is a large turret lathe. Orders have been received from three customers for the turret lathe. Question: How many lathes should be shipped from each factory to each customer?

36. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.36 Some Data Shipping Cost for Each Lathe ToCustomer 1Customer 2Customer 3 FromOutput Factory 1$700$900$80012 lathes Factory 280090070015 lathes Order Size10 lathes8 lathes9 lathes

37. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.37 The Distribution Network

38. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.38 Spreadsheet Formulation

39. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.39 Algebraic Formulation LetS ij = Number of lathes to ship from i to j (i = F1, F2; j = C1, C2, C3). Minimize Cost = $700S F1-C1 + $900S F1-C2 + $800S F1-C3 + $800S F2-C1 + $900S F2-C2 + $700S F2-C3 subject to Factory 1:S F1-C1 + S F1-C2 + S F1-C3 = 12 Factory 2:S F2-C1 + S F2-C2 + S F2-C3 = 15 Customer 1:S F1-C1 + S F2-C1 = 10 Customer 2:S F1-C2 + S F2-C2 = 8 Customer 3:S F1-C3 + S F2-C3 = 9 and S ij ≥ 0 (i = F1, F2; j = C1, C2, C3).

40. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.40 Continuing the Super Grain Case Study David and Claire conclude that the spreadsheet model needs to be expanded to incorporate some additional considerations. In particular, they feel that two audiences should be targeted — young children and parents of young children. Two new goals –The advertising should be seen by at least five million young children. –The advertising should be seen by at least five million parents of young children. Furthermore, exactly $1,490,000 should be allocated for cents-off coupons.

41. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.41 Benefit and Fixed-Requirement Data Number Reached in Target Category (millions) Each TV Commercial Each Magazine Ad Each Sunday Ad Minimum Acceptable Level Young children1.20.105 Parents of young children0.50.2 5 Contribution Toward Required Amount Each TV Commercial Each Magazine Ad Each Sunday Ad Required Amount Coupon redemption0$40,000$120,000$1,490,000

42. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.42 Spreadsheet Formulation

43. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.43 Algebraic Formulation LetTV = Number of commercials for separate spots on television M = Number of advertisements in magazines. SS = Number of advertisements in Sunday supplements. Maximize Exposure = 1,300TV + 600M + 500SS subject to Ad Spending:300TV + 150M + 100SS ≤ 4,000 ($thousand) Planning Cost:90TV + 30M + 30SS ≤ 1,000 ($thousand) Number of TV Spots:TV ≤ 5 Young children:1.2TV + 0.1M ≥ 5 (millions) Parents:0.5TV + 0.2M + 0.2SS ≥ 5 (millions) Coupons:40M + 120SS = 1,490 ($thousand) and TV ≥ 0, M ≥ 0, SS ≥ 0.

44. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.44 Types of Functional Constraints TypeForm*Typical InterpretationMain Usage Resource constraintLHS ≤ RHS For some resource, Amount used ≤ Amount available Resource-allocation problems and mixed problems Benefit constraintLHS ≥ RHS For some benefit, Level achieved ≥ Minimum Acceptable Cost-benefit-trade-off problems and mixed problems Fixed-requirement constraint LHS = RHS For some quantity, Amount provided = Required amount Distribution-network problems and mixed problems * LHS = Left-hand side (a SUMPRODUCT function). RHS = Right-hand side (a constant).

45. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.45 Save-It Company Waste Reclamation The Save-It Company operates a reclamation center that collects four types of solid waste materials and then treats them so that they can be amalgamated into a salable product. Three different grades of product can be made: A, B, and C (depending on the mix of materials used). Question: What quantity of each of the three grades of product should be produced from what quantity of each of the four materials?

46. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.46 Product Data for the Save-It Company GradeSpecification Amalgamation Cost per Pound Selling Price per Pound A Material 1: Not more than 30% of total Material 2: Not less than 40% of total Material 3: Not more than 50% of total Material 4: Exactly 20% of total $3.00$8.50 B Material 1: Not more than 50% of total Material 2: Not less than 10% of the total Material 4: Exactly 10% of the total 2.507.00 CMaterial 1: Not more than 70% of the total2.005.50

47. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.47 Material Data for the Save-It Company Material Pounds/Week Available Treatment Cost per PoundAdditional Restrictions 13,000$3.001. For each material, at least half of the pounds/week available should be collected and treated. 2. $30,000 per week should be used to treat these materials. 22,0006.00 34,0004.00 41,0005.00

48. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.48 Spreadsheet Formulation

49. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.49 Algebraic Formulation Let x ij = Pounds of material j allocated to product i per week (i = A, B, C; j = 1, 2, 3, 4). Maximize Profit = 5.5(x A1 + x A2 + x A3 + x A4 ) + 4.5(x B1 + x B2 + x B3 + x B4 ) + 3.5(x C1 + x C2 + x C3 + x C4 ) subject toMixture Specifications:x A1 ≤ 0.3 (x A1 + x A2 + x A3 + x A4 ) x A2 ≥ 0.4 (x A1 + x A2 + x A3 + x A4 ) x A3 ≤ 0.5 (x A1 + x A2 + x A3 + x A4 ) x A4 = 0.2 (x A1 + x A2 + x A3 + x A4 ) x B1 ≤ 0.5 (x B1 + x B2 + x B3 + x B4 ) x B2 ≥ 0.1 (x B1 + x B2 + x B3 + x B4 ) x B4 = 0.1 (x B1 + x B2 + x B3 + x B4 ) x C1 ≤ 0.7 (x C1 + x C2 + x C3 + x C4 ) Availability of Materials:x A1 + x B1 + x C1 ≤ 3,000 x A2 + x B2 + x C2 ≤ 2,000 x A3 + x B3 + x C3 ≤ 4,000 x A4 + x B4 + x C4 ≤ 1,000 Restrictions on amount treated:x A1 + x B1 + x C1 ≥ 1,500 x A2 + x B2 + x C2 ≥ 1,000 x A3 + x B3 + x C3 ≥ 2,000 x A4 + x B4 + x C4 ≥ 500 Restriction on treatment cost:3(x A1 + x B1 + x C1 ) + 6(x A2 + x B2 + x C2 ) + 4(x A3 + x B3 + x C3 ) + 5(x A4 + x B4 + x C4 ) = 30,000 and x ij ≥ 0 (i = A, B, C; j = 1, 2, 3, 4).

50. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.50 Formulating an LP Spreadsheet Model Enter all of the data into the spreadsheet. Color code (blue). What decisions need to be made? Set aside a cell in the spreadsheet for each decision variable (changing cell). Color code (yellow with border). Write an equation for the objective in a cell. Color code (orange with heavy border). Put all three components (LHS, ≤/=/≥, RHS) of each constraint into three cells on the spreadsheet. Some Examples: –Production Planning –Diet / Blending –Workforce Scheduling –Transportation / Distribution –Assignment

51. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.51 LP Example #1 (Product Mix) 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. Question: What product mix will maximize their total profit?

52. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.52 Algebraic Formulation LetB = Number of benches to produce, T = Number of tables to produce. Maximize Profit = $8B + $18T subject to Labor:3B + 6T ≤ 1,600 hours Wood:12B + 38T ≤ 9,000 pounds and B ≥ 0, T ≥ 0.

53. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.53 Spreadsheet Formulation

54. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.54 LP Example #2 (Diet Problem) A prison is trying to decide what to feed its prisoners. They would like to offer some combination of milk, beans, and oranges. Their goal is to minimize cost, subject to meeting the minimum nutritional requirements imposed by law. The cost and nutritional contents of each food, along with the minimum nutritional requirements are shown below. Milk (gallons) Navy Beans (cups) Oranges (large Calif. Valencia) Minimum Daily Requirement Niacin (mg)3.24.90.813.0 Thiamin (mg)1.121.30.191.5 Vitamin C (mg)3209345 Cost ($)2.000.200.25 Question: What should the diet for each prisoner be?

55. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.55 Algebraic Formulation Letx 1 = gallons of milk per prisoner, x 2 = cups of beans per prisoner, x 3 = number of oranges per prisoner. Minimize Cost = $2.00x 1 + $0.20x 2 + $0.25x 3 subject to Niacin:3.2x 1 + 4.9x 2 + 0.8x 3 ≥ 13 mg Thiamin:1.12x 1 + 1.3x 2 + 0.19x 3 ≥ 1.5 mg Vitamin C:32x 1 + 93x 3 ≥ 45 mg and x 1 ≥ 0, x 2 ≥ 0, x 3 ≥ 0.

56. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.56 Spreadsheet Formulation

57. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.57 George Dantzig’s Diet Stigler (1945) “The Cost of Subsistence” –heuristic solution. Cost = $39.93. Dantzig invents the simplex method (1947) –Stigler’s problem “solved” in 120 man days. Cost = $39.69. Dantzig goes on a diet (early 1950’s), applies diet model: –≤ 1,500 calories –objective: maximize (weight minus water content) –500 food types Initial solutions had problems –500 gallons of vinegar –200 bouillon cubes For more details, see July-Aug 1990 Interfaces article “The Diet Problem”, available for download at www.mhhe.com/hillier2e/articles

58. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.58 Least-Cost Menu Planning Models in Food Systems Management Used in many institutions with feeding programs: hospitals, nursing homes, schools, prisons, etc. Menu planning often extends to a sequence of meals or a cycle. Variety important (separation constraints). Preference ratings (related to service frequency). Side constraints (color, categories, etc.) Generally models have reduced cost about 10%, met nutritional requirements better, and increased customer satisfaction compared to traditional methods. USDA uses these models to plan food stamp allotment. For more details, see Sept-Oct 1992 Interfaces article “The Evolution of the Diet Model in Managing Food Systems”, available for download at www.mhhe.com/hillier2e/articles

59. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.59 LP Example #3 (Scheduling Problem) An airline reservations office is open to take reservations by telephone 24 hours per day, Monday through Friday. The number of reservation agents needed for each time period is shown below. A union contract requires that all employees work 8 consecutive hours. Question: How many reservation agents should work each 8-hour shift? Time Period Number of Agents Needed 12am – 4am11 4am – 8am15 8am – 12pm31 12pm – 4pm17 4pm – 8pm25 8pm – 12am19

60. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.60 Algebraic Formulation Letx 1 = agents who work 12am – 8am, x 2 = agents who work 4am – 12pm, x 3 = agents who work 8am – 4pm, x 4 = agents who work 12pm – 8pm, x 5 = agents who work 4pm – 12am, x 6 = agents who work 8pm – 4am. Minimize Number of agents = x 1 + x 2 + x 3 + x 4 + x 5 + x 6 subject to 12am–4am:x 1 + x 6 ≥ 11 4am–8am:x 1 + x 2 ≥ 15 8am–12pm:x 2 + x 3 ≥ 31 12pm–4pm:x 3 + x 4 ≥ 17 4pm–8pm:x 4 + x 5 ≥ 25 8pm–12am:x 5 + x 6 ≥ 19 and x 1 ≥ 0, x 2 ≥ 0, x 3 ≥ 0, x 4 ≥ 0, x 5 ≥ 0, x 6 ≥ 0.

61. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.61 Spreadsheet Formulation

62. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.62 Workforce Scheduling at United Airlines United employs 5,000 reservation and customer service agents. Some part-time (2-8 hour shifts), some full-time (8-10 hour shifts). Workload varies greatly over day. Modeled problem as LP: –Decision variables: how many employees of each shift length should begin at each potential start time (half-hour intervals). –Constraints: minimum required employees for each half-hour. –Objective: minimize cost. Saved United about $6 million annually, improved customer service, still in use today. For more details, see Jan-Feb 1986 Interfaces article “United Airlines Station Manpower Planning System”, available for download at www.mhhe.com/hillier2e/articles

63. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.63 LP Example #4 (Transportation Problem) A company has two plants producing a certain product that is to be shipped to three distribution centers. The unit production costs are the same at the two plants, and the shipping cost per unit is shown below. Shipments are made once per week. During each week, each plant produces at most 60 units and each distribution center needs at least 40 units. Distribution Center 123 Plant A$4$6$4 B$6$5$2 Question: How many units should be shipped from each plant to each distribution center?

64. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.64 Algebraic Formulation Letx ij = units to ship from plant i to distribution center j (i = A, B; j = 1, 2, 3), Minimize Cost = $4x A1 + $6x A2 + $4x A3 + $6x B1 + $5x B2 + $2x B3 subject to Plant A:x A1 + x A2 + x A3 ≤ 60 Plant B:x B1 + x B2 + x B3 ≤ 60 Distribution Center 1:x A1 + x B1 ≥ 40 Distribution Center 2:x A2 + x B2 ≥ 40 Distribution Center 3:x A3 + x B3 ≥ 40 and x ij ≥ 0 (i = A, B; j = 1, 2, 3).

65. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.65 Spreadsheet Formulation

66. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.66 Distribution System at Proctor and Gamble Proctor and Gamble needed to consolidate and re-design their North American distribution system in the early 1990’s. –50 product categories –60 plants –15 distribution centers –1000 customer zones Solved many transportation problems (one for each product category). Goal: find best distribution plan, which plants to keep open, etc. Closed many plants and distribution centers, and optimized their product sourcing and distribution location. Implemented in 1996. Saved $200 million per year. For more details, see 1997 Jan-Feb Interfaces article, “Blending OR/MS, Judgement, and GIS: Restructuring P&G’s Supply Chain”, downloadable at www.mhhe.com/hillier2e/articles

67. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.67 LP Example #5 (Assignment Problem) The coach of a swim team needs to assign swimmers to a 200-yard medley relay team (four swimmers, each swims 50 yards of one of the four strokes). Since most of the best swimmers are very fast in more than one stroke, it is not clear which swimmer should be assigned to each of the four strokes. The five fastest swimmers and their best times (in seconds) they have achieved in each of the strokes (for 50 yards) are shown below. BackstrokeBreaststrokeButterflyFreestyle Carl37.743.433.329.2 Chris32.933.128.526.4 David33.842.238.929.6 Tony37.034.730.428.5 Ken35.441.833.631.1 Question: How should the swimmers be assigned to make the fastest relay team?

68. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.68 Algebraic Formulation Letx ij = 1 if swimmer i swims stroke j; 0 otherwise t ij = best time of swimmer i in stroke j Minimize Time = ∑ i ∑ j t ij x ij subject to each stroke swum:∑ i x ij = 1 for each stroke j each swimmer swims 1:∑ j x ij ≤ 1 for each swimmer i and x ij ≥ 0 for all i and j.

69. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.69 Spreadsheet Formulation