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Modeling Using Linear Programming

OM2. SUPPLEMENTARY CHAPTER C. Modeling Using Linear Programming. DAVID A. COLLIER AND JAMES R. EVANS. Supplementary Chapter C. Modeling Using Linear Programming. l e a r n i n g o u t c o m e s.

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Modeling Using Linear Programming

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  1. OM2 SUPPLEMENTARY CHAPTER C Modeling Using Linear Programming DAVID A. COLLIER AND JAMES R. EVANS

  2. Supplementary Chapter C. Modeling Using Linear Programming l e a r n i n g o u t c o m e s LO1Explain how to recognize decision variables, the objective function, and constraints in formulating linear optimization models. LO2Describe how to use linear optimization models for OM applications. LO3Explain how to use Excel Solver to solve linear optimization models on spreadsheets.

  3. Supplementary Chapter C. Modeling Using Linear Programming h aller’sPub & Brewery is a small restaurant and microbrewery that makes six types of special beers, each having a unique taste and color. Jeremy Haller, one of the family owners who oversees the brewery operations, has become worried about increasing costs of grains and hops that are the principal ingredients and the difficulty they seem to be having in making the right product mix to meet demand and using the ingredients that are purchased under contract in the commodities market. Haller’s buys six different types of grains and four different types of hops.

  4. Supplementary Chapter C. Modeling Using Linear Programming Each of the beers needs different amounts of brewing time and is produced in 30-keg (4,350-pint) batches. While the average customer demand is 55 kegs per week, the demand varies by type. In a meeting with the other owners, Jeremy stated that Haller’s has not been able to plan effectively to meet the expected demand. “I know there must be a better wayof making our brewing decisions to improve our profitability.”

  5. Supplementary Chapter C. Modeling Using Linear Programming What do you think? Can you identify any examples when you needed to find a better way of planning, designing, or operating some system or process?

  6. Supplementary Chapter C. Modeling Using Linear Programming Quantitative models that seek to maximize or minimize some objective function while satisfying a set of constraints are called optimization models. Linear programming (LP) models are used widely for many types of operations design and planning problems that involve allocating limited resources among competing alternatives, and for supply chain management design and operations.

  7. Supplementary Chapter C. Modeling Using Linear Programming • Softwater Production Planning Problem • Pellets are produced in 40- and 80-pound bags. • Company has orders for 20,000 pounds • 4,000 pounds are currently in inventory • Limited amounts of packaging materials and packaging line time • Determine how many bags of each size to produce to maximize profit.

  8. Supplementary Chapter C. Modeling Using Linear Programming Decision Variables A decision variable is a controllable input variable that represents the key decisions a manager must make to achieve an objective. x1 = number of 40-pound bags produced x2 = number of 80-pound bags produced

  9. Supplementary Chapter C. Modeling Using Linear Programming Objective Function Suppose that Softwater makes $2 for every 40-lb. bag and $4 for every 80-lb. bag produced and sold. Max total profit = z = 2x1 + 4x2 [C.1] The constant terms in the objective function are called objective function coefficients.

  10. Supplementary Chapter C. Modeling Using Linear Programming Solutions Any particular combination of decision variables is referred to as a solution. Solutions that satisfy all constraints are referred to as feasible solutions. Any feasible solution that optimizes the objective function is called an optimal solution.

  11. Supplementary Chapter C. Modeling Using Linear Programming A Solution for the Softwater Problem • Suppose that Softwater decided to produce 200 40-pound bags and 300 80-pound bags. The profit would be • z = 2(200) + 4(300) • = 400 + 1,200 • = $1,600

  12. Supplementary Chapter C. Modeling Using Linear Programming Constraints A constraint is some limitation or requirement that must be satisfied by the solution. Suppose that each 40-pound bag requires 1.2 minutes of packaging time per bag and 80-pound bags require 3 minutes per bag. The total packaging time required is 1.2x1 + 3x2 Only 1,500 minutes of packaging time are available, so we have the constraint: 1.2x1 + 3x2 ≤ 1,500

  13. Supplementary Chapter C. Modeling Using Linear Programming Packaging Material Constraint Softwater has 6,000 square feet of packaging materials available; each 40-pound bag requires 6 square feet and each 80-pound bag requires 10 square feet. Since the amount of packaging materials used cannot exceed what is available, we have the constraint: 6x1 + 10x2 ≤ 6,000

  14. Supplementary Chapter C. Modeling Using Linear Programming • Aggregate Production Constraint • We need to produce a net amount of 16,000 pounds. Because the small bags contain 40 pounds of pellets and the large bags contain 80 pounds, we must impose this aggregate-demand constraint: • 40x1 + 80x2 ≥ 16,000

  15. Supplementary Chapter C. Modeling Using Linear Programming Nonnegativity Constraints We must prevent the decision variables from having negative values. Thus, we need the constraints: x1 and x2 ≥ 0

  16. Supplementary Chapter C. Modeling Using Linear Programming • Softwater Optimization Model • Max z = 2x1 + 4x2 (profit) • subject to • 1.2x1 + 3x2 ≤ 1,500 (packaging line) • 6x1 + 10x2 ≤ 6,000 (materials availability) • 40x1 + 80x2 ≥16,000 (aggregate production) • x1, x2 ≥ 0 (nonnegativity)

  17. Supplementary Chapter C. Modeling Using Linear Programming Linear Functions A function in which each variable appears in a separate term and is raised to the first power is called a linear function. The objective function and all constraints of the Softwater problem consist of linear functions. This is a requirement for a linear program and its solution procedure.

  18. Supplementary Chapter C. Modeling Using Linear Programming Production Scheduling Bollinger Electronics Company produces two electronic components for an airplane engine manufacturer. Demand for the next three months is:

  19. Supplementary Chapter C. Modeling Using Linear Programming Decision Variables xim denotes the production volume in units for product i in month m. Here i=1, 2, and m = 1, 2, 3; i = 1 refers to component 322A, i = 2 to component 802B, m = 1 to April, m = 2 to May, and m = 3 to June.

  20. Supplementary Chapter C. Modeling Using Linear Programming Objective Function Component 322A costs $20 per unit to produce and component 802B costs $10 per unit to produce. The production-cost part of the objective function is: 20x11 + 20x12 + 20x13 + 10x21 + 10x22 + 10x23

  21. Supplementary Chapter C. Modeling Using Linear Programming Objective Function To incorporate the relevant inventory costs into the model, let Iim denote the inventory level for product i at the end of month m. Inventory-holding costs are 1.5 percent of the cost of the product; that is, (.015)($20) = $0.30 per unit for component 322A, and (.015)($10) = $0.15 per unit for component 802B. The inventory-holding cost portion of the objective function can be written as: 0.30I11 + 0.30I12 + 0.30I13 + 0.15I21 + 0.15I22 + 0.15I23

  22. Supplementary Chapter C. Modeling Using Linear Programming Objective Function To incorporate the costs due to fluctuations in production levels from month to month, we need to define additional decision variables: Rm= increase in the total production level during month m compared with month m– 1 Dm = decrease in the total production level during month m compared with month m– 1

  23. Supplementary Chapter C. Modeling Using Linear Programming Complete Objective Function Min 20x11 + 20x12 + 20x13 + 10x21 + 10x22 + 10x23 + 0.30I11 + 0.30I12 + 0.30I13 + 0.15I21 + 0.15I22 + 0.15I23 + 0.50R1 + 0.50R2 + 0.50R3 + 0.20D1 + 0.20D2 + 0.20D3

  24. Supplementary Chapter C. Modeling Using Linear Programming Constraints First we must guarantee that the schedule meets customer demand. We have the basic equation: Ending inventory from previous month + Current production – Ending inventory for this month = This month’s demand Assume inventories at the beginning of the three-month scheduling period are 500 units for component 322A and 200 units for component 802B.

  25. Supplementary Chapter C. Modeling Using Linear Programming Constraints Month 1: 500 + x11–I11 = 1000 200 + x21–I21 = 1000 Month 2: I11 + x12–I12 = 3,000 I21 + x22–I22 = 500 Month 3: I12 + x13– I13= 5,000 I22 + x23–I23 = 3,000

  26. Supplementary Chapter C. Modeling Using Linear Programming Constraints Minimum Inventory Level: At least 400 units of component 322A and at least 200 units of component 802B: I13 ≥ 400 and I23 ≥ 200

  27. Exhibits C.1 and C.2 Additional Constraint Data Additional Constraint Data

  28. Supplementary Chapter C. Modeling Using Linear Programming Constraints Machine capacity: 0.10x11 + 0.08x21 ≤ 400 (month 1) 0.10x12+ 0.08x22 ≤ 500 (month 2) 0.10x13+1 0.08x23 ≤ 600 (month 3) Labor capacity: 0.05x11 + 0.07x21 ≤ 300 (month 1) 0.05x12+ 0.07x22 ≤ 300 (month 2) 0.05x13+ 0.07x23 ≤ 300 (month 3) Storage capacity: 2I11 + 3I21 ≤ 10,000 (month 1) 2I12+ 3I22 ≤ 10,000 (month 2) 2I13+ 3I23 ≤ 10,000 (month 3)

  29. Supplementary Chapter C. Modeling Using Linear Programming Constraints We must also guarantee that Rm and Dm will reflect the increase or decrease in the total production level for month m. Suppose the production levels for March were 1,500 units of component 322A and 1,000 units of component 802B. Then April production – March production = Change x11 + x21– 2,500 = Change x11 + x21– 2,500 = R1– D1 Similar constraints for May and June.

  30. Supplementary Chapter C. Modeling Using Linear Programming Constraints Production Smoothing Constraints: x11 + x21– R1 + D1 = 2,500 – x11– x21 + x12 + x22– R2 + D2 = 0 – x12– x22 + x13 + x23– R3 + D3 = 0

  31. Supplementary Chapter C. Modeling Using Linear Programming Blending Problems Grand Strand Oil Company produces regular-grade and premium-grade gasoline products by blending three petroleum components. The gasolines are sold at different prices, and the petroleum components have different costs. The firm wants to determine how to blend the three components into the two products in such a way as to maximize profits.

  32. Exhibit C.4 Petroleum Component Cost and Supply Data Regular-grade gasoline can be sold for $2.20 per gallon and the premium-grade gasoline for $2.40 per gallon. Current commitments to distributors require Grand Strand to produce at least 10,000 gallons of regular-grade gasoline.

  33. Exhibit C.5 Component Specifications for Grand Strand’s Products Data

  34. Supplementary Chapter C Modeling Using Linear Programming Decision Variables x1r = gallons of component 1 in regular gasoline x2r = gallons of component 2 in regular gasoline x3r = gallons of component 3 in regular gasoline x1p = gallons of component 1 in premium gasoline x2p = gallons of component 2 in premium gasoline x3p = gallons of component 3 in premium gasoline

  35. Supplementary Chapter C. Modeling Using Linear Programming Objective Function Max 2.20(x1r + x2r + x3r) + 2.40(x1p + x2p + x3p) – 1.00(x1r + x1p) - 1.20(x2r + x2p) - 1.64(x3r + x3p) By combining terms, we can then write the objective function as: Max 1.20x1r + 1.00x2r + 0.56x3r+ 1.40x1p + 1.20x2p + 0.76x3p

  36. Supplementary Chapter C. Modeling Using Linear Programming Constraints Component availability: x1r + x1p ≤ 5,000 (component 1) x2r + x2p ≤ 10,000 (component 2) x3r+ x3p ≤ 10,000 (component 3) Regular grade gasoline requirement: x1r + x2r+x3r ≥ 10,000

  37. Supplementary Chapter C. Modeling Using Linear Programming • Constraints • Component 1 must account for at most 30 percent of the total gallons of regular gasoline produced: • x1r /(x1r + x2r + x3r) ≤ 0.30 or x1r ≤ 0.30(x1r + x2r + x3r) • Rewrite this as: • 0.70x1r - 0.30x2r - 0.30x3r≤ 0 • Other specification constraints: • – 0.40x1r+ 0.60x2r– 0.40x3r ≤0 • – 0.20x1r – 0.20x2r + 0.80x3r ≤ 0 • – 0.75x1p – 0.25x2p– 0.25x3p ≤ 0 • – 0.40x1p+ 0.60x2p– 0.40x3p ≤ 0 • – 0.30x1p – 0.30x2p + 0.70x3p ≤ 0

  38. Supplementary Chapter C. Modeling Using Linear Programming Transportation Problem The transportation problem is a special type of linear program that arises in planning the distribution of goods and services from several supply points to several demand locations.

  39. Exhibits C.6 and C.7 Foster Generators Supply/Demand Data Foster Generators Supply/Demand Data

  40. Exhibit C.8 Foster Generators Transportation Cost per Unit Foster Generators Cost Data

  41. Supplementary Chapter C. Modeling Using Linear Programming Transportation Table

  42. Supplementary Chapter C. Modeling Using Linear Programming • Transportation LP Model • Min total cost = 3x11 + 2x12 + 7x13 + 6x14 + 7x21 + 5x22 + 2x23 + 3x24 + 2x31 + 5x32 + 4x33 + 5x34 • Subject to • Cleveland: x11 + x12 + x13 + x14 = 5,000. • Bedford: x21 + x22 + x23 + x24 = 6,000. • York: x31 + x32 + x33 + x34 = 2,500. • Boston: x11 + x21 + x31 = 6,000. • Chicago: x12 + x22 + x32 = 4,000 • St. Louis: x13 + x23 + x33 = 2,000 • Lexington: x14 + x24 + x34 = 1,500

  43. Supplementary Chapter C. Modeling Using Linear Programming LP Model for Crashing Decisions

  44. Supplementary Chapter C. Modeling Using Linear Programming Data

  45. Supplementary Chapter C. Modeling Using Linear Programming Decision Variables and Objective Function xi = start time of activity i yi = amount of crash time used for activity I Min 2,000yA + 1,000yB + 2,500yC + 1,500yD + 500yE

  46. Supplementary Chapter C. Modeling Using Linear Programming Constraints For each arc from activity i to activity j in the network, the start time for the following activity must be at least as great as the finish time for each immediate predecessor with crashing applied xj ≥ xi + normal time for activity i - yi

  47. Supplementary Chapter C. Modeling Using Linear Programming • Precedence Constraints • xB ≥ xA + 10 - yA • xD ≥ xB + 14 - yB • xC ≥ xB + 14 - yB • xE ≥ xD + 11 - yD • xE ≥ xC + 6 - yC • xF ≥ xE + 8 - yE

  48. Supplementary Chapter C. Modeling Using Linear Programming • Other Constraints • Maximum Crash Times: • yA ≤ 3 • yB ≤ 4 • yC ≤ 2 • yD ≤ 2 • yE ≤ 4 • Project Completion Time: • xF = 35

  49. Supplementary Chapter C. Modeling Using Linear Programming Using Excel Solver – Softwater Spreadsheet Model

  50. Supplementary Chapter C. Modeling Using Linear Programming Solver Model

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