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1 BA 555 Practical Business Analysis Linear Programming (LP) Introduction Examples LINDO and Excel-Solver Agenda

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2 Decision-making under Uncertainty Decision-making under uncertainty entails the selection of a course of action when we do not know with certainty the results that each alternative action will yield. This type of decision problems can be solved by statistical techniques along with good judgment and experience. Example: buying stocks/mutual funds.

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3 Decision-making under Certainty Decision-making under certainty entails the selection of a course of action when we know the results that each alternative action will yield. This type of decision problems can be solved by linear/integer programming technique. Example: A company produces two different auto parts A and B. Part A (B) requires 2 (2) hours of grinding and 2 (4) hours of finishing. The company has two grinders and three finishers, each of which works 40 hours per week. Each Part A (B) brings a profit of $3 ($4). How many items of each part should be manufactured per week?

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4 Steps in Quantifying and Solving a Decision Problem Under Certainty Formulate a mathematical model: Define decision variables, State an objective, State the constraints. Input the model to a LP/ILP solver, e.g., LINDO or EXCEL Solver. Obtain computer printouts and perform sensitivity analysis. Report optimal strategy.

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5 Example 1 (p. 61) A company produces two different auto parts A and B. Part A (B) requires 2 (2) hours of grinding and 2 (4) hours of finishing. The company has two grinders and three finishers, each of which works 40 hours per week. Each Part A (B) brings a profit of $3 ($4). How many items of each part should be manufactured per week?

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6 Solving a LP problem: LINDO or EXCEL Solver Install LINDO or EXCEL Solver (do at least one.) LINDO: http://www.lindo.com/. Go to DOWNLOAD HOMEPAGE. On the left-hand-side, chose LINDO FOR WINDOWS (not LINDO API, not LINGO.)http://www.lindo.com/ Its syntax is given on pp. 78 – 80 of the class packet. EXCEL Solver: Under Tools / Add-Ins. Check the SOLVER ADD-INS box. Click OK. It is supported by the textbook (Chapter 4, pp. 209 – 281)

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7 Example 2 Logistics (p.62) Per-Unit Shipping cost Warehouse Store AStore BStore CCapacity W1 $9$8$6100 W2 $7$4$3200 Requirement 14050110300

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8 Example 8 Purchasing (p.68)

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9 Example 3 Media Selection (p.63)

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10 Example 4 Portfolio Selection (p.64)

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11 Example 6 Blending (p.66) Ajax Fuels, Inc., is developing a new additive for airplane fuels. The additive is a mixture of three ingredients: A, B, and C. For proper performance, the total amount of additive (amount of A + amount of B + amount of C) must be at least 10 ounces per gallon of fuel. However, because of safety reasons, the amount of additive must not exceed 15 ounces per gallon of fuel. The mix or blend of the three ingredients is critical. At least 1 ounce of ingredient A must be used for every ounce of ingredient B. The amount of ingredient C must be greater than one- half the amount of ingredient A. If the costs per ounce for ingredients A, B, and C are $0.10, $0.03, and $0.09, respectively, find the minimum-cost mixture of A, B, and C for each gallon of airplane fuel.

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12 Example 5 Production Scheduling (p.65) Q1Q2Q3Q4 Demand2000400030001500 Capacity4000300020004000 Inv. Cost Per unit $250 $300 Prod. Cost Per unit $10,000$11,000$12,100$13,310 Initial inventory = 100 Ending inventory ≥ 500

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13 Example 7 Staff Scheduling (p.67)

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14 Variation: Staff Scheduling SHIFTMTWRFSASU 1X1 2X2 3X3 4X4 5X5 6AB Req’17131519141611 Different schedules Different benefits Full-time vs. part-time Etc.

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15 Example 9 Multi-period Financial Planning (p.69) Cash Flow at Time Period # Investment0123 A–10.510 B0–10.51 C–11.200 D–1001.9 E00–11.5 Constraints: Balance cash inflow and cash outflow at all time periods.

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