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Readings Readings Chapter 4 Linear Programming Applications in Marketing, Finance, and Operations Management

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Overview Overview

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Overview Production Scheduling Problems are Resource Allocation Problems when outputs are fixed and when outputs and inputs occur at different periods in time. The simplest problems consider only two time periods. Production Scheduling Problems with Dynamic Inventory help managers find an efficient low-cost production schedule for one or more products over several time periods (weeks or months). Workforce Assignment Problems are Resource Allocation Problems when labor is a resource with a flexible allocation; some labor can be assigned to more than one work center. Make or Buy Problems are Linear Programming Profit Maximization problems when outputs are fixed and when inputs can be either made or bought. Make or Buy Problems help minimize cost. Product Mix Problems are Resource Allocation Problems when outputs have different physical characteristics. Product Mix Problems thus determine production levels that meet demand requirements. Blending Problems with Weight Constraints help production managers blend resources to produce goods of a specific weight at minimum cost.

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Overview Tool Summary Do not make integer restrictions, and maybe the variables at an optimum will be integers. First Example: Pi = (integer) number of producers in month i. Use compound variables: First Example: Pi = number of producers in month i Use dynamic or recursive constraints: First Example: Define the constraint that the number of apprentices in a month must not exceed the number of recruits in the previous month: A2 - R1 < 0; A3 - R2 < 0 Constrain one variable to be a proportional to another variable: First Example: Define the constraint that each trainer can train two recruits: 2T1 - R1 > 0; 2T2 - R2 > 0 Use inventory variables: Second Example: P2 + I1–I2 = 150 (production-net inventory = demand)

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**Production Scheduling**

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**Production Scheduling**

Overview Production Scheduling Problems are Resource Allocation Problems when outputs are fixed and when outputs and inputs occur at different periods in time. Production Scheduling Problems thus help managers find an efficient low-cost production schedule for one or more products over several periods in the future (weeks or months). The manager determines the production levels that meet demand requirements, given limitations on production capacity, labor capacity, and storage space, while minimizing total production and storage costs. The simplest problems consider only two time periods.

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**The cost of storing a wheel from one month to the next is $2. **

Production Scheduling Question: Chip Foose is the owner of Chip Foose Custom Cars. Chip has just received orders for 1,000 standard wheels and 1,250 deluxe wheels next month and for 800 standard and 1,500 deluxe the following month. All orders must be filled. The cost of making standard wheels is $10 and deluxe wheels is $16. Overtime rates are 50% higher. There are 1,000 hours of regular time and 500 hours of overtime available each month. It takes .5 hour to make a standard wheel and .6 hour to make a deluxe wheel. The cost of storing a wheel from one month to the next is $2. Minimize total production and inventory costs for standard and deluxe wheels.

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**SI = number of standard wheels held in **

Production Scheduling Define the production variables: Determine the regular-time and overtime production quantities in each month for standard and deluxe wheels. Month Month 2 Wheel Reg. Time Overtime Reg. Time Overtime Standard SR SO SR SO2 Deluxe DR DO DR DO2 Define the inventory variables: Determine the inventory quantities for standard and deluxe wheels. SI = number of standard wheels held in inventory from month 1 to month 2 DI = number of deluxe wheels held in

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**Min (production cost per wheel) x (number of wheels produced) **

Production Scheduling Define the objective function: Minimize total production and inventory costs for standard and deluxe wheels. Min (production cost per wheel) x (number of wheels produced) + (inventory cost per wheel) x (number of wheels in inventory) Min 10SR1 + 15SO1 + 10SR2 + 15SO2 + 16DR1 + 24DO1 + 16DR2 + 24DO2 + 2SI + 2DI

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**(1) SR1 + SO1 = 1,000 + SI or SR1 + SO1 - SI = 1,000 Deluxe: **

Production Scheduling Define the production month 1 constraint on (units required) + (units stored). Standard: (1) SR1 + SO1 = 1,000 + SI or SR1 + SO1 - SI = 1,000 Deluxe: (2) DR1 + DO1 = 1,250 + DI or DR1 + DO1 - DI = 1,250

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**(3) SR2 + SO2 = 800 - SI or SR2 + SO2 + SI = 800 Deluxe: **

Production Scheduling Define the production month 2 constraint on (units required) + (units stored). Standard: (3) SR2 + SO2 = SI or SR2 + SO2 + SI = 800 Deluxe: (4) DR2 + DO2 = 1,500 - DI or DR2 + DO2 + DI = 1,500

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**Production Scheduling**

Define the constraint Reg. Hrs. Used Month 1 < Reg. Hrs. Avail. Month 1: (5) .5SR1 + .6DR1 < 1000 Define the constraint OT Hrs. Used Month 1 < OT Hrs. Avail. Month 1: (6) .5SO1 + .6DO1 < 500 Define the constraint Reg. Hrs. Used Month 2 < Reg. Hrs. Avail. Month 2: (7) .5SR2 + .6DR2 < 1000 Define the constraint OT Hrs. Used Month 2 < OT Hrs. Avail. Month 2: (8) .5SO2 + .6DO2 < 500

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**Interpretation: Total cost $67,500, no storage, production schedule: **

Production Scheduling Interpretation: Total cost $67,500, no storage, production schedule: Month Month 2 Reg. Time Overtime Reg. Time Overtime Standard Deluxe

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**Production Scheduling with Dynamic Inventory**

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**Production Scheduling with Dynamic Inventory**

Overview Production Scheduling Problems with Dynamic Inventory help managers find an efficient low-cost production schedule for one or more products over several time periods (weeks or months). A production scheduling problem is resource-allocation problem for each of several periods in the future. Complex problems consider more than two time periods, so there are many periods of inventory.

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**Production Scheduling with Dynamic Inventory**

Question: Wilson Sporting Goods produces baseballs. Wilson must decide how many baseballs to produce each month. It has decided to use a 6-month planning horizon. The forecasted demands for the next 6 months are 10,000; 15,000; 30,000; 35,000; 25,000; and 10,000. Wilson wants to meet these demands on time, knowing that it currently has 5,000 baseballs in inventory and that it can use a given month’s production to help meet the demand for that month. During each month there is enough production capacity to produce up to 30,000 baseballs, and there is enough storage capacity to store up to 10,000 baseballs at the end of the month, after demand has occurred.

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**Production Scheduling with Dynamic Inventory**

The forecasted production costs per baseball for the next 6 months are $12.50, $12.55, $12.70, $12.80, $12.85, and $12.95. The holding cost per baseball held in inventory at the end of the month is figured at 5% of the production cost for that month: $0.625, $0.6275, $0.635, $0.64, $0.6425, and $ The selling price for baseballs is not considered relevant to the production decision because Wilson will satisfy all customer demand exactly when it occurs – at whatever the selling price. Therefore, Wilson wants to determine the production schedule that minimizes the total production and holding costs.

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**The following constraints define inventories: **

Production Scheduling with Dynamic Inventory The decision variables are the production quantities for the 6 months, labeled P1 through P6. To keep quantities small, all quantities are in hundreds of baseballs. Constraints are easier to understand if we add variables I1 through I6 to be the corresponding end-of-month inventories (after meeting demand). For example, I3 is the number of baseballs left over at the end of month 3. The following constraints define inventories: P1 – I1 = (production–inventory = net demand) P2 + I1–I2 = 150 (production-net inventory = demand) P3 + I2–I3 = 300 (production-net inventory = demand) P4 + I3–I4 = 350 (production-net inventory = demand) P5 + I4–I5 = 250 (production-net inventory = demand) P6 + I5–I6 = 100 (production-net inventory = demand)

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**Finally, production and inventory storage are assumed non-negative.**

Production Scheduling with Dynamic Inventory There are obvious constraints are on production and inventory storage capacities: Pj 300 and Ij 100 for each month j (j = 1, …, 6). Finally, production and inventory storage are assumed non-negative.

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**Production Scheduling with Dynamic Inventory**

There

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**Production Scheduling with Dynamic Inventory**

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Workforce Assignment Workforce Assignment

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Workforce Assignment Overview Workforce Assignment Problems are Resource Allocation Problems when labor is one of the resources, and labor allocation has some flexibility; at least some labor can be assigned to more than one department or work center. Workforce Assignment Problems thus help when employees have been cross-trained on two or more jobs or, for instance, when sales personnel can be transferred between stores.

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**Type of Production Rate Wage Employee (Wings/Month) Per Month **

Workforce Assignment Question: National Wing Company (NWC) is gearing up for the new B-48 contract. NWC has agreed to supply 20 wings in April, 24 in May, and 30 in June. Wings can be freely stored from one month to the next. Currently, NWC has 100 fully-qualified workers. A fully qualified worker can either be placed in production or can train new recruits. A new recruit can be trained to be an apprentice in one month. After another month, the apprentice becomes a qualified worker. Each trainer can train two recruits. At the end of June, NWC wishes to have at least 140 fully-qualified workers. (Note: NWC must use firm-specific training. There is no outside market for fully-qualified workers.) The production rate and salary per employee type is listed below. Type of Production Rate Wage Employee (Wings/Month) Per Month Production $3,000 Trainer $3,300 Apprentice $2,600 Recruit $2,200 How should NWC optimize?

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Workforce Assignment Answer: Define the Decision Variables Pi = number of producers in month i (where i = 1, 2, 3 for April, May, June) Ti = number of trainers in month i (where i = 1, 2 for April, May) Ai = number of apprentices in month i (where i = 2, 3 for May, June) Ri = number of recruits in month i (where i = 1, 2 for April, May) Define the objective function Minimize total wage cost for producers, trainers, apprentices, and recruits for April, May, and June: Min 3000P T R1 + 3000P T A2+2200R2 + 3000P T A3+2200R3

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Workforce Assignment Define the constraint that total production in Month 1 (April) must equal or exceed contract for Month 1: .6P1 + .3T1 +.05R1 > 20 Define the constraint that total production in Months 1-2 (April, May) must equal or exceed total contracts for Months 1-2: .6P1 + .3T R1 + .6P2 + .3T2 + .4A R2 > 44 Define the constraint that total production in Months 1-3 (April, May, June) must equal or exceed total contracts for Months 1-3: .6P1+.3T1+.05R1+.6P2+.3T2+.4A2+.05R2+.6P3+.4A3 > 74 Define the constraint that the number of producers and trainers in a month (fully qualified workers) must not exceed the initial supply of 100, plus any apprentices employed in a previous month: (4) P1 + T1 < 100 P2 + T2 < 100 P3 + T3 < A2

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Workforce Assignment Define the constraint that the number of apprentices in a month must not exceed the number of recruits in the previous month that have not already become apprentices: A2 < R1; (8) A3 < (R1 - A2) + R2 Note: Constraint (8) allows a recruit from Month 1 to be laid off in Month 2, then rehired as an apprentice in Month 3. Define the constraint that each trainer can train two recruits: 2T1 - R1 > 0; (10) 2T2 - R2 > 0 Define the constraint that at the end of June, there are to be at least 140 fully qualified workers: (11) A2 + A3 > 140

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Workforce Assignment Interpretation: Total wage cost = $1,098,000, using the following workforce assignment: April May June July Producers Trainers Apprentices Recruits

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Make or Buy Make or Buy

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Make or Buy Overview Make or Buy Problems are Linear Programming Profit Maximization problems when outputs are fixed and when inputs can be either made or bought. Make or Buy Problems thus help production managers to minimize cost by comparing, for some inputs, the lower cost of manufacturing those inputs (rather then buying them) to the opportunity cost of the scarce resources used in manufacture.

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Make or Buy Question: The Janders Company markets business and engineering products. Janders is currently preparing to introduce two new calculators: one for the business market called the Financial Manager, and one for the engineering market called the Technician. Each calculator has three components: a base, an electronic cartridge, and a faceplate or top. The same base is used for both calculators, but the cartridges and tops are different. All components can be manufactured by the company or purchased from outside suppliers.

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Make or Buy Company forecasters indicate that 3000 Financial Manager calculators and 2000 Technician calculators will be demanded. However, manufacturing capacity is limited. The company has 200 hours of regular time manufacturing time and 50 hours of overtime that can be scheduled for the calculators. Overtime involves a premium at the additional cost of $9 per hour.

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**Manufacture cost (regular time) per unit**

Make or Buy Determine how many units of each component to manufacture and how many to buy given the cost, purchase price and manufacturing time requirements: Component Manufacture cost (regular time) per unit Purchase cost per unit Manufacturing Time Base $0.50 $0.60 1.0 minutes Financial cartridge $3.75 $4.00 3.0 minutes Technician cartridge $3.30 $3.90 2.5 minutes Financial top $0.65 Technician top $0.75 $0.78 1.5 minutes

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**Define decision variables BM = number of bases manufactured **

Make or Buy Answer: Define decision variables BM = number of bases manufactured BP = number of bases purchased FCM = number of Financial cartridges manufactured FCP = number of Financial cartridges purchased TCM = number of Technician cartridges manufactured TCP = number of Technician cartridges purchased FTM = number of Financial tops manufactured FTP = number of Financial tops purchased TTM = number of Technician tops manufactured TTP = number of Technician tops purchased OT = number of overtime hours

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**Manufacture cost (regular time) per unit**

Make or Buy Define the objective: Min 0.5BM + 0.6BP FCM + 4FCP+ 3.3TCM + 3.9TCP + 0.6FTM FTP TTM TTP + 9OT Component Manufacture cost (regular time) per unit Purchase cost per unit Manufacturing Time Base $0.50 $0.60 1.0 minutes Financial cartridge $3.75 $4.00 3.0 minutes Technician cartridge $3.30 $3.90 2.5 minutes Financial top $0.65 Technician top $0.75 $0.78 1.5 minutes

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**FCM + FCP = 3000 Financial cartridges **

Make or Buy Demand Constraints: BM + BP = Bases FCM + FCP = 3000 Financial cartridges TCM + TCP = 2000 Technician cartridges FTM + FTP = 3000 Financial tops TTM + TTP = 2000 Technician tops Overtime Resource Constraint: OT < 50 Manufacturing Time Constraint (right-side in minutes): BM + 3FCM + 2.5TCM + FTM + 1.5TTM < OT

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Make or Buy The Management Scientist can solve this 11-variable, 7-constraint linear program: BM = 5000 BP = 0 FCM = FCP = TCM = TCP = 0.000 FTM = 0.000 FTP = TTM = 0.000 TTP = OT = 0.000

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Product Mix Product Mix

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Product Mix Overview Product Mix Problems are Resource Allocation Problems when outputs have different physical characteristics. Product Mix Problems thus help managers determine the production levels that meet demand requirements, given limitations on production capacity and labor capacity, to maximize profit or minimize cost.

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Product Mix Question: Floataway Tours has $420,000 that can be used to buy new rental boats for hire during the summer. The boats can be bought from two different manufacturers. Floataway Tours would like to buy at least 50 boats, and would like to buy the same number from Sleekboat as from Racer to maintain goodwill. At the same time, Floataway Tours wishes to have a total seating capacity of at least 200. Maximum Expected Boat Builder Cost Seating Daily Profit Speedhawk Sleekboat $ $ 70 Silverbird Sleekboat $ $ 80 Catman Racer $ $ 50 Classy Racer $ $110 How should Floataway Tours optimize?

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Product Mix Define the Decision Variables x1 = number of Speedhawks ordered x2 = number of Silverbirds ordered x3 = number of Catmans ordered x4 = number of Classys ordered Define the Objective Function Maximize total expected daily profit: Max (Expected daily profit per unit) x (Number of units) Max 70x1 + 80x2 + 50x x4

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Product Mix Define the constraint to spend no more than $420,000: (1) 6000x x x x4 < 420,000 Define the constraint to buy at least 50 boats: (2) x1 + x2 + x3 + x4 > 50 Define the constraint that the number of boats from Sleekboat equals the number of boats from Racer: (3) x1 + x2 = x3 + x4 or x1 + x2 - x3 - x4 = 0 Define the constraint that seating capacity be at least 200: (4) 3x1 + 5x2 + 2x3 + 6x4 > 200

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**Interpretation: Expected daily profit is $5,040.00. **

Product Mix Interpretation: Expected daily profit is $5, Buy 28 Speedhawks from Sleekboat; buy 28 Classy’s from Racer. The minimum number of boats was exceeded by 6 (surplus for constraint #2). The minimum seating capacity was exceeded by 52 (surplus for constraint #4).

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**Blending with Weight Constraints**

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**Blending with Weight Constraints**

Overview Blending Problems with Weight Constraints help production managers blend resources to produce goods of a specific weight at minimum cost.

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**Blending with Weight Constraints**

Question: The Maruchan Corporation receives four raw grains from which it blends its Maruchan Ramen Noodle Soup. Maruchan advertises that each 8-ounce packet meets the minimum daily requirements for vitamin C, protein and iron. The following is the cost of each raw grain, the vitamin C, protein, and iron units per pound of each grain. Vitamin C Protein Iron Grain Units/lb Units/lb Units/lb Cost/lb Maruchan is interested in producing the 8-ounce mixture at minimum cost while meeting the minimum daily requirements of 6 units of vitamin C, 5 units of protein, and 5 units of iron.

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**Blending with Weight Constraints**

Answer: Define the decision variables. xj = the pounds of grain j (j = 1,2,3,4) used in 8-ounce mixture Define the objective. Minimize the total cost for an 8-ounce mixture: Min .75x x x x4

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**Blending with Weight Constraints**

Constrain the total weight of the mix to 8-ounces (.5 pounds): (1) x1 + x2 + x3 + x4 = .5 Constrain the total amount of Vitamin C in the mix to be at least 6 units: (2) 9x1 + 16x2 + 8x3 + 10x4 > 6 Constrain the total amount of protein in the mix to be at least 5 units: (3) 12x1 + 10x2 + 10x3 + 8x4 > 5 Constrain the total amount of iron in the mix to be at least 5 units: (4) 14x2 + 15x3 + 7x4 > 5

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**Blending with Weight Constraints**

Interpretation: The mixture costs Frederick’s 40.6 cents. Optimal blend: 0.099 lb. of grain 1 0.213 lb. of grain 2 0.088 lb. of grain 3 0.099 lb. of grain 4

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**BA 452 Quantitative Analysis**

End of Lesson A.9

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