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Aggregate Production Planning (APP)

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Aggregate Production Planning (APP) l Matches market demand to company resources l Plans production 6 months to 12 months in advance l Expresses demand, resources, and capacity in general terms l Develops a strategy for economically meeting demand l Establishes a companywide game plan for allocating resources

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Inputs and Outputs to Aggregate Production Planning Aggregate Production Planning Company Policies Financial Constraints Strategic Objectives Units or dollars subcontracted, backordered, or lost Capacity Constraints Size of Workforce Production per month (in units or $) Inventory Levels Demand Forecasts

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Strategies for Meeting Demand 1. Use inventory to absorb fluctuations in demand (level production) 2. Hire and fire workers to match demand (chase demand) 3. Maintain resources for high demand levels 4. Increase or decrease working hours (over & undertime) 5. Subcontract work to other firms 6. Use part-time workers 7. Provide the service or product at a later time period (backordering)

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Strategy Details l Level production - produce at constant rate & use inventory as needed to meet demand l Chase demand - change workforce levels so that production matches demand l Maintaining resources for high demand levels - ensures high levels of customer service l Overtime & undertime - common when demand fluctuations are not extreme

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Strategy Details l Subcontracting - useful if supplier meets quality & time requirements l Part-time workers - feasible for unskilled jobs or if labor pool exists l Backordering - only works if customer is willing to wait for product/services

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Level Production Time Production Demand Units

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Chase Demand Time Units Production Demand

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APP Example l The Bavarian Candy Company (BCC) makes a variety of candies in three factories worldwide. Its line of chocolate candies exhibits a highly seasonal pattern with peaks in winter months and valleys during the summer months. Given the costs and quarterly sales forecasts, determine whether a level production or chase demand production strategy would be more economically meet the demand for chocolate candies.

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APP Using Pure Strategies Hiring cost = $100 per worker Firing cost = $500 per worker Inventory carrying cost = $0.50 per kilogram per quarter Production per employee = 1,000 kilograms per quarter Beginning work force = 100 workers QuarterSales Forecast (kg) Spring80,000 Summer50,000 Fall120,000 Winter150,000

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Level Production Strategy SalesProduction QuarterForecastPlanInventory Spring80,000100,00020,000 Summer50,000100,00070,000 Fall120,000100,00050,000 Winter150,000100,0000 400,000140,000 Cost = 140,000 kilograms x $0.50 per kilogram = $70,000

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Chase Demand Strategy SalesProductionWorkersWorkersWorkers QuarterForecastPlanNeededHiredFired Spring80,00080,00080-20 Summer50,00050,00050-30 Fall120,000120,00012070- Winter150,000150,00015030- 10050 Cost = (100 workers hired x $100) + (50 workers fired x $500) = $10,000 + 25,000 = $35,000

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LP Formulation Define H t = # hired for period t F t = # fired for period t I t = inventory at end of period t P t = Production in period t W t = Workforce in period t Min Z = $100 (H 1 + H 2 + H 3 + H 4 ) + $500 (F 1 + F 2 + F 3 + F 4 ) + $0.50 (I 1 + I 2 + I 3 + I 4 )

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Subject to P 1 - I 1 = 80,000(1)Demand P 1 - I 1 = 80,000(1)Demand I 1 + P 2 - I 2 = 50,000(2)constraints I 2 + P 3 - I 3 = 120,000(3) I 3 + P 4 - I 4 = 150,000(4)

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Min Z = $100 (H 1 + H 2 + H 3 + H 4 ) + $500 (F 1 + F 2 + F 3 + F 4 )+ $0.50 (I 1 + I 2 + I 3 + I 4 ) Subject to P 1 - I 1 = 80,000(1)Demand I 1 + P 2 - I 2 = 50,000(2)constraints I 2 + P 3 - I 3 = 120,000(3) I 3 + P 4 - I 4 = 150,000(4) P 1 - 1,000 W 1 = 0(5)Production P 2 - 1,000 W 2 = 0(6)constraints P 3 - 1,000 W 3 = 0(7) P 4 - 1,000 W 4 = 0(8)

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Min Z = $100 (H 1 + H 2 + H 3 + H 4 ) + $500 (F 1 + F 2 + F 3 + F 4 )+ $0.50 (I 1 + I 2 + I 3 + I 4 ) Subject to P 1 - I 1 = 80,000(1)Demand I 1 + P 2 - I 2 = 50,000(2)constraints I 2 + P 3 - I 3 = 120,000(3) I 3 + P 4 - I 4 = 150,000(4) P 1 - 1,000 W 1 = 0(5)Production P 2 - 1,000 W 2 = 0(6)constraints P 3 - 1,000 W 3 = 0(7) P 4 - 1,000 W 4 = 0(8) W 1 - H 1 + F 1 = 100(9) Work force W 2 - W 1 - H 2 + F 2 = 0(10) constraints W 3 - W 2 - H 3 + F 3 = 0(11) W 4 - W 3 - H 4 + F 4 = 0(12)

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Min Z = $100 (H 1 + H 2 + H 3 + H 4 ) + $500 (F 1 + F 2 + F 3 + F 4 )+ $0.50 (I 1 + I 2 + I 3 + I 4 ) Subject to P 1 - I 1 = 80,000(1)Demand P 1 - I 1 = 80,000(1)Demand I 1 + P 2 - I 2 = 50,000(2)constraints I 2 + P 3 - I 3 = 120,000(3) I 3 + P 4 - I 4 = 150,000(4) P 1 - 1,000 W 1 = 0(5)Production P 2 - 1,000 W 2 = 0(6)constraints P 3 - 1,000 W 3 = 0(7) P 4 - 1,000 W 4 = 0(8) W 1 - H 1 + F 1 = 100(9) Work force W 2 - W 1 - H 2 + F 2 = 0(10) constraints W 3 - W 2 - H 3 + F 3 = 0(11) W 4 - W 3 - H 4 + F 4 = 0(12)

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LP Solution: Z = $32,000 H 1 = 0, F 1 = 20, I 1 = 0, P 1 =80000; H 2 = 0, F 2 = 0, I 2 = 30000, P 2 =80000; H 3 = 10, F 3 = 0, I 3 = 0, P 3 =90000; H 4 = 60, F 4 = 0, I 4 = 0, P 4 =150000;

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Summary: APP By Linear Programming Min Z = $100 (H 1 + H 2 + H 3 + H 4 ) + $500 (F 1 + F 2 + F 3 + F 4 )+ $0.50 (I 1 + I 2 + I 3 + I 4 ) Subject to P 1 - I 1 = 80,000(1)Demand I 1 + P 2 - I 2 = 50,000(2)constraints I 2 + P 3 - I 3 = 120,000(3) I 3 + P 4 - I 4 = 150,000(4) P 1 - 1,000 W 1 = 0(5)Production P 2 - 1,000 W 2 = 0(6)constraints P 3 - 1,000 W 3 = 0(7) P 4 - 1,000 W 4 = 0(8) W 1 - H 1 + F 1 = 100(9) Work force W 2 - W 1 - H 2 + F 2 = 0(10) constraints W 3 - W 2 - H 3 + F 3 = 0(11) W 4 - W 3 - H 4 + F 4 = 0(12) where H t = # hired for period t F t = # fired for period t I t = inventory at end of period t LP Solution: Z = $32,000 H 1 = 0, F 1 = 20, I 1 = 0, P 1 =80000; H 2 = 0, F 2 = 0, I 2 = 30000, P 2 =80000; H 3 = 10, F 3 = 0, I 3 = 0, P 3 =90000; H 4 = 60, F 4 = 0, I 4 = 0, P 4 =150000;

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APP By The Transportation Method ExpectedRegularOvertimeSubcontract QuarterDemandCapacity Capacity Capacity 19001000100500 215001200150500 316001300200500 430001300200500 Regular production cost per unit = $20 Overtime production cost per unit = $25 Subcontracting cost per unit = $28 Inventory carrying cost per unit per period = $3 Beginning inventory = 300 units

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Production Plan Strategy Variable PeriodDemand Reg Prodn Overtime Sub End Inv 190010001000500 215001200150250600 3 160013002005001000 4 300013002005000 Total7000480065012502100

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l Regular Production Cost = (4,800 * $20)=$96,000 l Overtime Production Cost = (650 * $25)= $16,250 l Subcontracting Cost = (1,250 * $28) = $35,000 l Inventory Cost = (2,100 * $3) = $ 6,300 l The Total Cost of the Plan = $153,550

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Linear Programming Formulation Let: l D t = units required in period t, (t = 1,…,T) l m = number of sources of product in any period l P it = capacity, in units of product, of source i in period t, (i = 1,…,m) l X it = planned quantity to be obtained from source i in period t l c it = variable cost per unit from source i in period t l h t = cost to store a unit from period t to period t+1 l I t = inventory level at the end of period t, after satisfying the requirement in period t

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Optimal Value (Z) = $153,550 l XR1 l XR2 l XR3 l XR4 l XO1 l XO2 l XO3 l XO4 l XS1 l XS2 l XS3 l XS4 = 1000 = 1200 = 1300 = 0 = 150 = 200 = 0 = 350 = 500

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Strategies for Managing Demand l Shift demand into other periods incentives, sales promotions, advertising campaigns l Offer product or services with counter- cyclical demand patterns create demand for idle resources

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Aggregate Planning for Services 1. Most services can’t be inventoried 2. Demand for services is difficult to predict 3. Capacity is also difficult to predict 4. Service capacity must be provided at the appropriate place and time 5. Labor is usually the most constraining resource for services

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Services Example l The central terminal at the Deutsche Cargo receives airfreight from aircraft arriving from all over Europe and redistributes it to aircraft for shipment to all European destinations. The company guarantees overnight shipment of all parcels, so enough personnel must be available to process all cargo as it arrives. The company now has 24 employees working in the terminal. The forecasted demand for warehouse workers for the next 7 months is 24, 26, 30, 28, 28, 24, and 24. It costs $2,000 to hire and $3,500 to lay off each worker. If overtime is used to supply labor beyond the present work force, it will cost the equivalent of $2,600 more for each additional worker. Should the company use a level capacity with overtime or a matching demand plan for the next six month?

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The Level Capacity with Overtime Plan

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The Matching Demand Plan

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l The cost of the Level Capacity with Overtime = $ 41,600 l The total cost of the Matching Demand plan = $12,000 + $21,000 = $33,000 l Hence, since the cost of matching demand plan is less than the level capacity plan with overtime and would be the preferred plan

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Aggregate Planning Example 1 l A manufacturer produces a line of household products fabricated from sheet metal. To illustrate his production planning problem, suppose that he makes only four products and that his production system consists of five production centers: stamping, drilling, assembly, finishing (painting and printing), and packaging. For a given month, he must decide how much of each product to manufacture, and to aid in this decision, he has assembled the data shown in Tables 1 and 2. Furthermore, he knows that only 2000 square feet of the type of sheet metal used for products 2 and 4 will be available during the month. Product 2 requires 2.0 square feet per unit and product 4 uses 1.2 square feet per unit.

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l TABLE 1 Production Data for Example 1 PRODUCTION RATES IN HOURS PER UNIT Production DEPARTMENT PRODUCT 1 PRODUCT 2 PRODUCT 3 PRODUCT 4 Hours DEPARTMENT PRODUCT 1 PRODUCT 2 PRODUCT 3 PRODUCT 4 Hours Available Available l Stamping l Stamping 0.03 0.15 0.05 0.10 400 l Drilling l Drilling 0.06 0.12 ----- 0.10 400 l Assembly l Assembly 0.05 0.10 0.05 0.12 500 l Finishing l Finishing 0.04 0.20 0.03 0.12 450 l Packaging l Packaging 0.02 0.06 0.02 0.05 400

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l TABLE 2 Product Data for Example 1 NET SELLING VARIABLE SALES POTENTIAL PRODUCT PRICE/UNIT COST/UNIT MINIMUM MAXIMUM PRODUCT PRICE/UNIT COST/UNIT MINIMUM MAXIMUM 1 l 1 $10 $6 1000 6000 2 l 2 25 15 ----- 500 3 l 3 16 11 500 3000 4 l 4 20 14 100 1000

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A Linear Program of Example 1: l Define x i be the number of units of Product i to be produced per month, i = 1, 2, 3, and 4.

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l SolutionLINGO Software Package l Solution of Example 1 using LINGO Software Package (get a free copy of this package from the web site at www.lindo.com): l Objective value: 42600.00 Variable Value Reduced Cost X1 l X1 5500.000 0.0000000 X2 l X2 500.0000 0.0000000 X3 l X3 3000.000 0.0000000 X4 l X4 100.0000 0.0000000

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Row Slack or Surplus Dual Price l PROFIT l PROFIT 42600.00 1.0000000 l STAMPING l STAMPING 0.0000000 0.0000000 l DRILLING l DRILLING 0.0000000 66.66666 l ASSEMBLY l ASSEMBLY 13.00000 0.0000000 l FINISHING l FINISHING 28.00000 0.0000000 l PACKAGING l PACKAGING 195.0000 0.0000000 l SHEETMETAL l SHEETMETAL 880.0000 0.0000000 l MINPROD1 l MINPROD1 4500.000 0.0000000 l MAXPROD1 l MAXPROD1 500.0000 0.0000000 l MAXPROD2 l MAXPROD2 0.0000000 2.0000000 l MINPROD3 l MINPROD3 2500.000 0.0000000 l MAXPROD3 l MAXPROD3 0.0000000 5.000000 l MINPROD4 l MINPROD4 0.0000000 -0.6666667 l MAXPROD4 l MAXPROD4 900.0000 0.0000000

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Ranges in which the basis is unchanged: Objective Coefficient Ranges Current Allowable Allowable Variable Coefficient Increase Decrease Variable Coefficient Increase Decrease l X1 l X1 4.000000 INFINITY INFINITY l X2 l X2 10.00000 INFINITY INFINITY l X3 l X3 5.000000 INFINITY INFINITY l X4 l X4 6.000000 INFINITY INFINITY

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Righthand Side Ranges: Row Current Allowable Allowable RHS Increase Decrease RHS Increase Decrease l STAMPING l STAMPING 400.0000 INFINITY 100.0000 l DRILLING l DRILLING 400.0000 INFINITY 3000.000 l ASSEMBLY l ASSEMBLY 500.0000 INFINITY 500.0000 l FINISHING l FINISHING 450.0000 INFINITY 5500.000 l PACKAGING l PACKAGING 400.0000 INFINITY 0.0 l SHEETMETAL l SHEETMETAL 2000.000 INFINITY 13.00000 l MINPROD1 l MINPROD1 1000.000 INFINITY 28.00000 l MAXPROD1 l MAXPROD1 6000.000 INFINITY 195.0000 l MAXPROD2 l MAXPROD2 500.0000 INFINITY 880.0000 l MINPROD3 l MINPROD3 500.0000 INFINITY 4500.000 l MAXPROD3 l MAXPROD3 3000.000 INFINITY 500.0000 l MINPROD4 l MINPROD4 100.0000 INFINITY 2500.000 l MAXPROD4 l MAXPROD4 1000.000 INFINITY 900.0000

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