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**Aggregate Production Planning**

(APP)

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**Aggregate Production Planning (APP)**

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

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**Inputs and Outputs to Aggregate Production Planning**

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

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

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

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Level Production Demand Production Units Time

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Chase Demand Demand Units Production Time

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APP Example 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**

Quarter Sales Forecast (kg) Spring 80,000 Summer 50,000 Fall 120,000 Winter 150,000 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

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**Level Production Strategy**

Sales Production Quarter Forecast Plan Inventory Spring 80, ,000 20,000 Summer 50, ,000 70,000 Fall 120, ,000 50,000 Winter 150, ,000 0 400, ,000 Cost = 140,000 kilograms x $0.50 per kilogram = $70,000

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**Chase Demand Strategy Sales Production Workers Workers Workers**

Quarter Forecast Plan Needed Hired Fired Spring 80,000 80, Summer 50,000 50, Fall 120, , Winter 150, , 100 50 Cost = (100 workers hired x $100) + (50 workers fired x $500) = $10, ,000 = $35,000

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LP Formulation Define Ht = # hired for period t Ft = # fired for period t It = inventory at end of period t Pt = Production in period t Wt = Workforce in period t Min Z = $100 (H1 + H2 + H3 + H4) + $500 (F1 + F2 + F3 + F4) + $0.50 (I1 + I2 + I3 + I4)

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**Subject to P1 - I1 = 80,000 (1) Demand**

Min Z = $100 (H1 + H2 + H3 + H4) + $500 (F1 + F2 + F3 + F4)+ $0.50 (I1 + I2 + I3 + I4) Subject to P1 - I1 = 80,000 (1) Demand I1 + P2 - I2 = 50,000 (2) constraints I2 + P3 - I3 = 120,000 (3) I3 + P4 - I4 = 150,000 (4)

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**P2 - 1,000 W2 = 0 (6) constraints P3 - 1,000 W3 = 0 (7)**

Min Z = $100 (H1 + H2 + H3 + H4) + $500 (F1 + F2 + F3 + F4)+ $0.50 (I1 + I2 + I3 + I4) Subject to P1 - I1 = 80,000 (1) Demand I1 + P2 - I2 = 50,000 (2) constraints I2 + P3 - I3 = 120,000 (3) I3 + P4 - I4 = 150,000 (4) P1 - 1,000 W1 = 0 (5) Production P2 - 1,000 W2 = 0 (6) constraints P3 - 1,000 W3 = 0 (7) P4 - 1,000 W4 = 0 (8)

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**W2 - W1 - H2 + F2 = 0 (10) constraints W3 - W2 - H3 + F3 = 0 (11) **

Min Z = $100 (H1 + H2 + H3 + H4) + $500 (F1 + F2 + F3 + F4)+ $0.50 (I1 + I2 + I3 + I4) Subject to P1 - I1 = 80,000 (1) Demand I1 + P2 - I2 = 50,000 (2) constraints I2 + P3 - I3 = 120,000 (3) I3 + P4 - I4 = 150,000 (4) P1 - 1,000 W1 = 0 (5) Production P2 - 1,000 W2 = 0 (6) constraints P3 - 1,000 W3 = 0 (7) P4 - 1,000 W4 = 0 (8) W1 - H1 + F1 = 100 (9) Work force W2 - W1 - H2 + F2 = 0 (10) constraints W3 - W2 - H3 + F3 = 0 (11) W4 - W3 - H4 + F4 = 0 (12)

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**Min Z = $100 (H1 + H2 + H3 + H4) + $500 (F1 + F2 + F3 + F4)+ $0**

Min Z = $100 (H1 + H2 + H3 + H4) + $500 (F1 + F2 + F3 + F4)+ $0.50 (I1 + I2 + I3 + I4) Subject to P1 - I1 = 80,000 (1) Demand I1 + P2 - I2 = 50,000 (2) constraints I2 + P3 - I3 = 120,000 (3) I3 + P4 - I4 = 150,000 (4) P1 - 1,000 W1 = 0 (5) Production P2 - 1,000 W2 = 0 (6) constraints P3 - 1,000 W3 = 0 (7) P4 - 1,000 W4 = 0 (8) W1 - H1 + F1 = 100 (9) Work force W2 - W1 - H2 + F2 = 0 (10) constraints W3 - W2 - H3 + F3 = 0 (11) W4 - W3 - H4 + F4 = 0 (12)

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LP Solution: Z = $32,000 H1= 0, F1= 20, I1= 0, P1=80000; H2= 0, F2= 0, I2= 30000, P2=80000; H3= 10, F3= 0, I3= 0, P3=90000; H4= 60, F4= 0, I4= 0, P4=150000;

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**Summary: APP By Linear Programming**

Min Z = $100 (H1 + H2 + H3 + H4) + $500 (F1 + F2 + F3 + F4)+ $0.50 (I1 + I2 + I3 + I4) Subject to P1 - I1 = 80,000 (1) Demand I1 + P2 - I2 = 50,000 (2) constraints I2 + P3 - I3 = 120,000 (3) I3 + P4 - I4 = 150,000 (4) P1 - 1,000 W1 = 0 (5) Production P2 - 1,000 W2 = 0 (6) constraints P3 - 1,000 W3 = 0 (7) P4 - 1,000 W4 = 0 (8) W1 - H1 + F1 = 100 (9) Work force W2 - W1 - H2 + F2 = 0 (10) constraints W3 - W2 - H3 + F3 = 0 (11) W4 - W3 - H4 + F4 = 0 (12) where Ht = # hired for period t Ft = # fired for period t It = inventory at end of period t LP Solution: Z = $32,000 H1= 0, F1= 20, I1= 0, P1=80000; H2= 0, F2= 0, I2= 30000, P2=80000; H3= 10, F3= 0, I3= 0, P3=90000; H4= 60, F4= 0, I4= 0, P4=150000;

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**APP By The Transportation Method**

Expected Regular Overtime Subcontract Quarter Demand Capacity Capacity Capacity 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**

Period Demand Reg Prodn Overtime Sub End Inv Total

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**Regular Production Cost = (4,800 * $20)=$96,000**

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

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**Linear Programming Formulation**

Let: Dt = units required in period t, (t = 1,…,T) m = number of sources of product in any period Pit = capacity, in units of product, of source i in period t, (i = 1,…,m) Xit = planned quantity to be obtained from source i in period t cit = variable cost per unit from source i in period t ht = cost to store a unit from period t to period t+1 It = inventory level at the end of period t, after satisfying the requirement in period t

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**Optimal Value (Z) = $153,550 XR1 XR2 XR3 XR4 XO1 XO2 XO3 XO4 XS1 XS2**

= 1000 = 1200 = 1300 = 0 = 150 = 200 = 350 = 500

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**Strategies for Managing Demand**

Shift demand into other periods incentives, sales promotions, advertising campaigns 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 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? Linear decision rule (LDR) payroll, staffing, over/undertime, inventory costs Search decision rule (SDR) find minimum cost combination of labor levels & production rates Management coefficients model uses regression analysis to improve consistency of planning 20

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**The Level Capacity with Overtime Plan**

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**The Matching Demand Plan**

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**The cost of the Level Capacity with Overtime = $ 41,600**

The total cost of the Matching Demand plan = $12,000 + $21,000 = $33,000 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**

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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**TABLE 1 Production Data for Example 1**

PRODUCTION RATES IN HOURS PER UNIT Production DEPARTMENT PRODUCT 1 PRODUCT 2 PRODUCT 3 PRODUCT Hours Available Stamping Drilling Assembly Finishing Packaging

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**TABLE 2 Product Data for Example 1**

NET SELLING VARIABLE SALES POTENTIAL PRODUCT PRICE/UNIT COST/UNIT MINIMUM MAXIMUM $ $

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**A Linear Program of Example 1:**

Define xi be the number of units of Product i to be produced per month, i = 1, 2, 3, and 4.

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Solution of Example 1 using LINGO Software Package (get a free copy of this package from the web site at Objective value: Variable Value Reduced Cost X X X X

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**Row Slack or Surplus Dual Price**

PROFIT STAMPING DRILLING ASSEMBLY FINISHING PACKAGING SHEETMETAL MINPROD MAXPROD MAXPROD MINPROD MAXPROD MINPROD MAXPROD

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**Ranges in which the basis is unchanged:**

Objective Coefficient Ranges Current Allowable Allowable Variable Coefficient Increase Decrease X INFINITY INFINITY X INFINITY INFINITY X INFINITY INFINITY X INFINITY INFINITY

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**Righthand Side Ranges:**

Row Current Allowable Allowable RHS Increase Decrease STAMPING INFINITY DRILLING INFINITY ASSEMBLY INFINITY FINISHING INFINITY PACKAGING INFINITY SHEETMETAL INFINITY MINPROD INFINITY MAXPROD INFINITY MAXPROD INFINITY MINPROD INFINITY MAXPROD INFINITY MINPROD INFINITY MAXPROD INFINITY

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