Aggregate Production Planning (APP)

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

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

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)

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

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

Level Production Demand Production Units Time

Chase Demand Demand Units Production Time

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.

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

Level Production Strategy Sales Production Quarter Forecast Plan Inventory Spring 80,000 100,000 20,000 Summer 50,000 100,000 70,000 Fall 120,000 100,000 50,000 Winter 150,000 100,000 0 400,000 140,000 Cost = 140,000 kilograms x $0.50 per kilogram = $70,000

Chase Demand Strategy Sales Production Workers Workers Workers Quarter Forecast Plan Needed Hired Fired Spring 80,000 80,000 80 - 20 Summer 50,000 50,000 50 - 30 Fall 120,000 120,000 120 70 - Winter 150,000 150,000 150 30 - 100 50 Cost = (100 workers hired x $100) + (50 workers fired x $500) = $10,000 + 25,000 = $35,000

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)

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)

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)

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)

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)

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;

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;

APP By The Transportation Method Expected Regular Overtime Subcontract Quarter Demand Capacity Capacity Capacity 1 900 1000 100 500 2 1500 1200 150 500 3 1600 1300 200 500 4 3000 1300 200 500 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

Production Plan Strategy Variable Period Demand Reg Prodn Overtime Sub End Inv 1 900 1000 100 0 500 2 1500 1200 150 250 600 3 1600 1300 200 500 1000 4 3000 1300 200 500 0 Total 7000 4800 650 1250 2100

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

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

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

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

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

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

The Level Capacity with Overtime Plan

The Matching Demand Plan

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

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.

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

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

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.

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

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

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

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