1. ©2005 McGraw-Hill/Irwin 3-1 Chapter 3. Aggregate Planning (Steven Nahmias)

2. ©2005 McGraw-Hill/Irwin 3-2 Hierarchy of Production Decisions Long-range Capacity Planning

3. ©2005 McGraw-Hill/Irwin 3-3 Planning Horizon Aggregate planning: Intermediate- range capacity planning, usually covering 2 to 12 months. Short range Intermediate range Long range Now2 months1 Year

4. ©2005 McGraw-Hill/Irwin 3-4 4 Aggregate Planning Strategies n Should inventories be used to absorb changes in demand during planning period? n Should demand changes be accommodated by varying the size of the workforce? n Should part-timers be used, or should overtime and/or machine idle time be used to absorb fluctuations? n Should subcontractors be used on fluctuating orders so a stable workforce can be maintained? n Should prices or other factors be changed to influence demand?

5. ©2005 McGraw-Hill/Irwin 3-5 Introduction to Aggregate Planning n Goal: To plan gross work force levels and set firm-wide production plans so that predicted demand for aggregated units can be met. Concept is predicated on the idea of an “aggregate unit” of production. May be actual units, or may be measured in weight (tons of steel), volume (gallons of gasoline), time (worker-hours), or dollars of sales. Can even be a fictitious quantity. (Refer to example in text and in slide below.) Concept is predicated on the idea of an “aggregate unit” of production. May be actual units, or may be measured in weight (tons of steel), volume (gallons of gasoline), time (worker-hours), or dollars of sales. Can even be a fictitious quantity. (Refer to example in text and in slide below.)

6. ©2005 McGraw-Hill/Irwin 3-6 Why Aggregate Planning Is Necessary n Fully load facilities and minimize overloading and underloading n Make sure enough capacity available to satisfy expected demand n Plan for the orderly and systematic change of production capacity to meet the peaks and valleys of expected customer demand n Get the most output for the amount of resources available

7. ©2005 McGraw-Hill/Irwin 3-7 Aggregation Method Suggested by Hax and Meal n They suggest grouping products into three categories: n items, families, and types. n Items are the finest level in the product structure and correspond to individual stock-keeping units. For example, a firm selling refrigerators would distinguish white from almond in the same refrigerator as different items. n A family in this context would be refrigerators in general. n Types are natural groupings of families; kitchen appliances might be one type.

8. ©2005 McGraw-Hill/Irwin 3-8 Aggregate Planning n Aggregate planning might also be called macro production planning. n Whether a company provides a service or product, macro planning begins with the forecast of demand. n Aggregate planning methodology is designed to translate demand forecasts into a blueprint for planning : - staffing and - staffing and - production levels - production levels for the firm over a predetermined planning horizon.

9. ©2005 McGraw-Hill/Irwin 3-9 Aggregate Planning n The aggregate planning methodology discussed in this chapter assumes that the demand is deterministic n This assumption is made to simplify the analysis and allow us to focus on the systematic and predictable changes in the demand pattern. n Aggregate planning involves competing objectives: - react quickly to anticipated changes in demand - react quickly to anticipated changes in demand - retaining a stable workforce - retaining a stable workforce - develop a production plan that maximizes profit over the planning horizon subject to constraints on capacity - develop a production plan that maximizes profit over the planning horizon subject to constraints on capacity

10. ©2005 McGraw-Hill/Irwin 3-10 Steps in Aggregate Planning n Prepare the sales forecast (Note that all producting planning activities begin with sales forecast) n Total all the individual product or service forecasts into one aggregate demand (if not homogeneous use labor- hours, machine-hours or sales dollars) n Transform the aggregate demand into worker, material and machine requirements n Develop alternative capacity plans n Select a capacity plan which satisfies aggregate demand and best meets the objectives of the organization.

11. ©2005 McGraw-Hill/Irwin 3-11 Overview of the Problem Suppose that D 1, D 2,..., D T are the forecasts of demand for aggregate units over the planning horizon (T periods.) The problem is to determine both work force levels (W t ) and production levels (P t ) to minimize total costs over the T period planning horizon.

12. ©2005 McGraw-Hill/Irwin 3-12 Important Issues in Aggregate Planning n Smoothing. Refers to the costs and disruptions that result from making changes in production and workforce levels from one period to the next (cost of hiring and firing workers). n Bottleneck Planning. Problem of not meeting the peak demand because of capacity restrictions. A bottleneck occurs when the capacity of the productive system is insufficient to meet a sudden surge in the demand. Bottlenecks can also occur in a particular part of the productive system due to the breakdown of a key piece of equipment or the shortage of a critical resource.

13. ©2005 McGraw-Hill/Irwin 3-13 Important Issues in Aggregate Planning n Planning Horizon. The planning horizon is the number of periods of demand forecast used to generate the aggregate plan. If the horizon is too short, there may be insufficient time to build inventories to meet future demand surges and if it is too long the reliability of the demand forecasts is likely to be low. (ın practice, rolling schedules are used) n Treatment of Demand. Assume demand is known. Ignores uncertainty to focus on the predictable/systematic variations in demand, such as seasonality.

14. ©2005 McGraw-Hill/Irwin 3-14 Relevant Costs n Smoothing Costs – changing size of the work force – changing number of units produced n Holding Costs – primary component: opportunity cost of investment in inventory n Shortage Costs – Cost of demand exceeding stock on hand.  Other Costs: payroll, overtime, subcontracting.

15. ©2005 McGraw-Hill/Irwin 3-15 Cost of Changing the Size of the Workforce Fig. 3-2

16. ©2005 McGraw-Hill/Irwin 3-16 Holding and Back-Order Costs Fig. 3-3 Back-orders Positive inventory Slope = C P Slope = C i $ Cost Inventory

17. ©2005 McGraw-Hill/Irwin 3-17 Aggregate Units The method is based on notion of aggregate units. They may be n Actual units of production n Weight (tons of steel) n Volume (gallons of gasoline) n Dollars (Value of sales) n Fictitious aggregate units(See example 3.1)

18. ©2005 McGraw-Hill/Irwin 3-18 Example of fictitious aggregate units. (Example 3.1) One plant produced 6 models of washing machines: Model # hrs. Price % sales A 55324.228532 K 42424.934521 L 98985.139517 L 38005.242514 M 26245.452510 M 38805.872506 Question: How do we define an aggregate unit here?

19. ©2005 McGraw-Hill/Irwin 3-19 Example continued n Notice: Price is not necessarily proportional to worker hours (i.e., cost): why? One method for defining an aggregate unit: requires:.32(4.2) +.21(4.9) +... +.06(5.8) = 4.8644 worker hours. This approach for this example is reasonable since products produced are similar. When products produced are heterogeneous, a natural aggregate unit is sales dollars. One method for defining an aggregate unit: requires:.32(4.2) +.21(4.9) +... +.06(5.8) = 4.8644 worker hours. This approach for this example is reasonable since products produced are similar. When products produced are heterogeneous, a natural aggregate unit is sales dollars.

20. ©2005 McGraw-Hill/Irwin 3-20 Prototype Aggregate Planning Example (this example is not in the text) The washing machine plant is interested in determining work force and production levels for the next 8 months. Forecasted demands for Jan-Aug. are: 420, 280, 460, 190, 310, 145, 110, 125. Starting inventory at the end of December is 200 and the company would like to have 100 units on hand at the end of August. Find monthly production levels. The washing machine plant is interested in determining work force and production levels for the next 8 months. Forecasted demands for Jan-Aug. are: 420, 280, 460, 190, 310, 145, 110, 125. Starting inventory at the end of December is 200 and the company would like to have 100 units on hand at the end of August. Find monthly production levels.

21. ©2005 McGraw-Hill/Irwin 3-21 Step 1: Determine “net” demand. (subtract starting inventory from period 1 forecast and add ending inventory to period 8 forecast.) MonthNet PredictedCum. Net Demand Demand Demand Demand 1(Jan)220220 2(Feb)280500 3(Mar)460960 4(Apr)190 1150 5(May)3101460 6(June)1451605 7(July)1101715 8(Aug)2251940

22. ©2005 McGraw-Hill/Irwin 3-22 Step 2. Graph Cumulative Net Demand to Find Plans Graphically

23. ©2005 McGraw-Hill/Irwin 3-23 Basic Strategies n Constant Workforce (Level Capacity) strategy: –Maintaining a steady rate of regular-time output while meeting variations in demand by a combination of options. n Zero Inventory (Matching Demand)strategy: –Matching capacity to demand; the planned output for a period is set at the expected demand for that period.

24. ©2005 McGraw-Hill/Irwin 3-24 Constant Workforce Approach n Advantages –Stable output rates and workforce n Disadvantages –Greater inventory costs –Increased overtime and idle time –Resource utilizations vary over time

25. ©2005 McGraw-Hill/Irwin 3-25 Zero Inventory Approach n Advantages –Investment in inventory is low –Labor utilization is high n Disadvantages –The cost of adjusting output rates and/or workforce levels

26. ©2005 McGraw-Hill/Irwin 3-26 Constant Work Force Plan Suppose that we are interested in determining a production plan that doesn’t change the size of the workforce over the planning horizon. How would we do that? Suppose that we are interested in determining a production plan that doesn’t change the size of the workforce over the planning horizon. How would we do that? One method: In previous picture, draw a straight line from origin to 1940 units in month 8: The slope of the line is the number of units to produce each month. One method: In previous picture, draw a straight line from origin to 1940 units in month 8: The slope of the line is the number of units to produce each month.

27. ©2005 McGraw-Hill/Irwin 3-27 Monthly Production = 1940/8 = 242.2 or rounded to 243/month. But: there are stockouts.

28. ©2005 McGraw-Hill/Irwin 3-28 How can we have a constant work force plan with no stockouts? Answer: using the graph, find the straight line that goes through the origin and lies completely above the cumulative net demand curve:

29. ©2005 McGraw-Hill/Irwin 3-29 From the previous graph, we see that cum. net demand curve is crossed at period 3, so that monthly production is 960/3 = 320. Ending inventory each month is found from: Month Cum. Net. Dem. Cum. Prod. Invent. 1(Jan)220 320 100 1(Jan)220 320 100 2(Feb)500 640 140 3(Mar)960 960 0 4(Apr.) 1150 1280 130 5(May) 1460 1600 140 6(June) 1605 1920 315 7(July) 1715 2240 525 8(Aug) 1940 2560 620

30. ©2005 McGraw-Hill/Irwin 3-30 But - may not be realistic for several reasons: n It may not be possible to achieve the production level of 320 unit/mo with an integer number of workers n Since all months do not have the same number of workdays, a constant production level may not translate to the same number of workers each month.

31. ©2005 McGraw-Hill/Irwin 3-31 31 To Overcome These Shortcomings: n Assume number of workdays per month is given (reasonable!) n Compute a “K factor” given by: K = number of aggregate units produced by one worker in one day

32. ©2005 McGraw-Hill/Irwin 3-32 Finding K n Suppose that we are told that over a period of 40 days, the plant had 38 workers who produced 520 units. It follows that: n K= 520/(38*40) =.3421 = average number of units produced by one worker in one day. = average number of units produced by one worker in one day.

33. ©2005 McGraw-Hill/Irwin 3-33 33 Computing Constant Work Force -- Realistically n Assume we are given the following # working days per month: 22, 16, 23, 20, 21, 22, 21, 22. –March is still the critical month. n Cum. net demand thru March = 960. n Cum # working days = 22+16+23 = 61. n We find that: –960/61 = 15.7377 units/day –15.7377/.3421 = 46 workers required –Actually 46.003 – here we truncate because we are set to build inventory so the low number should work (check for March stock out) – however we must use care and typically ‘round up’ any fractional worker calculations thus building more inventory

34. ©2005 McGraw-Hill/Irwin 3-34 34 Why again did we pick on March? n Examining the graph we see that that was the “Trigger point” where our constant production line intersected the cumulative demand line assuring NO STOCKOUTS! n Can we “prove” this is best?

35. ©2005 McGraw-Hill/Irwin 3-35 35 Tabulate Days/Production Per Worker Vs. Demand To Find Minimum Numbers Month# Work Days#Units/workerForecast Demand netMin # WorkersC. Net DemandC.Units/Worker Min # Workers Jan22.007.53220.0029.23220.007.5329.23 Feb16.005.47280.0051.15500.0013.0038.46 Mar23.007.87460.0058.46960.0020.8746.00 Apr20.006.84190.0027.771150.0027.7141.50 May21.007.18310.0043.151460.0034.8941.84 Jun22.007.53145.0019.271605.0042.4237.84 Jul21.007.18110.0015.311715.0049.6034.57 Aug22.007.53225.0029.901940.0057.1333.96

36. ©2005 McGraw-Hill/Irwin 3-36 36 What Should We Look At? n Cumulative Demand says March needs most workers – but will mean building inventories in Jan + Feb to fulfill the greater March demand n If we keep this number of workers we will continue to build inventory through the rest of the plan!

37. ©2005 McGraw-Hill/Irwin 3-37 Constant Work Force Production Plan Mo # wk days Prod. Cum Cum Nt End Inv Mo # wk days Prod. Cum Cum Nt End Inv Level Prod Dem Level Prod Dem Jan 22 346 346 220 126 Jan 22 346 346 220 126 Feb 16 252 598 500 98 Feb 16 252 598 500 98 Mar 23 362 960 960 0 Mar 23 362 960 960 0 Apr 20 315 1275 1150 125 Apr 20 315 1275 1150 125 May 21 330 1605 1460 145 May 21 330 1605 1460 145 Jun 22 346 1951 1605 346 Jun 22 346 1951 1605 346 Jul 21 330 2281 1715 566 Jul 21 330 2281 1715 566 Aug 22 346 2627 1940 687 Aug 22 346 2627 1940 687

38. ©2005 McGraw-Hill/Irwin 3-38 Addition of Costs n Holding Cost (per unit per month): $8.50 n Hiring Cost per worker: $800 n Firing Cost per worker: $1,250 n Payroll Cost: $75/worker/day n Shortage Cost: $50 unit short/month

39. ©2005 McGraw-Hill/Irwin 3-39 Cost Evaluation of Constant Work Force Plan n Assume that the work force at the end of Dec was 40. n Cost to hire 6 workers: 6*800 = $4800 n Inventory Cost: accumulate ending inventory: (126+98+0+...+687) = 2093. Add in 100 units netted out in Aug = 2193. Hence Inv. Cost = 2193*8.5=$18,640.50 n Payroll cost: ($75/worker/day)(46 workers )(167days) = $576,150 n Cost of plan: $576,150 + $18,640.50 + $4800 = $599,590.50

40. ©2005 McGraw-Hill/Irwin 3-40 Cost Reduction in Constant Work Force Plan (Mixed Strategy) In the original cum net demand curve, consider making reductions in the work force one or more times over the planning horizon to decrease inventory investment.

41. ©2005 McGraw-Hill/Irwin 3-41 Zero Inventory Plan (Chase Strategy) n Here the idea is to change the workforce each month in order to reduce ending inventory to nearly zero by matching the workforce with monthly demand as closely as possible. This is accomplished by computing the # of units produced by one worker each month (by multiplying K by #days per mo.) and then taking net demand each month and dividing by this quantity. The resulting ratio is rounded up to avoid shortages.

42. ©2005 McGraw-Hill/Irwin 3-42 42 An Alternative is called the “Chase Plan” n Here, we hire and fire (layoff) workers to keep inventory low! n We would employ only the number of workers needed each month to meet demand n Examining our chart (earlier) we need: »Jan: 30; Feb: 51; Mar: 59; Apr: 27; May: 43 Jun: 20; Jul: 15; Aug: 30 »Found by: (monthly demand)  (monthly pr. /worker)

43. ©2005 McGraw-Hill/Irwin 3-43 43 An Alternative is called the “Chase Plan” n So we hire or Fire (lay-off) monthly »Jan (starts with 40 workers): Fire 10 (cost $8000) »Feb: Hire 21 (cost $16800) »Mar: Hire 8 (cost $6400) »Apr: Fire 31 (cost $38750) »May: Hire 15 (cost $12000) »Jun: Fire 23 (cost $28750) »Jul: Fire 5 (cost $6250) »Aug: Hire 15 (cost $12000) n Total Personnel Costs: $128950

44. ©2005 McGraw-Hill/Irwin 3-44 n I got the following for this problem: Period # hired #fired 1 10 1 10 2 21 2 21 3 8 3 8 4 31 4 31 5 15 5 15 6 24 6 24 7 4 7 4 8 15 8 15

45. ©2005 McGraw-Hill/Irwin 3-45 45 An Alternative is called the “Chase Plan” n Inventory cost is essentially 165*8.5 = $1402.50 n Employment costs: $428325 n Chase Plan Total: $558677.50 n Betters the “Constant Workforce Plan” by: »599590.50 – 558677.50 = 40913 n But will this be good for your image? n Can we find a better plan?

46. ©2005 McGraw-Hill/Irwin 3-46 Disaggregating The Aggregate Plan n Disaggregation of aggregate plans mean converting an aggregate plan to a detailed master production schedule for each individual item (remember the hierarchical product structure given earlier: items, families, types). n Keep in mind that unless the results of the aggregate plan can be linked to the master production schedule, the aggregate planning methodology could have little value.

47. ©2005 McGraw-Hill/Irwin 3-47 Aggregate Plan to Master Schedule Aggregate Planning Disaggregation Master Schedule

48. ©2005 McGraw-Hill/Irwin 3-48 Optimal Solutions to Aggregate Planning Problems Via Linear Programming Linear Programming provides a means of solving aggregate planning problems optimally. The LP formulation is fairly complex requiring 8T decision variables(1.workforce level, 2. production level, 3. inventory level, 4. # of workers hired, 5. # of workres fired, 6. overtime production, 7. idletime, 8. subcontracting) and 3T constraints (1. workforce, 2. production, 3. inventory), where T is the length of the planning horizon. (See section 3.5, pg.125)

49. ©2005 McGraw-Hill/Irwin 3-49 Optimal Solutions to Aggregate Planning Problems Via Linear Programming Clearly, this can be a formidable linear program. The LP formulation shows that the modified plan we considered with two months of layoffs is in fact optimal for the prototype problem. Refer to the latter part of Chapter 3 and the Appendix following the chapter for details.

50. ©2005 McGraw-Hill/Irwin 3-50 Exploring the Optimal (L.P.) Approach n We need an Objective Function for cost of the aggregate plan (target is to minimize costs): –Here the c i ’s are cost for hiring, firing, inventory, production, etc –H T and F T are number of workers hired and fired –I T, P T, O T, S T AND U T are numbers units inventoried, produced on regular time, on overtime, by ‘sub-contract’ or the number of units that could be produced on idled worker hours respectively

51. ©2005 McGraw-Hill/Irwin 3-51 Exploring the Optimal (L.P.) Approach n This objective Function would be subject to a series of constraints (one of each type for each period) n ‘Number of Workers’ Constraints: n Inventory Constraints: n Production Constraints: Where: n t * k is the number of units produced by a worker in a given period of n t days

52. ©2005 McGraw-Hill/Irwin 3-52 Real Constraint Equation (rewritten for L.P.): n Employee Constraints: n Inventory Constraints:

53. ©2005 McGraw-Hill/Irwin 3-53 Real Constraint Equations (rewritten for L.P.): n Production Constraints:

54. ©2005 McGraw-Hill/Irwin 3-54 Real Constraint Equations (rewritten for L.P.): n Finally, we need constraints defining: –Initial Workforce size –Starting Inventory –Final Desired Inventory –And, of course, the general constraint forcing all variables to be  0