Chapter 3. Aggregate Planning (Steven Nahmias)

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Chapter 3. Aggregate Planning(Steven Nahmias)
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Hierarchy of Production Decisions
Long-range Capacity Planning
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Planning Horizon

• Aggregate planning Intermediate-range capacity
  planning, usually covering 2 to 12 months.

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Aggregate Planning Strategies

• Should inventories be used to absorb changes in
  demand during planning period?
• Should demand changes be accommodated by varying
  the size of the workforce?
• Should part-timers be used, or should overtime
  and/or machine idle time be used to absorb
  fluctuations?
• Should subcontractors be used on fluctuating
  orders so a stable workforce can be maintained?
• Should prices or other factors be changed to
  influence demand?

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Introduction to Aggregate Planning

• 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.)

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Why Aggregate Planning Is Necessary

• Fully load facilities and minimize overloading
  and underloading
• Make sure enough capacity available to satisfy
  expected demand
• Plan for the orderly and systematic change of
  production capacity to meet the peaks and valleys
  of expected customer demand
• Get the most output for the amount of resources
  available

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Aggregation Method Suggested by Hax and Meal

• They suggest grouping products into three
  categories
• items, families, and types.
• 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.
• A family in this context would be refrigerators
  in general.
• Types are natural groupings of families kitchen
  appliances might be one type.

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Aggregate Planning

• Aggregate planning might also be called macro
  production planning.
• Whether a company provides a service or product,
  macro planning begins with the forecast of
  demand.
• Aggregate planning methodology is designed to
  translate demand forecasts into a blueprint for
  planning
• - staffing and
• - production levels
• for the firm over a predetermined planning
  horizon.

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Aggregate Planning

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

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Steps in Aggregate Planning

• Prepare the sales forecast (Note that all
  producting planning activities begin with sales
  forecast)
• Total all the individual product or service
  forecasts into one aggregate demand (if not
  homogeneous use labor-hours, machine-hours or
  sales dollars)
• Transform the aggregate demand into worker,
  material and machine requirements
• Develop alternative capacity plans
• Select a capacity plan which satisfies aggregate
  demand and best meets the objectives of the
  organization.

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Overview of the Problem

• Suppose that D1, D2, . . . , DT are the forecasts
  of demand for aggregate units over the planning
  horizon (T periods.) The problem is to determine
  both work force levels (Wt) and production levels
  (Pt ) to minimize total costs over the T period
  planning horizon.

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Important Issues in Aggregate Planning

• 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).
• 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.

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Important Issues in Aggregate Planning

• 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. (in
  practice, rolling schedules are used)
• Treatment of Demand. Assume demand is known.
  Ignores uncertainty to focus on the
  predictable/systematic variations in demand, such
  as seasonality.

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Relevant Costs

• Smoothing Costs
• changing size of the work force
• changing number of units produced
• Holding Costs
• primary component opportunity cost of investment
  in inventory
• Shortage Costs
• Cost of demand exceeding stock on hand.
• Other Costs payroll, overtime, subcontracting.

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Cost of Changing the Size of the Workforce
Fig. 3-2
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Holding and Back-Order Costs
Fig. 3-3
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Aggregate Units

• The method is based on notion of aggregate units.
  They may be
• Actual units of production
• Weight (tons of steel)
• Volume (gallons of gasoline)
• Dollars (Value of sales)
• Fictitious aggregate units(See example 3.1)

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Example of fictitious aggregate units.(Example
3.1)

• One plant produced 6 models of washing machines
• Model hrs. Price
  sales
• A 5532 4.2 285 32
• K 4242 4.9 345 21
• L 9898 5.1 395 17
• L 3800 5.2 425 14
• M 2624 5.4 525 10
• M 3880 5.8 725 06
• Question How do we define an aggregate unit here?

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Example continued

• 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.

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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.

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Step 1 Determine net demand.(subtract
starting inventory from period 1 forecast and add
ending inventory to period 8 forecast.)

• Month Net Predicted Cum. Net
• Demand
  Demand
• 1(Jan) 220 220
• 2(Feb) 280 500
• 3(Mar) 460 960
• 4(Apr) 190 1150
• 5(May) 310 1460
• 6(June) 145 1605
• 7(July) 110 1715
• 8(Aug) 225 1940

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Step 2. Graph Cumulative Net Demand to Find Plans
Graphically
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Basic Strategies

• Constant Workforce (Level Capacity) strategy
• Maintaining a steady rate of regular-time output
  while meeting variations in demand by a
  combination of options.
• Zero Inventory (Matching Demand)strategy
• Matching capacity to demand the planned output
  for a period is set at the expected demand for
  that period.

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Constant Workforce Approach

• Advantages
• Stable output rates and workforce
• Disadvantages
• Greater inventory costs
• Increased overtime and idle time
• Resource utilizations vary over time

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Zero Inventory Approach

• Advantages
• Investment in inventory is low
• Labor utilization is high
• Disadvantages
• The cost of adjusting output rates and/or
  workforce levels

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Constant Work Force Plan

• Suppose that we are interested in determining
  a production plan that doesnt 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.

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Monthly Production 1940/8 242.2 or rounded to
243/month. But there are stockouts.
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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

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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
• 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

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But - may not be realistic for several reasons

• It may not be possible to achieve the production
  level of 320 unit/mo with an integer number of
  workers
• 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.

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To Overcome These Shortcomings

• Assume number of workdays per month is given
  (reasonable!)
• Compute a K factor given by
• K number of aggregate units produced by one
  worker in one day

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Finding K

• Suppose that we are told that over a period of 40
  days, the plant had 38 workers who produced 520
  units. It follows that
• K 520/(3840) .3421
• average number of units produced by one
  worker in one day.

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Computing Constant Work Force -- Realistically

• Assume we are given the following working days
  per month 22, 16, 23, 20, 21, 22, 21, 22.
• March is still the critical month.
• Cum. net demand thru March 960.
• Cum working days 221623 61.
• 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

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Why again did we pick on March?

• Examining the graph we see that that was the
  Trigger point where our constant production
  line intersected the cumulative demand line
  assuring NO STOCKOUTS!
• Can we prove this is best?

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Tabulate Days/Production Per Worker Vs. Demand To
Find Minimum Numbers
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What Should We Look At?

• Cumulative Demand says March needs most workers
  but will mean building inventories in Jan Feb
  to fulfill the greater March demand
• If we keep this number of workers we will
  continue to build inventory through the rest of
  the plan!

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Constant Work Force Production Plan

• Mo wk days Prod. Cum Cum Nt
  End Inv
• Level
  Prod Dem
• Jan 22 346 346
  220 126
• Feb 16 252 598
  500 98
• Mar 23 362 960
  960 0
• Apr 20 315 1275
  1150 125
• May 21 330 1605
  1460 145
• Jun 22 346 1951
  1605 346
• Jul 21 330 2281
  1715 566
• Aug 22 346 2627
  1940 687

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Addition of Costs

• Holding Cost (per unit per month) 8.50
• Hiring Cost per worker 800
• Firing Cost per worker 1,250
• Payroll Cost 75/worker/day
• Shortage Cost 50 unit short/month

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Cost Evaluation of Constant Work Force Plan

• Assume that the work force at the end of Dec was
  40.
• Cost to hire 6 workers 6800 4800
• Inventory Cost accumulate ending inventory
  (126980. . .687) 2093. Add in 100 units
  netted out in Aug 2193. Hence Inv. Cost
  21938.518,640.50
• Payroll cost
• (75/worker/day)(46 workers )(167days) 576,150
• Cost of plan 576,150 18,640.50 4800
  599,590.50

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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.

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Zero Inventory Plan (Chase Strategy)

• 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.

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An Alternative is called the Chase Plan

• Here, we hire and fire (layoff) workers to keep
  inventory low!
• We would employ only the number of workers needed
  each month to meet demand
• 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)

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An Alternative is called the Chase Plan

• 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)
• Total Personnel Costs 128950

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• I got the following for this problem
• Period hired fired
• 1 10
• 2 21
• 3 8
• 4 31
• 5 15
• 6 24
• 7 4
• 8 15

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An Alternative is called the Chase Plan

• Inventory cost is essentially 1658.5 1402.50
• Employment costs 428325
• Chase Plan Total 558677.50
• Betters the Constant Workforce Plan by
• 599590.50 558677.50 40913
• But will this be good for your image?
• Can we find a better plan?

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Disaggregating The Aggregate Plan

• 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).
• 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.

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Aggregate Plan to Master Schedule
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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)

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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.

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Exploring the Optimal (L.P.) Approach

• We need an Objective Function for cost of the
  aggregate plan (target is to minimize
  costs)
• Here the cis are cost for hiring, firing,
  inventory, production, etc
• HT and FT are number of workers hired and fired
• IT, PT, OT, ST AND UT 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

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Exploring the Optimal (L.P.) Approach

• This objective Function would be subject to a
  series of constraints (one of each type for each
  period)
• Number of Workers Constraints
• Inventory Constraints
• Production Constraints

Where nt k is the number of units produced by
a worker in a given period of nt days
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Real Constraint Equation (rewritten for L.P.)

• Employee Constraints
• Inventory Constraints

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Real Constraint Equations (rewritten for L.P.)

• Production Constraints

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Real Constraint Equations (rewritten for L.P.)

• 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