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Chapter 3. Aggregate Planning (Steven Nahmias). Hierarchy of Production Decisions. Long-range Capacity Planning. Long range. Intermediate range. Short range. Now. 2 months. 1 Year. Planning Horizon.

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Chapter 3. Aggregate Planning (Steven Nahmias)

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hierarchy of production decisions
Hierarchy of Production Decisions

Long-range Capacity Planning

planning horizon

Long range

Intermediate range




2 months

1 Year

Planning Horizon

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

aggregate planning strategies
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?
introduction to aggregate planning
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.)

why aggregate planning is necessary
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
aggregation method suggested by hax and meal
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.
aggregate planning
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.

aggregate planning1
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

steps in aggregate planning
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.
overview of the problem
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.

important issues in aggregate planning
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.
important issues in aggregate planning1
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. (ın 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.

relevant costs
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.
holding and back order costs

$ Cost

Slope = Ci

Slope = CP

Back-orders Positive inventory


Fig. 3-3

Holding and Back-Order Costs
aggregate units
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)
example of fictitious aggregate units example 3 1
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?

example continued
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.

prototype aggregate planning example this example is not in the text
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.

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

basic strategies
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.
constant workforce approach
Constant Workforce Approach
  • Advantages
    • Stable output rates and workforce
  • Disadvantages
    • Greater inventory costs
    • Increased overtime and idle time
    • Resource utilizations vary over time
zero inventory approach
Zero Inventory Approach
  • Advantages
    • Investment in inventory is low
    • Labor utilization is high
  • Disadvantages
    • The cost of adjusting output rates and/or workforce levels
constant work force plan
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?

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.

how can we have a constant work force plan with no stockouts
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:


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

but may not be realistic for several reasons
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.
to overcome these shortcomings
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

finding k
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/(38*40) = .3421

= average number of units produced by one worker in one day.

computing constant work force realistically
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 = 22+16+23 = 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
why again did we pick on march
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?
what should we look at
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!
constant work force production plan
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

addition of costs
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
cost evaluation of constant work force plan
Cost Evaluation of Constant Work Force Plan
  • Assume that the work force at the end of Dec was 40.
  • Cost to hire 6 workers: 6*800 = $4800
  • 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
  • Payroll cost:

($75/worker/day)(46 workers )(167days) = $576,150

  • Cost of plan: $576,150 + $18,640.50 + $4800 = $599,590.50
cost reduction in constant work force plan mixed strategy
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.

zero inventory plan chase strategy
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.
an alternative is called the chase plan
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)
an alternative is called the chase plan1
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

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
an alternative is called the chase plan2
An Alternative is called the “Chase Plan”
  • Inventory cost is essentially 165*8.5 = $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?
disaggregating the aggregate plan
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.
optimal solutions to aggregate planning problems via linear programming
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)

optimal solutions to aggregate planning problems via linear programming1
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.

exploring the optimal l p approach
Exploring the Optimal (L.P.) Approach
  • We need an Objective Function for cost of the aggregate plan (target is to minimize costs):
    • Here the ci’s 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
exploring the optimal l p approach1
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

real constraint equation rewritten for l p
Real Constraint Equation (rewritten for L.P.):
  • Employee Constraints:
  • Inventory Constraints:
real constraint equations rewritten for l p1
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