©2005 McGraw-Hill/Irwin
3-1
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: Intermediaterange capacity planning, usually covering
2 to 12 months.
Long range
Short
range
Now
Intermediate
range
2 months
1 Year
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3-4
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?
4
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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.)
Why Aggregate Planning Is
Necessary
©2005 McGraw-Hill/Irwin
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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©2005 McGraw-Hill/Irwin
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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3-8
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 laborhours, 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.
©2005 McGraw-Hill/Irwin
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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.
©2005 McGraw-Hill/Irwin
Important Issues in
Aggregate Planning


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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 Planning


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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.
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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.
Cost of Changing
the Size of the Workforce
Fig. 3-2
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©2005 McGraw-Hill/Irwin
Fig. 3-3
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$ Cost
Holding and Back-Order Costs
Slope = Ci
Slope = CP
Back-orders
Positive inventory
Inventory
©2005 McGraw-Hill/Irwin
3-17
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)
©2005 McGraw-Hill/Irwin
Example of fictitious aggregate units.
(Example 3.1)
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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.
©2005 McGraw-Hill/Irwin
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
1(Jan)
2(Feb)
3(Mar)
4(Apr)
5(May)
6(June)
7(July)
8(Aug)
Net Predicted
Demand
220
280
460
190
310
145
110
225
Cum. Net
Demand
220
500
960
1150
1460
1605
1715
1940
©2005 McGraw-Hill/Irwin
Step 2. Graph Cumulative Net Demand
to Find Plans Graphically
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2000
1800
1600
1400
1200
Cum Net Dem
1000
800
600
400
200
0
1
2
3
4
5
6
7
8
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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.
©2005 McGraw-Hill/Irwin
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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 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.
©2005 McGraw-Hill/Irwin
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Constant Workforce Plan (zero ending inv)
2000
1500
1000
500
0
1
2
3
4
5
6
7
8
Monthly Production = 1940/8 = 242.2 or rounded to
243/month.
But: there are stockouts.
©2005 McGraw-Hill/Irwin
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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:
Constant Work Force Plan With No Stockouts
3000
2500
2000
1500
1000
500
0
1
2
3
4
5
6
7
8
©2005 McGraw-Hill/Irwin
From the previous graph, we see that cum. net demand curve3-29
is crossed at period 3, so that monthly production is 960/3 =
320. Ending inventory each month is found from:
Month
Cum. Net. Dem.
1(Jan)
220
2(Feb)
500
3(Mar)
960
4(Apr.)
1150
5(May)
1460
6(June)
1605
7(July)
1715
8(Aug)
1940
Cum. Prod.
320
640
960
1280
1600
1920
2240
2560
Invent.
100
140
0
130
140
315
525
620
©2005 McGraw-Hill/Irwin
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.
3-30
©2005 McGraw-Hill/Irwin
3-31
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/(38*40) = .3421
= average number of units produced by
one worker in one day.
©2005 McGraw-Hill/Irwin
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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 = 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
33
©2005 McGraw-Hill/Irwin
3-34
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?

34
©2005 McGraw-Hill/Irwin
3-35
Tabulate Days/Production Per Worker Vs.
Demand To Find Minimum Numbers
Month
# Work Days
#Units/worker
Forecast Demand net
Min # Workers
C. Net Demand
C.Units/Worker
Min #
Workers
Jan
22.00
7.53
220.00
29.23
220.00
7.53
29.23
Feb
16.00
5.47
280.00
51.15
500.00
13.00
38.46
Mar
23.00
7.87
460.00
58.46
960.00
20.87
46.00
Apr
20.00
6.84
190.00
27.77
1150.00
27.71
41.50
May
21.00
7.18
310.00
43.15
1460.00
34.89
41.84
Jun
22.00
7.53
145.00
19.27
1605.00
42.42
37.84
Jul
21.00
7.18
110.00
15.31
1715.00
49.60
34.57
Aug
22.00
7.53
225.00
29.90
1940.00
57.13
33.96
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©2005 McGraw-Hill/Irwin
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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!

36
©2005 McGraw-Hill/Irwin
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Constant Work Force Production Plan
Mo
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
# wk days
22
16
23
20
21
22
21
22
Prod. Cum Cum Nt End Inv
Level Prod Dem
346
346
220
126
252
598 500
98
362
960
960
0
315
1275 1150
125
330
1605 1460
145
346 1951 1605
346
330
2281 1715
566
346
2627 1940
687
©2005 McGraw-Hill/Irwin
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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

©2005 McGraw-Hill/Irwin
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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: 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

©2005 McGraw-Hill/Irwin
Cost Reduction in Constant Work Force Plan
(Mixed Strategy)
3-40
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.
Plan Modified With Lay Offs in March and May
2000
1500
1000
500
0
1
2
3
4
5
6
7
8
©2005 McGraw-Hill/Irwin
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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.
©2005 McGraw-Hill/Irwin
An Alternative is called the “Chase
Plan”
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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)
42
©2005 McGraw-Hill/Irwin
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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©2005 McGraw-Hill/Irwin
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
I got the following for this problem:
Period
1
2
3
4
5
6
7
8
# hired
#fired
10
21
8
31
15
24
4
15
©2005 McGraw-Hill/Irwin
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?

45
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©2005 McGraw-Hill/Irwin
3-46
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.
©2005 McGraw-Hill/Irwin
3-47
Aggregate Plan to Master Schedule
Aggregate
Planning
Disaggregation
Master
Schedule
©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)
©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.
©2005 McGraw-Hill/Irwin
3-50
Exploring the Optimal (L.P.) Approach

We need an Objective Function for cost of the aggregate plan
(target is to minimize costs):
T
c
t 1
H
N H  cF N F  cI IT  cR PR  co OT  cu UT  cS ST 
– 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
©2005 McGraw-Hill/Irwin
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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: Wt  Wt 1  H t

Inventory Constraints:

Production Constraints:

 Ft
I t  I t 1  Pt  St  Dt
Pt  k nt Wt  Ot  U t
Where: nt * k is the number of units produced
by a worker in a given period of nt days
©2005 McGraw-Hill/Irwin
Real Constraint Equation (rewritten for
L.P.):

Wt
Employee Constraints:
 Wt 1  H t  Ft  0
Specifically:
W1  W0  H1  F1  0

Inventory Constraints:
P
t
 I t  I t 1  St  Dt
specifically:
P1  I1  I 0  S1  D1
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©2005 McGraw-Hill/Irwin
Real Constraint Equations (rewritten for
L.P.):

Production Constraints:
Pt  k nt Wt  Ot  U t  0
specifically:
P1  k n1 W1  O1  U1  0
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©2005 McGraw-Hill/Irwin
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
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