©2005 McGraw-Hill/Irwin
3-1
Chapter 3.
Aggregate Planning
(Steven Nahmias)
©2005 McGraw-Hill/Irwin
Hierarchy of
Production Decisions
Long-range Capacity Planning
3-2
©2005 McGraw-Hill/Irwin
3-3
Planning Horizon
Aggregate planning: Intermediaterange capacity planning, usually covering
2 to 12 months.
Long range
Short
range
Now
Intermediate
range
2 months
1 Year
©2005 McGraw-Hill/Irwin
3-4
Aggregate planning

Aggregate planning is intermediate-range
capacity planning used to establish
employment levels, output rates, inventory
levels, subcontracting, and backorders for
products that are aggregated, i.e.,
grouped or brought together. It does not
specifically focus on individual products but
deals with the products in the aggregate.
©2005 McGraw-Hill/Irwin
3-5
Concept of Aggregate Product


For example, imagine a paint company that
produces blue, brown, and pink paints; the
aggregate plan in this case would be expressed as
the total amount of the paint without specifying
how much of it would be blue, brown or pink.
Such an aggregate plan may dictate, for example,
the production of 100,000 gallons of paint during
an intermediate-range planning horizon, say
during the whole year. The plan can later be
disaggregated as to how much blue, brown, or
pink paint to produce every specific time period,
say every month.
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

3-6
©2005 McGraw-Hill/Irwin
3-7
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?
7
©2005 McGraw-Hill/Irwin
3-8
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 handout and in slide below.)
©2005 McGraw-Hill/Irwin
Aggregation Method Suggested by
Hax and Meal




3-9
Hax and Meal suggest grouping products into three
categories:
1. items, 2. families, and 3. types.
Items are the finest level in the product structure and
correspond to individual Stock-Keeping Units (SKU). 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.
©2005 McGraw-Hill/Irwin
3-10
Aggregate Units
The method is based on notion of aggregate
units. They may be
 Actual units of production
 Weight (tons of steel)
 Dollars (Value of sales)
 Fictitious aggregate units(See example 3.1)
©2005 McGraw-Hill/Irwin
Example of fictitious aggregate units.
(Example 3.1)
3-11
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?
©2005 McGraw-Hill/Irwin
3-12
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
3-13
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.

©2005 McGraw-Hill/Irwin
3-14
Aggregate Planning



The aggregate planning methodology discussed in this
chapter assumes that the demand is deterministic and
dynamic.
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
- retain a stable workforce
- develop a production plan that maximizes profit over
the planning horizon subject to constraints on
©2005 McGraw-Hill/Irwin
3-15
Nature of Demand
Demand
I. Deterministic

Static

Dynamic
II. Probabilistic

Stationary

Non-Stationary
 In aggregate production planning, we assume that
demand is deterministic and dynamic.

©2005 McGraw-Hill/Irwin
3-16
Costs in Aggregate Planning



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, idle cost,
subcontracting.
Cost of Changing
the Size of the Workforce
Fig. 3-2
©2005 McGraw-Hill/Irwin
3-17
©2005 McGraw-Hill/Irwin
Fig. 3-3
3-18
$ Cost
Holding and Back-Order Costs
Slope = Ci
Slope = CP
Back-orders
Positive inventory
Inventory
Overview of the Aggregate
Production Problem
©2005 McGraw-Hill/Irwin
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.
3-19
©2005 McGraw-Hill/Irwin
Prototype Aggregate Planning Example
(this example is not in the handout)
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.
3-20
©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.)
Month
1(Jan)
2(Feb)
3(Mar)
4(Apr)
5(May)
6(June)
7(July)
8(Aug)
Net Predicted
Demand
220(420-200)
280
460
190
310
145
110
225(125+100)
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
3-22
2000
1800
1600
1400
1200
Cum Net Dem
1000
800
600
400
200
0
1
2
3
4
5
6
7
8
©2005 McGraw-Hill/Irwin
3-23
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,
Chase) strategy:
– Matching capacity to demand; the
planned output for a period is set at the
expected demand for that period.
©2005 McGraw-Hill/Irwin
3-24
Level vs. Chase Strategy
©2005 McGraw-Hill/Irwin
3-25
Advantages and Disadvantages

Chase Strategy
–
–
–
–

Reduced inventory costs.
High levels of worker utilization.
Cost of fluctuating workforce levels.
Potential damage to employee morale.
Level Strategy
– Worker levels and production output are stable.
– High inventory costs.
– Increased labor costs.
©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?
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
3-27
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
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:
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 3-29
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.
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:
 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
31
©2005 McGraw-Hill/Irwin
3-32
Finding K

Suppose we are told that over a period of 40
days, 520 units were produced with 38
workers. It follows that:

K= 520/(38*40) = .3421 average number
of units produced by one worker in one day.
©2005 McGraw-Hill/Irwin
3-33
Computing Constant Work Force -Realistically

Assume we are given the following # of 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
33
calculations thus building more inventory
©2005 McGraw-Hill/Irwin
3-34
Why again did we pick on March?
Examining the graph we see that March was
the “Trigger point” where our constant
production line intersected the cumulative
demand line assuring NO STOCKOUTS!
 Can we “prove” this is the best?

34
©2005 McGraw-Hill/Irwin
3-35
Tabulate Days/Production Per Worker Versus
Demand To Find Minimum Numbers
Month
# Work Days
#Units/worker
Forecast Demand net
Cum. Net
Demand
Cum.Units/
Worker
Min #
Workers
Jan
22.00
7.53
220.00
220.00
7.53
29.23
Feb
16.00
5.47
280.00
500.00
13.00
38.46
Mar
23.00
7.87
460.00
960.00
20.87
46.00
Apr
20.00
6.84
190.00
1150.00
27.71
41.50
May
21.00
7.18
310.00
1460.00
34.89
41.84
Jun
22.00
7.53
145.00
1605.00
42.42
37.84
Jul
21.00
7.18
110.00
1715.00
49.60
34.57
Aug
22.00
7.53
225.00
1940.00
57.13
33.96
35
©2005 McGraw-Hill/Irwin
3-36
What Should We Look At?
Cumulative Demand says March needs
most workers – this can be interpretted as
building inventories in Jan + Feb to fulfill
the greater March demand
 However, if we keep this number of
workers we will continue to build inventory
through the rest of the plan!

36
©2005 McGraw-Hill/Irwin
3-37
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
3-38
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
3-39
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
3-41
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 month) 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”
3-42
Here, we hire and fire (layoff) workers to
keep inventory low!
 We would employ only the number of
workers needed each month to meet the
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
production/worker), for Jan= 220/(22*.3425)
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
43
3-43
©2005 McGraw-Hill/Irwin
3-44

Changing the Level of Work Force
Period
1
2
3
4
5
6
7
8
# hired
#fired
10
21
8
31
15
23
5
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
 It is better than 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
3-45
©2005 McGraw-Hill/Irwin
Example
3-46
Demand for Quantum Corporation’s action toy series follows a seasonal pattern –
growing through the fall months and culminating in December, with smaller peaks
in January (for after-season markdowns, exchanges, and accessory purchases) and
July (for Christmas-in-July specials).
MONTH
DEMAND (CASES)
MONTH
DEMAND (CASES)
January
1000
July
500
February
400
August
500
March
400
September
1000
April
400
October
1500
May
400
November
2500
June
400
December
3000
Each worker can produce on average 100 cases of action toys each month. Overtime is
limited to 300 cases, and subcontracting is unlimited. No action toys are currently in
inventory. The wage rate is $10 per case for regular production, $15 for overtime
production, and $25 for subcontracting. No stockouts are allowed. Holding cost is $1
per case per month. Increasing the workforce costs approximately $1,000 per worker.
Decreasing the workforce costs $500 per worker.
©2005 McGraw-Hill/Irwin
Example – Level Production
Input:
3-47
Beg. Wkrs 10
Regular
$10
Hiring
$1,000
Units/Wkr 100
Overtime
$15
Firing
$500
Beg. Inv.
Subk
$25
Inventory
0
Cost: $146,000
$1
Month
Demand
Reg
OT
Subk
Inv
#Wkrs
#Hired
#Fired
Jan
1000
1,000
0
0
0
10
0
0
Feb
400
1,000
0
0
600
10
0
0
Mar
400
1,000
0
0
1,200
10
0
0
Apr
400
1,000
0
0
1,800
10
0
0
May
400
1,000
0
0
2,400
10
0
0
Jun
400
1,000
0
0
3,000
10
0
0
Jul
500
1,000
0
0
3,500
10
0
0
Aug
500
1,000
0
0
4,000
10
0
0
Sept
1000
1,000
0
0
4,000
10
0
0
Oct
1500
1,000
0
0
3,500
10
0
0
Nov
2500
1,000
0
0
2,000
10
0
0
Dec
3000
1,000
0
0
0
10
0
0
Total
12,000
12,000
0
0
26,000
0
0
©2005 McGraw-Hill/Irwin
Example – Chase Demand
Input:
3-48
Beg. Wkrs 10
Regular
$10
Hiring
$1,000
Units/Wkr 100
Overtime
$15
Firing
$500
Beg. Inv.
Subk
$25
Inventory
0
Cost: $149,000
$1
Month
Demand
Reg
OT
Subk
Inv
#Wkrs
#Hired
#Fired
Jan
1000
1000
0
0
0
10
0
0
Feb
400
400
0
0
0
4
0
6
Mar
400
400
0
0
0
4
0
0
Apr
400
400
0
0
0
4
0
0
May
400
400
0
0
0
4
0
0
Jun
400
400
0
0
0
4
0
0
Jul
500
500
0
0
0
5
1
0
Aug
500
500
0
0
0
5
0
0
Sept
1000
1000
0
0
0
10
5
0
Oct
1500
1500
0
0
0
15
5
0
Nov
2500
2500
0
0
0
25
10
0
Dec
3000
3000
0
0
0
30
5
0
Total
12,000
12,000
0
0
0
26
6
©2005 McGraw-Hill/Irwin
3-49
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
3-50
Aggregate Plan to Master Schedule
Aggregate
Planning
Disaggregation
Master
Schedule
©2005 McGraw-Hill/Irwin
3-51
Rough-cut Capacity Planning

Aggregate planning is based on a general
production plan that deals with how much
capacity will be available and how it will be
allocated. A rough-cut capacity plan can
be developed to evaluate the work load that
a production plan imposes on work centers.
©2005 McGraw-Hill/Irwin
3-52
Below is a bill of labor which lists the hours required
in each department to make one unit of product.
PRODUCT
DEPARMENT
A
B
C
D
11
0.4
0.2
0.7
0.5
22
0.1
0.6
0.4
0.9
33
1.1
0.3
0.7
0.6
44
0.3
0.8
0.2
0.5
55
0.5
0.0
0.4
0.6
©2005 McGraw-Hill/Irwin
3-53
Master Production Schedule
MASTER PRODUCTION SCHEDULE
MONTH
PRODUCT
1
2
3
4
5
6
A
180
240
300
420
350
260
B
210
220
240
230
220
200
C
500
480
450
440
420
400
D
310
330
380
410
480
500
©2005 McGraw-Hill/Irwin
Develop capacity requirements
for the following combinations:
MONTH DEPARTMENT
2
4
6
3
33
11
55
22
3-54
©2005 McGraw-Hill/Irwin
3-55
The solution is outlined below:




240(1.1) + 220(0.3) + 480(0.7) + 330(0.6) = 864
hours (Month 2)
420(0.4) + 230(0.2) + 440(0.7) + 410(0.5) = 727
hours (Month 4)
260(0.5) + 200(0.0) + 400(0.4) + 500(0.6) = 590
hours (Month 6)
300(0.1) + 240(0.6) + 450(0.4) + 380(0.9) = 696
hours (Month 3)
©2005 McGraw-Hill/Irwin
3-56
Excess Demand

For example, we must have at least 864
hours available in Department 33 for Month
2 to meet capacity requirements. Suppose
that we only have 640 man hours available
in Department 33 in Month 2. Then, we can
use aggregate planning strategies such as
hiring, overtime, etc. to bring the capacity
up to the required amount of 864 man or
machine hours in order to comply with
master production schedule.
©2005 McGraw-Hill/Irwin
3-57
Work Force Size Planning

In aggregate planning the major objective is
to determine feasible and possibly optimal
production quantities and the corresponding
capacity (work force size) to accommodate
such production requirements. An example
of determining the appropriate work force
size follows.
©2005 McGraw-Hill/Irwin
3-58
Work Force Size Planning






Quarter
1
2
3
4
1 year
Forecasted Demand
(standard units of work)
6,000
4,500
4,200
5,500
20,200
©2005 McGraw-Hill/Irwin
3-59
Questions



a. Assume employees contribute 180 regular working
hours each month, and each unit requires 2 hours to
produce. How many employees will be needed during
Quarter 1 and Quarter 2?
b. What will be the average labor cost for each unit if the
company pays employees $10/hour and maintains for the
entire year a sufficient staff to meet the peak demand?
c. What percentage above the standard-hour cost is the
company's average labor cost per unit in this year due to
excess staffing for all but the peak quarterly period?
©2005 McGraw-Hill/Irwin
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Answers
 a.
Quarter 1:6000 x 2 = 12,000 hours
12,000 / (180 x 3 months) = 22.22 --->
23 employees
Quarter 2:
4,500 x 2 = 9,000
9,000/(180 x 3 months) = 16.66 ---> 17
employees
©2005 McGraw-Hill/Irwin
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Solution
b. (180 hours/employ.-month x 12 months x
23 employ. x $10/hour)/20200=
= 496,800/20200= $24.59/unit
c. 2 hours x $10/hour = $20/unit (standard
cost)
($24.59 -$20)/$20 = 0.23 or 23% higher
©2005 McGraw-Hill/Irwin
3-62
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 workers 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.
©2005 McGraw-Hill/Irwin
3-63
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
3-64
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
3-67
©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
3-68
©2005 McGraw-Hill/Irwin
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Example
CAPACITY
REGULAR
OVERTIME
SUBCONTRACT
DEMAND
INVENTORY
HOLDING COST
PERIOD 1
200
80
100
340
PERIOD 2
180
60
100
300
COST
$20 /unit
$25 /unit
$30 /unit
$3 /unit/period
©2005 McGraw-Hill/Irwin
LP formulation for the problem
above:
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Min 20X111 + 23X112 + 25X211 + 28X212 + 30X311 + 33X312 +
20X122 + 25X222 + 30X322
Subject To
X111 + X112 <=200
X211 + X212 <=80
X311 + X312 <=100
X122 <=180
X222 <=60
X322 <=100
X111 + X211 + X311 = 340
X112 + X212 + X312 + X122 + X222 + X322 = 300
©2005 McGraw-Hill/Irwin
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Solution


For example, for X312 the first subscript (3) stands for
the type of capacity (i.e. subcontract). The middle
subscript (1) stands for the supplying period or
production period (i.e. period 1). The last subscript (2)
stands for the receiving period or consumption period
(i.e. period 2).
The optimal values of the variables such as x111, x112,
x211, etc. would show how much regular capacity,
overtime capacity, etc. to use in each time period in
order to minimize the cost of the planned aggregate
production.