Chapter Three
Chapter Overνiew
Purpose
Το develop techniques for aggregating units of production, and determining
levels a nd workforce
levels based οη predicted demand for
Key Points
1. Aggregate units of production. Thi$ chapter could also have been called Macro
Production Planning, since the purpose of aggregating units ί$ to be able to
develop a top-down plan for the entire firm ΟΓ for some subset of the firm, such
as a product line ΟΓ a particular plant. For large firms producing a wide range of
products ΟΓ for firms providing a service rather than a product, determining
appropriate aggregate units can be a challenge. The most direct approach is to
eχpress aggregate units ίη some generic measure, such as dollars of sales, tons of
steel, ΟΓ gallons of paint. For a service, $uch as provided by a consulting firm ΟΓ a
law firm, billed hours would be a reasonable way of eχpressing aggregate units.
2. Aspects of aggregate p/anning. The following
aggregate planning:
are the
σιοει
• Smoothing. (05tS that arise from changing production
• Bott/enecks. Planning in anticipation
important feature5 of
and workforce levels.
• Treatment of demand. ΑΙΙ the mathematical model5 in thi5 chapter consider
demand Ιο be known, i.e, have zero forecast error.
3. Costs ίπ aggregate p/anning.
• Smoothing costs. The CO$tof changing production
and/or workforce
level5.
C05t of dollar5 inve5ted in inventory.
• Shortage costs. The CO$t$associated with back-ordered
ΟΓ
105t demand.
• Labor costs. These include direct labor costs on regular time, overtime,
5ubcontracting costs, and idle time cost$.
4. So/ving aggregate p/anning prob/ems. Approχimate solutions Ιο aggregate
planning problem5 can be found graphically, and eχact solutions via lίnear
programming. When solving problem5 graphically, the first 5tep is to draw a
124
5. The /inear decision ru/e. The aggregate planning concept had its ωοιε ίη the
work of Holt, Modigliani, Muth, and Simon (1960) who developed a model for
Pittsburgh Paints (presumably) to determine their workforce and production
level5. The model used quadratic approχimations for the costs, and obtained
simple lίnear equations for the optimal policies. This work 5pawned the later
interest in aggregate planning.
6. Mode/ing management behavior. Bowman (1963) considered linear decision rules
similar to tho$e derived by Holt, Modigliani, Muth, and Simon eχcept that he
suggested fitting the parameters of the model based on management's actions,
rather than prescribing optimal actions ba$ed on ωει minimization. Thi$ is one of
the few eχamples of a mathematical model used to describe human behavior ίπ
the conteχt of operations planning.
7. Oisaggregating aggregate p/ans. While aggregate planning is useful for providing
approχimate solutions for macro planning at the firm level, the que$tion is
whether these aggregate plans provide any guidance for planning at the lower
levels of the firm. Α disaggregation scheme is a means of taking an aggregate
plan and breaking it down to get more detailed plans at lower levels of the firm.
of peak demand periods.
• P/anning horizon. One mU$t cho05e the number of period5 considered
carefully. If too short, 5udden changes in demand cannot be anticipated. If Ιοο
long, demand foreca5ts become unreliable.
• Ho/ding costs. The opportunity
125
graph of the cumulative net demand curve. If the goal ί5 to develop a level plan
(i.e., one that has constant production ΟΓ workforce levels over the planning
horizon), then one matches the cumulative net demand curve as closely as possible
with a straight lίne. If the goal is to develop a zero-inventory plan (i.e., one that
minimizes holding and shortage costs), then one tracks the cumulative net
demand curve as cl05ely as possible each period. While lίnear programming
provide5 ωει optimal 50lutions, the method does not take into account
management policy, such Β5 avoiding hiring and firing as much as possible. For
a problem with a Tperiod planning horizon, the linear programming formulation
requires 8Tvariables and 3Τ constraints. For long planning horizons, this can
become quite tedious. Another issue that must be dealt with is that the solution to
a linear program ί5 noninteger. Το handle this problem, one would either have to
specify that the problem variables were integers (which could make the problem
computationally unwieldy) ΟΓ develop some suitable rounding procedure.
Aggregate Planning
suitable production
aggregate units.
Aggr,ga,e Planning
As we go througlllife, we tnake both Illicro and macro decisions. Micro decisions tnight
be \vhat Ιο eat forbreakfast, whatroute Ιο take towork, what auto service ιο use, orwhich
movie to rent. Macro decisions are the kind t!Jat change the course of one's life: where Ιο
lίνe, what Ιο major ίη, whicll job Ιο take, wllom Ιο marry. Α cotnpany also must make
both micro and macro decisions every day. Ιη this cllapter we explore decisions made at
the macro leve!, such as planning companywide workforce and production leνels.
Aggregate planning, which might al50 be called macro production planning,
addresses the problem of deciding how many employees the firm Sllould retain and, for a
manufacturing firm, the quantity and the mix ofproducts to be produced. Macro planning
is ηο! lίmίted Ιο manLIfacturing firms. Service organizations must determine employee
staffing needs as well. For example, airlines must plan staffing levels for flight attendants
and pilots, and !lospitals must plan staffing levels for nurses. Macro planning strategies
are a fundamental part ofthe firm's overall business strategy. Some firms operate οη the
philosophy that costs can be controlled only by making frequent changes ίn tl1e size
andlor composition of the workforce. The aerospace industry ίη Califomia ίη the 1970s
adopted this strategy. As government contracts shifted fron1 one producer Ιο another, so
126
Chapter Three
3.1
Aggregalι! Plαnning
did the technical workforce. Other firιns have.a reputation for retaining employees, even"
ίη bad times. Until recently, ωΜ andAT&Twere two we1J-known examples.
Whether a firm provides a service οτ produces a product, macro planning begins
with the forecast of demand. Techniques for demand forecasting were presented ίη
Chapter 2. How responsive the firιn can be το anticipated changes ίη the demand depends οιι several factors. These factors include the genera], strategy the firm may have
regarding retaining workers and ifs οοτυηιίτυισιιε-το existing empIoyees.
As noted ίη Chapter 2, demand forecasts afe generallY'wrong because there is aImost
always a random οοπιρουεφι ofthe demand that cannot be predicted exactly ίη advance.
The aggregate pIanning methodology discussed ίη this chapter requires the assumption
that demand is deterministic, οτ known ίη advance. This assumption is made Ιο simpIify
the analysis and alJow us το focus οτιthe systematic orpredictable changes ίη the demand
pattem, rather than οτι the unsystematic οτ random changes. Inventory management
subject to randomness is treated ίη detail ίη Chapter 5.
TraditionalJy, most manufacturers have chosen ιο retain primary production ίηhouse. Some components might be purchased from outside suppliers (see the discussion ofthe make-or-buy problem ίη Chapter Ι), but the Ρήmary product is traditionalJy
produced by the firm. Henry Ford was one ofthe firstAmerican manufacturers το design
a completely vertically integrated business. Ford even owned a stand of rubber trees so
ίι wouId υοι have το purchase rubber.for tires.
That philosophy is undergoing a dramatic change, however. Ιιι dynamic environments,
firms are finding that they canbe more flexible ifthemanufacturing
is outsourced; that is,
ifit is done οτι a subcontract basis. One exampIe is Sun Microsystems. a Califomia-based
producer of computer workstations. Sun, a market leader, adopted the strategy of focusing οιι product innovation and desigiJ rather than οιι manufacturing. It has developed close
ties ιο contract manufacturers such as San Jose-based Solecιron Corporation. winner of
the Βaldήge Award for Quaiity. Subcontracting its primary maηufactuήηg function has
allowed Sun Ιο be more flexible and Ιο focus οη innovation ίη a rapidly changing market.
Aggregate plaBning involves competing objectives. One objective is Ιο react quickly
ιο anticipated changes ίη demand, which would require making frequent and potential1y
large changes in the size ofthe labor force. Such a strategy has been called a chase strategy. This may 1Jecost effective, but could be a ροοτ loηg-ruη business strategy. Workers
who are laίd off may ηο! be avaiIable when business tums around. For this reason, the firm
may wish Ιο adopt the objective of retaining a stable workforce. However, this strategy
often results ίη large bUΊldups of inventory duήηg Ρeήοds of low demand. Service firms
may incur substantial debt tomeetpayrolJs in slowperiods. Α third objective is Ιο develop
a production plan for the firm that maximizes profit over the planning hοήΖοn subject Ιο
constraints οη capacity. When profit maximization is the primary objective, explicit costs
ofmaking'changes
mustbe factored ίηΙο the decision process.
Aggregate planning methodology is designed Ιό translate demand forecasts ίηΙο a
blueprint for planning staffing and production levels for the firm over a predetermined
planning horizon. Aggregate planning methodology is ηοΙ lίmίted ιο top-Ievel planning. Although generaJ]y considered Ιο be a macro planning ιοοl for determining
overalJ workforce and production leνels, large companies may find aggregate planning
useful at the plant level as well. Production planning may be viewed as a hierarchical
process ίη which purchasing, production, and staffing decisions must be made at severallevels ίn the firm. Aggregate planning methods may be applied at almost any level,
although tl1e conceptis
one of maήagίηg grotιps of items rather than single items.
The inventory planning methods discussed ίη Chapters 4 and 5 are geared toward
single-item control.
Aggregar< UnitsofPrαlucιion
127
This chapter reviews several techniques for determining aggregate plaίIs. Some.6f
these ετο heuήstίό (i.e., approximate) and some are optimal. We hope tοcοήΎeΥ ιο the
reader ετι understahding of the issues involved ίη aggregate planning'i'a kή6wleage"of
the basic tools available for providing solutions, and an appreciation of the dif!icιJltίes
associated with imp(ementing aggregate plans ίτι the real world.
,~
,
"
AGGREGAtE
.,0
'}'
,,"0
,"~;
\
"1"
,
,'~:r~"';CI'~::,
:
ONITS'"OFpRobuttION.
The agg~ega:te planning approach is predicated οα the e~istence of an aggregate υηίι
ofproduction. When the types ofitems produced are similar, an aggregate production
υηίι can couespond Ιο 'aΠ "av~rage" item, but if many different·types of items are produced, ίι would be more ερρτορτίειε το consider aggregate units ίη terms of weight
(tons of steel), νοίυπισ (gallons of gasoline), amount of work required (worker-years
ofprogramming time), οτ dollar value (value ofinventory ίτι dollars). What the approρτίετε aggregating scheme, should be is πο! always obvious. Ιι depends ωι the context
ofthe'pafticular
planning problem'and the leνel of aggregation required.
Example3.1
Α plant managerworking lor a large national appliarice lίrm is considering implementing an aggregate planning system ιο determine the worklorce and production levels in his plant. This particular plant produces six models ΟΙ washing machines,The characteristics ΟΙ the machines are
Model Number
Α5532
Κ4242
Ι9898
Ι3800
Μ2624
Μ3880
Nlιmber of Worker-Hours
Required to Produce
4.2
4.9
5.1
5.2
5.4
5.8
, SeIIing Price
$285
345
395
425
525
725
The plant manager must decide οπ the particular aggregation scheme Ιο use. One possibilίΙΥ is to deline an aggregate υπίΙ as'on'e dollar ΟΙ όυιρυΙ Unlortunately, the selling prices ΟΙ the
various models ΟΙ washing machines are ποΙ consistent with th·e number ΟΙ worker-hours required Ιο produce them. The ratio ΟΙ the,selling price.divided by the worker-hours is $67,86 for
Α5532 and $125,00 ΙΟΓ M::J880, (The company bas~s its pricing on the faet that the less
expensive models have a higher sales volume.) The manager notices that the percentages οΙ
the total number οΙ sales Ιοτ these si\<models have been fairly constant, with values οΙ 32 percent lοr Α5532, 21 percent lοr Κ4242, 17 percent Ιοτ ~9898, 14 percent for L3800, 1Ο percent
lοr Μ2624, and 6 percent for Μ3880. He decides to define an aggregate uni! ΟΙ production as
a lίctίtίoυs washing machine requiring (.32)(4.2) + (.21)(4.9) + (.17)(5.1) + (.14)(5.2) +
(.10)(5.4) + (.06)(5.8) = 4.856 hours ΟΙ labor time. He can obtain ~ales forecasts for aggregate
produetion units ίπ essentially the same way by multiplying the appropriate Iractions by the
lorecasts lor υπίι sales ΟΙ each type ΟΙ machine.
The apptΌach used by the plant manager ίο Example 3.1 was possible because of the
relative similarity of the jJroducts prodtIced. However, defining an aggregate υη1ι of
production at a higher leνel ofthe firm is more difficult. Ιη cases ίη which the firrn produces a large νaήety ofprodncts, a natural aggregate υπίι is sales dollars. Althollgh, as
we saw ίη the' example, tl1ίs will 110!11ecessarily tral1slate Ιο tbe same nnmber of units
of prodnction 'for each item, ίι wiH generally provide a good approximatio11 for planning at tl1e highest leνel of a firm that produces a diverse product lίηe.
128
Chapter Three
Aggr,gaιc Planning
3.2
FIGURE 3-1
TI,e /1ierarchy of
129
ροτίοσε of time, stIcb as weeks, quarters, ΟΓyears. An important featιιre of aggregate
planning is that the,de,mands are treated as known constants (that is, the forecast error is
assumed ιο be zero). Thereasons for making this assumption will be d,iscussed later.
The goal ofaggregate planning is το determine aggregate production qua.ntities and
the leνels of resources required το achieve these production goals. Ιτι practice, this
translates to nnding the number ofworkers that should be employed and the number of
aggregate tInits ιο be produced ίη each of the planning periods J, 2, ... , Τ. The objective ο[ aggregate planning is to balance the advantages of producing to tneet demand
as closely as possible against the disrtIptions caused by changing the levels of producιίοο and/or the workforce levels.
The primary issnes related ιο the aggregate planning problem include
produclion pIanning
dccisiol1S
Aggregate planning (and the associated problem of disaggregating the aggregate
plans orconverting them ίπΙο detailed master schedules) is closely related το hierarchical
production planning (ΗΡΡ) championed by Hax and Meal (1975). ΗΡΡ considers
workforce sizes and production rates at a variety oflevels ofthe nrm as opposed 10 simply the τορ level, as ίπ aggregate planning. For aggregate planning purposes, Hax and
Meal recommend the foJlowing hierarchy:
Ι. /tenls. These are the nnal products ιο be delivered to the οιιετοηιετ, Απ item is often
referred 10 as an SKU (for stockkeeping ιαιίι) and represents tl1e nnest leνel of detail
ίπ the ρτοοιιοτ είτυοιυτε.
2. FamiIie:,·. These are denned as a group of items that share a οοηιπιοη mantIfactιιring
setιιp cosΙ
3. Types. Types are groups of families with prodtIction quantities that are determined
by a single aggregate production ρlαη.
lπ Exalnple 3.1, items would
Α family might be all washing
Hax-Meal aggregation scheme
the aggregation method shotIld
and prodllCt Ιine.
correspond 10 individuallnodels ofwashing machines.
nJachines, and a type might be large appliances. Τhe
wiII πο! necessarily work ίη every sitιιation.ln general,
be consistent with the nrιn's organizational strιtcΙUre
[η Figure 3-1 we present a sCl1ematic of the aggregate planning fιlnction and its
place ίπ tl1e hierarchy ofproduction planning decisions.
3.2
Overview of the AggregaLe ]Jlαnning IJroblem
OVERVIEW OF ΤΗΕ AGGREGATE PLANNING PROBLEM
Having now denned the appropriate aggregate ιιηίι [or the leνel of the firm [or which αη
aggregate plan is ιο be determined, we assume that there exists a forecast ofthe deInand
for a specined planning horizon, expressed ίη terms of aggregate prodtIction units. Let
DJ, D2, ... , Dr be the demand forecasts for the next Tplanning periods. [π most applications, a planning period is a month, altl10tIgh aggregate plans can be developed for other
Ι. SmooIhing, Smoothing refers το costs that resιιlt from changing production and
workforce levels from ουε period το the ηοχΙ Two ofthe key components of smoothing
costs ΗΓοthe costs that result frol11 hiring and nring workers. Aggregate planning
methodology reql1ires the ερεοίϋοειίοτι of these costs, which may be difficult to εειίmate. Firing workers could have fal'-reaching consequences and costs that may be difficult to evaluate. Firms that hire and nre freqtIently develop a poor public image. This
could adversely affect sales and discourage potential employees frOI11joining the company. Furthermore, workers that are laίd off might not simply wait around for busiiιess
ιο pick up. Firing workers can have a detrimental effect οη the futιιre size ofthe labor
force ifthose workers obtain employment ίτι other indusιries. Finally, tnost companies
ΗΤοsimply πο! at liberty Ιο hire and fire at will. Labor agreements restrict the freedom
of management το freely alter workforce leνels. However, ίι is διίll valtlable for management ιο be aware of the cost trade-offs associated with varying workforce leνels
and the attendant savings Ιτι ίnνentOlΎ costs.
2. Bottleneck probIenIs. We use the term bottleneck ιο refer το the inability of tl1e
system το respond το stIdden changes ίπ demand as a restIlt of capacity restrictions. For
example, a bottleneck cotIld arise when the forecast [οr demand ίη οιιε month is unusually high, and the ρΙΗΩΙdoes υοτ have slIfficient capacity το meet that demand. Α
breakdown of a vital piece of equipment also οοιιί« result ίn a bottleneck.
3. P/anlling horizon. TI,e number of periods for which the deI11and is to be forecasted, and hence the ntImber of periods for which workforce and inventory levels are
Ιο be determined, mlIst be specified ίη advance. The choice oftl1e value ofthe forecast
horizon, Τ, can be significant ίη determining the usefulness of the aggregate plan. If
Τ is 100 sI.nall, then cnrrent production leνels might ηο! be adequate for I11eeting the
demand beyond the horizon length. Jf Τ ίδ 100 large, ίι is likely that the forecasts far
ίηΙο the fιιΙUτo wiII prove inaccurate. If futιιre demands turn out Ιο be very different
from the forecasts, then clIrrent decisions indicated by the aggregate plan could be ίπcorrecΙ Another isstre involving the planning horizon is the end-oJ-hoI'izon effect. For
example, the aggregate plan might recommend that the inventory at the end ofthe horiΖοn be drawn to ΖοΓΟίη order to minimize 110lding costs. This cotIld be a ροοτ strategy,
especiaIIy if demand increases at that time. (However, this particIIlar problem can be
avoided by adding a constraint specifying mil1imnnJ ending inventory leνels.)
Ιπ practice, rolling schedules are almost always tIsed. This means that at the time of
the next decision, a new forecast of demand is appended Ιο the fornJer fΟΓecasts and old
forecasts ll1ight be revised Ιο reflect new information. The new aggregate plan may recommend different production and workforce levels fol' the cwTent period. than were
recomlnended one peTίod ago. When only the decisions fol' the CΙΙΓrentplanning period
need Ιο be illlplemented imJnediately, II,e schedule shotIld be viewed as dYl1anlic ra!her
than static.
130
Chapter Three
Aggr,gaιe Ρ/αηηίηι
3.3
Co,ιs ίη Aggr,gaιe Plaηning
131
Although rolling schedules are common, ίι is possible that because of production
lead times, the schedule must be frozen for a certain number of planning periods. This
means that decisions over some collection of future periods cannot be altered. The
most direct means of dealing with frozen horizons is simply Ιο label as period Ι the first
period ίη which decisions are τιοτ frozen.
οι
4. Treatment
demand. As noted above, aggregate planning methodology requires
the εεειυυριίου that demand is known with certainty. This is simultaneotIsly a weakness and a strength of the approach. Τι is a weakness because ίι ignores the possibiIity
(and, ίη fact, likeIihood) offorecast errors. As noted ίη the discussion of forecasting
techniques ίη Chapter 2, ίτ is virtually a certainty that demand forecasts are wrong.
Aggregate planning does αοι provide any buffer against unanticipatedforecast
errors.
However, most inventory nιodels that allow for random demand require that the average demand be constant over ιίσιο. Aggregate planning aIIows tlle manager Ιο Ιοοιιε οη
the systematic changes that are generaJly αοτ present ίη models that assnme random
demand. ΒΥ assuming deterministic denland, the effects of seasonal fluctuations and
business cycles can be incorporated ίητο the planning function.
3.3
COSTS ΙΝ AGGREGATE PLANNING
As with most ofthe optimization problems considered ίη production managelnent, the .
goal ofthe analysis is to choose the aggregate plan that IIlinimizes cost. Ιι is important
to identify and measure those specific costs that are affected by the planning decision.
Ι. SI1700t!lil1g cσsfs. Smoothing costs are those costs that accrue as a result of
cllanging the production levels from one period το tlle next. Ιη the aggregate pIanning
context, the most salient snιoothing cost is the cost of changing the size of the workforce. Tncreasing the size of the workforce requires time and expense to advertise ροsitions, interview prospectiνe employees, and train new hires. Decreasing the size of
the workforce means that workers ιτιιιετ be laίd off. Severance pay is thus one cost of
decreasing the size of the workforce. Other costs, somewhat 11arder ιο measnre, are
(α) the costs ofa decIine ίη worker morale that lllay result and (b) tlle potential for de-'.·
creasing the size of the labor ροοl ίη tlle future, as workel"s who are laίd off acquire jobs
with otller firnlS ΟΓ ίη otller ίηdustΓίes.
Most of the models that we consider assume that the costs of increasing and deCl"easing the size ofthe workforce are lίnear functions ofthe numbeI' ofemployees that
are hired οτ fired. That is, there is a constant dollar amount charged for each employee'
hired ΟΙ' fired. TIle assumption of linearity is probably reasonable ιιρ Ιο a ροίηι. As the
supply of labor becoInes scarce, there may be additional costs reqLIired 10 hire more
wοrkeΓS, and tlle costs of laying off workers may go ιιρ substantially if the nuιnber όf
workers la.ίd off is Ιοο large. Α typical cost function fol" changing the size of the
workforce appears ίη Figure 3-2.
2. Ho/ding costs. Holding costs are the costs tllat accrue as a Γesιιlt ofllaving capital
tied ιιρ ίη inventoI)'. Jf the firm can decrease its inventory, tlle money saved coLIld be ίπ-.
vested elsewJlere with a return that will vary witll tlle indnstry and with the specific
company. (Α more complete discussion ofholding costs is deferred to ChapteI" 4.) Holding costs are almost always assLImed Ιο be lίηear ίη the number ofunits being 11eldat a
partictIlar ροίηΙ ίη time. We will assunle for the Ρuφοses of the aggregate planning
analysis that the holding cost is eΧΡΓessed ίπ terms of dollars per ιιηίι held per planning
period. We also will assume that Ilolding costs are charged against the inventorY..
remaining οη hand at tlle end of the planning ΡeΓίοd. This assLI1Ilption is Inade fqr'
convenience onJy. Holding costs could be charged against starting inventory
inventory as well.
ΟΓ
average
3. Shortage costs. Holding costs are charged against the aggregate inventory as
long as ίι is positive. Ια some situations ιι may be necessary to incuI" shortages, \vhich
ετο represented by a negative level of inventory. Shortages can occnr when forecasted
denland exceeds the capacity of the production facility ΟΓ when denlands are higher
than anticipated. For the purposes of aggregate planning, ίτ is generally assunled that
excess demand is backlogged and filled ίη a future period. Ιπ a highly competitive situation, however, ίι is possible that excess demand is lost and the customer goes elsewllere. This case, which is known as lost sales, is Inore appropriate ίη the management
of single items and is more common ίη a retail than ίη a manufacturing context.
As with holding costs, shortage costs are generally assumed to be lίnear. Convex
fllnctions also can accllrateIy describe shortage costs, but lίηear flInctions seem Ιο be
the most common. Figure 3-3 shows a typical holding/shortage cost function.
4. Regular time cost~·. These costs involve tlle cost of prodLIcing one nηίι of output
during regular working hours. Included ίη this category are the actual payroll costs of
regular employees working οη Γegular time, the direct and ind.irect costs ofmateriaJs,
and other mannfactιIring expenses. When aII production is carried οιι! οη regular time,
reguIar payroII costs become a "slInk cost," because the number ofunits produced lllust
equal the number ofunits demanded over any planning horizon ofsufficient length. Jf
there is ηο overtime οτ workeI" idIe time, reglllar payroll costs do ηο! have Ιο be
included ίη the evaluation of different strategies.
5. OIIeI·tinze al1d .S1Ibcontracting costs. Overtime and subcontracting costs are the
costs of production of units ηο! produced οη regular time. Overtime refers Ιο producιίοη by regular-time eInployees beyond the normal workday, and subcontracting I'efers
Ιο the production of iteIns by an outside SUΡΡlίeΓ.Again, ίι is generally assumed that
botll of these costs aJ'e lίηear.
6. ldIe time costs. The complete formulatioIl of the aggregate planning problem
also includes a cost for underutilization ofthe workforce, οτ idle tifRe. Ιη most contexts,
,132 'Chapter
Three
,,3.4, "Α j>rororyp<.prρblim
Aggregαιe PIαnning
133
5. :A'large-manufacturer
ofhousehold.consumer
goods ίιι considering ihtegτating ετιaggregateop!anning nΊodel ,ίηιο its rήanιJfacturing 'strategy. Twe· of th.e, company
-νίοε presitlehtsdisagre~
strongly' astot~e'~atΊie ofthe ~pproach. What arguments
. m'ighteach 'of the'Yide:presi'dents',use'fo'sUpport 'his or'Her 'ροίιιτ of view?
'
FIGURE-3=3
Holding and backorder costs
~ '-,
:~ ι"
j~\
T'~
_c(_"----·;"Ξ{·.;;;(ι,j""v~..
.• Ά,;
'·"ί"'ί);;::Jι~-'
.;.' ,
"
6. .Descrioe' th~ follqwing costs and discu'ssΊhe difficu1ties.that arise ίη attempting
'measure them in area1 operating enviromnent.
"
.-
.
)~,
10
'
~;;':-
α.: Smοοthίl1gιcosΙs
b.' Hdldirig costs
σ, ,PayroJ[ costs
7. OiSCtIS§tJle followiiig statenienl: '''Sirlce''we ιise a roiling production
really doii'theed το know tlIe demaridbey~nd'next
πιοητίι,"
schedιtle, Ι
8, St. Clair County Hospital is attempting το asse~k iis needs [or nurses over the coming four montbs (January to April), The need for nurses depends οη both the
riumbers aηd',tneΊΥΡes of patients ϊη the hospita1. Based οο a study conducted by
constIltants, th§nosp1tal has determined that the following ratios of nurses ιο
pati en ts are required:
Patient Type
the idle time cost is zero, as the direct co~ts of i'dle 'time wbϋld' be taken into account ϊη
labor costs and lower production levels, However, idle titιie could have other consequences' for the firm. For exaIi1ple, i,fthe aggregate units aτe ίπρυτ to another process,
idle time οτι theliIie cott1d, resul!'in hi'gher'costs ιο tlle subsequent ρτοοσεε. Ιτι such
cases, one woUld exp!icitly include εροείυνο idle οοει.
When plannitιg is dohe ata re!ativelY'high Ieve! ofthe fiτm: the effects of intangib!e
factors are mbre ρτουοιιαοεο. Any sο!utίόή td the aggregate p!anning prob!ern obtained
from'a cos't-based ιποde!'musΙbe considereίI carefully in'the context of.companypol'icy. Απ optimal so!ution to a t1iathematica!Ίri.ode!'migHt resultin a policy that requires
fTequent hiring and' firing of p~rsoimel. Such a po!icy may' be infeasib!e because of
prior contract agreements, οτ uiJdesirab!'e~because of the potential negative effects οτι
the firm's public image.
Problems for Sections 3.1-3.3.
'Numbers of'Nurses
(Required per Patient
Major surgery
ΜίηΟΓ surgery
Maternity
C ritical care
Other
Forecasts
'·i·Jan.
Feb.
,Mar.
28
12
22
75
80
21
25
43
45'
94
16
45
90
60
73
0.4
0,1
0,5
0,6
0.3
Patient
."',1'"
"
a. How many nurses shollld be working eachmonth
'ΑΡΓ, ";'
18
32
26
30
77
το most close!y match patient
forecasts?
"ψ"ι 'ι:11
b, Suppose the hospita1 does not want το change its'policy of πο! increasing tlle
nursing staff slze by more than 1Ο percentjn any montll'. Suggest.a sehedule, of
nurse staffing over the four months that meets this requirement and a1so meets
the need for n\Jrses each month.
ι
Ι. What does the term aggregαte unit ofpl-oduction mean? 00 aggregate production
units always correspond to actual items?,Oo they ever? Discuss,
2. What isthe aggtegation
scheme recominended
by'Haxarid
3, Oiscuss the following
problem:
terms and their re!ationship
'
3.4
Α PROTOTYPE PROBLEM
'Ι,,,
το the aggregate
planning
a. Smoothing
b, Bottlenecks
c. Capacity
d. Planning horizon
4. iA local macHine shop eIllploys 60 workers whb have a Y'ariety of skills, The shop
accepts one-time orders and a!SΟ'maίήtaίη's a nuTnber ofregu1ar clients, Discuss some
ofthe difficulties with using the aggregate p!anning methodology in this context.
""
'"
,
"
:
",
One can obtairi adeiIl\hte,,,sollltions for many aggregate pla.ιining problems by hand or by
using relativel'y straightfoτwaτd graphical techniques. Linear progranuning is a means
of ob!aining (nearly)·optimal solutions. We illustrate the different solution techniques
with the'fo]Jowing examp!e.
'
Meal?
Example3.2
Densepack is Ιο plan workforce and production levels for the six-month period January ΙΟ June,
The firm produces a lίη-e of disk drives for mainframe computers that are plug compatible with
several computers produced by major πianufacturers, Forecast demands over the' next siχ
months for a ρartίcύ(ar lίne of drives produced ίη the Milpitas, California, plant are 1,280, 640.
900, 1,200, 2,OOO,and.1 ,400, ..There are currently(end of December) 300 workers employed
ίη the Milpitas plant. Ending inventory'in December is eχpected to be 500 units, and the firm
would Iike to have 600 units οη hand at the end of June,
134
Chapter Three
AggregQte Plιtnning
. 3.4
There are several ways ιο incorporate the starting and,the eriding inventory constraints ίη
the formulation. The most convenient iSsimply το modify the values οΙ the predicted dema
Define net predicted demand ίπ period'l ·as.the predicted demand minus initial inventor:y.
there is a minimum ending inventory constraint, then this .amount should be added Ιο the ,c
mand ίπ period Τ. Minimum buffer inventorie.s also can be handled by modifying the predicte
demand. ΙΙ there is a minimum buffer inventory ίπ eve'r'yperiod, this amount should be added't,
the first period's demand. Ιι there is a minimum buffer inventory ίπ only one period, this amourit
should be added το that period's demand and subtracted from the next period's demand. Actual.
ending inventories should be computed using the original demand pattern, however.
.
Returning to ουΓ example, we define the net predicted demand for January as 780 (1,>280.'"",
500) and the net predicted demand for June as 2,000 (1,400 + 600). ΒΥconsidering net de~
mand, we may make the simplifying assumption that starting and ending inventories are both
zero. The net predicted demand and the net cυmulative demand for the six months January to
June are as follows:
Month
January
February
March
ΑΡΓίl
May
June
Net Predicted
Demand
Net CumuIative
Demand
780
640
900
1,200
2,000
2,000
780
1,420
2,320
3,520
5,520
7,520
Α
Prow,:Ype'Probwn
135
Ιπ order to illustrate the costtra.dg,offs οΙvarίρ.u.s ρ(ος!ucψ<n plans, ,we V\(ίΙΙassume' il!1·,the
example that there are~only three.cos'ts to be considered: costof hiring workers, cost ol'fi-ring
workers, and cost οί holding inveniory. De.fine·
.
(Η
t;
= Co~t.of, hi~ing οοε worker = $500,
= C;st6ffiring o~J,w~rker = $f':ooo"
, (, = Cost.of holding oπ~. unit οΙ inventory
(F
for one month
=
$8Ό.
We require a means()f translating ~~gregate production ίπ units to wo'rkforce levels. B~c'ause
ποΙ all months have an equal number οΙ working days, we will us.e a day as an indivisible υπίί of
measure .arid define
'
Κ = Number οΙ aggregate. units produced by one .worker ίπ one day.
In the past, the plant manager observed that over 22 working days,
cbnstant at 76 workers, the firm produced 245 disk drives. That means
duction ratewas 245/22 = 11.1364drives per daywhen there were 76
plant. It follows that one .worker produced an'average'of 11.1364/76
day. Hence, Κ = 0.14653 lοr this example:
with the workforce level
that οπ average the ΡΓΟworkers employed atthe
= 0.14653 drive ίπ one
We will evaluate two alternatiνe plans ,[ΟΤ managing the. worl<.force that represent
two essentially opposite management
strategies. Plan Ι· is to change the workforce
eacll month iη order Ιο produce enough un.its to most closely,matcll tJ1e demand pattem.
This is known as a zero il1vel1tory ρΙαη. Plan 2 is το maintain the minimun1 constant
workforce necessary to satisfy thenet demand. This is known as the constant wοrkjΌΙ'ce
ρΙαη.
The cumulative net demand is pictured ίπ Figure 3-4. Α production plan is the specification
ΟΙ the production levels for each month. ΙΙ shortages are not permitted, then cυmulative
production must be at least as great as cυmulative demand each period. Ιπ addition Ιο the
cumulative net demand, Figure 3-4 also shows one feasible production plan.
Eνaluation cίf a Chase Strategy (Zero Inνentory Plan)
Here, we will deνeIop a 'production, plan Ιοτ Densepack that minimizes
the leνels of
inνentory the firnl must hold during the six-month planning horizon. Table 3-1 sumIllarizes the ίιιρυι ίηίοτυιειίωι [or the calculations al1d Sl10WStl1e minimum number of
workers required ίη each month.
Ol1e obtail1s the entries ίη the final cOll1mn ofTable 3-1, the m.inimum number of
workers required each Illonth, by dividing the forecasted netdeIlland
by the number of
units produced per worker. The νalue of this'ratio is then rounded tιpwαrd ιο tlle next
FIGURE 3-4
Α feasibIe aggregate
plaη for Densepack
5
6
~Ι;{
3.4
136
Chapter Three
Α P,oto<ype P,oblem
137
Aggreg",e Pi<ιnning
ΤΑΒΙΕ 3-3
Computation ofthe
Minimum Workforce
Required by
Densepack
ΤΑΒΙΕ 3-4
higher integer. We.mustround upwardtoguarantee
that shortages do πο! occur. As an example, consider the month ofJanuary. Foτming the ratio 780/2.931 gives 266.12, which
ίε τottnded ιιρ Ιο 267 workers. The number of working days each month depends ιιροο a
variety of factors, such as paid holidays and worker schedules. The reduced number of
days ίπ June is due το.a planned shutdowη ofthe plant ίη the last week of Ιυτιε.
Recall that the nttmber'ofworkers
employed at the end of December is 300. Ηiήηg .
and fiήng workers each month το match forecast demand as close!y as possible resttlts
ίπ the aggregate plan given ίπ Table 3-2.
The number of units prodLIced each month (τοίυαιυ F ίπ Table 3-2) is obtaίned by
the following formula:
'ι
Number of
units ρτοσυοοσ
Number
Average number of
.
of workers χ aggregate LInltSproduced ιτι
a month by a slngle worker
rounded το the nearest integer.
The tota! cost of this ρτοοιιοιίοη p!an is obtained by mtt!tip!ying the totals at the
bottom of Table 3-2 by tbe ερρτορτίειο costs. Ροτ this example, the total cost of hίήng,
firing, and .ho!ding is (755)(500) + (145)(1,000) + (30)(80) = $524,900. This cost
must now be adjusted το ίnc!LIdethe cost ofholding Ιοτ the ending inventory of 600 units,
which was netted ου! of the demand for Ιιιυε, Hence, the tota! cost of this plan is .
524,900 + (600)(80) = $572,900. Note that the initial inventory of 500 units does τιοτ
enter ίηΙο the ca!culations because ίι will be netted ουι dttring the month of January.
Ιι is usually impossible το achieve zero inventory at tl1e end of each planning period
because i(is ποΙ possible το employ a fractiona! number of workers. Ροτ this reason:,
there will almost always be some inventory remaining at the end of each period ίπ
addition Ιο the inventory required Ιο be οτι hand at the end of the planning horizon.
Ιι is possible that ending inventory ίπ one οτ more periods could bui!d ιιρ to a ροίυτ
w.here the size of the workforce could be reduced by one οτ more workers. Ιτι this
example there is sufficient inventory οη hand ιο reduce the workforce by one worker ίπ
the months of both March and May. Check that the resu!ting plan hίΓes a total of
753 woτkers and fires a tota! of 144 workers and has a total of on!y 13 units of inventory. The cost ofthis modified plan comes Ιο $569,540.
Evaluation of the Constant Workforce
Plan
Now assume that t!le goa! is ιο e!iminate complete!y the need for hiring and firing
during the planning horizon. !π order to guarantee tI1at sl10rtages do ποΙ occur i,n any
JIIventory Lcvels for
Constant Workforce
Schedule
period, ίι is necessary Ιο coιnpute the minimum workforce required for everγ month ίη
the planning horizon. Ροτ January, the net cumulative demand is 780 and there are
2.931 units produced per worker, resulting ιτι a τυίυίπιιυ» workforce of267 ίη JanLIary.
There are exact!y 2.931 + 3.5 Ι 7 = 6.448 units produced pel" worker ίη January and
February combined, which have a cumulative demand of 1,420. Hence, !,420/6.448 =
220.22 = 221 workers are required to coverboth January and February. Continuing to
form the ratios of the cumulative net demand and the cumulative number of units
prodLIced per worker for each month ίιι the horizon results ίη Table 3-3.
The minimum number of workers required for the entire sΊX-month planning period
is the Π1aΧίmum entry ίη coluι'nn D ίτι Table 3-3, which is 411 workers. It is on!y a
coincidence tl1at the maximum ratio οccuπed ίη the final period.
Becatlse tllere are 300 wo.rkers employed at the end of DeceΠ1ber, the constant
workforce plan requires hiring 111 workers at the beginning of January. Νο further hiring
and firing ofworkers are required. The inventory leνels that resu!t from a constant work·
force of 411 wo.rkers appear ίη Table 3-4. The monthly production leνels ίη column C ΟΙ
the table are obtained by mtIltiplying the number ofunits prodnced perworker eacll Π10ntt
by the fixed workforce size of 41 Ι workers. The total of the ending ίnνentoΙΊ leνels ί~
5,962 + 600 = 6,562. (Recall the 600 units that were nette<1οιι! of the deιnand for JlIne ..
Hence tl1e total inventory costofthis plan is (6,562)(80) = $524,960. Το this we add t11(
cost of increasing the workforce froιn 300 ιο 411 ίn JanLIary, which is (Ι 11)(500) =
$55,500, giving a total costofthis plan o.f$580,460. This is sσmewhat higherthan the cos
oftlle zero inventory plan, which was $569,540. Howeνer, because costs ofthe ιwo plan
are close, ίι is like!y that the coιnpany would prefer the constant workforce plan ίη orde
to avoid any unaccounted for costs of making freqLIent changes ίn the workforce.
138
Chapter Three
3.4
Aggregare Planrtίng
Mixed Strategies and Additionalfonstraints -
",·1
<. .~
,:i-tπ!~:
Η,1
The zero inventory plan ana constant workforce strategies just treated ate l?uie,.strafe1
gies: they are designed to achieve one objective: With τυοτε flexibility,sιhal1'm6dific~tions can result ίή, dJamaticaliy lower. c6.8t~:;One -migbt que~tion tbe interest }Ίι ίiiιinιrai
calculations cbnsidefing tbat aggregate~planhing 'prohlems can be formuJafJd "and
solved optimally by Jinear programming.".
...
'.
,.'.
.'
Manua). ,αi\cιi1atίοns· eiίhance ίntuitίοrίandιihderstandίήg.
Cbmj:>liters aredumb. It ~\./
easy to overlook a critical constraint (π objectlve when using a computer. Ιι 's inιportant Ιρ ,
bave feel for the rίght soJution beforesolvmg a probJem οτι a computer, so that gJaring'
mistakes are obvious. Απ important skiH, largely ignored these days, is being able to do'a
ballpark calculation ίn one's Ilead before pulling ου! the calculator οτ computer.
Figure 3-4 shows the constant workforce strategy for Densepack. The hatched area
represents tbe inventory carried ίη each month. Suppose we alJow a sίngJe change ίη the
production rate during the six months. Can you identify a strategy fi:om the figύre tI1at
substantially reduces inventor)ι without pennitting shortages?
' ,
",ι,,·, '
Graphically, the problemis
το cover ihe cuΊnulative net demaήtl clIrve with ~ό
straight lines, rather than one straigbt JiJ;1'e.
This can be accompJished by driving'fbe net
inventory ιο zero at the end of perίod 4'(April). Το do so,we'need ιο produce enolIg]j ,
ίη each oftbe months January through Αρτll ιο meet the cumuJative net demand each
month. 'Πιετ means we need το produce3,~20/4
= 880 tJnits ίη each of the .first four
months, Ιπ Figιιre 3-4, the line connecting'the origin το 'the cumulative net dem'and in'
Aprillies wholly above the ουιυιιίειίνε net detnand curve:for tl1e ρτίοτ rtιonths. Ifthe
g.raph is accurate, that means that there shouJd be τιο shortages occurring ίη these
lnonths, The May and Ιιυιε production is then set το 2,000, exactly matching the net
demand ίη these lnonths, With this policy we obtain
Cumulatiνe
NetDemand
.
ΑΡΓίl
May
June':
As we will see ίη Section 3.5, this poJicy .turns·out ιο be optimal for the Densepack
problelll,
:.,
The graphical solution method also can be used when additional constraints are pres-;,
ent. For exanΊPIe, suppose that the production capacity ofthe plant is only 1,800 units "._....
'
per montb. Tben the policy is infeasible in May and June. Ιη this case the constraint ;1Ω\ΊΙ.' Ι,;"~ Γ
means tl1at the slope of the cumulative· ,production curν,e is bounded by ] ,800, One
solHtion ίη this case would be to produce 980 ίη each of tlTe first four months and
1,800 units ίη each of the last two months. Another constraint might be that the maximun1 change fromone τηοηΙlι ιο the next be ηο more than 750 units. Suggest a producιίοη plan ιο meet this constraint.
, TlTes,eare a,few'examp.les of constraintsthat mighι.arise ίη using aggregate planning
methodology, As. the constraints become:. more compJex, finding good solutions
graphically becomes tnore difficult: Fortunatelyl·most cbnstraints of this natl1re can be'
incorporated easily into the lίιιear programming formuJations of aggregate planning
probJeJns.,
ί"l-'
"1
.
"
;:,
"ι . .ι.,·
,~
Ι;
i.~'·
,~-,~I
''1';'':
:τι
\;t~iT ;~;{)
'J ••
';,
,Ι,'
.
,'Γ
,
ιω\ , 'j:or;~asted
Deman,d'
, (t~ousands of, packages)
jf.
300,
120
'1
2
3
4
5
20o'
110
135
, ,A'ssume that ea,ch wοrker,staΥsΌη.the.,jόb, for at!IEast one year, and that,Grey
'currently has three workers οα the payroiI. Heestimates that he wψ have 20,000
.packages on'lfantl'at',the
ofthe ς:urrent y~at: A:ssuiiι~'that, οε theaverage,
ach
worker i,spaid $25,000 per year and is r~s'poiιsible for prodlIcing 30~000 packages.
In"elitO'ry' .costs 'havevl)eeh 'estiInated td"be" 4"ce'ιιts pei package per year, and
shortages fι~e'1)ot aI10\Ved;f' η .:<,
"Ι
'
i
.
Base<;loiι' the effort( of interνiewing and, training new workers, Farmer Grey εεtimates tha.t ίι ,costs $,500 for, each worker hired, Severance pay amounts to $1,000
per worke,r. " _
end
ι::
.. α" Assutning, that sho'rtages are ηοί allowed, determine
the minimum constant
workforce that he will Jieed over the next five years.
Cumulatiνe
Production
880
1,760
2,640
3,520
5,520
7,520
!-\..-
~,:ι
Η; _
~ ,ι
b. Evaluate tbe cost of the pJan found ίη part (α), ,
tl"
January
February
March
'ι:
Pro",rype,ProbJem 139
, 9,. Harold Grey QWΠS)i sm3lI fann ίn .the Salinas Valleythat grows apricots. The. apri; ;"cots,are·dried -οιι the .premise,s and sold· to a ,number,of large supermarket chfiins.
·Based οτι past,~xperϊe\lce,and comllli,tted contracts,h'e,esiimates
that s<ilesΌver the
next five years ihthotlsanos of',I'ackages wil\be as follows:
a
Month
;4;'
'~·ι
Α
10, F,0r'the datagi~eh
ίη ProbJem 9, ιraph the cumulative net demand.
α. Graphically detennine a production plan that changes the production rate exactly
once during tp.e five years, and evιΗuate the cost of that plan.
b, Graphicany deter,mine a product~on plan. that changes the prodlIction
exactly twice dlIrίng the five years, and evallIate the cost of that plan.
rate
11. Αη ilnplicit assurnption made in Problem 9 was that dried aprίcots unsold a( the
end of a year couJd be sold ίη subsequ~nt years, Suppose that apncots unsold at the
end of any year π;ιust be discarde,d, Assume a disposal· cost of $0.20 per package.
Resolve Problelll 9 under these conditions.
12. The personnel departrnent of the Α&Μ Corporation wants to know how many
workers will be needed each month for the next six-month production period, The
following is a monthly demand forecast for the six-month peribd.
Month
July
August
September
October
November
December'
Forecasted' Demand
1,250
1,100
950
909
1,000
1,150
3.5
140
Chapter
Three
Soi"rionσ{Aggregaιe PlanningProblems.1ry
ΙίneΙJi"ΡrΟgTamrhίng
~41
AιweRaι, PlanninR
α. DeterrniI\e. the mίnίmιιm ..constarit:workforGe:i,t)Ίat",will
The lnventoτy οτι harid at tlJe endΌf Ιιπιε was 500 units ..:τheocompany, wants 10
malntaln a mlnimum inventory of 300 tInits each month 'and w~~ld ΊίΊce to have
400 unlts 'οη hatιd at the end of Deeember.!.Each ιιηίί requires five employee-honrs
to ptodnce .•.thereare'20
workingdayseach
mbilth,iιnd each emp1oyeeyνorks an
eight-honr day. Tlie workforceat the end·of June was'·35 workers.
" ';~',
'ο
α. What' ls the' minimum constant workf~rce r~;quired to meet demand over the
next four months?
Ά
b. Αεειισιε that CΙ, = ω cents pec cο(}kίe'Ρer,ιmοnt11;,cΗ
Eva1uate the cost ofthe p1an detlved ·ίη part (α),
=
$100, and cF
=
$206. '
3.5
Days
J'anuary
February
22
March
"6
21
ApriI
19;
[ΊΛaΥ
23
20
June
July
August
September
October
November
December
24
12
19'
22
20
16
'τ,'·,·'.:".-'"
.
,Ι'
-,
ΑGG~~,G~Τ~,~ΡίΑΝΝljΝGPRO'B,LEMS
SOLUTlON OF
Βγ LINEAR PROGRAMMING
_ι~,
\f
Linear prograJI)ffii.ng ί5,a τοσο uS,e<!.
to ,de~cribe a generaJ,class of optimization problems.
The objecti:ve is tp,determine va1ues 9fn n.onnegative rea~variables ίη order το maximize
ΟΓminlmize ε.Ιίαεετ .function of these.yariab1es ψat ls subject.to m lίnear constralnts of
,ιηese variab1es.1 Th,e primary ~dvanta.ge iniformulating a,problem as a 1inear program ls
that ορτίπιε! so!tItlons can be found 'Very efficiently QY'the'si:nιplex method.
When all costf\mctions are lίnear, there ls a linear programming formulation of the
g~ne~a1:aggregat~IPlanning problem.· Because of the efficie)lcy of commerclal 1inear
,ΡrοgramΠ1ίng οοαοε, this means t4aι'(essentίaIlΥ) optima1 sqlutions can be obtained for
2
Cost
Predicted Demand
(ίπ $10,000)
340
380
220
100
490
625
375
310
175
145
120
165
]nventory holding costs are based οη a 25 percent anntIal lnterest charge. !t'is'
anticipated that there will be 675 workers οη the payroll at the end of tlle cnrrent
year and inventories will amοunΙlο $120,000. The firm wotIld like to 11aveat least
$100,000 of inventory at the end of December next year. 1I is estimated tllat eabll
WΟΓkeraccounts for an average of$60,000 ofproduction per year (asstIme that one·
year conslsts of 250 working daYs). The cost of hirlng a new worker ls $200, and
the cost of !aYlng off a worker ls $400.
Parame~ers
and Given Information
The following νείιιεε ar~ asslImed to'be
known:
τ
,CH
= ~ost'of,hiring
one.worker,,,",
,C,p = ,Cost:of firing ουε worker; πΙ"
C[
Month
,ί
very large problems
14. Α local semlconductor firm, Superchip, ls p.1anning lts workforce and productlon
1evels ονετ' the ne~t year. Tlie firm' fnakes a'vIιribty όf'm'icrΟΡrocessοrs and uses sa1es .
doJlars as lts aggregate ρτοσυοιίοιι measure: Βεεοο ση orders recelved and sales ,.
forecasts provided by the marketing department, the estlmate of dol1ar sales for the
next year by month is as follows:
.Production
'r,'
>" .
b. Determine the' prodnction plan that' meets demand but does αοι hire οτ fire
workers dύπng the six-monih period.
13. Mr. Meadows Cookie COJnpany makes a variety of chocolate chip cookies ίη tbe
plant ίη Albion, Michigan. Based ωι orders recelved and forecasts ofbuying babits,
ίτ ls estimated that the demand for the next [οιιτ months is 850, 1,260,51 Ο, and 980,
expressed ίη thousands of cookies. During a 46-day period when there were
120 workers; the-company produced 1.7 mil1ion cookies. Assume that the'ntImber '
of workdays over the [οιιτ months are respectively 26, 24, 20, and 16. There are
currently 100 workers employed, aiid thete ls υσ startlng inventory of cookies.,
;'1- ,\1'":-'
] 5. Fot the datainProblem~14, determine the cost oftheplaiιthat changes theworkforce
size each period to most '~lb!ie1y;n~tch the demaιid.
16. Graph the cumιι1atίve,net'delnand fofthe SUΡerchίΡ data of.Problem ]4. Graphically deterrnin"e' a, Ρrοdιιctίοή;ρ1aπ'that
changes production levels υο mόre that!
tllree tirnes, and deterrnlne the tota1cost of that p1an,
α, Determine,a minimum ίηv~~!ότy jxoduction plan (i.e., one that al10ws arbitrary
hiring andfiring).
meet the predicted
demand fot the comlng ~e&r·,i";.,,, ; ,
";,,
b. Evaluate.the cost όftheρlan deterh1ined Ια -part (αΙ.
= Οοει of holding one υηίι of· stock for one period,
cR = Cost Ο[ΡrοducίηgC;~eΉ~ίt~~
re~lar
time,
Co = lncr~nie,htal cost of,prod~cing o'η~ unit οτι oyertime,
Cu = Idle cost per unit,o~prodtIctjon,
.. ,"
Cs = Cost to subcontriιct one υηίι ofproduction,
'.,
ηι = Number οfΡrοdιιctiοn
daΥs.lrtΡeι'iορ.I",
Κ = Number of aggregate units produced by oIίe worker ίη one day,
10 = 1nitial;inyentory
οη haI)d,atthe.start
o~ the p.lannipg 110τίΖΟΠ,
Wo = !nitia1 workforce at the start of the planning horlzon,
Dt = Fοreca'stΌf demand ίη period t.
The cost parameters a1so may be tirne dependent; thatls, they may change wlth t.Timedependent costparameters could be ύsefυ1 for Π10deiing changes ίη thecosts οfhίήng
or firing due, for example, to shortages ίπ tl1e labor ροοl, οτ changes ίη the costs of ρτοduction and/or storage due to slJortages ίη tiie supply of resources, οτ cllanges ίη interest
rates.
, An overνiew
of Iinear programming
2 The qualifier
is intluded
point Iater.
can be found
because rounding ma{ιjive
ih Supplement
suboptimal
1, which foIIows this chapter.
solutions.
There wiII be more about this
142
Chapter
Three
Aggrcgatf
Planning
3.5
Ρι = Production
ι,
Ιενεί ίη period Ι,
Ιι = Inventory leve] ίη period
ι,
Ηι = Number ofworkers
hired ίη period ι,
FI = Number ofworkers
fired ίη period
υ
ι,
τ
ΟΙ = Overtime production ίη units,
υι = Worker
of Aggre~αte PIanning Problems.by Linear Ρrο.ι:rammίη,(!' 143
lη addition Ιο these constraints, lίnear programming requires that all problem variables
be nonnegative. These constraints and the nonnegativity constraints are the minimum that
must be present ίη any formtllation. Notice that (Ι), (2), and. (3) οοτιετίιιιιε 3 Tconstraints,
rather than 3 constraints, where Tis the length ofthe forecast horizon.
The formu]ation also reqnires specification of the initial inventory, 10, and the ίηίtia] workforce, Wo, and may inc]ude specification of the ending inventory ιτι th.e final
period, lτ.
The objective function inclndes all the costs defined earlier. 'Πιο lίnear programming
forιnll]ation is to choose values ofthe probleIn variables W" Ρι, 1" Η" Ft, Ο"
ι, and S,lo
Problem Variables
The following are the problem variables:
WI = Workforce ]eve] ίη period
Soluιion
Σ (cHH
Minimize
idle time ίη units ("undertime"),
+ cpF, + ctl, + cRP, + coOI + cυU, + C8S,)
I
'=1
SI = Number of units st1bcontracted from ontside.
snbject 10
The overtime and id]e time variables are determined ίη the following way. The
term Knt represents the number of units produced by one worker ίη period t, so that
Kn,W, would be the number ofunits produced by the entire workforce ίη period ι. However, we do υοι require that Kn,W, = Ρ,. If Ρ, > Kn,Jflt, then the number of υυίιε ρτοdnced exceeds what the workforce can ρτοσυοε οη regular time. This Ineans that the difference is being produced οη overtime, so that the number ofunits produced ωι overtime
is exactly ΟΙ = Ρ, - KntW,. If Ρι < KntWt, then the workforce is producing less than ίι
shotIld be οη regular time, which means that there is worker idle time. The idle time is
measured ίη units ofproduction rather than ίη time, and is given by ι = Kn,W, - Ρι.
υ
W, = W,_ 1 + Ηι - F,
(conservation
Ρ, = Kn,WI
+ Ο, -
for Ι :s t :s Τ
of workforce),
U,
for Ι :s t :s Τ
(production and workforce)
1ι = Ιι- ι
+
Ρ,
+ SI
- Dt
for ] :s t:s Τ (inventory balance),
Η" F" Ι,, Οι, U" S" W" Ρι ~ Ο (nonnegativity),
(Α)
(Β)
(C)
(D)
ρίιιε any additiona] constraints that define the values of starting inventory, starting
workforce, ending inventory, οτ any otheI' variab]es with νείιιεε that arefixed ίη advance.
Problem Constraints
Rounding the Variables
Three sets of constraints are required for the ]jnear programming formιιlation. They are
included το ensure that conservation of labor and conservation of units are satisfied.
Ι. Conservation of workforce constraints.
WI
=
Wt-1
Number = Number
of workers
of workers
ίη t
ίη Ι - Ι
+ Ηι
+ Number
hired
ίη Ι
Ft
- Number
fired
ίη Ι
for Ι :s t :s Τ.
2. Conservation of units constraints.
Ιι
=
lι- ι
[nventory = Inventory
ίη ι
ίη ι - Ι
+
+
Ρι
Number
ofunits
produced
ίη Ι
3. Constraints relating production
Ρ,
Nuιnber
of units
prodtIced
ίη t
= KntW,
= Number
of units
produced
by regtIlar
workforce
ίη ι
+
+
+
+
Dt
SI
Number
- Demand
ίη ι
ofunits
subcontracted
ίη ι
for 1 :s ι :s Τ.
levels Ιο workforce levels.
Ο,
U,
NuInber - Number
of units
of units
prodtIced
of idle
οη overproduction
time ίη t
ίη t
fol' ] :s ι :s Τ.
Ιn general, the optimal values ofthe problem variables wiII τιοτ be integers. However,
fractional va]ues for many of the variables do ιιοι make sense. These variables include
tlle size of the workforce, tlle number of workers hired each ρειίοτί, and the number of
workers fired each period, and είεο may include the ηUΠ1ber of units produced each
period. (Ιι is possible that fractional numbers of units could be produced ίη εουιε
applications.) One way Ιο deal with this problem is το require ίη advance ιίιετ some οτ
aII ofthe problem variab]es assume οηlΥ integer values. Unfortunately,
tlliS makes the
solution algoritllm considerably more CΟΠ1Ρ]eχ.The resulting ΡrοbleΠ1, known as an ί ηteger lίnear ΡrοgraΠ1Π1ίηg prob]em, requires mtIch Π10re computational effort Ιο solve
than does ordinary Jinear prograInming. For a moderate-sized pΓOb]em, solving the
prob]em as an integer ]jnear program is certa.inly a reasonabJe alternative.
Ifan integer programming code is l1navailable οτ ifthe problem is simply Ιοο Jarge
Ιο solve by integer programlning, linear programmingstill
provides a workable solution. However, after the lίnear programIning soltItion is obtained, some of the problem
variables must be rounded Ιο integer valtIes. Simply rounding off each variable Ιο the
closest integer may lead Ιο an infeasibJe solution and/or one ίη which prodllction and
workforce levels are inconsistent. It is ηο! obviolls wllat is t]le best way to round the
variables. We recotnmend the following conservative approach: rollfid tlle values ofthe
numbers of workers ίη each period ι to W" the next ]arger integer. Once the va]ues of
W/ are determined, the va]ues ofthe other variables, Η/, Ft, and Ρι, can be fOl1nd along
with the cost of the resulting plan.
Conservative rounding will always result ίη a feasible sollltion, but will ΓarelΥ
give the optimal soltItion. The conservative solution generally can be improved by
trial-and-error experimentation.
144
Chapter Three
3.5
Aggrcgaιe Plann,ng
SQluιionο[ Aggregare Planning Problems by Unear ProgramminR
14!
There iS τιο guarantee that if a problem can be formulated as a lίnear program, the
final solution makes sense ίη the context of the problem. Ιτι the aggregate planning.
problem, ίι does αοι make sense that there should be both oνertime production and idle
time ίη the same period, and ίι does τιοι make sense that workers should be hired and.
fired ίη the same period. This means tllat either one οτ both of tlle variables ΟΙ and U;
must be zero, and either one οτ both of the variables Η, and F, must be zero [οτ each t/:.
Ι ,s; t ,s; Τ. This [eqnirement can be included explicitly ίη the probleIn formtIlation by ~i,
adding the constraints
' .
O,U, = Ο
for Ι ,s; t,s; Τ,
H,F, = Ο
for Ι
,s;
ι,,;; Τ,
since iftlle prodnct oftwo νariables is zero ίι means τίιετ at least one ηιυει be zero. Unfortunately, these constraints are τιοι Ιίαεετ, as they involve a prodnct of probleni
variables. However, ίι turns ου! that ίι is αοτ necessary Ιο explicitly inclnde thes~
constraints, becanse the optimal solntion ιο a lίπear programming problem always οοcurs at an extreme ροίτιτ oftlle feasible regioll. It can be showll that every extreme pdint .
solntion autotnaticaIIy has tlliS property. Ifthis were ιιοτ the case, the lillear prograrrt:
ming solution wotIld be Il1eaningless.
'.
Extensions
Linear programming also can be used to solve somewhat more general νersions ofthe
aggregate planning problem. Uncertainty of demand can be acconnted for indirectly by
asstIming ιίιετ there is a minilllnm bnffer illventory Β, each period. lπ that case we
wotIld include the constra ints
for Ι ,s; ι,,;; Τ.
[,~B,
'Πιε constants Ξ, would have ιο be specified ίπ advallce. Upper bonnds ου the number of workers hired and the nutnber of workers fired each period could be included in',;t.C!
a similar way. Capacity constraints οπ the amount of production each period cduld'
easily be represented by the set of οοηειτείτιιε:
P,,s; C,
for Ι ,s; t ,s; Τ.
The lίπear programming formnlation introduced ίπ tlliS section asstIIned that inνen-:~j'
tory leνels wotIld neνer go negative. However, ίn some cases ίι Inight be desirable oτ~
even necessary to allow demand Ιο exceed slIpply, [οτ example, if forecast demand :
exceeded prodnction capacity over some set of planning periods. lπ ol'der ιο treat
backlogging όf excess demalld, the invelltory level [Ι must be expressed as the djffer:
ence between two nonnegatiνe variables, say [,+ and 1,-, satisfying
[Ι
= [/ - [,-'
1/ ~ Ο,
lπ the developmellt ofthe lίπear programming model, we stated the requirement that
all the cost functions ηιυετ be lίπear. This is ιιοι strictly correct. Linear programming
also can be nsed when the cost flInctions are C017vex piecewi8e-li17ear !ul1ctions.
Α οουνοκ function is one with an illcreasillg slope. Α piecewise-linear function is
one that is composed of straight-line segments. Hence, a convex piecewise-linear
functioll is a function composed of straight lίnes that Ιιενε increasing slopes. Α typical
exalnple is presented ίπ Figure 3-5.
[η practice, ίι is likely that some ΟΓ all of the cost Ιιιαοιίωιε for aggregate planning
are convex. For example, if Figure 3-5 τερτοεεητε the cost of hiring workers, then tlle
marginal cost of IJiring οιιε additiollal worker increases with the number of workers
that Ιιενο already οεετι Ilired. This is probably more accnrate than asstIIning that the
cost of hiring ουε additional worker is a οοπετωιτ jlldependent of tlle nnmber of
workers previously hired. As more workers are hired, the available labor ροοl shrirιks
and lnore effort mIIst be expended Ιο hire tlle remailling available workers.
lη order Ιο see exactly how convex pίecewίse-~jllear functiolls would be incorporated
ίnΙο the lίnear ΡrοgΓaιnιnίπg formlllatioll, we wil1 consider a very silnple case. Snppose
that the cost oflliring new workers is represented by the function pictured ίπ FίgUΓe3-6.
Accordillg Ιο the figure, ίι costs CHI Ιο Ilire each worker υπιίΙ Η* workers are hired, and
ίι costs C/12 for each worker hired beyond Η* \vorkers, with CHI < Cfl2. The variable Η"
the number ofworkers hired ίll period ι, mnst be expressed as the stlln of two variables:
ζ ~Ο.
The holding cost wonld 1l0Wbe charged against [/ and the penalty cost for back οτ'
ders (say cp) agaillst [,-. However, notice that for the solIItioll to be sensible, ίι mnst be
true that [/ and 1,- are ηοΙ botll positive ίη the same period t. As witlJ the oνertitne and
idle tilllC and the hirillg alld. firing variables, the properties of lίπear programming wil1
guaralltee that this holds wi Ιπου! having Ιο explicitly illclude the constraint [/ ς = ο:
ίπ the fOI·mlIlation.
Η, = Ηι,
+ Η2,.
Interpret Ηιι as the nUlllber ofworkers hired up Ιο Η* and Η2, as tlle llutnbeI' of
workers hired beyolld Η* ίπ period Ι. The cost οΓ hiring is now reΡΓesented ίη the
objective fUllction as
τ
Σ
I=Ι
(CI'IIHII
+ CInH2'),
146
Chapter Three
3.6
."'κgregare Plαnning
SoI<ing Aggregαte Planning Problemsby,Lineαr
Progrcmm'ng' Ah. EXDmple
147
be.J;Iloreefρcient for)argeproblem~ aηq <eOUldpI9y,idesoMions whep, some ofthe cost
functions are nonlinear. Α paper that details a transportation-type procedure ιυοτε. effic
cient than thesimplex method for solving aggregate planning problems is Erengιic and
FIGURE 3-6
Convex piecewiseIinear- hiring cost
function
>':
Tufekci(1988
SOιVING AGGREGATE Pi:.ANNiNGPROBLEMSBYLINEAR
PROGRAMMING: ANE~AMPΙE
We ,wiHdemonstrate the use oflinear programming'by:(Jnding
theoptimal solution ιο
the example presented ίn Sectlon 3.4. As there is υο subcontracting,.overtime,
στ id1e
time aHowed, and the cost coefficients are constant with respect το time, the objective
function is simply
" ·6
Σ Ηι + 1,000Σ
Minimize. ( 500
,t=l
6
.,.
Ρ, +80
ι=\
6
ΣI
ι=ι
t).
The boundary conditions .comprise .the specifications of .:the initial inventoτy οί
500 unlts, .theinitlal wprkforce of 300 work:ers, and. the. ending inventory of 600 units.
These are besthanp1edby including a separate additional constralnt for each boundary
and the additional
condition.
The constraints are~.όbtaίηed by substltutlng t "=1, ... , 6 ίηΙο Equations (Α), (Β).
and.(C). The·fuH set:qf.constraints expressed ίη st~ndard Ιίτιεετ programmίng forma1
(with all problemvariables
the 1eft-hand side and nonnegatlve constants ση thε
right-hand side) is as fol1ows:
ση
constraints
Ht ='Ηιι + H2t
Ο :5 Ηιι:5 Η*
Ο
:5
H2t
must also be included.
Ιη order for the final solution (α make'sense, itcan never be the case that Ηιι < Η*,
and H2t > Ο for Some t. (Why?) Ηοινενοτ; because lίnear programming searches for ιΜ
minimum costsolution,
ίι will force Ηιι ΙΟ its maximum value before aIIowing H2t Ιο
become positive, since CHl < CH2' This is the reason that the cost functions must be
convex. This approach can easily be extended το more than two linear segments and to
any of the other cost functions ρτεεέιιι ιιι the objective function. 'Πιε technique is
known as separable convex programming and is discussed ίη greater detail ίn Hillier
and Lieberman (1990).
.
.'
Other SοlutίοnΜetl1όds
Bowman (1956) suggested a transportation form.ulation ofthe aggregate planning problem when back orders are ηοι permitted. Several authors have explored solving aggregate
pIanning problems with specially tailored algorithms. These algorithms could be faster
[ατ solving large aggregate planning problems than a generallinear programming code
such as υΝΟο. Ιτι addition, some of these formulations allow for certain types of nonlinear costs. For example, if there are economies of scale ίn production, the production
cost function is likely Ιο be a concave function ofthe number of units produced. Concave
cost Ιϊυιοιίοηε are not as amenable το linear programming formnlatio11s as convex funcc
tions. We believe tllat a general-purpose linear programrning package is sufficient for
most reasonable-sized problems that one is likely ιο encounter ίη the real wOl·ld. Ηοινever, the reader sho.uld be aware that there are alternative solution techniques that conld
·= Ο,
W1
-
Wo - Ηι + Fι
W2
-
Wl
-
W] - W2-
.=
Η2 + F2
Η]
Ο,
+ F3\=0,
W4 - W3 - Η4 + F4 = Ο,
Ws - W4 - Hs + F'i;= Ο,
W6 -
-
Ws
Η6
+ Ψ6
(Α
= Ο;'
+ 10 = 1,180,
+ I1 =640,
Ρ3 - Ι] + 12 = 9ΡΟ,
Ρ4 - 14 + /3= 1;200,
Ps - 15 + 14= 1,000,
Ρ6 - 16 + 15 = I?~OO;
Ρι - I1
Ρ2 - /2
Pj
-
Ρ2
-
3.517W2 = Ο,
Ρ3
-
2.638W] = Ο,
Ρ4
-
3.81Or1l4 = ο,
2.931 Wl = Ο,
Ρ5 - 3.224W5 =0,
Ρ6 -
(Β
2.198W6 = Ο;
(C
148
Chapter
Three
ΑglζΓeφιe
Plonning
3.6
WI,·
.. ,
W6, Ρι,· .. , Ρ6, lι,· .. ,/6, FJ,
•••
,
F6, Ηι, ... ,Η6;ΞΞ:
SoIving Am,rregate [)!anning Problems b-yLinear Programming:
An ExαmfJle
149
Ο;
Wo = 300,
10 = 500,
/6 = 600.
We Ιιενε solved this lίnear progranl using the LlNDO system deve]oped by Schrage
(Ι 984). The output ofthe LINDO program is given ίη Table 3-5. Ιη the ειιρρίοηιεητ οο
linear programming, we a]so treat so]ving linear programs with ExceJ. The Exce!
ΤΑΒΙΕ 3-5
Linear Programming Solution of Densepack Problem
Solver, of course, gives the same resυlts. The value of the objective function at the ορtimal solution is $379,320.90, Wllich is considerably less than that achieved with eith.eI"
the zero inventory plan ΟΓ the constant workforce plan. Ηωνενετ, this cost is based ου
fractional values ofthe variables. The actnal cost wiJJ be slightly higher after rounding.
FoJJowing the τουικίίιη; procedure recommended earlier, we will τοιιικί alJ the valιιοε of W, το the next higher integer. That gives WI = ... = W4 = 273 and Ws = W6 =
738. This determines the νείαοε ofthe other problem variables. This means that the fίrm
shonld fire 27 workers ία JanlIary and hire 465 \vorkers ίη May. The complete solution is
given ίη Table 3-6.
Again, because co!umn Η ίη Table 3-6 corresponds το net demand, we add th.e
600 ιυυτε of ending inventory ίη June, giving a total inventory of 900 + 600 = 1,500
units. Hence, the total cost ofthis plan is (500)(465) + (1,000)(27) + (80)(1,500) =
$379,500, \vhich represents a sHbstantial savings over both tlle zero inventory plan and
the constant workforce plan.
'Πιε restIlts of the !inear programming ana!ysis suggest another plan that might
be more suitable for the company. Βεοειιεε the optimal strategy is το decrease the
workforce ίη .fanuary and blIild ίι back ιιρ again ίη May, a reasonab]e alternative
might be to πο! fire tlle 27 workers ίπ JanlIary and ιο Ιυτε fewer workers ίη May.
]η this case, the most efficient method for finding tJle οοσεοι nl1mber of wοrkeΓS
to hire ίπ May is ιο simp]y re-solve the lίnear pτogram, bl1t without the variables
FI, ... , F6, as αο firing of workers means that these variables are forced ιο zero.
(lf νου wish το avoid reentering the problem ίπΙο the computer, simply append the
old formυlation with the οοηετιείατε Ρι = Ο, Fz = Ο, ... , F6 = Ο.) The optima!
nl1mber of workers to hire ίη May turns οιιι το be 374 if ηο workers are fίred, and
the cost ofthe plan is approximately $386,120. This is only slightly more expensive
than the optimal plan, and has the important advantage of ποι requiring the firing of
any workers.
Problems for Sections 3.5 and 3.6
17. Formnlate Problem 13 as a Jinear program.
include all tlle reql1ired constraints.
Be sure ιο define aJJ variables alld
1 SO
Chapter
Three
Aggre~αte ΡΙαπnίιιι
3.6
18. Problem 17 was solved οα the computer. The lίnear ΡrogΓammίng ουιρυι of the
ΡrοgΓam is as follows:
SoI'Iinf:" Aggregαιc I)kInning
α. Write down tl1e complete
ΡrοblCΠL~by Uneαr Programfning:
lίnear programnling
'forιntIlation
Απ Example
ofthe
151
revised
problem.
b. Solve the revised problem using a lίηear programming code. What difference
does the new 11iringcost function make ίn the εοίυτίοη?
LΡ ΟΡΤ1ΜυΜ
FOUND
ΑΤ STEP
OBJECT1VE
1)
FUNCT10N
{~:~.{~)~~1;:,
.~~:;:
14
*22.
VALUE
36185.0600
VAR1ABLE
Η1
Η2
Η3
Η4
F1
F2
F3
F4
11
12
13
W1
w2
W3
W4
Ρ1
Ρ2
Ρ3
Ρ4
14
VALUE
6.143851
64.310690
.000000
116.071400
.000000
.000000
87.662330
.000000
.000000
.000000
.000000
106.143900
170.454500
82.792210
198.863600
850.000000
1260.000000
510.000000
980.000000
.000000
REDUCED
.000000
.000000
300.000000
.000000
300.000000
300.000000
.000000
300.000000
59.415580
189.285700
31.006490
.000000
.000000
.000000
.000000
.000000
.000000
.000000
.000000
120.292200
COST
Month
1
2
3
4
5
6
Belts
Handbags
Attache Cases
22
20
19
24
21
17
2,500
2,800
2,000
3,400
3,000
1,600
1,250
680
1,625
745
835
375
240
380
110
75
126
45
α. Using worke!" hours as an aggregate
measlIre of ρτοτίυοτίοη, convert the forecasted demands ιο demands ίη terms of aggregate units.
b. Determine another plan that achieves a lower cost than that determined ίη
part (α) while still retaining feasibility (tl1at is, having a nonnegative amount
of inventory οη hand at the end of each period).
b. What wotIld be the size of the workforce needed το satisfy the deInand for
the coming six months οη Γegular time only? Would ίι be to the conιpany's
advantage το bring the permanent workforce ιφ to this level? Why οτ WllYτιοι?
α. Formulate Problem 9 as a lίnear program.
c. Determine a production plan that meets the demand lIsing only regular-time
employees and tl1e total cost of that plan.
b. Solve the problem usi.ng a linear programming code. Round the variables ίη
the resulting solution and determine ιίιε cost of tl1e plan you obtain.
*20.
Number of
Working Days
The belts reqιIire an aνerage of two Ιιοιιτ» to ρτοοιιοο, the handbags three
Ιιουτε, and the attache cases six hours. ΑΙΙ the workers have the skill ιο work ση
any itetD. LeatheΓ-ΑlΙ11as 46 employees who each has a share ίη the firm and canσοι be fired. There are an additional 30 locals that are available and can be hired
for short periocls at higher cost. Regular employees earn $8.50 per hour οιι reglIlar time and $14.00 per hoιIf ου overtime. ReglIlar time comprises a εενεη-Ιιουτ
workday, and the regular employees will work as much overtime as is available.
The additional workers are hired for $11.00 per Ιιουτ and are kept οτι the payrol\
for at least one flIll mOI1t11.Costs of hίι-ίηg and fiι-ίng are negligible.
Because of the competitive natnre of the indlIstry, LeatheΓ-ΑIΙ does τιοτ want
to incur any demand back orders.
α. ΒΥ rounding the values of the variables using the conservative approach described ίη Section 3.5, find a feasible solution το the problem. Determine the
cost of the resulting plan.
19.
Leather-AII ρτοσιιοεε a lίne of handmade leather ρτοσυαε. Αι the present time,
the company is producing οηlΥ belts, handbags, and attache cases. 'Πιε predicted
demand for tl1ese three types of items over a six-montll planning horizon is as
follows:
α. Ροιπιιιίειε
d. Determine a ρτοοιιοιίοιι plan that ιιιίίίεσε only additional employees to absorb
excess demand and the cost of that plan.
Ρτοοίεπι 14 as a linear program.
b. Solve the problem using a lίηear programlning
and determine the cost oftl1e resulting plan.
code. RolInd off the εοίυιίωι
21. Consider Mr. Meadows Cookie Company, described ίη Problem 13. Suppose that
the cost ofl1iring workers each period is $100 for each worker ιιηιίl20 workers are
hired, $400 for each worker when between 21 and 50 workers are hired, and $700
for each wot·keI"hired beyond 50.
*23.
a. Formulate the problem of optiIllizing Ιeather-ΑIΙ's hiring schedule as a lineaI·
program. Defil1e all problem variables and includ.e whatever constraints are
necessary.
b. Solve the problem formulated
ίη part (α) using a lίnear progran1m ing
code. Ronnd all the relevant variables and detern1ine the cost of the
reslIlting plan. Compare Υουτ restIlts with those obtained ίη Problem 22(c)
and (ά).
152
3.7
Chapter Three
3.8
Aggτegaιe Pkιnning
ΤΗΕ LINEAR DEClSION RULE
Ατι interesting alternative approach το solving the aggregate planning problem has been
suggested by Holt, Modigliani, Muth, and Simon (1960). They assume that all the rele~
vant costs, including inventory costs and the costs of changing production leνels and
numbers of wοrkeΓS, are represented by quadratic functions. That is, the total cost over
the T-period plannil1g 1101'ίΖοηcan be written ίη the form
Σ [clW,
κ
+
C2(W, -
W,_t)2
+ C3(P,
- Kn,W,)2
+
C4P,
+ cs(l,
/, = J,_I
+ Ρ, -
D,
Ρ, =
- C6)2)
subject το
for 1 s t S Τ.
The valtIes of the constants Ct, C2, ... , C6 must be deteτmined for each parιicιιlar
application. Expressing the cost functions as quadratic functions rather than sitnple
lίnear fUl1ctions has some advantages over the lίnear programming approach. As quad- .
ratic functions are differentiable, the standard ruIes of calct1lus can be used ιο
determine optimaI solutions. The optimal solution will occur where the first partiaJ
derivatives with respect to each of the probIenl variables are zero. When quadIatic
functions are differentiated, they yield first-order equations (\inear εσιιετίοαε), which
are easy ιο solve. Also, quadratic functions give a more εοοιιτετε ερριωωηετίοιι ιο
general nonlinear functiol1S than do lίηear functions. (Any nonlinear flInction may be
approxinJated by a Taylor series ωφετιείοιι ίη Wllich the first two terms yield a qlIadratic approximation.)
However, the quadratic appIΌach aIso has one serious disadvantage. It is that
qIIadratic fllnctions (that is, parabolas) are symmetric (see FiglIre 3-7). Hence, the cost
, Number-units held
(bl
Cost of chonging
ίπνθηΙΟΓΥleνels
Σ (a"D +
n=O
I
II
+ bWt-t
+ cJ,-l +
d).
The terms απ, b, c, and d are constants that depend οτι the cost parameters. WI has a είπιilar form. The οοωριιιετίοαεί advantages of the lίηear decision rule are less itnportant
now than they were when this analysis was first done because of the widespread availability of comptlters today. Because lίnear programming provides a much more flexible
framework foτ formulating aggregate planning probleIns, and reasonably large Ιίιιεετ
ΡΓοgrams can be solved even οτι persohal computers, one would have ιο conclude that
lίηear programming is a preferable solution technique.
Ιιshould be recognized, however, that the texl by Holt, ModigJiani, Muth, and Simon
represents a laηdmark work ίη the application of quantitive methods ιο production planning problems. The authors developed a solution method that reslllts ίn a set of for.mulas that are easy το iIllplement, and they actually undertook the implementation of the
method. The work details the application of the approach 10 a large manufacturer
of hotIsehold paints ίη the Pittsburgh area. The analysis was actually implemented ίη
the company, but a subsequent visit Ιο the firm indicated that serious problems arose
Wllen tlle lίηear decision nlle was foIIowed, primarily because ofthe fiτm's policy of ποι
firing workers when the model indicated that they should be fiΓed. (See Vollmann,
Berry, and Whybark, 1992, ρ. 627.)
MODELING
FIGURE 3-7
Quadratic cost functions lIsed ίη deriving t!le Iίnear decision ru!e
(01 Cost of chonging
workforce leνels
1S3
ofhiring a given number ofworkers must be the same as thecost offiring the same number ofworkers, and tIle cost ofprodllcing a given nlllflber ofunits οη overtime τιιιιετ be
ιίιε same as the cost attached το the same number ofunits ofworker idle time. The probIem can be overcome somewhat by υοι setting the center of symmetry of the cost funcιίοη at ΖθΓΟ, but the basic problem of symmetry still prevails.
The most appealing feature ofthis approach is the simple form ofthe optimal policy.
For exanιple, the optimal production level ίη period t, Ρ" has the form
τ
,=ι
Mod,/ingManag,ment Bεhαoίor
MANAGEMENT
BEHAVIOR
Bowman (1963) developed θη ίηteΓestίηg technique for aggregate planning that should
be applicable Ιο otheI' types of problems as well. His idea is Ιο construct a sensible
model [οτ controIIing prodllction leνels and fit the ΡaraΠ1eters of tlle lnodel as closely
as possible Ιο the actllal ρτίοτ decisions made by management. Ιη this way the model
reflects the judgment and the experience ofmanagers and avoids a number ofthe problems that arise when using more traditional modeling methods. One such problem is
deternlining the accllracy of the assumptions required bythe lnodel. Another problem
that is avoided is the need to deteflfline vaIues of ίηριιι par31nelers that could be
diflicult to meastlre.
Let us consider the problem of producing a single product over Τ planning periods.
Suppose that Dt, ... , Dr are tlle forecasts of demand [or the next Τ periods and Ihat
production levels ΡΙ, ... , Pr must be determined. The silnplest I'easonable decision
rule is
Ρ, = Dt
[οτ Ι ,;; ι s Τ.
Trying Ιο match prodllction exactly Ιο demand will generally result ίη a production
sClledule that is 100 enatic. Production smootlling can be accomplished tIsing a decision rule of the form
Ρι = Dt
+ α(Ρι-ι
- Dt),
154 Chapter Three Aggrega", Ρ/αππίππ
3,9;])jsαggregaIing,Aggτeg~ιe
pian;
155 ~;
1,\";'1
whefe α iS a'sϊl1οΌthίng fa'ctor for pToduction,lτav.iJi'gthe property Ο S α S ι, (The,
effect of smoothing here ί5 the sameias,that of exptJnential.smoothing,
di,scus5ed ίη
detail ίη Chapter 2.) When α equals zero, this rule is 'identical to tlIe, -οτιε jU5t,presenti:d, When α =; Ι, the ρτοσυοτίοη. iJf,period ι'ί); exactly the 'same as the ρτοάυοιίοτι
ίπ period ,Ι -' Ι. 'Πιε choice of α provides a meanS'ο[φΙacίng
a relative weightootι
matching production with' demand Verst!'S'holdingproduction
constant from period
το perJod.
i'
- ,
.
Ιη addition το smoothing production, the firm al50 mJght be interested ίπ keeping ίτιventory leνels near εοτυε target leνel, say ΙΝ. Α model that smooths both inventory and
production simultaneously is
.
+ a(Pt-I""D,)
+ β(IΝ
-
h
-,
.
,
,
.
W,
the
where Ο S β S Ι measures
relatlve ~~ight pl~ced o~ ~m~CΊthing of inventory.
Finally, the model should incorporate ctemand forecasts., Ιιι this way, productίon
leνels can be increased ,στ decieased.ίή :anticipation of ε. change ίn the pattem of
demands. Α rule that includes smoothirig' of production' 'and inventory as well as forecasts of future demand is
'
I~" ,"~
= .οιαο, + .0088Dt+J.+
,
.
the foclιu]a
.'
..i'.'
'ι
" .
:007.1DH2,+ ,0054D'+3
j
+ α(Ρι-1
- D,)
+ β(ΙΝ
,0042D'+4
~" ::",>'1
<.1, i:''-'
Ά
;,,'1·1;
';~"J>,j;'
11'
~.'r ',(;
..Suj>pose tfιat Ιοτεσεετ demands for the corning
months are 150, L64, 185,
193, 167, )43,,\26,93,84,72,50,66.
Th,r current nιunber of employees is ]80,
and the cnrrent in~entory .Ievel ls 45 aggregate units,
α. Compute the "alues όf the aggregate productionleνeland
Σ αt-i+ID
+
.003IDt+5 + ,0023D,+6 + ,0016D'+7 + ,0012DH8
+, ,0009Dt+9 + 1,0006Όι+Ίο +.0005'Dt+
+ ,743W,_1 ...: ,0111-1 - 2,09:'
Ι+"
Ρι =
- ,009D'+8
- .46411_,1,+ 153,
~,~.Γ ~'-~_
.".
+
4
f
:008D,+6 -.OIQDI+7
- .068b,t9~:Ob7D;tlO'~:005Di+11
+.993W,rl
.013Dt
<"_",
,-Ι
- ,022D,+s:'"
1,-1),
+
. η= :463D;,+"'.':Δ4DΜ+;111Dι!t{+,Ο46Dι+':j'+
and the w6rkfofce level ί~ -peHod Ι froin
Ρι ='0,'
f
111'{,
24. .Πισ fΟrm,οfΦe bptίmal prod\1ction rule derived bY,Holt, Modig]iani, Muth, ii1ι~
,11." SinlfJI!( 196Q) fόr,-the case; pK):hepaint coIηpanY,indicates
that the ρτοτίυοιίοη Ιενο!
",';.,," inper;cid t ShofiTd,be computed from the'formula
ό
the number ofworkers
that the company should be usiηg ίη the curtent ρετίοά.
-/'-1),
ί=!
b, Repeat the οείουίετίου you performed ίη part (α) bnt ignore a1l terms with a
coefficient ιίιει Ιιεε an absolute va]ne·less thiιn οτ equal to ,01 when calculatlng
P,,-and less than
equal to ,005 when calculating W" How much difference
dbyou obSerνe ίη th'e,ariswer?-- ι,
'
,ι
This decision rule is arrived at by ,a straightforward common-sense analysis of
what a good production rule should be. It is interesting ιο note how similar this rule is'
to the lίnear decislon rule discussed ίn Section 3.7, which was derived from a mathematical model. Undoubtedly, the previous work οα the lίnear decision rule insplred
the form ofthis rule, Ιι.ίε often the case.that-the most-valuable feature οί-ε model ,js,
lndicating the best !oi'm' of ah optiriιal ;i:rategy, and ηΌι necessariΙΥ tli~
stra'feg~
itself,
This particulat model for Ρι requires determina!ion of αl, , , , , αn, α, β, and [Ν'
Bowman suggests that the νa]ues ofthese paranιeter~ be deterriιined by retrospectively
obserνing the system for a'reasonable period of time, and fitting the parameters το the
actual history of management actlons di.ιiil1g th~t tinιe bsing a'technique such as least
squares. Ιη this way the model,becomes a reflectioiι ofpast management behavior,
Bowτnah cbmpares the 'actiJal experience' of 11 'nuffiber of companies with the
experience thaΊ they wόuΙd' have had ιising his approach, and shows that ίη most cases ~"
there would have'been 'a substantial cost f~diJction, The modei merely emnlates man-~'·
agement behavior, so why shonld ίι outperform actu<il mariagement experience? The
theory is thata πIodel derived ίη this 'manner ί!>a reflection of managemeiιt making
"
ratlonal decisions, as most of the tlme the system Is stable, However, if a sudden unl1sl1alevent oc'curs, such as a fuuch~higher-than-anticipated
demand οτ a su,dden decline
ίη productlve capacity due, for example, ιο the breakdown of a machine οτ the loss of
key personηel, ίι ί5 likely that many managers would tend Ιο overreact. However, the
model would recommend production leνels that ar~ consistent,with past declsion making, Hence, the use of a simple but coηsistent model {οτ making decisions would keep
management ποιή paiιicking ίη an unusuaT situation. Althougli' the approacllis concep- ,
tually appealing, there are few reports ίη the literatιireof successful implementations of
such a techniqne,
jj~sr
στ
σ. Siίppose thatthe cιίrreίιtdem~ήd is re~lized ιο be Ι ~~"and the new forecast one
""year ahead is ,72'.'Usirig tlie approxim~tidn suggestediti part (b), deterlnίne'the
new values fQ,!the aggregate productίop ievel and the-nιιmber of workers. [Ηίιιι:
ί:
Yoil'fiίιist cΌm'ρute ,the'he~ι:'-QaΙUeγ,..:'{and 'assume1:h1it'the new value of Wt-I
thevaΙuecόΜΡutedίnΡart'(a):],
',γ,'
ι,
25"
is
Usirigthe f6116wίήg ν~lίΙ~~&[fuanagenieJι{~όefficίeiitSj'i'οr
Bowman's smoothed
, , . ,. -'
('Ι'
<,' • '.:','
"ι"."
';;'_~/'
\
','"
,
--'---',." . ι
Ρ'i'οd!!'ctίοήmodel, ~.eteilniiJe the pl'Oductί?iilevel for which Harold Grey (described
in Problem, 9) shoίlld plan ίη the cοmiήg year: .A:.ssumetHat the cunent production
level i~ Ί :Sb;Odbpackag~~,' /
,,'
'Ι,ι,"
,t"
,οι ';" ,3416,;
αs,= ,0023,'
,~)I
α2 =,Ι2l.1"α3
. α = ;6",
=
.0?56",
β = 3,
,!j
α4 = .0663,
ΕΝ = 40,000,
26, α, Discnss the similarities and the differences betwee~ th'e linear decislon rule of
Holt, Modiglianl, Muth, and Simon and Bowman's management coefficients
approach, '
b. Discuss the relatίve advantages and disadvantages
of linear decision rules
versus lineaτ program,ming for solving aggregate planning problems,
DISAGGREGATlNG AGGREGATE PLANS
As we saw earlier ίη ,this chapter, aggregate p]anning Inay b" done at several leνels of
the fiπn. Example 3,] considered a single plant, and showed how one might define
an aggregate unit to includ,e the slx different items produced at that plant. Aggregate
~·!·rl, .,.-,~.----~
156
Chapter Three Aggregate P/anning
planning might be done for a single plant, a product family, a group of families, οτ fl
the firm as a whole..
There are two views of production planning: bottom-up οτ ιop-down. The bott~
υρ approach means that one would start with individual item production plans. Th
plans could then be aggregated up the chain of products το produce aggregate plans.
The top-down approach, which is the one treated ίπ this chapter, is ιο start with an aggregate plan at a high leνel. These plans would then have Ιο be "disaggregated" Ιο proσυοε detailed production plans at the plant and individual item leνels.
Ιι is σοι clear that disaggregation is an ίεευε ίη a11οίτοιυυετεηοεε.
[f the aggregate
plan is tIsed only for πιεοτο planning purposes, and τιοτ for planning at the detaillevel,
then one need υοτ worry about disaggregation. However, ifit is important that individtIal item ρτοάυοτίοιι plans and aggregate plans be consistent, then ίτ might be necessary
to consider disaggregation schemes.
The disaggregation problem is similar to the classic problelJl of resource allocation.
Consider 110W resources are allocated ίη a university, for example. Α tIniversity receives τενευυεε from τιιίιίοτι, gifts, interest οπ the endowment, and research grants,
Costs include salaries, maintenance, and capital expenditures. Once an annual budget ,"
is determillecl, each school (arts and science, engineering, business, law, etc.) and each
btIdget center (staff, maintenance, buildings and grotInds, etc.) woLIld have Ιο be allocated its share. Budget centers would have to allocate funds Ιο each oftheir stIbgrotIps.
For example, each school would allocate funds το individual departments, and departments wotIld allocate funds ιο faculty and staff ίη that department.
Ιη tlle manufacturing context, a disaggregation scheme isjLIst a means of allocating aggregate units to individtIal items. JtIst as funds are allocated οιι several leνels ίη the university, aggregate units might have το be disaggregated at severallevels ofthe firm. This
is the idea behind hierarchical ρτοουοτίου planning championed by several researchers at
ΜΙΤ and reported ίπ detail ίη Bitran and Hax (1981) and Hax and Candea (1984).
We will disctIss one possible approach ιο the disaggregatioll problem from Bitran
and Hax (1981). StIppose that Χ" represents the numbel' of aggregate υυίιε οί ρτοουοτίοιι [οτ a particular planning period. FtIrther, ευρροεε that Χ* represents an aggrega"
ιίοτι of η different items (Υι, Υ2, ... , Υιι)' The qtIestion is how το divide up (i.e., disaggregate) Χ* anlong the n items. We know that holding costs are already included ίη the
determination of Χ", so we need τιοτ incltIde them again ίπ the disaggregation scheme.
Suppose that Kj represents the fixed cost of setting υρ Ιοτ ριοοιιοιίου of~, and λ) is the
annual usage rate for item j. Α reasonable optimization criterion ίη this context is το
choose Υι, Υ2, ... , Yn το millimize the average annual cost of setting up [οτ ρτοοιιοιίοη.
As we will see ίη Chapter 4, the average annual setup cost [οτ item j is KjA/~.
Hence, disaggregation
requires solving the following mathematical programming
problem:
J
Minimize
KjAj
ΣΥ
j=1
}
subject to
J
Σ~=x*
j=I
and
ajs,~s,bi
[οτ Ι s, j s, J.
ρ.
;π
, ':Ι:ii;i<,-:
':.""
",
The upper and lower bounds οη Yj account [οτ possible side constraints οη the producτίοιι leνel [οτ itemj.
Α number of feasibility issues need ιο be addressed before the family τυη sizes Υί
are further disaggregated ίυτο lots for individual ίτειυε. The objective is ιο schedule
the lots for individtIal itelns within a family so tllat they run out at the scheduled setup
time for the family. Ιη this way, items within the same family call be produced within
the same productioll setup.
The concept of disaggregating the aggregate plan along organizational lilles ίn a
fashion that is collsistent with the aggregation SC!lenle is an appea.1ing one. Whetller ΟΓ
ηοΙ the methods discussed ίn this section provide a WΟΓkable link between aggregate
plans and detailed item schedules remains to be seen.
Anotheτ appτoach Ιο the disaggregation problem has been explored by Chung and
Krajewski (1984). They develop a mathem.atical progra.mming fOllnulation of the
problem. lnputs Ιο the program include aggregate plans [οΙ' each product [ωηίΙΥ· This
includes setup time, setup status, total production leveI for the family, inνentory leνel,
workforce leνel, overtime, and regular time availability. Tlle goal of the analysis is to
specify lot sizes and timing ofpJ'Oduction runs [οτ each individual item. consi.stent with
the aggΓegate information [οτ the product family. AltllOUgh sllch a fOlllll1latiol1
158
Chapter Three
3.11
Aggr'gaIe Planniiιg
PrαccicαlConsideranons
159
•.~,.).,:\;;1'
Ρrόνid~s"~'~οt~~tίaJ link between the aggregate plan and the master production SChi
ule, the resulting mathematical program requires many inputs and can result ίτι a very "
large mixed integer problem that could be very time-consuming to solve. For these re~:.;
sohs, firms are unlikely to use such methods when there are large numbers of ίndίνΓdual items ίη each product family.
Strategies for market penetration, local content rules, and quotas constrain product
flow across borders.
Ρτοουοι designs n1ay vary by national market.
Centralized control of n1ιιltinationa! enterprises creates difficlllties [οτ several reasons, and decentralized control requires coordination.
CtIJtural, language, and skil1 differences can be significant.
Problems for Section 3.9
27. What does "disaggregation
of aggregate plans" mean?
28, Discuss the following quotation made by a production manager: ''Aggregate plahning is useless ιο me because the results have nothing to do with my master
production schedule."
3.1 Ο
PRODUCTlON
PLANNING
'ί'
ΟΝ Α GLOBAL SCALE
Globalization of manufactu.ring operations is commonplace. Many major οοτροτε-"
tions are now classified as multinationals; manufacturing and distribution activities
routinely cross intemational borders. With the globalization of botl1 sources of pτo~
duction and markets, firms must rethink production p!anning strategies. One issue explored ίη this chapter was smoothing of production plans over τίηιε; costs of increas~
ing οτ decreasing workforce levels (and, hence, production leve!s) play a major τοίο '
ίπ the optimization of any aggregate plan. When formu!ating g!obal production
strategies, other sJnoothing issues arise. Exchange rates, costs of direct labor, and tax .
structure are just some of the differences among οουτπτίεε that πιιιετ be factored ίηΙΟ
a global strategy.
Why the increased interest ίπ global operations? [η shorι, cost and competitiveness.
According to McGrath and BequiI!ard (Ι 989):
The benefits of a properIy executed international manufacturing strategy can be νΟΓΥ
substantial. Α well deνeloped strategy can haνe a direct impact οη rhe financiaI performance
and uitimateiy be reflected ίη increased profirability. Ιη Ihe elecIronic industry, rhere are
exampIes of companies attributing 5% Ιο 15% reduction ίη cosI of goods sol<l, 10% Ιο 20%
increase ίη saIes. 50% Ιο Ι50% improνement ίη asset UΙilizaIion, and 30% Ιο 100% increase
ίη inventory tu.rnover Ιο their intemationaIization ofmanufacturing.
Determining optimal globalized manufacturing strategies is clearly a daunting problem for any multinational firm. One can formulate and so!ve mathematical models
similar το the lίnear programming formulations of the aggregate planning mode!s
presented ίπ this chapter, but the results of these models must always be balanced
against judgment and experience. Cohen et al. (1989) consider such a model. They assume multiple products, plants, markets, raw materials, vendors, vendor supply CO!ltract alternatives, τίιηε periods, and countrίes. Their forn1ulation is a large-scale mixed
integer, nonlinear program.
One issue τιοι treated ίη their model is that of exchange rate fluctuations and their effect οη both pricing and production planning. Pricing, ίη parιicu!ar, is traditional1y
done by adding a πιετκιιρ το υπίι costs ίπ the home maΓket. This cornpletely ignores
the issue of excl1ange rate fluctuations and can lead. 10 unreasonable prices ίο sorne
countries. For example, this issue has arisen at Caterpillar Tractor (CaterpillaI" Tractor
Company, 1985). [π this case, dealers all over the WΟΓldwere bil!ed ίπ U.S. dollars
based οη U.S. production costs. When the dol1ar was strong relative ιο other cιιrrencies,
ΤΟΙθίΙprices cl1arged το overseas customers were ιιοι competitive ίπ local markets.
Caterpil1aI" folInd itself losing market share abroad as a result. Ιη the early Ι980s the
firnl switched το a 10caΙΙy cornpetitiνe pricing strategy το counteract this problern.
The ηοιίοιι that l11anufacturing capacity can be used as a hedge against exchange
rate fluctuations 11as been explored by several researchers. Kogut and Kulatilaka
(Ι 994), [οτ example, develop a rnathematical model [οτ determining when ίι ίδ optirnal
Ιο switch prodllction frorn one locatίon to another. Since the cost of s\vitching is
assuIned to be positive, there mnst be a snfficiently large difference ίπ exchange rates
before switching is recoll1mended. As an example, they consider a situation where a
firl11can produce ίΙδ product ίη either the United States οτ GerιηaπΥ. Ifproduction is
clIrrently being done ίπ οπο locatίon, the model provides a lηeans of detern1ining if ίι
is econon1ical ιο switch locatίons based οη tlle ΓeΙatίνe strengths ofthe eUIΌand dollar.
Wllile stIch lηοdels are ίn the early stages of developlnent, they provide a n1eans of
("ationalizing international prodnction planning strategies. SiIl1ilar issues have been
explored by Ηucl1ΖeΓιneίer and Cohen (1996) as \vel1.
Cohen et al. (1989) outline some ofthe issues that a firm must consider when planning
production leve!s οπ a wor!dwide basis. These inc!ude
Τπ order ιο achieve t!1e kinds of economies of sca!e required Ιο be competitive
today, mlIltinationa! plants and vendors must be managed as a g!obal systen1.
Duties and tariffs are based οη material flows. Their impact n1ust be factored ίπιο
decisions regarding shipments of raw lηaterίal, intermediate prodllct, and finished
product across national boundaries.
Exchange rates fllIctuate randomly and affect production costs and pricing decisions
ίπ countries where the product is produced and sold.
Corporate tax rates vary widely from one country to another.
Global sourcing ιηust take ίηΙο account longer lead tίιηes, lower unit costs, and
access to new technologies.
PRACTlCAL CONSIDERATIONS
Aggregate planning can be a valuable aid ίπ plaoning prodlIction and workforce levels for
a company, and provides a Ineans of absorbing deιηand fluctuations by srnootlling workforce and plΌduction leνels. There are a number of advantages for planning οπ the aggregate lονοΙ oveI"the detai! level. One is that the cost ofpreparing forecasts and deteΓInining
productivity and cost ΡaraιηeteΓS οη an individllal iten1 basis can be prohibitive. T11ecost
of data collection and ίnρυ! ίδ probabJy a 1110ΓΟ
significant drawback of ]aΓge-scale Il1atheιηatίcal ΡlΌgΓaιηmίng fΟΠl1l1latίοns of detailed production plans than is the cost of
actllally perfoΓIl1iI1gtl1e COIl1plltations.Α second advantage of aggΓegate planning is the
relative improvement ίη forecast aCCUΓaCΥ
that cal1 be achieved by aggregating items. As
160
Chapter
Three
Aggregalc P"'""ing
3.13
\ve saw ίη Chapter 2 οα forecasting, aggregate forecasts are generally more accurate than
3
indivi<lual forecasts. Finn.lly, an aggregate planning tramework allows the manager, ΙΟ
see the "big picture" and υοι be unduly influenced by specifics.
With all these advantages, one would think that aggregate planning would have an
important place ίη the planning of the production activities of most companies. How.:,.
ever, this does ηοι appear το be the case. 'Πιετε are a ntImber of reasons for tl1e lack or
interest ίη aggregate planning methodology. First is the difficlIlty of properly defining art
aggregate ιιηίι οί ρτοσιιοιίοιι. There appears το be ηο simple way ofspecifying how ίηdividtIal itenls shotIld be aggregated that will work ίη all situations. Secolld, once an ag- .
gregating scheme is developed, cost estimates and demand forecasts for aggregate units
are required. It is difficult enough το obtain accurate cost and deInand information for
real units. Obtaining such information οα θη aggregate basis cotIld be considerably
more difficult, depending ου tl1e aggregation schetne tbat is used. TlΊίrd, aggregate planning models rarely reflect the political and operational realities of the environnlent ίη
which the company is operating. Assuιning that workfoioce levels can be changed easily
is probably αοι very realistic for most companies. Finally, as Silver and Peterson (Ι 985)
suggest, managers do τιοι want ιο rely οη a mathematical Inodel for answers ιο the
extremely sensitive and important issues that are addressed ίη this analysis.
Another issue is whether ΟΤ αοι the goals of aggregate planning analysis can be
achieνed through methods other than those discussed ίη this chapter. Schwarz and
Johnson (Ι 978) clainUhat the cost savings achieved lIsing a lίηear decision rL1lefor
aggregate planning could be obtained by improved Inanagement ofthe aggregate inventory alone. They substantiate their hypothesis using the data reported ίη Holt, Modigliani,
Muth, and Simon (Ι 960), and show that fortl1e case ofthe paintcompany virtual1y all the
cost savings ofthe lίηear decision rule were a result ofincreasing the bIIfFer inventory and
ηο! ofmaking major changes ίη the size oftl1e labor force. This single comparison does
ηοΙ verify the authors' 11ypothesis ίη general, but suggests that better inventory management, using the methods Ιο be discussed ίη Chapters 4 and 5, could return a large portion .
ofthe benefits that a company might realize with aggregate planning.
Even with tlΊίs loηg lίst of disadvantages, the mathematical models discussed ίη this
chapter can, and should, serve as θη aid to production planners. AlthoLIgh optimal solutions Ιο a mathematical model may ηοι be true optimal solutions Ιο the problenl that they
address, they do provide insight and could reveal altel'natives tl1at would ηο! otl1erwise
be evident.
3.12
HISTORICAL NOTES
The aggregate planning problem was conceived ίη ηη important series οΓ papers that appeared ίη the mid-1950s. The first, Holt, Modigliani, and Simon (1955), discussed the
structLIΓeoftl1e problenl and introduced the quadratic cost approach, and tl1e later sttIdy
ofHoJt, Modigliani, and Muth (Ι 956) concentrated οη the coInplltational aspects ofthe
model. Α complete description ofthe metl10d and its npplication Ιο production planning
Γοτ a ρηίηι coInpany is presented ίη Holt, Modigliani, Muth, and Simon (1960).
That prodIIction planning problems could be formIllated as linearpΓOgrams appears Ιο
have been known ίη the early 1950s. Bowman (1956) discussed tl1euse οΓa transportation
model for production planning. The particular lίηear programming forιnlIlation of tlle
aggregate planning problem discllssed ίη Section 3.5 is' essentially tl1e same as the one
3
More precisely. ίl the individual
lorecast
error lor the aggregate
lorecasts are πο! perfectly
correlated,
then the standard
will be less than the sum ΟΙ the standard deviations
ofthe
deviation
indlvidual
ΟΙ the
Summary
161
deνeloped by Hansmann and Hess (Ι 960). Otl1er lίηearprogrammίηg formulations oftl1e
production planning problem generally involve nlu1tiple products ΟΓmore complex cost
structures (see, for example, Newson, Ι975η and Ι975b) ..
More recent work ου the aggregate planning problem has focused οιι aggregation
and disaggregation issues (Axsater, 1981; Bitran and Hax, 198 Ι; and Ζοίίει, 1971), the
incorporation oflearning curves ίnΙο lίnear decision rules (Ebert, 1976), extensions ιο
al10w for multiple prodncts (Bergstrom and Smith, 1970), and inclusion of marketing
and/or financial variables (Damon and Schramm, 1972, and Leitch, 1974).
Taubert (1968) considers a tecllnique he refers Ιο as the search decision rule. 'Πιο
method requires developing a computer simulation model ofthe system and searcl1ing
the response surface using standard search techniques Ιο obtain a (υοι necessarily ορtimal) solution. Taubert's approach, which was described ίη detail ίη BlIffa and Tanbert
(1972), gives results that are comparable Ιο those of Holt, ModigJίani, Μιιιο, and
Simon (1960) for the case ofthe paint company.
Kamien and Li (1990) developed a mathematicalInodel
ιο examine the efFects of
subcontl'acting ου aggregate planning decisions. The authors show tllat under certain
circumstances ίι is preferred το producing in-house, and proνides an additional πιεεηε
of smootl1ing production and workforce levels.
Determining οριίωε! production levels Γοι'all products Ρrοdιιced by a large firm can be ηη
σιοτυιουε undertaking. AggI'egate pIannil1g addresses this problem by assuIning that individIIal items can be grouped together. Ho\vever, finding an effective aggregating scheme can
be difficult. The aggregating scheme chosen must be consistent with the structure of tlte
organization and the natιιre of the products produced. One particular aggregating scheΠJe
that 11asbeen sttggested is itenIs,jamiIies, and types. ltems (or stockkeeping ttnits), representing the finest leνel of detail, wοιιld be identified by separate part numbers ..Families are
groups of items that share a commollIllanufacturing setιιΡ, and types are natural grΟΙιΡίngs
of families. This particular aggregation scheme is ΓθίΓIΥgeneral, but there is ηο guarantee
that ίι wiJl work ίη every application.
/η aggregate planning one assun1es deteπιιίnίstίc den,and. DeIlland forecasts over a
specified planning hOI'izon are required ίηριιΙ This assLImprion is ηο! made for the sake of
realisIll, bllt ιο allow tl1eanalysis ιο focus οη the changes ίη the demand that are systematic
rather than random. Tl1e goal of the analysis is to determine the nllmber of workers tl1at
should be employed eaCI1peI'iod and the nunlber of aggregate lInits that SI10uldbe produced
each period.
Theobjective is to Ininimize costs ofproduction, ρηΥΓΟΙΙ,
holding, and cl1anging the sizeof
the workforce. The costs ofmakillg changes are generaIIy referred Ιο assnlOoI/,il1g C08tS.Most
ofthe aggregate planning Inodels discussed ίη this chapter assume that all the costs are Iineaι"
jitnctions. This means tl1at tl1e cost of hiιing an additional worker is the same as the cost of
hiring the previous work.er,and the cost ofholding αη additional ιιηίι of inventory is the same
as the cost ofholding the previous ιιηίι ofinvenιory. This assttmption is probably a reasonable
approximation for Inost real systems. Jt is unlikely that the primary probleIn with applying
aggregate planning metllodology Ιο a real sitIIation will be tllat the shape ofthe cost functions
is incorrect; ίι is more likely thatthe primary difficulty will be ίη corτectly estimating the costs
and otl1errequired ίηριιΙ information.
One can fίnd approximate solutions ιο aggregate planning problems by gΓaΡΙ1ίcaΙ
nletl1ods. Α prodLIction schedule satisfies demand ίΓ the graph of cIImulative prodlIction
lίes above the graph of cumulative dCΠland.As long as all costs aI'e lίηear, optimal solιιtίοns4 can be found by lineaι' pl'lJg1'OrιIIning. Because solIItions of lίηear programIning
forecast errors.
4
That is. optimal
subject ιο rounding
errors.
162
Chapter Three
AggτegaIe
I'lιιnning
probIems
may be fractions,
way ιο round
appropriateness
must
Linear
ΟΓ
all oftbe vaήabΙes
ίε αοι aIways obvious.
be considered
οοετε and constraints
intangibIe
the probIem
must be rounded
Once
ίη the context
solution
of the pIanning
that might be difficult
ιο integers.
an οριίοιεί
probIem,
its
as there are
ίο include directiy ίη
impossible
ΟΓ
The best
ί$ found,
formulation.
also can be used το determine
programming
individual
Inight
some
the variables
item levei, but the size ofthe
Inake such
progranlming
an approach
fornJuiations
resulting
unreaIistic
ρτοσυοιίοιι
optimai.
problem
ievels
at the
and the ίnρυι data requirements
for large οοιυρεηίεε.
However,
detailed
Iinear
could be used at the faΠ1ίΙΥ level ratller than the individual
item
level for Iarge firms.
EarIy
work
by Holt,
fUl1ctions be quadratic
ροΙίCΥ is a simple
assumption
Modigliani,
functions.
lίnear
of quadratic
Muth,
and
With quadratic
function,
(1960)
required
ιοο
is probably
that
the form of the resulting
as the lineαI' decision
known
cost functions
Simon
costs,
optimal
πι/e. However,
το be
restrictive
the cost
accurate
for
the
πιοει
reaL ρτοοίσιηε.
lnanαgenIent coefficients modei fits a reasonable matllematical
model of the
το the actual decisions made by management over a period of time. This approach
Bowman's
system
Ilas the advantage
managemel1t
ofincorporating
ρετεοτυιεί
management's
υιτο the
intuition
to be COl1sistent ίπ future decision
model,
and encoLIrages
making.
ρίωιε will bc of lίttle ιιεο ΙΟ the fim) if they cannot be coordinated
witli
(ί.ε., the master ρτοτίιιοτίου schedllle). The probLcm of dis'agg,.,i"
gαting aggregαte ρ/απ!>' is a difficu1t one, bllI one that must be addressed, ί f the aggregate
pIans ετε ιο have νείιιε το the firιn. There Ιιενε beel1 some mathematical
programnling
Aggregate
detailed
item schedllles
formlIIations
of the disaggregating
gation schenles
problem
have yet το be proven
\.
/' •.-. ~
,.7 ::. ~" ;iI'
"
)
J ΙΛ ι.,
f
,. ίΊ:,·Η
Η,,'
\Ί.-
ίπ tl1e lίterature.
but these disaggre-
~~,;
·':ι,ι::. ': :4' ι ι~':,ι:~I
ι.
'
,An aggr'~gil\e, planning
P,10de( j.s"beifJg"c~?~i,ίle!~d
f9r the follo\,:!ng' applicaiions.
Suggest an agg'regate measιire Όf Ρrόductiοi:ι and discuss tl)(: difficulties
ofapplying
Ά 'aggl'eg,\~e
'; '.d.' Planhing
":'>,1
,.<'.'
.rI,
29.
~,ι
::t; :/.,"
. ,1 'Ι" ;(>
Additional
Problems ου:
. Agg'l-egate
Pla:,rt~iiig:1H
suggested
ίη practice.
.
pLaruling in~ach'c~~e',:,::,~':,K~'
for tlιeΆsίΖe 0'ft11e',facult),i'jh:a
·1). "DeteΙ'li1ίηίng<\ί15rkfοr6eΨeqiiιre'i"'~J1'tg'fδr
j
';'~>';~';,'",ι} :'-' ,,'Ι,
university:';1
1i'trave!
ii'geilcy.'!/
. c. Planning workt;orc$, a~d ~rodUc\t}?,~.,!~y~ls ,ί~~,~sn;~~D\:e's'sing plant ihat proctuces
_ canned sardiιi.e~; ~c~ovies,
kippers,a~~'Smbk!;d
b);'sters.
_
';
"'_l: '_,:1
';' .•
,;:ζ~.:~.J_ ' .."J~~'
ι, ι •• ,[:'
.
..'
30. Α plan! )ηanager ,emploxed bya 'I1ational ;produ~er of kitc!)en; products is 'planning
j'
wοι;kfοrceleveΙs
ιιη average
~
f,or the' next three monihs.
of [our labor
Αιι .aggregate
hours. ηοreC,astdernaίιds'
un,j,t of ρτοτίιιοτίωι
[οτ th,e thι;ee-mοί1tli
follows:
~,
Forecasted
, Demand
(ίn aggreunits)
',
gate
1"Ι,
' ;,',
,(1,2;400
\1
;..ι
,;
,,~;.
requires
110τίΖοη 'art: as
)jΙ,.Ί
,
.
11
,
,iI ,Ι
'
Additi011αl Problcms
164 Chapter Three Aggregace Pωηηίη,~
σ, Deteπnine the mlnlmuln constant workforce p]anfor Parls Palnt atid the cost
p]an, Assume that stock-outs arc υοι al1o\ved.
c. If.Paιis were able 10 back-order excess den1and al a οοει of$2 per ga]]on pe~
ter, detcrιnlne a mi~imuln constant workforcep]an that Ιιοκίε Ιοεε lnventory t
the,plan you foundinpati (α), but incurs stock-outs iηquaΓteΓ 2. Detcrminc the'
of Ilte new plan.
//~
}\υΙ-
(37,
\,'-
t]1is ρίεη.
Month
ΑΡΓίl
May
ho
Foπnu]ate Ρτοοίεηι 32 as a ]jnear program, Assume tl1at stock-outs
al1o\ved,
June
July
August
September
.
35, σ, Formulate Prob]cln 33 as a linearprogram.
b. S()lvc the ]inear progra111.Round the variables arid determlne the cost oftlte reslJlt
plan,
'
Ι:,:
36, The Μτ, Meadows Cookie Company can obtain accurate forecasts fOr 12.tn
based οη fιrm οπίετε. These forecasts and the nnmber of workdayspermon\h
a
fol1o\vs:
Demand Forecast
cookies)
(ίη thousandsof
1
2
3
4
5
6
7
8
9
10
11
12
850
1,260
510
980
770
850
1,050
1,550
1,350
1,000
970
680
14
21
23
24
21
13
During a 46-dayperiod
when there \vere Ι 20\vorkers,
the fir111
1,700,000 cookies. AssunJc tl1at there arc 100 workers employed at the hMinninO"
IJ10nth 1 and zero startlnginventory,
ά.. Fίnd the minlmum constant
workforce
needed ιο meet montl1]y denland.
11
22
20
23
16
20
Forecasted Demand
(ίπ hundreds of cases)
85
93
122
176
140
63
α. Based οτι tl1iSintorιnatlon, find the ιιιίηίωΙΙΙl1 constant workforce plan for Yeasty
over the six l11ontI1s,and detcrmine I1iring, firing, and holding οοειε associatcd witl1
that ρίεα,
b. Suppose tl131ίι takes 01lCηιοιιι]: to ΙΓαίη a nclv WΟΓker,How \νίΙΙ that affect youι·
solution?
Workdays
26
24
20
18
22
23
Production
Days
As of ΜaΓcΙ131, Yeasty 11ad86 workers οη the payroll. Over a pcriod of 26 WΟΓking days \vhen tl1ere were 10Οworkers ου tl1e payrol], Yeasty produced 12,000 cases
οfbeCΓ. 'Πιο cost ιο hirc each \vorkcr is $125 and tl1e cost of laying off each \vorkeI"is
$300. Holding costs ευιουυι Ιο 75 cents per case per ηιοητίι.
As of Marclt 31, Yeasty cxpccts ιο have 4,500 cases ofbecI" ίη stock, and ίι i'ants to
maintain Η nlininnlm bttffer invcntory of ] ,000 cases each month. Ιτ plans Ιο start
Οοιοοετ with 3.000 cases οη hand.
'~"'!;X
b. So]ve the linear program, Round.the variables and determine the cost ofthe resιJlti
Month
165
11eYeasty ΒΓe\viηg Coιnpany produces ε popular Jocal beeI· known as Ιτοη Stomach.
Becr saJes are solnew]lal seasonal, and Yeasty is planl1ing its ρτοουοϋοη and worktoI·ce
levels οο MarcJ1 3 J for the next six ll1onths. The dell1and forecasts aI'c as follo\vs:
33. Consider the problem of Parls Palnt presented inProblem 32. Suppose that the
Ιιεε the capacity ιο elnploy a maxlmum of 370 workers, Suppose that regu]ar,ti
employee costs are $12.50 per Ιιοιιτ, Assume sevcn-hour days, five-day wceks, a
four-week months. Overtime is paid οη a time-and-a-ha]f basls. Subcontracti~g
avai]able θ! a cost of$7 per gallon ofpaint produced. Overtlme is ]imited Ιο three
per employce per day, and πο more than 100,000 gaJlons can be subcontracted.
any quarter. Deteτmine a policy that mects the demand, and the cost of that ρ()Ιί
(Assume ειοοκ-οιιιε are nο! aJlowed.)
plan.
l'/anηing
c. Construct a ρlηη that changes the Ιενε! 01" ιίιο workforce το ιυοει montl11Y
del11and as c]osely as possible, Ιη designing the logic for your calcu]atiol1s,
be sure that invcntory does υοι go negative ίη any month. Dctermine the cost of
b. Determinc the zcro invetιtdry p]an thathiresaιid
fires wοτkers eac]l
match deInand as c]osc]y as ,possib]e and thc cost of that p]an.
G.
A,ggreJ.:"are
δ, Assuιne cI = $0.1 Οper cookie per month, cll = $] 00, al1d cp = $200. Add colulnns
that give t]le ουυιυίετίνε on-h.and invenlory and invel1tory cost. W]lat is the total
cost of the constnnt workforce plan?
Assume that Parls currently has 80,000 gaJlons ofpaint ίη inventory and wou]d
the year with a11lnventory of at ]east 20,000 gal1()ns.
ιο end
34.
οη
('. Suppose tl1at the maxiI11tI111
numbeI" ot' WΟΓkerstl1at the colnpany can expcct to
bc able 10 hire ίn one ιηοηιΙ1 is 10. How wil1 that Hffect youl" solution to
ρπτΙ (α)?
Ρι
vt'
c!. DeΙeΓnJiηe the zcro invcl1tolOY
ρΙπη and t11Ccost of that ρΙηη. [Υοιι ΙΙΙΗΥ ignoI·c tl1C
conditions ίη pHrts (b) and (c').)
I~
\ 38.
α)
Fonl1tIlHtc 111CΡΓοblenι of p]anning Ycasty"s procluction leve]s, as dcscribed ίn
''-..J' PΓOb1em37, as a linear prognlll1. [Υοιι ιηaΥ ignΟΓCthe conditions ίη parts
(b)
and (ι-).)
6, Solvc tllCrcstIltlng JincaI"ριυgram. RotIIld tl1Cnppropriat.e vHriabJes Hnd detcrn1inc
tl1Ccost οΓ ΥΟΙΙΓsoltItion.
c. Suppose Yeasty does ηο! wish to .fire3l1Y\Vorkers. \Vl1at ls IJ1Coptimal Ρ]ΗΙΙ sub.iect
1001is cOl1stralnt?
39. Α local canning COll1fJanyselJs c3nned vegelables to a supermarket chain ίn tl1C
Minneapolis al'ea, Α typical case ot' canned vegetables reqtIlres an average οΓ 0,2 clay
of labor to pJ"Oducc. The aggregate inventory οη hand "ι 111eend of .func i5
166
Chapter Three
AIW<garc
['lanning
Appendix 3-Α 167
800 cases. The demand for thc vegetables can be accLIrately predictccl for about
18 I110nths based ΟΙ1 o,-ders received by the firn1. The predicted demands for the
next 18 months arc as foIIows:
Month
July
Augu5t
September
October
November
December
January
February
March
Forecasted
Demand
(hundreds
of cases)
23
28
42
26
29
58
19
17
25
Workdays
21
14
20
23
18
10
20
14
20
Month
Month
29
33
31
20
16
33
35
28
50
20
22
21
18
14
20
23
18
10
aggregate plan and
b. ΟενεΙορ a spreadsheet ιο find a ρlηη that hires and. fi[es workers I110nthly ίπ
ordcr το n1inimize inventory costs. Dctern1ine the total cost of ΙΙ13Ι Ρl3η as
\veII.
c. Graph ιΙ1' ctImulativc nct deI11and and, using thc grapl1,find ,1ΡI,lη that cl1anges (he
workforce level exactly once during tl1e 18 l110nths and 11as" 10,,'e1"cost than either
0[t11C plans considcI'ed ίπ parts (11) and (b).
::f~
40. Consider tl1e lίnear ρτοοικιίοη rule of Ηοίι, Modigliani, Muth, and Sil11on, appcaring
ίη Problem 24. Design 3 spreadsheet το οουιρυιο Ρ, and W, [ΓΟΙ11deI11and Ιοτοοοετε. Ιηζ
particular, supposc that ει tl1e end ofDccember
1995 the workforce \Vas ΗΙ 180 and the
inventory was ει 45 aggregate units.
a. Using the forecasts
of demand for the nlOnths J"Hnuary tllfolIgh DeccIllber 1996
given ίη ProblcIll 24, find tl1e recommended levels of aggregate production and.:
workforce size for January.
',
*b. Design ΥΟΙΙΓ sprcadsheet
so tl1al ίι can cffίcicntly
tIρdate recoI11I11cnded.·
levels as demands ετο realizcd. Ιη partictIlar, ειιρροεο ιίιετ the dCI11l1ndfo;'
JantIary was aCttIally 175 and tl1e forccast for January 1997 was 130. WJlat
are the I'ecoI11Illcndcd levels οΓ aggregate ρτοουοτίο» and wοrkfΟΓCC size for
February? (Ηίαι: Stol"C the denland forccasts and coetΊicicnts
ίη separate,
οοίιυυτιε and modify ιίιο conlptIting formtl1a as nc\v data a1"C appcnded Ιο
the old.)
.
*c. Πιε actιιal realizations of dCll1and and new forecasts of demancl for tl1CreI11ainder'
of 1996 are given ίη the followiI1g table. Detcrmine updatcd valnes fOΓ Ρ, and W,
each Inonth.
148
200
190
180
130
95
120
68
88
75
60
160
175
120
190
230
280
210
110
95
92
77
Glossary of Notation for Chapter 3
Tl1e ΓΊτη1currently has 25 \vorkcI·s. The cost ofhiring and training a new worker is
$ 1,000, and ιίιε cost το lay off onc worker is $1,500. The firn1 cslin1ates a cost of$2.80
Ιο store a casc of vegetables for a ηιοηιίι. Tl1ey woLIld like ιο have 1,500 cascs ίη
inventoIOYat the cnd oftl1C 18-ιηοηιl1 planning horizon.
a. Develop a spreadsheet ιο find the minimum constant workforce
detcrmine the total cost oftl1at plan.
New
Forecast (1997)
February
March
ΑΡΓίl
May
June
July
August
September
October
Novelnber
December
Forecasted
Demand
(hundreds
of cases)
April
May
June
July
Augu5t
September
October
November
December
Actual
Demand (1996)
" = SIll00111iI1!!const,IIlI fOl"JJl"OCIUCliOIl
"'1d clcIllanc] usccl ίη BOWHl<II1'sl11odel.
..,
β
= SIllootl1in!! COI1stant [οι· ίΙ1veη1ΟΓΥtlSccI ίη 80"'Ι11:1Ι1'5 ιυοοε].
ΙΊ.·
('11
=
=
('1
= Cost οι' 110lding one ιιηίι ο! stock fOΓουε [1CI·iod.
Cost" οΓ ΙίΙ'ίηι; one WΟΓΙCCΓ.
Cosl οΓ 11ίΓίηι;one \vol·kcI·.
Ι'()
=
('1'
= Pcnalty σοει ·foΓbacl< o,-dcr5.
('Ι(
Cosl οί' ριυdιιcίηg
= Cost
οί' ιποcJιιcίng
onc Lιπίι 011ονειιίιυο.
OI1C ιιnίι
οιι
I'egLII.H
ιίυιο.
(".Ι'
= Cost 10 sιιbcοηlΓΠCI ουε ιιnίι οί ρ.οοικιίωι.
('ΙΙ
= I(ile cost ]JCI'ιιηίι
(1Ι'
]JI"OCltICriOI1.
D, = FΟΓCC'lSΙοΙ" (lcInal1ll ίη JJCI'ioclι.
F, = NUInbel' οΓ wOl"kcrs fiI'cd ;η JJeriod r.
Ν, = NlIIllbcI" of ,vorkcl's 11il'ccIίη pcγjod ι.
Ι,. = Il1vcnt"ory Icvel ίl1 (JCI'iod ι.
Κ = Nun1ber o]',tggreg<1te units lποdιιccd Ι'Υ OI1C\VOI"kCI'ίl1 OI1Cday.
λl
= ΑηηΙΙ<11dCIl1Hnc!ΙΟΓt':1I11ily.i(rcfcI' to Scclion
17, = NtllllbcΓ σΓ IJΓOdtIct"iOJldnys ίl1 I,eriod /.
Ο, = OvcrtiIllC 1"'oclιrctioI1 ίη units.
Ρ,
=
Ριυουοιίο»
levcl ίη I'CΓίοcl ι.
;), = NtIInbcI" οΓ ιιηίι, SlιbCΟΠΙΓacle(1lίΌΠ1otIlsidc.
r =
Νυιιιίιο. οΙ" I'C,-iocls ίη 111C111,1Ι1nίlψ110ΓίΖοη.
U, = \VΟΓkCΓi(llc tiI11Cίη ιιηίι, ("lII1CICI·tiI11C").
Η/,
=
'ΝΟΓkΓΟΓCCIcvel ;n pCI'iocl ι.
3.9).
168
Chapter Three
"'\~σr('~αιι>Ρlnηl1ίη~
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51.2
Α PROTOTYPE LlNEAR PROGRAMMING
ExampleS/.I
PROBLEM
Sidneyville manufactures household and commercial furnishings. The Office Division produr
two desks, ΓΟllΙορ and regular. Sidneyville constructsthe
desks ίπ Its plant outside IVledfo
OregoΓl, from a selection οΙ woods. The woods are cut ΙΟ a uniform thickness οΙ 1 IncrI. For t
reason, one measures the wood ίπ unlts of square feet. One rolltop desk requires 10 \quare ιι
ΟΙ pine, 4 square leet ΟΙ cedar, and 15 square feet οΙ maple. One regular de\k requI