Aggregate PlanningChapter Overνiew

Purpose

Το develop techniques for aggregating units of production, and determiningsuitable production levels a nd workforce levels based οη predicted demand foraggregate units.

Key Points

1. Aggregate units of production. Thi$ chapter could also have been called MacroProduction Planning, since the purpose of aggregating units ί$ to be able todevelop a top-down plan for the entire firm ΟΓ for some subset of the firm, suchas a product line ΟΓ a particular plant. For large firms producing a wide range ofproducts ΟΓ for firms providing a service rather than a product, determiningappropriate aggregate units can be a challenge. The most direct approach is toeχpress aggregate units ίη some generic measure, such as dollars of sales, tons ofsteel, ΟΓ gallons of paint. For a service, $uch as provided by a consulting firm ΟΓ alaw firm, billed hours would be a reasonable way of eχpressing aggregate units.

2. Aspects of aggregate p/anning. The following are the σιοει important feature5 ofaggregate planning:

• Smoothing. (05tS that arise from changing production and workforce levels.• Bott/enecks. Planning in anticipation 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 Ιοοlong, demand foreca5ts become unreliable.

• Treatment of demand. ΑΙΙ the mathematical model5 in thi5 chapter considerdemand Ιο be known, i.e, have zero forecast error.

3. Costs ίπ aggregate p/anning.

• Smoothing costs. The CO$tof changing production and/or workforce level5.• Ho/ding costs. The opportunity C05t of dollar5 inve5ted in inventory.• Shortage costs. The CO$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 lίnearprogramming. When solving problem5 graphically, the first 5tep is to draw a

124

Chapter Three Aggr,ga,ePlanning 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 planninghorizon), then one matches the cumulative net demand curve as closely as possiblewith a straight lίne. If the goal is to develop a zero-inventory plan (i.e., one thatminimizes holding and shortage costs), then one tracks the cumulative netdemand curve as cl05ely as possible each period. While lίnear programmingprovide5 ωει optimal 50lutions, the method does not take into accountmanagement policy, such Β5 avoiding hiring and firing as much as possible. Fora problem with a Tperiod planning horizon, the linear programming formulationrequires 8Tvariables and 3Τ constraints. For long planning horizons, this canbecome quite tedious. Another issue that must be dealt with is that the solution toa linear program ί5 noninteger. Το handle this problem, one would either have tospecify that the problem variables were integers (which could make the problemcomputationally unwieldy) ΟΓ develop some suitable rounding procedure.

5. The /inear decision ru/e. The aggregate planning concept had its ωοιε ίη thework of Holt, Modigliani, Muth, and Simon (1960) who developed a model forPittsburgh Paints (presumably) to determine their workforce and productionlevel5. The model used quadratic approχimations for the costs, and obtainedsimple lίnear equations for the optimal policies. This work 5pawned the laterinterest in aggregate planning.

6. Mode/ing management behavior. Bowman (1963) considered linear decision rulessimilar to tho$e derived by Holt, Modigliani, Muth, and Simon eχcept that hesuggested fitting the parameters of the model based on management's actions,rather than prescribing optimal actions ba$ed on ωει minimization. Thi$ is one ofthe few eχamples of a mathematical model used to describe human behavior ίπthe conteχt of operations planning.

7. Oisaggregating aggregate p/ans. While aggregate planning is useful for providingapproχimate solutions for macro planning at the firm level, the que$tion iswhether these aggregate plans provide any guidance for planning at the lowerlevels of the firm. Α disaggregation scheme is a means of taking an aggregateplan and breaking it down to get more detailed plans at lower levels of the firm.

As we go througlllife, we tnake both Illicro and macro decisions. Micro decisions tnightbe \vhat Ιο eat forbreakfast, whatroute Ιο take towork, what auto service ιο use, orwhichmovie to rent. Macro decisions are the kind t!Jat change the course of one's life: where Ιοlίνe, what Ιο major ίη, whicll job Ιο take, wllom Ιο marry. Α cotnpany also must makeboth micro and macro decisions every day. Ιη this cllapter we explore decisions made atthe 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 amanufacturing firm, the quantity and the mix ofproducts to be produced. Macro planningis ηο! lίmίted Ιο manLIfacturing firms. Service organizations must determine employeestaffing needs as well. For example, airlines must plan staffing levels for flight attendantsand pilots, and !lospitals must plan staffing levels for nurses. Macro planning strategiesare a fundamental part ofthe firm's overall business strategy. Some firms operate οη thephilosophy that costs can be controlled only by making frequent changes ίn tl1e sizeandlor composition of the workforce. The aerospace industry ίη Califomia ίη the 1970sadopted this strategy. As government contracts shifted fron1 one producer Ιο another, so

126 Chapter Three 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 beginswith the forecast of demand. Techniques for demand forecasting were presented ίηChapter 2. How responsive the firιn can be το anticipated changes ίη the demand de-pends οιι several factors. These factors include the genera], strategy the firm may haveregarding retaining workers and ifs οοτυηιίτυισιιε-το existing empIoyees.

As noted ίη Chapter 2, demand forecasts afe generallY'wrong because there is aImostalways a random οοπιρουεφι ofthe demand that cannot be predicted exactly ίη advance.The aggregate pIanning methodology discussed ίη this chapter requires the assumptionthat demand is deterministic, οτ known ίη advance. This assumption is made Ιο simpIifythe analysis and alJow us το focus οτι the systematic orpredictable changes ίη the demandpattem, rather than οτι the unsystematic οτ random changes. Inventory managementsubject to randomness is treated ίη detail ίη Chapter 5.

TraditionalJy, most manufacturers have chosen ιο retain primary production ίη-house. Some components might be purchased from outside suppliers (see the discus-sion ofthe make-or-buy problem ίη Chapter Ι), but the Ρήmary product is traditionalJyproduced by the firm. Henry Ford was one ofthe firstAmerican manufacturers το designa completely vertically integrated business. Ford even owned a stand of rubber trees soίι 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-basedproducer of computer workstations. Sun, a market leader, adopted the strategy of focus-ing οιι product innovation and desigiJ rather than οιι manufacturing. It has developed closeties ιο contract manufacturers such as San Jose-based Solecιron Corporation. winner ofthe Βaldήge Award for Quaiity. Subcontracting its primary maηufactuήηg function hasallowed 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 quicklyιο anticipated changes ίη demand, which would require making frequent and potential1ylarge changes in the size ofthe labor force. Such a strategy has been called a chase strat-egy. This may 1Jecost effective, but could be a ροοτ loηg-ruη business strategy. Workerswho are laίd off may ηο! be avaiIable when business tums around. For this reason, the firmmay wish Ιο adopt the objective of retaining a stable workforce. However, this strategyoften results ίη large bUΊldups of inventory duήηg Ρeήοds of low demand. Service firmsmay incur substantial debt tomeetpayrolJs in slowperiods. Α third objective is Ιο developa production plan for the firm that maximizes profit over the planning hοήΖοn subject Ιοconstraints οη capacity. When profit maximization is the primary objective, explicit costsofmaking'changes mustbe factored ίηΙο the decision process.

Aggregate planning methodology is designed Ιό translate demand forecasts ίηΙο ablueprint for planning staffing and production levels for the firm over a predeterminedplanning horizon. Aggregate planning methodology is ηοΙ lίmίted ιο top-Ievel plan-ning. Although generaJ]y considered Ιο be a macro planning ιοοl for determiningoveralJ workforce and production leνels, large companies may find aggregate planninguseful at the plant level as well. Production planning may be viewed as a hierarchicalprocess ίη which purchasing, production, and staffing decisions must be made at sev-erallevels 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 towardsingle-item control.

3.1 Aggregar<UnitsofPrαlucιion 127

This chapter reviews several techniques for determining aggregate plaίIs. Some.6fthese ετο heuήstίό (i.e., approximate) and some are optimal. We hope tοcοήΎeΥ ιο thereader ετι understahding of the issues involved ίη aggregate planning'i'a kή6wleage"ofthe basic tools available for providing solutions, and an appreciation of the dif!icιJltίesassociated with imp(ementing aggregate plans ίτι the real world.

,~ ,,"0 ,

, " .,0 '}' ,"~; \ "1" ,'~:r~"';CI'~::, :

AGGREGAtE ONITS'"OFpRobuttION.

Example3.1

The agg~ega:te planning approach is predicated οα the e~istence of an aggregate υηίιofproduction. When the types ofitems produced are similar, an aggregate productionυηίι can couespond Ιο 'aΠ "av~rage" item, but if many different·types of items are pro-duced, 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-yearsofprogramming time), οτ dollar value (value ofinventory ίτι dollars). What the appro-ρτίετε aggregating scheme, should be is πο! always obvious. Ιι depends ωι the contextofthe'pafticular planning problem'and the leνel of aggregation required.

Α plant managerworking lor a large national appliarice lίrm isconsidering implementing an ag-gregate planning system ιο determine the worklorce and production levels in his plant. This par-ticular plant produces six models ΟΙ washing machines,The characteristics ΟΙ the machines are

Model NumberNlιmber of Worker-Hours

Required to Produce , SeIIing Price

Α5532Κ4242Ι9898Ι3800Μ2624Μ3880

$285345395425525725

4.24.95.15.25.45.8

The plant manager must decide οπ the particular aggregation scheme Ιο use. One possibil-ίΙΥ is to deline an aggregate υπίΙ as'on'e dollar ΟΙ όυιρυΙ Unlortunately, the selling prices ΟΙ thevarious models ΟΙ washing machines are ποΙ consistent with th·e number ΟΙ worker-hours re-quired Ιο produce them. The ratio ΟΙ the,selling price.divided by the worker-hours is $67,86 forΑ5532 and $125,00 ΙΟΓ M::J880, (The company bas~s its pricing on the faet that the lessexpensive models have a higher sales volume.) The manager notices that the percentages οΙthe total number οΙ sales Ιοτ these si\<models have been fairly constant, with values οΙ 32 per-cent lοr Α5532, 21 percent lοr Κ4242, 17 percent Ιοτ ~9898, 14 percent for L3800, 1Ο percentlοr Μ2624, and 6 percent for Μ3880. He decides to define an aggregate uni! ΟΙ production asa 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 aggregateproduetion units ίπ essentially the same way by multiplying the appropriate Iractions by thelorecasts lor υπίι sales ΟΙ each type ΟΙ machine.

The apptΌach used by the plant manager ίο Example 3.1 was possible because of therelative similarity of the jJroducts prodtIced. However, defining an aggregate υη1ι ofproduction at a higher leνel ofthe firm is more difficult. Ιη cases ίη which the firrn pro-duces a large νaήety ofprodncts, a natural aggregate υπίι is sales dollars. Althollgh, aswe saw ίη the' example, tl1ίs will 110!11ecessarily tral1slate Ιο tbe same nnmber of unitsof prodnction 'for each item, ίι wiH generally provide a good approximatio11 for plan-ning at tl1e highest leνel of a firm that produces a diverse product lίηe.

128 Chapter Three Aggr,gaιc Planning

FIGURE3-1TI,e /1ierarchy ofproduclion pIanningdccisiol1S

Aggregate planning (and the associated problem of disaggregating the aggregateplans orconverting them ίπΙο detailed master schedules) is closely related το hierarchicalproduction planning (ΗΡΡ) championed by Hax and Meal (1975). ΗΡΡ considersworkforce sizes and production rates at a variety oflevels ofthe nrm as opposed 10 sim-ply the τορ level, as ίπ aggregate planning. For aggregate planning purposes, Hax andMeal recommend the foJlowing hierarchy:

Ι. /tenls. These are the nnal products ιο be delivered to the οιιετοηιετ, Απ item is oftenreferred 10 as an SKU (for stockkeeping ιαιίι) and represents tl1ennest leνel of detailίπ the ρτοοιιοτ είτυοιυτε.

2. FamiIie:,·.These are denned as a group of items that share a οοηιπιοη mantIfactιιringsetιιp cosΙ

3. Types. Types are groups of families with prodtIction quantities that are determinedby a single aggregate production ρlαη.

lπ Exalnple 3.1, items would correspond 10 individuallnodels ofwashing machines.Α family might be all washing nJachines, and a type might be large appliances. ΤheHax-Meal aggregation scheme wiII πο! necessarily work ίη every sitιιation.ln general,the aggregation method shotIld be consistent with the nrιn's organizational strιtcΙUreand prodllCt Ιine.

[η Figure 3-1 we present a sCl1ematic of the aggregate planning fιlnction and itsplace ίπ tl1e hierarchy ofproduction planning decisions.

3.2 OVERVIEW OF ΤΗΕ AGGREGATE PLANNING PROBLEM

Having now denned the appropriate aggregate ιιηίι [or the leνel of the firm [or which αηaggregate plan is ιο be determined, we assume that there exists a forecast ofthe deInandfor a specined planning horizon, expressed ίη terms of aggregate prodtIction units. LetDJ, D2, ... , Dr be the demand forecasts for the next Tplanning periods. [π most applica-tions, a planning period is a month, altl10tIgh aggregate plans can be developed for other

3.2 Overview of the AggregaLe ]Jlαnning IJroblem 129

ροτίοσε of time, stIcb as weeks, quarters, ΟΓyears. An important featιιre of aggregateplanning is that the,de,mands are treated as known constants (that is, the forecast error isassumed ιο be zero). Thereasons for making this assumption will be d,iscussed later.

The goal ofaggregate planning is το determine aggregate production qua.ntities andthe leνels of resources required το achieve these production goals. Ιτι practice, thistranslates to nnding the number ofworkers that should be employed and the number ofaggregate tInits ιο be produced ίη each of the planning periods J, 2, ... , Τ. The objec-tive ο[ aggregate planning is to balance the advantages of producing to tneet demandas closely as possible against the disrtIptions caused by changing the levels of produc-ιίοο and/or the workforce levels.

The primary issnes related ιο the aggregate planning problem include

Ι. SmooIhing, Smoothing refers το costs that resιιlt from changing production andworkforce levels from ουε period το the ηοχΙ Two ofthe key components of smoothingcosts ΗΓοthe costs that result frol11 hiring and nring workers. Aggregate planningmethodology reql1ires the ερεοίϋοειίοτι of these costs, which may be difficult to εειί-mate. Firing workers could have fal'-reaching consequences and costs that may be dif-ficult to evaluate. Firms that hire and nre freqtIently develop a poor public image. Thiscould adversely affect sales and discourage potential employees frOI11joining the com-pany. Furthermore, workers that are laίd off might not simply wait around for busiiιessιο pick up. Firing workers can have a detrimental effect οη the futιιre size ofthe laborforce ifthose workers obtain employment ίτι other indusιries. Finally, tnost companiesΗΤοsimply πο! at liberty Ιο hire and fire at will. Labor agreements restrict the freedomof management το freely alter workforce leνels. However, ίι is διίll valtlable for man-agement ιο be aware of the cost trade-offs associated with varying workforce leνelsand the attendant savings Ιτι ίnνentOlΎ costs.

2. Bottleneck probIenIs. We use the term bottleneck ιο refer το the inability of tl1esystem το respond το stIdden changes ίπ demand as a restIlt of capacity restrictions. Forexample, a bottleneck cotIld arise when the forecast [οr demand ίη οιιε month is un-usually high, and the ρΙΗΩΙdoes υοτ have slIfficient capacity το meet that demand. Αbreakdown 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 fore-casted, and hence the ntImber of periods for which workforce and inventory levels areΙο be determined, mlIst be specified ίη advance. The choice oftl1e value ofthe forecasthorizon, Τ, can be significant ίη determining the usefulness of the aggregate plan. IfΤ is 100 sI.nall, then cnrrent production leνels might ηο! be adequate for I11eeting thedemand beyond the horizon length. Jf Τ ίδ 100 large, ίι is likely that the forecasts farίηΙο the fιιΙUτo wiII prove inaccurate. If futιιre demands turn out Ιο be very differentfrom the forecasts, then clIrrent decisions indicated by the aggregate plan could be ίπ-correcΙ Another isstre involving the planning horizon is the end-oJ-hoI'izon effect. Forexample, the aggregate plan might recommend that the inventory at the end ofthe hori-Ζοn 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 beavoided by adding a constraint specifying mil1imnnJ ending inventory leνels.)

Ιπ practice, rolling schedules are almost always tIsed. This means that at the time ofthe next decision, a new forecast of demand is appended Ιο the fornJer fΟΓecasts and oldforecasts ll1ight be revised Ιο reflect new information. The new aggregate plan may rec-ommend different production and workforce levels fol' the cwTent period. than wererecomlnended one peTίod ago. When only the decisions fol' the CΙΙΓrentplanning periodneed Ιο be illlplemented imJnediately, II,e schedule shotIld be viewed as dYl1anlic ra!herthan static.

130 Chapter Three Aggr,gaιe Ρ/αηηίηι

Although rolling schedules are common, ίι is possible that because of productionlead times, the schedule must be frozen for a certain number of planning periods. Thismeans that decisions over some collection of future periods cannot be altered. Themost direct means of dealing with frozen horizons is simply Ιο label as period Ι the firstperiod ίη which decisions are τιοτ frozen.

4. Treatment οιdemand. As noted above, aggregate planning methodology requiresthe εεειυυριίου that demand is known with certainty. This is simultaneotIsly a weak-ness 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 forecastingtechniques 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 aver-age demand be constant over ιίσιο. Aggregate planning aIIows tlle manager Ιο Ιοοιιε οηthe systematic changes that are generaJly αοτ present ίη models that assnme randomdemand. ΒΥ assuming deterministic denland, the effects of seasonal fluctuations andbusiness 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 importantto identify and measure those specific costs that are affected by the planning decision.

Ι. SI1700t!lil1gcσsfs. Smoothing costs are those costs that accrue as a result ofcllanging the production levels from one period το tlle next. Ιη the aggregate pIanningcontext, the most salient snιoothing cost is the cost of changing the size of the work-force. Tncreasing the size of the workforce requires time and expense to advertise ρο-sitions, interview prospectiνe employees, and train new hires. Decreasing the size ofthe workforce means that workers ιτιιιετ be laίd off. Severance pay is thus one cost ofdecreasing 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 jobswith otller firnlS ΟΓ ίη otller ίηdustΓίes.

Most of the models that we consider assume that the costs of increasing and de-Cl"easing the size ofthe workforce are lίnear functions ofthe numbeI' ofemployees thatare 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 thesupply of labor becoInes scarce, there may be additional costs reqLIired 10 hire morewοrkeΓS, and tlle costs of laying off workers may go ιιρ substantially if the nuιnber όfworkers la.ίd off is Ιοο large. Α typical cost function fol" changing the size of theworkforce appears ίη Figure 3-2.

2. Ho/ding costs. Holding costs are the costs tllat accrue as a Γesιιlt ofllaving capitaltied ιιρ ίη inventoI)'. Jf the firm can decrease its inventory, tlle money saved coLIldbe ίπ-.vested elsewJlere with a return that will vary witll tlle indnstry and with the specificcompany. (Α more complete discussion ofholding costs is deferred to ChapteI" 4.) Hold-ing costs are almost always assLImed Ιο be lίηear ίη the number ofunits being 11eldat apartictIlar ροίηΙ ίη time. We will assunle for the Ρuφοses of the aggregate planninganalysis that the holding cost is eΧΡΓessed ίπ terms of dollars per ιιηίι held per planningperiod. 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'

3.3 Co,ιs ίη Aggr,gaιe Plaηning 131

convenience onJy. Holding costs could be charged against starting inventory ΟΓ averageinventory as well.

3. Shortage costs. Holding costs are charged against the aggregate inventory aslong as ίι is positive. Ια some situations ιι may be necessary to incuI" shortages, \vhichετο represented by a negative level of inventory. Shortages can occnr when forecasteddenland exceeds the capacity of the production facility ΟΓ when denlands are higherthan anticipated. For the purposes of aggregate planning, ίτ is generally assunled thatexcess demand is backlogged and filled ίη a future period. Ιπ a highly competitive sit-uation, however, ίι is possible that excess demand is lost and the customer goes else-wllere. This case, which is known as lost sales, is Inore appropriate ίη the managementof 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. Convexfllnctions also can accllrateIy describe shortage costs, but lίηear flInctions seem Ιο bethe 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 outputduring regular working hours. Included ίη this category are the actual payroll costs ofregular 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 lllustequal the number ofunits demanded over any planning horizon ofsufficient length. Jfthere is ηο overtime οτ workeI" idIe time, reglllar payroll costs do ηο! have Ιο beincluded ίη the evaluation of different strategies.

5. OIIeI·tinze al1d .S1Ibcontracting costs. Overtime and subcontracting costs are thecosts of production of units ηο! produced οη regular time. Overtime refers Ιο produc-ιίοη by regular-time eInployees beyond the normal workday, and subcontracting I'efersΙο the production of iteIns by an outside SUΡΡlίeΓ.Again, ίι is generally assumed thatbotll of these costs aJ'e lίηear.

6. ldIe time costs. The complete formulatioIl of the aggregate planning problemalso includes a cost for underutilization ofthe workforce, οτ idle tifRe. Ιη most contexts,

,132 'Chapter Three Aggregαιe PIαnning

FIGURE-3=3Holding and back-order costs

the idle time cost is zero, as the direct co~ts of i'dle 'time wbϋld' be taken into account ϊηlabor costs and lower production levels, However, idle titιie could have other conse-quences' for the firm. For exaIi1ple, i,f the aggregate units aτe ίπρυτ to another process,idle time οτι theliIie cott1d, resul!'in hi'gher'costs ιο tlle subsequent ρτοοσεε. Ιτι suchcases, one woUld exp!icitly include εροείυνο idle οοει.

When plannitιg is dohe ata re!ativelY'high Ieve! ofthe fiτm: the effects of intangib!efactors are mbre ρτουοιιαοεο. Any sο!utίόή td the aggregate p!anning prob!ern obtainedfrom'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 requiresfTequent hiring and' firing of p~rsoimel. Such a po!icy may' be infeasib!e because ofprior contract agreements, οτ uiJdesirab!'e~because of the potential negative effects οτιthe firm's public image.

Problems for Sections 3.1-3.3. ι

Ι. What does the term aggregαte unit ofpl-oduction mean? 00 aggregate productionunits always correspond to actual items?,Oo they ever? Discuss,

2. What isthe aggtegation scheme recominended by'Haxarid Meal?3, Oiscuss the following terms and their re!ationship το the aggregate planning

problem: '

a. Smoothingb, Bottlenecksc. Capacityd. Planning horizon

4. iA local macHine shop eIllploys 60 workers whb have a Y'ariety of skills, The shopaccepts one-time orders and a!SΟ'maίήtaίη's a nuTnber ofregu1ar clients, Discuss someofthe difficulties with using the aggregate p!anning methodology in this context.

,,3.4, "Α j>rororyp<.prρblim 133

5. :A'large-manufacturer ofhousehold.consumer goods ίιι considering ihtegτating ετι-aggregateop!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? '

~ '-, :~ 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 10

'measure them in area1 operating enviromnent. '" )~, . . - ~;;':-α.: Smοοthίl1gιcosΙsb.' Hdldirig costsσ, ,PayroJ[ costs

7. OiSCtIS§tJle followiiig statenienl: '''Sirlce''we ιise a roiling production schedιtle, Ιreally doii'theed το know tlIe demaridbey~nd'next πιοητίι,"

8, St. Clair County Hospital is attempting το asse~k iis needs [or nurses over the com-ing four montbs (January to April), The need for nurses depends οη both theriumbers aηd',tneΊΥΡes of patients ϊη the hospita1. Based οο a study conducted byconstIltants, th§nosp1tal has determined that the following ratios of nurses ιοpati en ts are required:

'Numbers of'Nurses ."',1'"Patient Forecasts

Patient Type (Required per Patient '·i·Jan. Feb. ,Mar. 'ΑΡΓ, ";'

Major surgery 0.4 28 21 16 18ΜίηΟΓ surgery 0,1 12 25 45 32Maternity 0,5 " 22 43 90 26Critical care 0,6 75 45' 60 30Other 0.3 80 94 73 77

a. How many nurses shollld be working eachmonth το most close!y match patientforecasts? "ψ"ι 'ι:11

b, Suppose the hospita1 does not want το change its'policy of πο! increasing tllenursing staff slze by more than 1Ο percentjn any montll'. Suggest.a sehedule, ofnurse staffing over the four months that meets this requirement and a1so meetsthe need for n\Jrses each month.

3.4 Α PROTOTYPE PROBLEM

Example3.2

'Ι,,, "" '" , " : ",One can obtairi adeiIl\hte,,,sollltions for many aggregate pla.ιining problems by hand or byusing relativel'y straightfoτwaτd graphical techniques. Linear progranuning is a meansof ob!aining (nearly)·optimal solutions. We illustrate the different solution techniqueswith the'fo]Jowing examp!e. '

Densepack is Ιο plan workforce and production levels for the six-month period January ΙΟ June,The firm produces a lίη-eof disk drives for mainframe computers that are plug compatible withseveral computers produced by major πianufacturers, Forecast demands over the' next siχmonths 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 employedίη the Milpitas plant. Ending inventory'in December is eχpected to be 500 units, and the firmwould Iike to have 600 units οη hand at the end of June,

FIGURE 3-4Α feasibIe aggregateplaη for Densepack

134 Chapter Three AggregQte Plιtnning

There are several ways ιο incorporate the starting and,the eriding inventory constraints ίηthe formulation. The most convenient iSsimply το modify the values οΙ the predicted demaDefine 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 ,cmand ίπ period Τ. Minimum buffer inventorie.s also can be handled by modifying the predictedemand. 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 amouritshould 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 bothzero. The net predicted demand and the net cυmulative demand for the six months January toJune are as follows:

Net Predicted Net CumuIativeMonth Demand DemandJanuary 780 780February 640 1,420March 900 2,320ΑΡΓίl 1,200 3,520May 2,000 5,520June 2,000 7,520

The cumulative net demand is pictured ίπ Figure 3-4. Α production plan is the specificationΟΙ the production levels for each month. ΙΙ shortages are not permitted, then cυmulativeproduction must be at least as great as cυmulative demand each period. Ιπ addition Ιο thecumulative net demand, Figure 3-4 also shows one feasible production plan.

5 6

~Ι;{

. 3.4 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·,theexample that there are~only three.cos'ts to be considered: costof hiring workers, cost ol'fi-ringworkers, and cost οί holding inveniory. De.fine· .

(Η = Co~t.of, hi~ing οοε worker = $500,

(F = C;st6ffiring o~J,w~rker = $f':ooo", (, = Cost.of holding oπ~. unit οΙ inventory for one month = $8Ό.

t;

We require a means()f translating ~~gregate production ίπ units to wo'rkforce levels. B~c'auseποΙ all months have an equal number οΙ working days, we will us.e a day as an indivisible υπίί ofmeasure .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, with the workforce levelcbnstant at 76 workers, the firm produced 245 disk drives. That means that οπ average the ΡΓΟ-duction ratewas 245/22 = 11.1364drives per daywhen there were 76 workers employed attheplant. It follows that one .worker produced an'average'of 11.1364/76 = 0.14653 drive ίπ oneday. Hence, Κ = 0.14653 lοr this example:

We will evaluate two alternatiνe plans ,[ΟΤ managing the. worl<.force that representtwo essentially opposite management strategies. Plan Ι· is to change the workforceeacll 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 constantworkforce necessary to satisfy thenet demand. This is known as the constant wοrkjΌΙ'ceρΙαη.

Eνaluation cίf a Chase Strategy (Zero Inνentory Plan)Here, we will deνeIop a 'production, plan Ιοτ Densepack that minimizes the leνels ofinνentory the firnl must hold during the six-month planning horizon. Table 3-1 sum-Illarizes the ίιιρυι ίηίοτυιειίωι [or the calculations al1d Sl10WStl1e minimum number ofworkers required ίη each month.

Ol1e obtail1s the entries ίη the final cOll1mn ofTable 3-1, the m.inimum number ofworkers required each Illonth, by dividing the forecasted netdeIlland by the number ofunits produced per worker. The νalue of this'ratio is then rounded tιpwαrd ιο tlle next

136 Chapter Three Aggreg",e Pi<ιnning

higher integer. We.mustround upwardtoguarantee that shortages do πο! occur. As an ex-ample, consider the month ofJanuary. Foτming the ratio 780/2.931 gives 266.12, whichίε τottnded ιιρ Ιο 267 workers. The number of working days each month depends ιιροο avariety of factors, such as paid holidays and worker schedules. The reduced number ofdays 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 resttltsίπ the aggregate plan given ίπ Table 3-2.

The number of units prodLIced each month (τοίυαιυ F ίπ Table 3-2) is obtaίned bythe following formula: 'ι

Number ofunits 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 costmust 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 τιοτenter 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 periodbecause 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 ίπaddition 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 ροίυτw.here the size of the workforce could be reduced by one οτ more workers. Ιτι thisexample there is sufficient inventory οη hand ιο reduce the workforce by one worker ίπthe months of both March and May. Check that the resu!ting plan hίΓes a total of753 woτkers and fires a tota! of 144 workers and has a total of on!y 13 units of inven-tory. The cost ofthis modified plan comes Ιο $569,540.

Evaluation of the Constant Workforce PlanNow assume that t!le goa! is ιο e!iminate complete!y the need for hiring and firingduring the planning horizon. !π order to guarantee tI1at sl10rtages do ποΙ occur i,n any

ΤΑΒΙΕ 3-3Computation oftheMinimum WorkforceRequired byDensepack

ΤΑΒΙΕ 3-4JIIventory Lcvels forConstant WorkforceSchedule

3.4 Α P,oto<ype P,oblem 137

period, ίι is necessary Ιο coιnpute the minimum workforce required for everγ month ίηthe planning horizon. Ροτ January, the net cumulative demand is 780 and there are2.931 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 andFebruary 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 toform the ratios of the cumulative net demand and the cumulative number of unitsprodLIced 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 periodis the Π1aΧίmum entry ίη coluι'nn D ίτι Table 3-3, which is 411 workers. It is on!y acoincidence tl1at the maximum ratio οccuπed ίη the final period.

Becatlse tllere are 300 wo.rkers employed at the end of DeceΠ1ber, the constantworkforce plan requires hiring 111 workers at the beginning of January. Νο further hiringand 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 ΟΙthe table are obtained by mtIltiplying the number ofunits prodnced perworker eacll Π10nttby 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 cosoftlle zero inventory plan, which was $569,540. Howeνer, because costs ofthe ιwo planare close, ίι is like!y that the coιnpany would prefer the constant workforce plan ίη ordeto avoid any unaccounted for costs of making freqLIent changes ίn the workforce.

138 Chapter Three Aggregare Planrtίng

",·1

Mixed Strategies and Additionalfonstraints - <. .~ ,:i-tπ!~:The zero inventory plan ana constant workforce strategies just treated ate l?uie,.strafe1gies: 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 ίiiιinιraicalculations cbnsidefing tbat aggregate~planhing 'prohlems can be formuJafJd "andsolved 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 a 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'aballpark calculation ίn one's Ilead before pulling ου! the calculator οτ computer.

Figure 3-4 shows the constant workforce strategy for Densepack. The hatched arearepresents tbe inventory carried ίη each month. Suppose we alJow a sίngJe change ίη theproduction rate during the six months. Can you identify a strategy fi:om the figύre tI1atsubstantially reduces inventor)ι without pennitting shortages? ' , ",ι,,·, '

Graphically, the problemis το cover ihe cuΊnulative net demaήtl clIrve with ~όstraight lines, rather than one straigbt JiJ;1'e.This can be accompJished by driving'fbe netinventory 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 eachmonth. 'Πιετ means we need το produce3,~20/4 = 880 tJnits ίη each of the .first fourmonths, 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. Iftheg.raph is accurate, that means that there shouJd be τιο shortages occurring ίη theselnonths, The May and Ιιυιε production is then set το 2,000, exactly matching the netdemand ίη these lnonths, With this policy we obtain

MonthCumulatiνe

NetDemandCumulatiνeProduction

8801,7602,6403,5205,5207,520

JanuaryFebruaryMarchΑΡΓίlMayJune':

As we will see ίη Section 3.5, this poJicy .turns·out ιο be optimal for the Densepackproblelll, :.,

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,;"~ Γmeans tl1at the slope of the cumulative· ,production curν,e is bounded by ] ,800, OnesolHtion ίη this case would be to produce 980 ίη each of tlTe first four months and1,800 units ίη each of the last two months. Another constraint might be that the maxi-mun1 change fromone τηοηΙlι ιο the next be ηο more than 750 units. Suggest a produc-ιίοη plan ιο meet this constraint.

, TlTes,eare a,few'examp.les of constraintsthat mighι.arise ίη using aggregate planningmethodology, As. the constraints become:. more compJex, finding good solutionsgraphically becomes tnore difficult: Fortunatelyl·most cbnstraints of this natl1re can be'incorporated easily into the lίιιear programming formuJations of aggregate planningprobJeJns.,

3.4 Α Pro",rype,ProbJem 139

Η,1 ;4;' 'ι: !-\..-~,:ι'~·ι Η; _ ί"l-' "1 ,~-,~I :τι 'J •• \;t~iT ;~;{)

, 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 thenext five years ihthotlsanos of',I'ackages wil\be as follows:

. . " ;:, "ι ..ι.,· ,~Ι; i.~'· ''1';'': ';, ,Ι,' . ,'Γ ,

jf.

~,ι

ιω\ , 'j:or;~asted Deman,d', (t~ousands of, packages)

'12345

300,12020o'110135

tl"

, ,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 end ofthe ς:urrent y~at: A:ssuiiι~'that, οε theaverage, ι::achworker 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, andshortages fι~e'1)ot aI10\Ved;f' η .:<, "Ι ' i .

Base<;loiι' the effort( of interνiewing and, training new workers, Farmer Grey εε-timates tha.t ίι ,costs $,500 for, each worker hired, Severance pay amounts to $1,000per worke,r. " _

.. α" Assutning, that sho'rtages are ηοί allowed, determine the minimum constantworkforce that he will Jieed over the next five years.

b. Evaluate tbe cost of the pJan found ίη part (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 exactlyonce during tp.e five years, and evιΗuate the cost of that plan.

b, Graphicany deter,mine a product~on plan. that changes the prodlIction rateexactly twice dlIrίng the five years, and evallIate the cost of that plan.

11. Αη ilnplicit assurnption made in Problem 9 was that dried aprίcots unsold a( theend of a year couJd be sold ίη subsequ~nt years, Suppose that apncots unsold at theend 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 manyworkers will be needed each month for the next six-month production period, Thefollowing is a monthly demand forecast for the six-month peribd.

Month Forecasted' Demand

JulyAugustSeptemberOctoberNovemberDecember'

1,2501,100

950909

1,0001,150

140 Chapter Three AιweRaι, PlanninR

The lnventoτy οτι harid at tlJe endΌf Ιιπιε was 500 units ..:τheocompany, wants 10malntaln a mlnimum inventory of 300 tInits each month 'and w~~ld ΊίΊce to have400 unlts 'οη hatιd at the end of Deeember.!.Each ιιηίί requires five employee-honrsto ptodnce .•.thereare'20 workingdayseach mbilth,iιnd each emp1oyeeyνorks aneight-honr day. Tlie workforceat the end·of June was'·35 workers.

α, Determine,a minimum ίηv~~!ότy jxoduction plan (i.e., one that al10ws arbitraryhiring andfiring). 'ο

b. Determine the' prodnction plan that' meets demand but does αοι hire οτ fireworkers dύπng the six-monih period.

13. Mr. Meadows Cookie COJnpany makes a variety of chocolate chip cookies ίη tbeplant 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 were120 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 arecurrently 100 workers employed, aiid thete ls υσ startlng inventory of cookies.,

α. What' ls the' minimum constant workf~rce r~;quired to meet demand over thenext four months? Ά

b. Αεειισιε that CΙ, = ω cents pec cο(}kίe'Ρer,ιmοnt11;,cΗ = $100, and cF = $206. 'Eva1uate the cost ofthe p1an detlved ·ίη part (α),

Α local semlconductor firm, Superchip, ls p.1anning lts workforce and productlon1evels 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 thenext year by month is as follows:

14.

Month.Production

DaysPredicted Demand

(ίπ $10,000)

J'anuaryFebruaryMarchApriI

[ΊΛaΥJuneJulyAugustSeptemberOctoberNovemberDecember

22"62119;2320241219'222016

340380220100490625375310175145120165

]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 cnrrentyear 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 eabllWΟΓkeraccounts 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, andthe cost of !aYlng off a worker ls $400.

3.5 Soi"rionσ{Aggregaιe PlanningProblems.1ry ΙίneΙJi"ΡrΟgTamrhίng ~41

3.5

α. DeterrniI\e. the mίnίmιιm ..constarit:workforGe:i,t)Ίat",will meet the predicteddemand fot the comlng ~e&r·,i";.,,, ; , ";,,

b. Evaluate.the cost όftheρlan deterh1ined Ια -part (αΙ. . ,Ι'" ';~', ;'1- ,\1'":-' 'r,' ,ί 'τ,'·,·'.:".-'" -,

] 5. Fot the datainProblem~14, determine the cost oftheplaiιthat changes theworkforcesize each period to most '~lb!ie1y;n~tch the demaιid. >" .

16. Graph the cumιι1atίve,net'delnand fofthe SUΡerchίΡ data of.Problem ]4. Graph-ically deterrnin"e' a, Ρrοdιιctίοή;ρ1aπ'that changes production levels υο mόre that!tllree tirnes, and deterrnlne the tota1cost of that p1an,

SOLUTlON OF ΑGG~~,G~Τ~,~ΡίΑΝΝljΝGPRO'B,LEMSΒγ LINEAR PROGRAMMING

_ι~, \f

Linear prograJI)ffii.ng ί5,aτοσο 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 maximizeΟΓminlmize 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 lsthat 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 theg~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

very large problems2

Cost Parame~ers and Given InformationThe following νείιιεε ar~ asslImed to'be known:τ

,CH = ~ost'of,hiring one.worker,,,",

,C,p = ,Cost:of firing ουε worker; πΙ"C[ = 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.Time-dependent costparameters could be ύsefυ1 for Π10deiing changes ίη thecosts οfhίήngor firing due, for example, to shortages ίπ tl1e labor ροοl, οτ changes ίη the costs of ρτο-duction and/or storage due to slJortages ίη tiie supply of resources, οτ cllanges ίη interest

rates.

, An overνiew of Iinear programming can be found ih Supplement 1, which foIIows this chapter.

2 The qualifier is intluded because rounding ma{ιjive suboptimal solutions. There wiII be more about this

point Iater.

Ι. Conservation of workforce constraints.

WI = Wt-1 + Ηι - FtNumber = Number + Number - Number

of workers of workers hired firedίη t ίη Ι - Ι ίη Ι ίη Ι

2. Conservation of units constraints.

for Ι :s t :s Τ.

142 Chapter Three Aggrcgatf Planning 3.5 Soluιion of Aggre~αte PIanning Problems.by Linear Ρrο.ι:rammίη,(!' 143

Problem VariablesThe following are the problem variables:

WI = Workforce ]eve] ίη period ι,Ρι = Production Ιενεί ίη period Ι,Ιι = Inventory leve] ίη period ι,

Ηι = Number ofworkers hired ίη period ι,FI = Number ofworkers fired ίη period ι,ΟΙ = Overtime production ίη units,υι = Worker idle time ίη units ("undertime"),

SI = Number of units st1bcontracted from ontside.

lη addition Ιο these constraints, lίnear programming requires that all problem variablesbe nonnegative. These constraints and the nonnegativity constraints are the minimum thatmust be present ίη any formtllation. Notice that (Ι), (2), and. (3) οοτιετίιιιιε 3Tconstraints,rather than 3 constraints, where Tis the length ofthe forecast horizon.

The formu]ation also reqnires specification of the initial inventory, 10, and the ίηί-tia] workforce, Wo, and may inc]ude specification of the ending inventory ιτι th.e finalperiod, lτ.

The objective function inclndes all the costs defined earlier. 'Πιο lίnear programmingforιnll]ation is to choose values ofthe probleIn variables W" Ρι, 1" Η" Ft, Ο" υι, and S,lo

τMinimize Σ (cHHI + cpF, + ctl, + cRP, + coOI + cυU, + C8S,)

'=1snbject 10

The overtime and id]e time variables are determined ίη the following way. Theterm Knt represents the number of units produced by one worker ίη period t, so thatKn,W, would be the number ofunits produced by the entire workforce ίη period ι. How-ever, we do υοι require that Kn,W, = Ρ,. If Ρ, > Kn,Jflt, then the number of υυίιε ρτο-dnced exceeds what the workforce can ρτοσυοε οη regular time. This Ineans that the dif-ference is being produced οη overtime, so that the number ofunits produced ωι overtimeis exactly ΟΙ = Ρ, - KntW,. If Ρι < KntWt, then the workforce is producing less than ίιshotIld be οη regular time, which means that there is worker idle time. The idle time ismeasured ίη units ofproduction rather than ίη time, and is given by υι = Kn,W, - Ρι.

W, = W,_ 1 + Ηι - F, for Ι :s t :s Τ(conservation of workforce),

Ρ, = Kn,WI + Ο, - U, for Ι :s t :s Τ(production and workforce)

(Α)

(Β)

1ι = Ιι- ι + Ρ, + SI - Dt for ] :s t:s Τ (inventory balance), (C)

(D)Η" F" Ι,, Οι, U" S" W" Ρι ~ Ο (nonnegativity),

ρίιιε any additiona] constraints that define the values of starting inventory, startingworkforce, ending inventory, οτ any otheI' variab]es with νείιιεε that arefixed ίη advance.

Problem ConstraintsThree sets of constraints are required for the ]jnear programming formιιlation. They areincluded το ensure that conservation of labor and conservation of units are satisfied.

Ιι = lι- ι + Ρι +[nventory = Inventory + Number +

ίη ι ίη ι - Ι ofunitsproduced

ίη Ι

SINumberofunits

subcontractedίη ι

Dt- Demand

ίη ι

for 1 :s ι :s Τ.

Rounding the VariablesΙn 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 includetlle size of the workforce, tlle number of workers hired each ρειίοτί, and the number ofworkers fired each period, and είεο may include the ηUΠ1ber of units produced eachperiod. (Ιι is possible that fractional numbers of units could be produced ίη εουιεapplications.) One way Ιο deal with this problem is το require ίη advance ιίιετ some οτaII ofthe problem variab]es assume οηlΥ integer values. Unfortunately, tlliS makes thesolution algoritllm considerably more CΟΠ1Ρ]eχ.The resulting ΡrοbleΠ1, known as an ίη-teger lίnear ΡrοgraΠ1Π1ίηg prob]em, requires mtIch Π10re computational effort Ιο solvethan does ordinary Jinear prograInming. For a moderate-sized pΓOb]em, solving theprob]em as an integer ]jnear program is certa.inly a reasonabJe alternative.

Ifan integer programming code is l1navailable οτ ifthe problem is simply Ιοο JargeΙο solve by integer programlning, linear programmingstill provides a workable solu-tion. However, after the lίnear programIning soltItion is obtained, some of the problemvariables must be rounded Ιο integer valtIes. Simply rounding off each variable Ιο theclosest integer may lead Ιο an infeasibJe solution and/or one ίη which prodllction andworkforce levels are inconsistent. It is ηο! obviolls wllat is t]le best way to round thevariables. We recotnmend the following conservative approach: rollfid tlle values ofthenumbers of workers ίη each period ι to W" the next ]arger integer. Once the va]ues ofW/ are determined, the va]ues ofthe other variables, Η/, Ft, and Ρι, can be fOl1ndalongwith the cost of the resulting plan.

Conservative rounding will always result ίη a feasible sollltion, but will ΓarelΥgive the optimal soltItion. The conservative solution generally can be improved bytrial-and-error experimentation.

3. Constraints relating production levels Ιο workforce levels.

Ρ, = KntW, + Ο, - U,Nuιnber = Number + NuInber - Numberof units of units of units of units

prodtIced produced prodtIced of idleίη t by regtIlar οη over- production

workforce time ίη t ίη tίη ι

fol' ] :s ι :s Τ.

144 Chapter Three Aggrcgaιe Plann,ng

There iS τιο guarantee that if a problem can be formulated as a lίnear program, thefinal 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 idletime 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, = ΟH,F, = Ο

for Ι ,s; t,s; Τ,

for Ι ,s; ι,,;; Τ,

since iftlle prodnct oftwo νariables is zero ίι means τίιετ at least one ηιυει be zero. Un-fortunately, these constraints are τιοι Ιίαεετ, as they involve a prodnct of problenivariables. However, ίι turns ου! that ίι is αοτ necessary Ιο explicitly inclnde thes~constraints, becanse the optimal solntion ιο a lίπear programming problem always οο-curs 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. '.

ExtensionsLinear programming also can be used to solve somewhat more general νersions oftheaggregate planning problem. Uncertainty of demand can be acconnted for indirectly byasstIming ιίιετ there is a minilllnm bnffer illventory Β, each period. lπ that case wewotIld include the constra ints

[,~B, for Ι ,s; ι,,;; Τ.

'Πιε constants Ξ, would have ιο be specified ίπ advallce. Upper bonnds ου the num-ber 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 treatbacklogging 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, 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 betrue that [/ and 1,- are ηοΙ botll positive ίη the same period t. As witlJ the oνertitne andidle tilllC and the hirillg alld. firing variables, the properties of lίπear programming wil1guaralltee that this holds wi Ιπου! having Ιο explicitly illclude the constraint [/ ς = ο:ίπ the fOI·mlIlation.

3.5 SQluιionο[ Aggregare Planning Problems by Unear ProgramminR 14!

lπ the developmellt ofthe lίπear programming model, we stated the requirement thatall the cost functions ηιυετ be lίπear. This is ιιοι strictly correct. Linear programmingalso 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 isone that is composed of straight-line segments. Hence, a convex piecewise-linearfunctioll is a function composed of straight lίnes that Ιιενε increasing slopes. Α typicalexalnple is presented ίπ Figure 3-5.

[η practice, ίι is likely that some ΟΓ all of the cost Ιιιαοιίωιε for aggregate planningare convex. For example, if Figure 3-5 τερτοεεητε the cost of hiring workers, then tllemarginal cost of IJiring οιιε additiollal worker increases with the number of workersthat Ιιενο already οεετι Ilired. This is probably more accnrate than asstIIning that thecost of hiring ουε additional worker is a οοπετωιτ jlldependent of tlle nnmber ofworkers previously hired. As more workers are hired, the available labor ροοl shrirιksand lnore effort mIIst be expended Ιο hire tlle remailling available workers.

lη order Ιο see exactly how convex pίecewίse-~jllear functiolls would be incorporatedίnΙο the lίnear ΡrοgΓaιnιnίπg formlllatioll, we wil1 consider a very silnple case. Snpposethat 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, andίι 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:

Η, = Ηι, + Η2,.

Interpret Ηιι as the nUlllber ofworkers hired up Ιο Η* and Η2, as tlle llutnbeI' ofworkers hired beyolld Η* ίπ period Ι. The cost οΓ hiring is now reΡΓesented ίη theobjective fUllction as

τΣ (CI'IIHII + CInH2'),I=Ι

FIGURE 3-6Convex piecewise-Iinear- hiring costfunction

146 Chapter Three ."'κgregarePlαnning

and the additional 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 ιΜminimum costsolution, ίι will force Ηιι ΙΟ its maximum value before aIIowing H2t Ιοbecome positive, since CHl < CH2' This is the reason that the cost functions must beconvex. This approach can easily be extended το more than two linear segments and toany of the other cost functions ρτεεέιιι ιιι the objective function. 'Πιε technique isknown as separable convex programming and is discussed ίη greater detail ίn Hillierand Lieberman (1990). . .'

Other SοlutίοnΜetl1όdsBowman (1956) suggested a transportation form.ulation ofthe aggregate planning prob-lem when back orders are ηοι permitted. Several authors have explored solving aggregatepIanning problems with specially tailored algorithms. These algorithms could be faster[ατ solving large aggregate planning problems than a generallinear programming codesuch as υΝΟο. Ιτι addition, some of these formulations allow for certain types of non-linear costs. For example, if there are economies of scale ίn production, the productioncost function is likely Ιο be a concave function ofthe number of units produced. Concavecost Ιϊυιοιίοηε are not as amenable το linear programming formnlatio11s as convex funcctions. We believe tllat a general-purpose linear programrning package is sufficient formost reasonable-sized problems that one is likely ιο encounter ίη the real wOl·ld. Ηοιν-ever, the reader sho.uld be aware that there are alternative solution techniques that conld

3.6 SoI<ing Aggregαte Planning Problemsby,Lineαr Progrcmm'ng' Ah. EXDmple 147

be.J;Iloreefρcient for)argeproblem~ aηq <eOUldpI9y,idesoMions whep, some ofthe costfunctions are nonlinear. Α paper that details a transportation-type procedure ιυοτε. efficcient than thesimplex method for solving aggregate planning problems is Erengιic andTufekci(1988>':

SOιVING AGGREGATE Pi:.ANNiNGPROBLEMSBYLINEARPROGRAMMING: ANE~AMPΙE

We ,wiHdemonstrate the use oflinear programming'by:(Jnding theoptimal solution ιοthe example presented ίn Sectlon 3.4. As there is υο subcontracting,.overtime, στ id1etime aHowed, and the cost coefficients are constant with respect το time, the objectivefunction is simply

(

" ·6 6 .,. 6

Minimize. 500 Σ Ηι + 1,000Σ Ρ, +80 Σ It).,t=l ι=\ ι=ι

The boundary conditions .comprise .the specifications of .:the initial inventoτy οί500 unlts, .theinitlal wprkforce of 300 work:ers, and. the. ending inventory of 600 units.These are besthanp1edby including a separate additional constralnt for each boundarycondition.

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 ση the 1eft-hand side and nonnegatlve constants ση thεright-hand side) is as fol1ows:

W1 - Wo - Ηι + Fι·= Ο,W2 - Wl - Η2 + F2.= Ο,W] - W2- Η] + F3\=0,

W4 - W3 - Η4 + F4 = Ο,Ws - W4 - Hs + F'i;= Ο,W6 - Ws - Η6 + Ψ6 = Ο;'

(Α

Ρι - I1 + 10 = 1,180,

Ρ2 - /2 + I1 =640,

Ρ3 - Ι] + 12 = 9ΡΟ,Ρ4 - 14 + /3= 1;200,

Ps - 15 + 14= 1,000,

Ρ6 - 16 + 15 = I?~OO;

(Β

Pj - 2.931 Wl = Ο,

Ρ2 - 3.517W2 = Ο,Ρ3 - 2.638W] = Ο,Ρ4 - 3.81Or1l4 = ο,Ρ5 - 3.224W5 =0,

Ρ6 - 2.198W6 = Ο;

(C

148 Chapter Three ΑglζΓeφιe Plonning 3.6 SoIving Am,rregate [)!anning Problems b-yLinear Programming: An ExαmfJle 149

WI,· .. , W6, Ρι,· .. , Ρ6, lι,· .. ,/6, FJ, ••• , F6, Ηι, ... ,Η6;ΞΞ: 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 οοlinear 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 ορ-timal 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 ουfractional values ofthe variables. The actnal cost wiJJ be slightly higher after rounding.

FoJJowing the τουικίίιη; procedure recommended earlier, we will τοιιικί alJ the val-ιιοε 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 fίrmshonld fire 27 workers ία JanlIary and hire 465 \vorkers ίη May. The complete solution isgiven ίη Table 3-6.

Again, because co!umn Η ίη Table 3-6 corresponds το net demand, we add th.e600 ιυυτε of ending inventory ίη June, giving a total inventory of 900 + 600 = 1,500units. 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 andthe constant workforce plan.

'Πιε restIlts of the !inear programming ana!ysis suggest another plan that mightbe more suitable for the company. Βεοειιεε the optimal strategy is το decrease theworkforce ίη .fanuary and blIild ίι back ιιρ again ίη May, a reasonab]e alternativemight 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 wοrkeΓSto hire ίπ May is ιο simp]y re-solve the lίnear pτogram, bl1t without the variablesFI, ... , 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 theold 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, andthe cost ofthe plan is approximately $386,120. This is only slightly more expensivethan the optimal plan, and has the important advantage of ποι requiring the firing ofany workers.

Problems for Sections 3.5 and 3.617. Formnlate Problem 13 as a Jinear program. Be sure ιο define aJJ variables alld

include all tlle reql1ired constraints.

LΡ ΟΡΤ1ΜυΜ FOUND ΑΤ STEP 14 {~:~.{~)~~1;:,.~~:;: *22.

OBJECT1VE FUNCT10N VALUE1 ) 36185.0600

VAR1ABLE VALUE REDUCED COSTΗ1 6.143851 .000000Η2 64.310690 .000000Η3 .000000 300.000000Η4 116.071400 .000000F1 .000000 300.000000F2 .000000 300.000000F3 87.662330 .000000F4 .000000 300.00000011 .000000 59.41558012 .000000 189.28570013 .000000 31.006490W1 106.143900 .000000w2 170.454500 .000000W3 82.792210 .000000W4 198.863600 .000000Ρ1 850.000000 .000000Ρ2 1260.000000 .000000Ρ3 510.000000 .000000Ρ4 980.000000 .00000014 .000000 120.292200

1 SO Chapter Three Aggre~αte ΡΙαπnίιιι

18. Problem 17 was solved οα the computer. The lίnear ΡrogΓammίng ουιρυι of theΡrοgΓam is as follows:

α. ΒΥ rounding the values of the variables using the conservative approach de-scribed ίη Section 3.5, find a feasible solution το the problem. Determine thecost of the resulting plan.

b. Determine another plan that achieves a lower cost than that determined ίηpart (α) while still retaining feasibility (tl1at is, having a nonnegative amountof inventory οη hand at the end of each period).

19. α. Formulate Problem 9 as a lίnear program.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. α. Ροιπιιιίειε Ρτοοίεπι 14 as a linear program.b. Solve the problem using a lίηear programlning code. RolInd off the εοίυιίωι

and determine the cost oftl1e resulting plan.

21. Consider Mr. Meadows Cookie Company, described ίη Problem 13. Suppose thatthe cost ofl1iring workers each period is $100 for each worker ιιηιίl20 workers arehired, $400 for each worker when between 21 and 50 workers are hired, and $700for each wot·keI"hired beyond 50.

3.6 SoI'Iinf:"Aggregαιc I)kInning ΡrοblCΠL~by Uneαr Programfning: Απ Example 151

α. Write down tl1e complete lίnear programnling 'forιntIlation ofthe revisedproblem.

b. Solve the revised problem using a lίηear programming code. What differencedoes the new 11iringcost function make ίn the εοίυτίοη?

Leather-AII ρτοσιιοεε a lίne of handmade leather ρτοσυαε. Αι the present time,the company is producing οηlΥ belts, handbags, and attache cases. 'Πιε predicteddemand for tl1ese three types of items over a six-montll planning horizon is asfollows:

Number ofMonth Working Days Belts Handbags Attache Cases

1 22 2,500 1,250 2402 20 2,800 680 3803 19 2,000 1,625 1104 24 3,400 745 755 21 3,000 835 1266 17 1,600 375 45

The belts reqιIire an aνerage of two Ιιοιιτ» to ρτοοιιοο, the handbags threeΙιουτε, and the attache cases six hours. ΑΙΙ the workers have the skill ιο work σηany itetD. LeatheΓ-ΑlΙ11as 46 employees who each has a share ίη the firm and can-σοι be fired. There are an additional 30 locals that are available and can be hiredfor short periocls at higher cost. Regular employees earn $8.50 per hour οιι reglI-lar time and $14.00 per hoιIf ου overtime. ReglIlar time comprises a εενεη-Ιιουτworkday, 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 wantto incur any demand back orders.

α. Using worke!" hours as an aggregate measlIre of ρτοτίυοτίοη, convert the fore-casted demands ιο demands ίη terms of aggregate units.

b. What wotIld be the size of the workforce needed το satisfy the deInand forthe coming six months οη Γegular time only? Would ίι be to the conιpany'sadvantage το bring the permanent workforce ιφ to this level? Why οτ WllYτιοι?

c. Determine a production plan that meets the demand lIsing only regular-timeemployees and tl1e total cost of that plan.

d. Determine a ρτοοιιοιίοιι plan that ιιιίίίεσε only additional employees to absorbexcess demand and the cost of that plan.

*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 arenecessary.

b. Solve the problem formulated ίη part (α) using a lίnear progran1m ingcode. Ronnd all the relevant variables and detern1ine the cost of thereslIlting plan. Compare Υουτ restIlts with those obtained ίη Problem 22(c)and (ά).

152 Chapter Three Aggτegaιe Pkιnning

3.7 ΤΗΕ LINEAR DEClSION RULE

Ατι interesting alternative approach το solving the aggregate planning problem has beensuggested 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 andnumbers of wοrkeΓS, are represented by quadratic functions. That is, the total cost overthe T-period plannil1g 1101'ίΖοηcan be written ίη the form

τΣ [clW, + C2(W, - W,_t)2 + C3(P, - Kn,W,)2 + C4P, + cs(l, - C6)2),=ι

subject το

/, = J,_I + Ρ, - D, for 1 s t S Τ.

The valtIes of the constants Ct, C2, ... , C6 must be deteτmined for each parιicιιlarapplication. Expressing the cost functions as quadratic functions rather than sitnplelίnear fUl1ctions has some advantages over the lίnear programming approach. As quad- .ratic functions are differentiable, the standard ruIes of calct1lus can be used ιοdetermine optimaI solutions. The optimal solution will occur where the first partiaJderivatives with respect to each of the probIenl variables are zero. When quadIaticfunctions are differentiated, they yield first-order equations (\inear εσιιετίοαε), whichare easy ιο solve. Also, quadratic functions give a more εοοιιτετε ερριωωηετίοιι ιοgeneral nonlinear functiol1S than do lίηear functions. (Any nonlinear flInction may beapproxinJated by a Taylor series ωφετιείοιι ίη Wllich the first two terms yield a qlIad-ratic approximation.)

However, the quadratic appIΌach aIso has one serious disadvantage. It is thatqIIadratic fllnctions (that is, parabolas) are symmetric (see FiglIre 3-7). Hence, the cost

FIGURE 3-7Quadratic cost functions lIsed ίη deriving t!le Iίnear decision ru!e

, Number--units held

(01 Cost of chongingworkforce leνels

(bl Cost of chongingίπνθηΙΟΓΥleνels

3.8 Mod,/ingManag,ment Bεhαoίor 1S3

ofhiring a given number ofworkers must be the same as thecost offiring the same num-ber ofworkers, and tIle cost ofprodllcing a given nlllflber ofunits οη overtime τιιιιετ beιίιε same as the cost attached το the same number ofunits ofworker idle time. The prob-Iem can be overcome somewhat by υοι setting the center of symmetry of the cost func-ιίοη 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

κΡ, = Σ (a"DI+II + bWt-t + cJ,-l + d).

n=O

The terms απ, b, c, and d are constants that depend οτι the cost parameters. WI has a είπι-ilar form. The οοωριιιετίοαεί advantages of the lίηear decision rule are less itnportantnow than they were when this analysis was first done because of the widespread avail-ability of comptlters today. Because lίnear programming provides a much more flexibleframework foτ formulating aggregate planning probleIns, and reasonably large ΙίιιεετΡΓοgrams can be solved even οτι persohal computers, one would have ιο conclude thatlίηear programming is a preferable solution technique.

Ιιshould be recognized, however, that the texl by Holt, ModigJiani, Muth, and Simonrepresents a laηdmark work ίη the application of quantitive methods ιο production plan-ning problems. The authors developed a solution method that reslllts ίn a set of for.mu-las that are easy το iIllplement, and they actually undertook the implementation of themethod. The work details the application of the approach 10 a large manufacturerof hotIsehold paints ίη the Pittsburgh area. The analysis was actually implemented ίηthe company, but a subsequent visit Ιο the firm indicated that serious problems aroseWllen tlle lίηear decision nlle was foIIowed, primarily because ofthe fiτm's policy of ποιfiring workers when the model indicated that they should be fiΓed. (See Vollmann,Berry, and Whybark, 1992, ρ. 627.)

MODELING MANAGEMENT BEHAVIOR

Bowman (1963) developed θη ίηteΓestίηg technique for aggregate planning that shouldbe applicable Ιο otheI' types of problems as well. His idea is Ιο construct a sensiblemodel [οτ controIIing prodllction leνels and fit the ΡaraΠ1eters of tlle lnodel as closelyas possible Ιο the actllal ρτίοτ decisions made by management. Ιη this way the modelreflects the judgment and the experience ofmanagers and avoids a number ofthe prob-lems that arise when using more traditional modeling methods. One such problem isdeternlining the accllracy of the assumptions required bythe lnodel. Another problemthat is avoided is the need to deteflfline vaIues of ίηριιι par31nelers that could bediflicult 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 Ihatproduction levels ΡΙ, ... , Pr must be determined. The silnplest I'easonable decisionrule is

Ρ, = Dt [οτ Ι ,;; ι s Τ.

Trying Ιο match prodllction exactly Ιο demand will generally result ίη a productionsClledule that is 100 enatic. Production smootlling can be accomplished tIsing a deci-sion rule of the form

Ρι = Dt + α(Ρι-ι - Dt),

154 Chapter Three Aggrega", Ρ/αππίππ

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 ίηdetail ίη Chapter 2.) When α equals zero, this rule is 'identical to tlIe, -οτιε jU5t,pre-senti:d, When α =; Ι, the ρτοσυοτίοη. iJf,period ι'ί); exactly the 'same as the ρτοάυοιίοτιίπ period ,Ι -' Ι. 'Πιε choice of α provides a meanS'ο[φΙacίng a relative weightootιmatching production with' demand Verst!'S'holdingproduction constant from periodτο perJod. i' - , .

Ιη addition το smoothing production, the firm al50 mJght be interested ίπ keeping ίτι-ventory leνels near εοτυε target leνel, say ΙΝ. Α model that smooths both inventory andproduction simultaneously is

Ρι ='0,' + a(Pt-I""D,) + β(IΝ - 1,-1),

where Ο S β S Ι measures the relatlve ~~ight pl~ced o~ ~m~CΊthingof 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 ofdemands. Α rule that includes smoothirig' of production' 'and inventory as well as fore-casts of future demand is

Ι+"Ρι = Σ αt-i+IDj + α(Ρι-1 - D,) + β(ΙΝ -/'-1),

ί=!

,ιThis 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 mathe-matical model. Undoubtedly, the previous work οα the lίnear decision rule insplredthe 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~ jj~srstra'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 retrospectivelyobserνing the system for a'reasonable period of time, and fitting the parameters το theactual history of management actlons di.ιiil1g th~t tinιe bsing a'technique such as leastsquares. Ιη this way the model,becomes a reflectioiι ofpast management behavior,

Bowτnah cbmpares the 'actiJal experience' of 11 'nuffiber of companies with theexperience 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? Thetheory 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 un-l1sl1alevent oc'curs, such as a fuuch~higher-than-anticipated demand οτ a su,dden declineίη productlve capacity due, for example, ιο the breakdown of a machine οτ the loss ofkey personηel, ίι ί5 likely that many managers would tend Ιο overreact. However, themodel would recommend production leνels that ar~ consistent,with past declsion mak-ing, Hence, the use of a simple but coηsistent model {οτ making decisions would keepmanagement ποιή paiιicking ίη an unusuaT situation. Althougli' the approacllis concep- ,tually appealing, there are few reports ίη the literatιireof successful implementations ofsuch a techniqne,

3,9;])jsαggregaIing,Aggτeg~ιe pian; 155 ~;1,\";'1

111'{, f

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

. . η= :463D;,+"'.':Δ4DΜ+;111Dι!t{+,Ο46Dι+':j'+ .013Dt+4h -, . , ,-Ι <"_", f

- ,022D,+s:'" :008D,+6 -.OIQDI+7 - ,009D'+8- .068b,t9~:Ob7D;tlO'~:005Di+11 .+.993W,rl - .46411_,1,+ 153,

and the w6rkfofce level ί~ -peHod Ι froin the foclιu]a, . ~,~.Γ ~'-~_.". ' .' ..i'.' 'ι " .

W, = .οιαο, + .0088Dt+J.+ :007.1DH2,+ ,0054D'+3 + ,0042D'+4+ .003IDt+5 + ,0023D,+6 + ,0016D'+7 + ,0012DH8+, ,0009Dt+9 + 1,0006Όι+Ίο +.0005'Dt+ 11'+ ,743W,_1 ...: ,0111-1 - 2,09:'

I~" ,"~ , ~" ::",>'1 <.1, i:''-' Ά ;,,'1·1; ';~"J>,j;' ~.'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 the number ofworkers

that the company should be usiηg ίη the curtent ρετίοά.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 calculatlngP,,-and less than στ equal to ,005 when calculating W" How much differencedbyou obSerνe ίη th'e,ariswer?-- ι, '

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:Yoil'fiίιist cΌm'ρute ,the'he~ι:'-QaΙUeγ,..:'{and 'assume1:h1it'the new value of Wt-I isthevaΙuecόΜΡutedίnΡart'(a):], ',γ,' ι ,

25" Usirigthe f6116wίήg ν~lίΙ~~&[fuanagenieJι{~όefficίeiitSj'i'οr Bowman's smoothed, , . ,. -' ('Ι' <,' • '.:',' ,t" ,~)I "ι"." ';;'_~/' \ ','" , --'---',." . ι

Ρ'i'οd!!'ctίοήmodel, ~.eteilniiJe the pl'Oductί?iilevel for which Harold Grey (describedin Problem, 9) shoίlld plan ίη the cοmiήg year: .A:.ssumetHat the cunent productionlevel i~ Ί :Sb;Odbpackag~~,' / ,,' 'Ι,ι,"

,οι ';" ,3416,; α2 =,Ι2l.1"α3 = .0?56", α4 = .0663,αs,= ,0023,' . α = ;6", β = 3, ,!j ΕΝ = 40,000,

26, α, Discnss the similarities and the differences betwee~ th'e linear decislon rule ofHolt, Modiglianl, Muth, and Simon and Bowman's management coefficientsapproach, '

b. Discuss the relatίve advantages and disadvantages of linear decision rulesversus lineaτ program,ming for solving aggregate planning problems,

ί:

DISAGGREGATlNG AGGREGATE PLANS

As we saw earlier ίη ,this chapter, aggregate p]anning Inay b" done at several leνels ofthe fiπn. Example 3,] considered a single plant, and showed how one might definean aggregate unit to includ,e the slx different items produced at that plant. Aggregate

ajs,~s,bi [οτ Ι s, j s, J.

ρ. ;π , ':Ι:ii;i<,-: ':."" ",

~·!·rl, .,.-,~.----~

156 Chapter Three Aggregate P/anning

planning might be done for a single plant, a product family, a group of families, οτ flthe 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. Thplans 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 ag-gregate plan at a high leνel. These plans would then have Ιο be "disaggregated" Ιο pro-συοε detailed production plans at the plant and individual item leνels.

Ιι is σοι clear that disaggregation is an ίεευε ίη a11οίτοιυυετεηοεε. [f the aggregateplan 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 individ-tIal item ρτοάυοτίοιι plans and aggregate plans be consistent, then ίτ might be necessaryto 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 re-ceives 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 eachbtIdget center (staff, maintenance, buildings and grotInds, etc.) woLIld have Ιο be allo-cated its share. Budget centers would have to allocate funds Ιο each oftheir stIbgrotIps.For example, each school would allocate funds το individual departments, and depart-ments wotIld allocate funds ιο faculty and staff ίη that department.

Ιη tlle manufacturing context, a disaggregation scheme isjLIst a means of allocating ag-gregate units to individtIal items. JtIst as funds are allocated οιι several leνels ίη the uni-versity, aggregate units might have το be disaggregated at severallevels ofthe firm. Thisis the idea behind hierarchical ρτοουοτίου planning championed by several researchers atΜΙΤ 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 Bitranand Hax (1981). StIppose that Χ" represents the numbel' of aggregate υυίιε οί ρτοουο-τίοιι [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., disag-gregate) Χ* anlong the n items. We know that holding costs are already included ίη thedetermination 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 theannual usage rate for item j. Α reasonable optimization criterion ίη this context is τοchoose 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 programmingproblem:

J

Σ~=x*j=I

The upper and lower bounds οη Yj account [οτ possible side constraints οη the produc-τίοιι leνel [οτ itemj.

Α number of feasibility issues need ιο be addressed before the family τυη sizes Υίare further disaggregated ίυτο lots for individual ίτειυε. The objective is ιο schedulethe lots for individtIal itelns within a family so tllat they run out at the scheduled setuptime for the family. Ιη this way, items within the same family call be produced withinthe same productioll setup.

The concept of disaggregating the aggregate plan along organizational lilles ίn afashion 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 aggregateplans and detailed item schedules remains to be seen.

Anotheτ appτoach Ιο the disaggregation problem has been explored by Chung andKrajewski (1984). They develop a mathem.atical progra.mming fOllnulation of theproblem. lnputs Ιο the program include aggregate plans [οΙ' each product [ωηίΙΥ· Thisincludes 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 tospecify lot sizes and timing ofpJ'Oduction runs [οτ each individual item. consi.stent withthe aggΓegate information [οτ the product family. AltllOUgh sllch a fOlllll1latiol1

J KjAjMinimize ΣΥ

j=1 }

subject to

and

158 Chapter Three Aggr'gaIe Planniiιg•.~,.).,:\;;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 ίndίνΓd-ual items ίη each product family.

Problems for Section 3.927. What does "disaggregation of aggregate plans" mean?28, Discuss the following quotation made by a production manager: ''Aggregate plah- 'ί'

ning is useless ιο me because the results have nothing to do with my masterproduction schedule."

3.1 Ο PRODUCTlON PLANNING ΟΝ Α GLOBAL SCALE

Globalization of manufactu.ring operations is commonplace. Many major οοτροτε-"tions are now classified as multinationals; manufacturing and distribution activitiesroutinely cross intemational borders. With the globalization of botl1 sources of pτo~duction and markets, firms must rethink production p!anning strategies. One issue ex-plored 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 productionstrategies, 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 ίηΙΟa 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 performanceand uitimateiy be reflected ίη increased profirability. Ιη Ihe elecIronic industry, rhere areexampIes 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% increaseίη inventory tu.rnover Ιο their intemationaIization ofmanufacturing.

Cohen et al. (1989) outline some ofthe issues that a firm must consider when planningproduction 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 competitivetoday, 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 ίπιοdecisions regarding shipments of raw lηaterίal, intermediate prodllct, and finishedproduct across national boundaries.Exchange rates fllIctuate randomly and affect production costs and pricing decisionsίπ 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, andaccess to new technologies.

3.11 Prαccicαl Consideranons 159

Strategies for market penetration, local content rules, and quotas constrain productflow across borders.Ρτοουοι designs n1ay vary by national market.Centralized control of n1ιιltinationa! enterprises creates difficlllties [οτ several rea-sons, and decentralized control requires coordination.CtIJtural, language, and skil1 differences can be significant.

Determining optimal globalized manufacturing strategies is clearly a daunting prob-lem for any multinational firm. One can formulate and so!ve mathematical modelssimilar το the lίnear programming formulations of the aggregate planning mode!spresented ίπ this chapter, but the results of these models must always be balancedagainst judgment and experience. Cohen et al. (1989) consider such a model. They as-sume multiple products, plants, markets, raw materials, vendors, vendor supply CO!l-tract alternatives, τίιηε periods, and countrίes. Their forn1ulation is a large-scale mixedinteger, nonlinear program.

One issue τιοι treated ίη their model is that of exchange rate fluctuations and their ef-fect οη both pricing and production planning. Pricing, ίη parιicu!ar, is traditional1ydone by adding a πιετκιιρ το υπίι costs ίπ the home maΓket. This cornpletely ignoresthe issue of excl1ange rate fluctuations and can lead. 10 unreasonable prices ίο sornecountries. For example, this issue has arisen at Caterpillar Tractor (CaterpillaI" TractorCompany, 1985). [π this case, dealers all over the WΟΓldwere bil!ed ίπ U.S. dollarsbased 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 thefirnl switched το a 10caΙΙy cornpetitiνe pricing strategy το counteract this problern.

The ηοιίοιι that l11anufacturing capacity can be used as a hedge against exchangerate fluctuations 11as been explored by several researchers. Kogut and Kulatilaka(Ι 994), [οτ example, develop a rnathematical model [οτ determining when ίι ίδ optirnalΙο switch prodllction frorn one locatίon to another. Since the cost of s\vitching isassuIned to be positive, there mnst be a snfficiently large difference ίπ exchange ratesbefore switching is recoll1mended. As an example, they consider a situation where afirl11can produce ίΙδ product ίη either the United States οτ GerιηaπΥ. Ifproduction isclIrrently being done ίπ οπο locatίon, the model provides a lηeans of detern1ining if ίιis 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 beenexplored by Ηucl1ΖeΓιneίer and Cohen (1996) as \vel1.

PRACTlCAL CONSIDERATIONS

Aggregate planning can be a valuable aid ίπ plaoning prodlIction and workforce levels fora company, and provides a Ineans of absorbing deιηand fluctuations by srnootlling work-force and plΌduction leνels. There are a number of advantages for planning οπ the aggre-gate lονοΙ oveI"the detai! level. One is that the cost ofpreparing forecasts and deteΓIniningproductivity and cost ΡaraιηeteΓS οη an individllal iten1 basis can be prohibitive. T11ecostof data collection and ίnρυ! ίδ probabJy a 1110ΓΟsignificant drawback of ]aΓge-scale Il1ath-eιηatίcal ΡlΌgΓaιηmίng fΟΠl1l1latίοnsof detailed production plans than is the cost ofactllally perfoΓIl1iI1gtl1e COIl1plltations.Α second advantage of aggΓegate planning is therelative improvement ίη forecast aCCUΓaCΥthat cal1 be achieved by aggregating items. As

160 Chapter Three Aggregalc P"'""ing

\ve saw ίη Chapter 2 οα forecasting, aggregate forecasts are generally more accurate thanindivi<lual forecasts.3 Finn.lly, an aggregate planning tramework allows the manager, ΙΟsee the "big picture" and υοι be unduly influenced by specifics.

With all these advantages, one would think that aggregate planning would have animportant 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 orinterest ίη aggregate planning methodology. First is the difficlIlty of properly defining artaggregate ιιηίι οί ρτοσιιοιίοιι. There appears το be ηο simple way ofspecifying how ίη-dividtIal 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 unitsare required. It is difficult enough το obtain accurate cost and deInand information forreal units. Obtaining such information οα θη aggregate basis cotIld be considerablymore difficult, depending ου tl1e aggregation schetne tbat is used. TlΊίrd, aggregate plan-ning models rarely reflect the political and operational realities of the environnlent ίηwhich the company is operating. Assuιning that workfoioce levels can be changed easilyis 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 theextremely sensitive and important issues that are addressed ίη this analysis.

Another issue is whether ΟΤ αοι the goals of aggregate planning analysis can beachieνed through methods other than those discussed ίη this chapter. Schwarz andJohnson (Ι 978) clainUhat the cost savings achieved lIsing a lίηear decision rL1leforaggregate planning could be obtained by improved Inanagement ofthe aggregate inven-tory 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 thecost savings ofthe lίηear decision rule were a result ofincreasing the bIIfFer inventory andηο! ofmaking major changes ίη the size oftl1e labor force. This single comparison doesηοΙ verify the authors' 11ypothesis ίη general, but suggests that better inventory manage-ment, 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 thischapter can, and should, serve as θη aid to production planners. AlthoLIgh optimal solu-tions Ιο a mathematical model may ηοι be true optimal solutions Ιο the problenl that theyaddress, they do provide insight and could reveal altel'natives tl1at would ηο! otl1erwisebe evident.

3.12 HISTORICAL NOTES

The aggregate planning problem was conceived ίη ηη important series οΓpapers that ap-peared ίη the mid-1950s. The first, Holt, Modigliani, and Simon (1955), discussed thestructLIΓeoftl1e problenl and introduced the quadratic cost approach, and tl1e later sttIdyofHoJt, Modigliani, and Muth (Ι 956) concentrated οη the coInplltational aspects ofthemodel. Α complete description ofthe metl10d and its npplication Ιο production planningΓοτa ρηίηι coInpany is presented ίη Holt, Modigliani, Muth, and Simon (1960).

That prodIIction planning problems could be formIllated as linearpΓOgrams appears Ιοhave been known ίη the early 1950s. Bowman (1956) discussed tl1euse οΓa transportationmodel for production planning. The particular lίηear programming forιnlIlation of tlleaggregate planning problem discllssed ίη Section 3.5 is' essentially tl1e same as the one

3 More precisely. ίl the individual lorecasts are πο! perfectly correlated, then the standard deviation ΟΙ thelorecast error lor the aggregate will be less than the sum ΟΙ the standard deviations ofthe indlvidualforecast errors.

3.13 Summary 161

deνeloped by Hansmann and Hess (Ι 960). Otl1er lίηearprogrammίηg formulations oftl1eproduction planning problem generally involve nlu1tiple products ΟΓmore complex coststructures (see, for example, Newson, Ι975η and Ι975b) ..

More recent work ου the aggregate planning problem has focused οιι aggregationand disaggregation issues (Axsater, 1981; Bitran and Hax, 198 Ι; and Ζοίίει, 1971), theincorporation oflearning curves ίnΙο lίnear decision rules (Ebert, 1976), extensions ιοal10w for multiple prodncts (Bergstrom and Smith, 1970), and inclusion of marketingand/or financial variables (Damon and Schramm, 1972, and Leitch, 1974).

Taubert (1968) considers a tecllnique he refers Ιο as the search decision rule. 'Πιοmethod requires developing a computer simulation model ofthe system and searcl1ingthe response surface using standard search techniques Ιο obtain a (υοι necessarily ορ-timal) 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, andSimon (1960) for the case ofthe paint company.

Kamien and Li (1990) developed a mathematicalInodel ιο examine the efFects ofsubcontl'acting ου aggregate planning decisions. The authors show tllat under certaincircumstances ίι is preferred το producing in-house, and proνides an additional πιεεηεof smootl1ing production and workforce levels.

Determining οριίωε! production levels Γοι'all products Ρrοdιιced by a large firm can be ηησιοτυιουε undertaking. AggI'egate pIannil1g addresses this problem by assuIning that indi-vidIIal items can be grouped together. Ho\vever, finding an effective aggregating scheme canbe difficult. The aggregating scheme chosen must be consistent with the structure of tlteorganization and the natιιre of the products produced. One particular aggregating scheΠJethat 11asbeen sttggested is itenIs,jamiIies, and types. ltems (or stockkeeping ttnits), repre-senting the finest leνel of detail, wοιιld be identified by separate part numbers ..Families aregroups of items that share a commollIllanufacturing setιιΡ, and types are natural grΟΙιΡίngsof families. This particular aggregation scheme is ΓθίΓIΥgeneral, but there is ηο guaranteethat ίι wiJl work ίη every application.

/η aggregate planning one assun1es deteπιιίnίstίc den,and. DeIlland forecasts over aspecified planning hOI'izon are required ίηριιΙ This assLImprion is ηο! made for the sake ofrealisIll, bllt ιο allow tl1eanalysis ιο focus οη the changes ίη the demand that are systematicrather than random. Tl1e goal of the analysis is to determine the nllmber of workers tl1atshould be employed eaCI1peI'iod and the nunlber of aggregate lInits that SI10uldbe producedeach period.

Theobjective is to Ininimize costs ofproduction, ρηΥΓΟΙΙ,holding, and cl1anging the sizeofthe workforce. The costs ofmakillg changes are generaIIy referred Ιο assnlOoI/,il1g C08tS.Mostofthe aggregate planning Inodels discussed ίη this chapter assume that all the costs are Iineaι"jitnctions. This means tl1at tl1ecost of hiιing an additional worker is the same as the cost ofhiring the previous work.er,and the cost ofholding αη additional ιιηίι of inventory is the sameas the cost ofholding the previous ιιηίι ofinvenιory. This assttmption is probably a reasonableapproximation for Inost real systems. Jt is unlikely that the primary probleIn with applyingaggregate planning metllodology Ιο a real sitIIation will be tllat the shape ofthe cost functionsis incorrect; ίι is more likely thatthe primary difficulty will be ίη corτectly estimating the costsand otl1errequired ίηριιΙ information.

One can fίnd approximate solutions ιο aggregate planning problems by gΓaΡΙ1ίcaΙnletl1ods. Α prodLIction schedule satisfies demand ίΓ the graph of cIImulative prodlIctionlίes above the graph of cumulative dCΠland.As long as all costs aI'e lίηear, optimal sol-ιιtίοns4 can be found by lineaι' pl'lJg1'OrιIIning. Because solIItions of lίηear programIning

4 That is. optimal subject ιο rounding errors.

\1

162 Chapter Three AggτegaIe I'lιιnning

probIems may be fractions, some ΟΓ all oftbe vaήabΙes must be rounded ιο integers. The bestway ιο round the variables ίε αοι aIways obvious. Once an οριίοιεί solution ί$ found, itsappropriateness must be considered ίη the context of the pIanning probIem, as there areintangibIe οοετε and constraints that might be difficult ΟΓ impossible ίο include directiy ίηthe probIem formulation.

Linear programming also can be used το determine optimai. ρτοσυοιίοιι ievels at theindividual item levei, but the size ofthe resulting problem and the ίnρυι data requirementsInight Inake such an approach unreaIistic for large οοιυρεηίεε. However, detailed Iinearprogranlming fornJuiations could be used at the faΠ1ίΙΥ level ratller than the individual itemlevel for Iarge firms.

EarIy work by Holt, Modigliani, Muth, and Simon (1960) required that the costfUl1ctions be quadratic functions. With quadratic costs, the form of the resulting optimalροΙίCΥ is a simple lίnear function, known as the lineαI' decision πι/e. However, theassumption of quadratic cost functions is probably ιοο restrictive το be accurate for πιοειreaL ρτοοίσιηε.

Bowman's lnanαgenIent coefficients modei fits a reasonable matllematical model of thesystem το the actual decisions made by management over a period of time. This approachIlas the advantage ofincorporating management's intuition υιτο the model, and encoLIragesmanagemel1t ρετεοτυιεί to be COl1sistent ίπ future decision making.

Aggregate ρίωιε will bc of lίttle ιιεο ΙΟ the fim) if they cannot be coordinated witlidetailed item schedllles (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 aggregatepIans ετε ιο have νείιιε το the firιn. There Ιιενε beel1 some mathematical programnlingformlIIations of the disaggregating problem suggested ίπ tl1e lίterature. but these disaggre-gation schenles have yet το be proven ίη practice.

,.7 ::. ~" ;iI' " ":'>,1

\. /' •.-. ~ ::t; :/.," ~,ι ~~,;

. ,1 'Ι" ;(> . ·':ι,ι::. ': :4' j ι ι~':,ι:~I ι. '

29. ,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

) f Ά 'aggl'eg,\~e pLaruling in~ach'c~~e',:,::,~':,K~' ';'~>';~';,'",ι} :'-' ,,'Ι, .'; '.d.' Planhing for tlιeΆsίΖe 0'ft11e',facult),i'jh:a university:';1 ,

·1). "DeteΙ'li1ίηίng<\ί15rkfοr6eΨeqiiιre'i"'~J1'tg'fδr 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. _,.<'.' Η,,' '; j' ~ "'_l: '_,:1 ';' .• ,;:ζ~.:~.J_ ' .."J~~' ι, ι •• ,[:' . ..'J ΙΛ ι., \Ί.- 30. Α plan! )ηanager ,emploxed bya 'I1ational ;produ~er of kitc!)en; products is 'planning

wοι;kfοrceleveΙs f,or the' next three monihs. Αιι .aggregate un,j,t of ρτοτίιιοτίωι requires 'ιιη average of [our labor hours. ηοreC,astdernaίιds' [οτ th,e thι;ee-mοί1tli 110τίΖοη 'art: asfollows:

11

AdditionalProblems ου:

. Agg'l-egatePla:,rt~iiig:1H ,. ίΊ:,·Η

.rI,

)jΙ,.Ί

, ~, ' ,

Forecasted, Demand(ίn aggre-gate units) ,iI ,Ι

1"Ι, ' ;,',,(1,2;400

;..ι

,; ,,~;.

164 ChapterThree Aggregace Pωηηίη,~

Assume that Parls currently has 80,000 gaJlons ofpaint ίη inventory and wou]dιο end the year with a11lnventory of at ]east 20,000 gal1()ns.σ, Deteπnine the mlnlmuln constant workforce p]anfor Parls Palnt atid the cost

p]an, Assume that stock-outs arc υοι al1o\ved.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.c. If.Paιis were able 10back-order excess den1and al a οοει of$2 per ga]]on pe~

ter, detcrιnlne a mi~imuln constant workforcep]an that Ιιοκίε Ιοεε lnventory tthe,plan you foundinpati (α), but incurs stock-outs iηquaΓteΓ 2. Detcrminc the'of Ilte new plan.

33. Consider the problem of Parls Palnt presented inProblem 32. Suppose that theΙιεε the capacity ιο elnploy a maxlmum of 370 workers, Suppose that regu]ar,tiemployee costs are $12.50 per Ιιοιιτ, Assume sevcn-hour days, five-day wceks, afour-week months. Overtime is paid οη a time-and-a-ha]f basls. Subcontracti~gavai]able θ! a cost of$7 per gallon ofpaint produced. Overtlme is ]imited Ιο three hoper 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.)

34. G. Foπnu]ate Ρτοοίεηι 32 as a ]jnear program, Assume tl1at stock-outsal1o\ved,

'~"'!;Xb. So]ve the linear program, Round.the variables and determine the cost ofthe resιJlti

plan. .

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.tnbased οη fιrm οπίετε. These forecasts and the nnmber of workdayspermon\h afol1o\vs:

MonthDemand Forecast

(ίη thousandsof cookies) Workdays

123456789

101112

8501,260

510980770850

1,0501,5501,3501,000

970680

262420182223142123242113

During a 46-dayperiod when there \vere Ι20\vorkers, the fir1111,700,000 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.

}\υΙ-

Additi011αl Problcms οη A,ggreJ.:"are l'/anηing 165

δ, Assuιne cI = $0.1 Οper cookie per month, cll = $] 00, al1dcp = $200. Add colulnnsthat give t]le ουυιυίετίνε on-h.and invenlory and invel1tory cost. W]lat is the totalcost of the constnnt workforce plan?

c. Construct a ρlηη that changes the Ιενε! 01" ιίιο workforce το ιυοει montl11Ydel11and 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

//~ t]1is ρίεη.

(37, 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 MarcJ13 J for the next six ll1onths. The dell1and forecasts aI'c as follo\vs:

MonthProduction

DaysForecasted Demand

(ίπ hundreds of cases)

ΑΡΓίlMayJuneJulyAugustSeptember

112220231620

8593

12217614063

As of ΜaΓcΙ131, Yeasty 11ad86 workers οη the payroll. Over a pcriod of 26 WΟΓk-ing days \vhen tl1ere were 10Οworkers ου tl1e payrol], Yeasty produced 12,000 casesοfbeCΓ. '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 tomaintain Η nlininnlm bttffer invcntory of ] ,000 cases each month. Ιτ plans Ιο startΟοιοοετ with 3.000 cases οη hand.

Ρι vt'

α. Based οτι tl1iSintorιnatlon, find the ιιιίηίωΙΙΙl1 constant workforce plan for Yeastyover the six l11ontI1s,and detcrmine I1iring, firing, and holding οοειε associatcd witl1that ρίεα,

b. Suppose tl131ίι takes 01lCηιοιιι]: to ΙΓαίηa nclv WΟΓker,How \νίΙΙ that affect youι·solution?

('. Suppose tl1at the maxiI11tI111numbeI" ot' WΟΓkerstl1at the colnpany can expcct tobc able 10 hire ίn one ιηοηιΙ1 is 10. How wil1 that Hffect youl" solution to

ρπτΙ (α)?

c!. DeΙeΓnJiηe the zcro invcl1tolOYρΙπη and t11Ccost of that ρΙηη. [Υοιι ΙΙΙΗΥ ignoI·c tl1CI~ conditions ίη pHrts (b) and (c').)

\ 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 Hnddetcrn1inctl1Ccost οΓ ΥΟΙΙΓsoltItion.

c. Suppose Yeasty does ηο! wish to .fire3l1Y\Vorkers. \Vl1at ls IJ1Coptimal Ρ]ΗΙΙ sub.iect1001is cOl1stralnt?

39. Α local canning COll1fJanyselJs c3nned vegelables to a supermarket chain ίn tl1CMinneapolis al'ea, Α typical case ot' canned vegetables reqtIlres an average οΓ0,2 clayof labor to pJ"Oducc. The aggregate inventory οη hand "ι 111eend of .func i5

166 Chapter Three AIW<garc ['lanningAppendix 3-Α 167

800 cases. The demand for thc vegetables can be accLIrately predictccl for about18 I110nths based ΟΙ1 o,-ders received by the firn1. The predicted demands for thenext 18 months arc as foIIows:

ActualDemand (1996)

NewForecast (1997)Month

FebruaryMarchΑΡΓίlMayJuneJulyAugustSeptemberOctoberNovelnberDecember

14820019018013095

12068887560

160175120190230280210110959277

Forecasted ForecastedDemand Demand

(hundreds (hundredsMonth of cases) Workdays Month of cases)

July 23 21 April 29 20Augu5t 28 14 May 33 22September 42 20 June 31 21October 26 23 July 20 18November 29 18 Augu5t 16 14December 58 10 September 33 20January 19 20 October 35 23February 17 14 November 28 18March 25 20 December 50 10

::f~

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$2.80Ιο store a casc of vegetables for a ηιοηιίι. Tl1ey woLIld like ιο have 1,500 cascs ίηinventoIOYat the cnd oftl1C 18-ιηοηιl1 planning horizon.

a. Develop a spreadsheet ιο find the minimum constant workforce aggregate plan and ..,detcrmine the total cost oftl1at plan.

b. ΟενεΙορ a spreadsheet ιο find a ρlηη that hires and. fi[es workers I110nthly ίπordcr 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 (heworkforce level exactly once during tl1e 18 l110nths and 11as" 10,,'e1"cost than either0[t11C plans considcI'ed ίπ parts (11) and (b).

40. Consider tl1e lίnear ρτοοικιίοη rule of Ηοίι, Modigliani, Muth, and Sil11on, appcaringίη Problem 24. Design 3 spreadsheet το οουιρυιο Ρ, and W, [ΓΟΙ11deI11and Ιοτοοοετε. Ιηζparticular, supposc that ει tl1e end ofDccember 1995 the workforce \Vas ΗΙ 180 and theinventory was ει 45 aggregate units.

a. Using the forecasts of demand for the nlOnths J"Hnuary tllfolIgh DeccIllber 1996given 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. WJlatare the I'ecoI11Illcndcd levels οΓ aggregate ρτοουοτίο» and wοrkfΟΓCC size forFebruary? (Ηίαι: Stol"C the denland forccasts and coetΊicicnts ίη separate,οοίιυυτιε and modify ιίιο conlptIting formtl1a as nc\v data a1"C appcnded Ιοthe 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.

Glossary of Notation for Chapter 3" = 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.· = Cost" οΓ ΙίΙ'ίηι; one WΟΓΙCCΓ.

('11 = Cosl οΓ 11ίΓίηι;one \vol·kcI·.

('1 = Cost οι' 110lding one ιιηίι ο! stock fOΓουε [1CI·iod.

Ι'() = Cosl οί' ριυdιιcίηg onc Lιπίι 011ονειιίιυο.

('1' = Pcnalty σοει ·foΓbacl< o,-dcr5.

('Ι( = Cost οί' ιποcJιιcίng 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'ίl1OI1Cday.

λl = ΑηηΙΙ<11dCIl1Hnc!ΙΟΓt':1I11ily.i(rcfcI' to Scclion 3.9).

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.

168 Chapter Three "'\~σr('~αιι>Ρlnηl1ίη~

ι "

BibliographyAllel1. S. 1.. αικ! Ε. W. Schllstcr. "Ρτεοιίσε! ΡτοοικτίωιSCllCdlIlil1g,νίιl, Capacity C0l15trail1tSal1d DYl1aIllicDCrn::lI1CI:F,ιιηίΙΥ PlanIling and Oisaggrcgation." ΡΙΌ(lιιι·tίοllαIlΙII,1\Ιf!nlVΙ:v 1\40Ilαg~nιel1t.J()ιιπιaI3S. 110. 4 (Ι 994),ρρ. 15-20.Ax~[ltel-. S. "Άggι'cg,ΙΙiοl1 οΓ Prodttct Dcιta for HienlΓcllic~llPI'o(ltIctiol1l'I,,"ning:' OlJeIvIioII.I· Πρ.ι·ρ",.ι·!, 29 (Ι 98 Ι).ρρ.744-56.BergstroIll, G. Ι .. and Β. Ε. Sl1lirh. "'Mulri-Itcm Productiol1Plallnillg-AIl Extension ofthe HMMS RLIles." MllJlagenIentS6enC'e 16 (Ι 970). ρρ, 6 Ι4-29.BiIran, G. R., [Ιllι1 Α. I-Iax. '"Oisaggregation nηcl ResοιιrceΑlΙοcηιίοl, Using ("nvex Knapsack Problcills with BoIInde<1VaI'iablc5:' Manag,'nIeIII 8cieIIC"e 27 (198 Ι ), ρρ. 431-4 Ι.BO\VI11al1.Ε. Η. "Ρτοοιιοιίοη Schedulil1g by the ΤΙ'ηηΨΟΓΙη-ιίοιι Mctllod of Lil1ear Programl1lil1g." Opel'OIionsΠΡ.,.ιll'ι!ι 4 (Ι 956). ρρ. 100-103.Βο\\ιιηί1l1. Ε. Η. "CUl1sistcncy <JIld ΟΡιίlηalίtΥ ίnM~lIHlgeri,11 D~ι:ίsίοl1 M"lkil1g." Mal111genIel1f $ι:;eιιι'e 9Ι 19(3). ρρ. 310-2 Ι.

Βιι !ΤΗ, Ε. S .. ί:llld W. Η. TalIbcrt. Ρι-υdιιc:!;υη-lιινentOl:\}

::':ω'ιel/ιχ: PlallIIing αll(1 Cοιι"υ/. Rev. ed. Νειν York:McGrH\\·-Hill/lI"\vil1. 1972."Cωcφίllal' Τπ:lctΟΓ CοιηΡί1nΥ." f-Iarvard BllSil1CSSSCl1001("cIse :1~5-·276 ( 1985).(ΊH!Π~, ΙΌ, HJld L. J. Knljcwski. "Plnnning Horizol1s fol'M:.I.stcl·Ρωιluctίοn SClleιiuling.·' '/ollI-naΙ (!f'OfJf!I-ali()n.~·

Μ"ιιageιιι<'ιιι. August 1984, ρρ. 389-406.C\)11CI1, Μ. Λ.; Μ. Ι. Fisllcr: UI1(1.1. J~likΙlJllar. ''InteJΊ1HtiOIl:IIΜnnΙIΙ:1CΙΙΙl,ίl1g <lllιl Distl'iblItioll Nct\\lorks:' [η j\ιIΙΙlla~ι:ίJ1g

111!(!1'11ι.ιιίοI1ΙΙI Λ4UΙΙΙψΚ!ll1'ίl1g, ed. Κ. FCΓdο\νs, ρρ. 67-93.Am,rcrιl,IIn: Nl)I·th Ηοίίεικ], 1989.Dnmon. \\1. \\/.. ""'1 I~.SCllΓanll11."Λ SimIIltnneotιsl)cι:ί ...•ίφη Μοοο! 1'01' Ρωιίικηοη. Markcting. nn<1 r-Inrll1ce,"Μalιagt'll/(:IΙΙ.5"C"ίι:Iι,ι: 19(1972).ρρ. 161-72.E.l1ert, It. J. ·Άgl:.!l'\~g<\ιl' PI~ltlning \vith Le;Πllilι~ (ΊIΓvef}!'()ιΙιιcιίvίΙΥ·" :HCII1ί1J.ξeIIIι..,II 5"L'it:l1C'(! 23 (1976),IΨ 171-82.Εn:ιιgιιc. S., ,111(1S. ΤιιΓckcί. '".Δι ΤrίιtlSΡοrtaιίοl1 ΤΥρε-"gHreg:.lιc "IΌ(IlICtiOI1 Mo(Iel with Bounds 011 Ιl1vellιulγ•.ΙΙΙι! (3:Ick.onlel·ill,g." ΙΊιnφt'οn JυΙΙΠJa! (?(OfJ(!I'CII;OIl.'"

!ic.'·"(f/"("!' 35 (Ι ')~X). [ψ 414-25Ηanslη:ll1l'J. Ε, ,11)(1 S. W. Hcss. "Α Lillcar Progrίlll1lnillgAjJpl'();Icll tu Ρωι!ιιctίοn <ll1dEmploYIl1cnI Sclιcdιllίlψ.'·Μalιa~ΨJΙΙΙ(!ΙΙΙ Τ(!ι'/ιιιοl(ψ,.\' Ι (Ι ~(0). ρρ. 46-5 Ι .Hax, Α. C .. :.1llιiD. ('3nLlcu. PJ-ol!lfuion Οl1ι!lnl'ΡΙΙfOl'11

MUl1agelIIellf. Eng!c\vo()(I Clifl's, NJ: Prelllice 1-1:.111,19R4.Βαχ, Α. ( .. and Ι-Ι.C".Mc"l. "HienIΓcI1ic,,1IlItcgI'atioll ofΡrοcΙιlctiοlι Ρlut1l1ίιψ 311(1 SCI1C(!IIling." Ιπ ΤIΛιf8 ,)'/lI(/ίι~<"; ίπΜα.ncιgel1ιeI115'ι'ίeΙl(·Ι'. V()llIIllC Ι, ΙοχiΔ·lίc.\·, cd. Μ. GeisleI·.Ne\vYork: EIsevicr. 1975.

ΗίllίΟΓ,F. S.. al1dG .. /. LicbclΊll,lIl. !nIIvdIIcIioll Ιο OperaιioHρ.5~αι·(·!ι. 5th cd. Sal1FI'al1cisco:Holdcl1 Day. 1990.Ηοlι, C. C".; F. Mo<ligliani; and J. F. MlItll. "ΟΟΓίvηιίοηof a.Linear Decisiol1 RlIlc for Ρτοιίικιίοη alld Εηιρίονηιοητ."MallGgeIIIenl Sι:ίι'nce 2 (Ι 956). ρρ. 159-77.Ηοlι. ο. ο. F. Modiglialli; .1.F. Μυτη: ,,"d Η. Α. δίηιοη.P!olll1ing Ρτο(JIΗ;tίοl1, IJ7VCl1fοτίe,\·. (,/I1ΙΙ J#)/'~/i.)f'(:(!. EnglewoodCΙiffs, NJ: Prentice ΗΗΙΙ,)960.Ηοίι, (. C.; F. MocIiglilllli; alld Η. Α. Sinlon. ''Α LincarDccision Rule for EmploYIncl1tand Ριοσυοιίοο SCIICdulil1g."Μaιιαgeιιιenι SC'ienc~ 2 () 955), ρρ. )-30.HHcl1zerιneier. Α .. anιl Μ. Α. COllCtl. "Vι:llιιίπg Ορετειιοιιε!Flexibility under Excllangc 1~"IcRisk." Ορο/ηιίοιι.ι·lΙe.<eαπh 44 (1996). ρρ. 100-13.Κεηυεη, Μ. Ι., and L. Ιί. "SubcontI'acting, Coordination,Flexibility, and Production Smoot'hil1g ίπ AggregateI'lanning.·· MaIIαgeIJ1el7I S"C"ίeIIl'e 36 ( 1990). ΙΨ )352-63.Koguι, Β.. ill1dΝ. KIIlatilaka. "'Orerating Flcxibility, GlobalΜΗnιιfactυrίng. ancl tI1COption VnlLlc οΓ[l MLlltillationa!Net\vork." Mαlla.fζειιιoιιι S'cieII('e 40 (Ι 994). ρρ. 123-39.Ιοίιεl1, R. Α. "M,trkcting St"IIcgy al1dΟριίιηηl ProdtIctionSchccilIlc.'· .ι,'/αιιαgeιιιeιιι S'l'iellc:e 20 (1974). ρρ. 903-1 Ι.McGratll, Μ. Ε.. "n(1 R. Β. Boqιιίllηrιl. "lnterl1atiollalΜaπιιfactιιrίng Strategies [ιl1ιi InfrastnlcfιII'al C'onsidcrationsίπ ιίιο Electronics InιΙιlstlγ" Ιη Μallagiιl,Ι!.ΙlΙf~ΠΙafίΟl1α!Μαnιι{aι·ιιιι·ίιιg. cd. Κ. Fc"lows, ρρ. 23-40. AIllsterdam:ΝΟΓΙΙ,Hollal1d, Ι9R9.Νοννεοη. Ε. F. Ρ. "Μulιί-Ιιοιl1 ΙΟΙ Sizc SclIclIlIlil1gbyHeuristic, Part Ι: \ινίιlι Fixctl ReSΟΙIΓces." IVlal1a,r;efl/f!1I1

.)("/·eιιce 21 (1975a), ρρ. 1186-93.Νεινεοιι, Ε. F. Ρ. "Multi-ItClll ί.οι 5ize ScIleduling byΗeΙΙΓίstic, Ρ,ιη 2: \ViIh Fixed RCSΟllΓCCS." :'vIl1na~ψηllenι8"ί(,1κρ 2 Ι ( Ι975b). ρρ. Ι Ι94- 1205.SCllr<.1gC, Ι. LineaI: Inf(!gc'l; υιιιl Q/((,/llnlfic: Ρ,-Ο,Ψ'(l1ΙΙJl1ίrιg,,·ίι!ι LlNDo. ΡηΙο ΑJtο, (Α: Scicntillc PI"ess. 1984.Sch,v"rl, Ι. Β.. ,tnd R. Ε.. lol,n50ll. ''Απ Apρl"ais,,1οΓ tllcElnpir.icnl Perfol'nlaI1ce οΓ Il1C Lillcal' Decisioll RLIle torAggregate Planning." IvfolllIgenIenl $ι-ίell('(! 24 ( Ι 978),ρρ.844-49.SiIver, Ε. Α .. :ll1d R. PCIeι-sun. O('ι'ί,~'ί()1/ 5:\'.\'T(~nι.\-.ι;)I'

117\'enfOJ:I' Λιfalιagι!171ι1111 Οllι! Ρι'οι!lι('/ίΟ/1 P/lIllIIillg. 211(1 ed.Ne,vYork: JOlll1Wilcy & SOI1,. 19R5.THubert. \ν. Η. ".ι\ Searcll rJCCiSioll f.ιιιιc f('II'II1CAggrcgaIeSCIlc(Iolillg Pr()blclll." Λ4ωΙ(Ψ.i!IIΙΡΙΙ! S'ι'ίι.:ιιι:e 14 (1968).ρρ. Β343-59.νΟΙΙΙll"ηl1.Τ. Ε.. \V. Ι. ΒοιτΥ; ,,"d D. <.:. Wllyb"I"k. Μωιιι/;ιι'-

III,.ίιlJ~, fJIGfJl1ing, ΙΙΙΙ(! ('ΟI/IΙ'ο! ,~\'.\'!(,IJI'.\'.31'(1 α!. Nc\\' York:McGrn,v-Hill/lrwin, Ι~92.Zollcl', Κ. "ΟριίΙl1ίΊ! r)ίsa~gΓCΗα.ιίοιι oJ'Aggn:gίJte ΡrοdοctίοllP/"I1S:' /\ι/αιια.ψιIιCI1Ι &ί",,('(' 17 (Ι 971). [ψ Β533-49.

Linear Programming51.1 INTRODUCTlON

Lineat· "rognll11ming is a tllathematical tecllnique for solving a broac! class of οριιιιίΖΗιίοη problenlS. These problems reqLIire maximizing οτ tllininlizing a lίηCHΓ fl1II(ιίου οΓ 11 Γeal variables subject to 111 constraints. One can fOΓllll1late ,Ind solve a IΗΓgnlJtllber ofreal problems with lίneaΓ ΡωgΓaιιιιιιίηg. Α ΡaΓιίallίst includes

Ι. SClledLIling οί ρειεοιιυε].2. SeveΓal varieties of blending pΓObleIlls incllIding cattle feed, salIsages, ice crean

and stecl nlaking.3. Inventory σοιιιιο! and ριυοιιοιιοι: planning.4. Distribution and logistics ρτοοίεηιε.5. Assigtltllenl problems.

ProbleIlls witll thousands οfvaΓίables and tllous,ιnds οΓ constraints are easily solvabοιι οοιηρυιειε ΙΟΙΙΗΥ.Linear pΓOgnll11Illing was developed to solve !ogistics ρτοοίεηcillI'ing World War 11.George Dantzig, a Illatl1enllttiCi<1n etllployed by the RAND (.οφnΙΙίοη Η! the tiIIle, c!eveloped a εοίυιίοη pΓOcedore 11Clabeled the Sitllplex Method. 'Πιtl1C nletl10d tuΓηed ου! 10 be so efficient ΓΟΓsolving l<trge probleIlls qιtick ΙΥ W,IS a $ΙΙprise evetl (ο its developer. That fact, οοιιρίαί \vith tl1e sitllltltaneolls developIllent of"!!electronic οοπιριιτει, established lίneaΓ progt'atllJ11ing as .111ίιυροτιωιι 1001 ίη 10gistin1anagemeni. The εικσοεε oflinear ΡΓΟgΓanιιιιίng in indLIstry spa\vned the cleveloptlleoJ"tlle disciplines οfΌΡeratίοns t'eseat-c11 (InC!IllHnageIllent science. "Πιο Sirnplex Metlll11,IS\Vitl1Stoocl tl1e test oftime. Only ίπ recent ΥeaΓS 113SHnotller Illetllod been clevelop,ιΙ1ηΙ potentialIy could be n10re efficient 111<1nthe Sίιιιρleχ Metl10d ΙΟΓsolving very lar~specially stt"UctLIred, lίnear prograIlls. TI1iS Illethod, known as Κ,ΙΤΙΙΙθrkar's Algoι'itllll1,natlled Γοι' tl1e Bell Labs Il1atl1etllatician Wl10 conceivec! ίι

lη Section S 1.2 we consider a typical tll,ltllIfactlIt'il1g ΡΙΌbΙem ιlιηΙ 've f'OΓIlllII,1te ΗΙsolve lIsing lίne,ιι' prograIlln1ing. Later \ve CΧΡIΟΓe110\VOl1e solves sn1aII probletlls (ti"is, having ι:xnctly two llecisiotl vaΓίables) grapI1ic<tlly, ΗηιΙ 110W one solves Ι,ΙΓpI'obletlls using a COIllplIter.

51.2 Α PROTOTYPE LlNEAR PROGRAMMING PROBLEM

ExampleS/.I Sidneyville manufactures household and commercial furnishings. The Office Division produrtwo desks, ΓΟllΙορand regular. Sidneyville constructsthe desks ίπ Its plant outside IVledfoOregoΓl, from a selection οΙ woods. The woods are cut ΙΟ a uniform thickness οΙ 1 IncrI. For treason, one measures the wood ίπ unlts of square feet. One rolltop desk requires 10 \quare ιιΟΙ pine, 4 square leet ΟΙ cedar, and 15 square feet οΙ maple. One regular de\k requI