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International Journal of Operations & Production ManagementEmerald Article: Multiple-stage production planning research: history andopportunities

Natalie C. Simpson, S. Selcuk Erenguc

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To cite this document: Natalie C. Simpson, S. Selcuk Erenguc, (1996),"Multiple-stage production planning research: history and

opportunities", International Journal of Operations & Production Management, Vol. 16 Iss: 6 pp. 25 - 40

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Multiple-stage production planning without joint costs or limitedresourcesSpecial ized multiple-stage product str uctures and production planning

environmentsSome of the earliest history of multiple-stage production planning lies in thedevelopment of algorithms intended for single-stage, single-item lot-sizingenvironments. Lot sizing itself dates back to 1915, when F.W. Harris[1]introduced the economic order quantity (EOQ) for single-item lot sizing withconstant demand. In 1958, the Wagner-Whitin algorithm[2] appeared in theliterature, providing a method of finding an optimal solution to a single-item,fixed-horizon, time-varying demand problem with concave costs, in whichbacklogging of demand is not allowed. Surprisingly, a 1974 survey by theAmerican Production and Inventory Control Society[3] found no respondentsusing it, apparently preferring to sacrifice optimality in favour of avoiding thecomputational inconvenience of the dynamic programming approach of the

Wagner-Whitin procedure. Examples of the heuristics available include theSilver-Meal algorithm[4] and the incremental part-period algorithm[5], whoserelative simplicity make them attractive alternatives.

One of the earliest MSPP-WO heuristics was developed by Schussel[6],through adaptation of a single-item method. Schussels economic lot release sizemodel (ELRS) found optimal solutions to single-item problems with constantdemand and incorporated such complications as warehousing costs andproduction costs partially described by learning curves. While the ELRSproduced an optimal solution in the single-item planning environment, aniterative adaptation of the ELRS was proposed as a heuristic procedure formultiple-stage pure serial production. (Within a pure serial product structure,each item has, at most, one predecessor and one successor.) Schusselsprocedure, appearing in the literature in 1968, represents the first application of

a multiple-pass heuristic to multiple-stage production planning. Continuing thework on the constant demand MSPP problem, Crowston et al.[7] provided adynamic programming formulation to find an optimal solution to the moregeneralized many predecessors/one successor product structure case. Crowstonet al.[8] described and tested three heuristic procedures for this problem, usingthe dynamic programming method to find benchmark solutions to six testproblems. Near optimality was observed, and computation time of theheuristics were a third to a fifth that of dynamic programming.

Schwarz and Shrage[9] also studied the more generalized multiple-stageproduct structure confined to constant demand. In this investigation, a branchand bound methodology was developed to find optimal solutions, systemmyopic policies, or the development of a solution by optimizing each stage

locally (considering only itself and its successor), were also investigated as analternative heuristic procedure. These policies, which are much easier tocalculate than those found through the branch-and-bound method, were foundto be near optimal for the 500 test problems generated.


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Some of the earliest work in the MSPP-WO problem with time-varying, ordynamic, demand was done by Zangwill[10]. In this 1966 paper, a dynamicprogramming algorithm was employed to identify an optimal solution to an

acyclic network model, representing a specialized system of facilities that caneach receive raw materials or the output of any upstream facility, and can eachsupply market demand or forward output to any downstream facility.Zangwill[11] also applied the network approach to a serial production modelwhich permitted backlogging of end item demand.

Generalized mult iple-stage product str uctures and production planning

environments

While many of the early efforts to develop MSPP-WO techniques were confinedto particular planning environments (for example, constant demand or specificstructures), later investigations began addressing more generalized problems.Relaxing the constant demand assumption of earlier work, Crowston andWagner[12] utilized dynamic programming to develop two algorithms for

computing the optimal solution to the many predecessor/one successor productstructure case. Later, Afentakiset al.[13] employed Lagrangian relaxation and abranch and bound procedure to obtain optimal solutions to generalizedmultiple-stage problems.

Arguably, it was the computational burden of the optimization of evenmoderately sized problems which generated much of the interest in MSPPheuristics. Earlier investigations in the 1970s, such as Biggs[14] and Biggs etal.[15], studied the performance of single-item lot sizing methods when applieddirectly to multiple-stage problems. One of the largest of such studies was thatof Veral and LaForge[16] in 1985. The incremental part period algorithm (IPPA),lot-for-lot ordering, the Wagner-Whitin algorithm, economic order quantities(EOQ) and periodic order quantities (POQ) lot-sizing policies were each applied

to 3,200 problems representing a variety of environmental factors. While theWagner-Whitin algorithm does not guarantee an optimal solution to a multiple-stage problem, this procedure still provided the benchmark against which thetotal cost performance of the other heuristics was measured. This study bothestablished the utility of lot sizing in the multiple-stage environment (lot-for-lotplanning consistently produced the poorest solutions) and the merit of IPPArelative to the other heuristics. Benton and Srivastava[17] published a similarstudy, but incorporated a rolling horizon into the experiment design andconsidered the use of full value added versus marginal value added holdingcosts employing the single-level techniques at each stage of a multiple-stageproblem. One of the interesting results of this study was the suggestion that thevarious single-item policies might be best used in combinations, such asapplying the Wagner-Whitin algorithm to the end item, while revising the

schedule of component items with some other single item method, such as theleast total cost (LTC) approach.

As the relative merits of various single-item planning techniques becameestablished in the literature, the development of improved methods began to


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emerge. One of the earliest such improved techniques for MSPP-WO problemsare those of Blackburn and Millen[18], appearing in the literature in 1982. Toenhance the performance of existing techniques, the investigators analysed the

sources of potential error when applying a single-level lot-sizing procedure to amultiple-stage environment. From this they derived a series of simple heuristicsto modify the cost factors of the problem, approximating the interdependenciesbetween levels. The performance of the cost modification models was evaluatedon randomly generated problems with 12 period planning horizons and up tofive component items. Unmodified, the single-level Wagner-Whitin algorithmsperformance ranged from an average of 4 per cent to an average of near 12 percent short of optimality (dependent on product structure), while the best costmodification procedure improved its average cost penalty to 1 per cent or less.

More recently, Coleman and McKnew[19] developed a heuristic lot-sizingalgorithm from the assumption that the failure of previous heuristics to performwell can stem from two sources: inability to find an optimal solution at thesingle item level and non-quantification of item interdependencies. The core of

their improved algorithm is the technique for order placement and sizing, orTOPS[20], originally developed for single-item lot sizing. From this STIL isderived, the Sequential TOPS with incremental look-down algorithm. Tested,STIL averaged within 1 per cent of the optimal solution (to find the optimalsolution, the authors employed McKnew et al.[21] integer programmingformulation).

Multiple-stage production planning with joint costs and no capacityconstraintsWhen the ordering of at least one member of a group of items incurs a fixedcharge, these items share a joint ordering cost, and may be said to belong to thesame joint cost family. Just as single-item production planning is the simplest

case of multiple-stage planning, single joint cost, single-level planning is thesimplest case of multiple-stage joint cost planning. Here, only one cost is sharedjointly by a group of items, and these items are not related by predecessor/successor relationships. The dynamic programming techniques developed byZangwill[10], mentioned earlier, also comprise some of the earliest workapplicable to this problem. Kao[22] developed an alternative dynamicprogramming formulation, reducing the state space but still requiringformidable computational effort. In addition, the author presented an iterativeheuristic for this planning environment, testing it with four two product, 12-period problems.

An alternate approach to single-level, multiple items with joint ordering costswas developed by Veinott[23], requiring the enumeration of all possibleproduction patterns satisfying certain known Wagner-Whitin style

properties. (Although possibly neglected in practice, a major contribution of theWagner-Whitin algorithm[2] is its accompanying proof that, in some optimalsolution to its single item problem, production in a period implies zero endinginventory for the previous period. This principle has since been extended to


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MSPP-WO and MSPP-JC problems and incorporated into many procedures,heuristic and otherwise.) Recently, Erenguc[24] proposed a branch and boundalgorithm for the single-family problem, avoiding the enumeration of all

possible schedules required by Veinott[23]. It was this algorithm that wasutilized by Chung and Mercan[25] to find benchmark solutions used to evaluatea proposed heuristic technique, which averaged within 1 per cent of optimalvalue over most of their problem sets.

When joint ordering costs are introduced into MSPP environments the issueof family-member order co-ordination is added to the difficulty of predecessor/successor order co-ordination. A joint ordering cost between items within aproduct structure may be the result of the fact that those items are physicallyidentical; that is, they represent the same component required by more than oneparent item. This particular form of joint cost we call component commonality,and many past investigations incorporating joint costs are confined to this case.(This problem we will indicate with MSPP-JC:CC.)The earliest method yielding optimal solutions to MSPP problems possibly

confounded by component commonality is that of Steinberg and Napier[26],appearing in 1980. The authors modelled the problem as a constrainedgeneralized network with fixed charged arcs and side constraints. The network,in turn, can be formulated as a mixed-integer programming problem, andsolved for optimality. While this formulation accommodates a wide range ofenvironmental conditions typically ignored by competing approaches,computation is extremely burdensome. The largest test problem attemptedcontained three end products sharing some common components, and a 12-period planning horizon. The formulation of such a problem contains 273integer variables alone and was reported to have required over 15 minutes ofCPU time on a Honeywell 6600 to solve.

Afentakis and Gavish[27] extended earlier work[13] to develop a branch andbound algorithm for the SMPP-JC:CC problem. Bounds on the optimal solutionwere derived from Lagrangian relaxation and subgradient optimization.Similar to Steinberg and Napier[26], the largest test problems involved three enditems, a 12-period planning horizon, and five to 15 item families within five to40 item problem structures. Computation time on an IBM 3032 ranged from onesecond to nearly two minutes, which compares favourably to the networkapproach of Steinberg and Napier[26].

To aid future experiment design, Collier[28] developed the Degree ofCommonality Index, a measurement of the component commonality present inone or more product structures. The Degre of Commonality Index indicates theaverage number of common parent items per distinct components present in aproblem, and can be calculated for a subassembly, a single end item, or a familyof items. The author employed simulation to establish a relationship between

degree of commonality and aggregate safety stock levels in an uncertainoperating environment. Simulation was also employed in the recent study byHo[29] to evaluate the performance of six single-level lot-sizing rules in amultiple stage with common components, rolling horizon environment. Unlike


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Collier[28], the degree of commonality present in the simulation was not amongthe factors varied between experiments. Rather, a measure of systemnervousness, or the frequency of rescheduling triggered by the rolling horizon

effect, was incorporated directly into the total cost expression of the multiple-stage problem.

One of the earlier improved MSPP-JC:CC heuristics was presented byGraves[30] in 1981. Gravess algorithm is a multi-pass adaptation of theWagner-Whitin approach, which modifies cost parameters at each iteration, torepresent the opportunity costs of decisions made in the prior iteration. On itsintroduction, the algorithm was tested over a relatively modest set of testproblems, involving five items, 12 planning periods and no componentcommonality. Surprisingly, it remains largely untested in the publishedliterature.

Bookbinder and Koch[31] studied the cost modification methods ofBlackburn and Millen[18], and proposed an extension to the MSPP-JC:CCproblem in 1990. While the original heuristics were confined to assembly-style

structures (no joint costs), the extension requires more generalized structures tobe partitioned into the largest possible assembly-style sub-structures.Blackburn and Millen's cost modifications can then be applied to sub-structures. This extended heuristic was featured among eleven tested in one ofthe largest multiple-stage heuristic studies to appear in the literature recently.

The study of Sumet al .[32] compared the performance of Bookbinder andKochs extension of Blackburn and Millen's cost modification method to tensimpler single-item based techniques, such as lot-for-lot ordering, Wagner-Whitin, and economic order quantity. While the Bookbinder and Kochsheuristic, on average, found the best solutions to 180 problems, the varyinglevels of commonality highlighted the fact that the degree to which the moresophisticated method outperformed the simpler methods decreases as the level

of component commonality increases.As product structures become increasingly generalized and complex,heuristic procedures based on problem simplification become more attractive.One such approach is the use of component clusters, dividing the originalproduct structure into groups of components required to order simultaneously.Once the clusters have been identified (based on the criteria that simultaneousordering of member components would be more economical), the ordering andholding costs of member components can be aggregated into cluster costs, andthese clusters can be manipulated in the search for a low-cost solution. Suchheuristic techniques have been explored by Roundy[33,34]. The extensivecomputational testing over 5,040 ten-period, five-cluster problems, showed thetwo heuristics of Roundy[34] to be within 0.7 per cent and 1.3 per cent ofoptimal, respectively. While highly generalized otherwise, these heuristics are

confined to problems in which external demand occurs for only one component.A recent mathematical formulation of the multiple-stage planning problem is

McKnewet al.s[21] zero-one integer programming model. This formulation canaccommodate joint costs of the most generalized form by linking all items that


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share a joint ordering cost with common ordering variables. The authors statethe formulation is useful for determining the optimal solution to medium-sizedresearch problems, and claim the LP relaxation would always be integer. This

claim was based on a proof demonstrating that the constraint matrix wastotally unimodular. Rajagopalan[35], however, showed this conclusion wasbased on invalid row operations. Thus, the claim of LP relaxations alwaysyielding integer solutions in [21] was shown to be unsupported.

A new taxonomy of multiple-stage planning heuristicsReviewing the literature strongly suggests that MSPP heuristics owe much oftheir origins to earlier, single-item techniques. Many improved heuristics havesought improvement over single-level procedures through certain strategies ofadaptation to multiple-stage environments. Examples of these strategiesinclude:

the creation of a multiple pass technique, seeking a better solution

procedure through iterative corrections of an essentially single-passprocedure[6,30];

modification of multiple-stage problem parameters, such as holding andordering costs, to reflect the parent/child item interactions that singleitem procedures would otherwise not consider[18,34];

modification of a single-item algorithm itself, incorporating someadditional routine which revises the actions of the single-item technique,to reflect the parent/child item interactions that single-item procedureswould otherwise not consider[19].

Recognition of these strategies has led us to propose a new taxonomy formultiple-stage planning heuristics, illustrated in Figure 1. First, all heuristicscan be divided into two groups: single-pass and multiple-pass techniques. Most

of the oldest multiple-stage planning techniques reside in the single-passcategory: SPSLAs, or single-pass, single-level algorithms. These are theapplications of single-item techniques to multiple-stage problems. The varietyof techniques in this category is owed to the number of candidate single-itemalgorithms, whether such an algorithm is applied sequentially throughmultiple-stage product levels or in one single aggregated pass, and the choice ofcost parameters employed in the calculations (full or marginal costs). In eitherthe single-pass or multiple-pass category, improved heuristics can be furtherdivided according to which approach is adopted to enhance their performancerelative to the SP-SLAs. One approach, denoted SP-MSL-MP, is to use anunmodified single-item procedure, but to modify the parameters of the multiple-stage problem to enhance the performance of the single-item procedure.

Blackburn and Millen[18] and Bookbinder and Koch[31] both employed thisapproach by proposing that the holding and ordering costs of each item be pre-processed according to a given formula, and an SP-SLA applied to the revisedmultiple-stage problem. The algorithm of Graves[30] employs a similar


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approach, but within a multiple-pass framework, which classifies it as a MP-MSL-MP. The cluster-based approach of Roundy[34] is another example of anSP-MSL-MP, but the product structure itself is modified to enhance theperformance of some sequentially applied SP-SLA. An alternate approach toadapting single-item techniques to a multiple-stage environment is the directrevision of the technique, incorporating some expanded routine that addressesthe multiple-stage nature of the problem. STIL, the algorithm of Coleman andMcKnew[19], is a single-pass example of this approach, or a SP-MSL-MA.Schussel's algorithm[6], one of the oldest published multiple-stage planningtechniques, is a multiple-pass example of a modified single-item technique, oran MP-MSL-MA.The final categories in the taxonomy are SP-OML and MP-OML algorithms,

denoting single-pass and multiple-pass original multiple-level algorithms.These algorithms do not trace their development back to efforts originallydirected at single-item problems. Rather, such algorithms would have beencreated directly from the structure of the multiple-stage problem.

Multiple-stage heuristics

Single pass (SP) algorithms Multiple pass (MP) algorithms

Directapplications of

single-levelalgorithms(SP-SLA)

Modifiedapplication ofsingle-levelalgorithms(SP-MSL)

Directapplication ofsingle-level

algorithms withmodified problem

parameters(MP-MSL-MP)

Modifiedapplication ofsingle-levelalgorithms

(MP-MSL-MA)

Examples:Cumulative Wagner-WhitinSequential Wagner-Whitin

Cumulative Silver-MealSequential Silver-Meal

etc.

Original multiple-level algorithms

(SP-OML and MP-OMSL)

Example:Graves[30]

Example:Schussel[6]

(Constant demand)

Directapplication ofsingle levelalgorithms

with modifiedproblem

parameters(SP-MSL-MP)

Examples:Blackburn and Millen[18]

(Cost modification)Roundy[34]

(Cluster-based)

Example:Coleman and McKnew[19]

(STIL)

Modifiedapplication ofsingle-levelalgorithms

(SP-MSL-MA)

Figure 1.A proposed taxonomyfor multiple-stageproduction planningheuristics


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Applying the MSPP heuristic taxonomy to published heuristic studiesreveals another interesting opportunity. Table I lists several of the heuristicstudies discussed previously, classifying those heuristics included according to

the taxonomy. This table illustrates the popularity of SP-SLAs in multiple-stageheuristic studies: all but one of the studies incorporated at least one heuristic ofthat description. In addition, none of the studies surveyed provided acomparison of improved heuristics of varying classifications. That is, animproved heuristic was typically evaluated by comparing its performance toless sophisticated SP-SLAs[19,30,31], or to heuristic procedures of the sameclassification (the three cluster-based SP-MSL-MPs of Roundy[34]), or both[18].Some measure of a heuristic's performance relative to optimality was utilized inless than half of the studies. As a result, the relative merits of the variousapproaches identified by the taxonomy introduced here remain untested.

Multiple-stage production planning with limited resourcesManufactur ing frameworks

In their 1987 review of the literature, Bahl et al.[36] state:

a casual look at practitioner-oriented literature such as theProduction and In ventor yManagementjournal andAPICS Conference Proceedingsstrongly suggests that most real-lifeenvironments are MLCR (Multiple Level Constrained Resource) problems.

The introduction of capacity constraints to multiple-stage planning bringsformidable complications, however. Lot sizing with capacity constraints and

significant set-up times has been shown to be computationally complex, or NPhard, even in the case of the single-level, single-resource problems[37,38]. Thework by Zangwill[10], previously discussed for its contribution to early MSPP-WO research, represents one of the earliest formulations of the single-level

Optimal solution orStudy Year Heuristics included lower bound provideda

Graves[30] 1981 1 MP-MSL-MP vs 2 SP-SLA Yes

Blackburn and Millen[18] 1982 4 SP-MSL-MP vs 3 SP-SLA Yes

Veral and Laforge[16] 1985 5 SP-SLA No

Brenton and Srivastava[17] 1985 6 SP-SLA No

Bookbinder and Koch[31] 1990 1 SP-MSL-MP vs 1 SP-SLA No

Coleman and McKnew[19] 1991 1 SP-MSL-MA vs 4 SP-SLA Yes

Roundy[34] 1993 3 SP-MSL-MP Yes

Ho[29] 1993 6 SP-SLA No

Sumet al.[32] 1993 1 SP-MSL-MP vs 10 SP-SLA No

Note:aYes indicates that either the optimal solution or a lower bound on the optimal solutionwas found for each experiment. In the case of the Coleman and McKnew[19] study, the optimalsolution was found for a sub-set of the experiments

Table I.Types of heuristics

included in ninepublished

multiple-stageheuristic studies


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(multiple items, but no predecessor/successor relationships) limited-resourceproblem with time-varying demand[36]. A wealth of single-level limited-resource heuristic procedures has followed, many based on the logic of the

Silver-Meal heuristic[4]. Starting with the single-pass heuristic of Eisenhut[39](which does not guarantee a feasible solution), refinements and variations onthis approach include Lambrecht and Vanderveken[40], Dogramci et al.[41], andDixon and Silver[42]. Maes and Wassenhove[43] conducted comparative testingof the three heuristics previously mentioned, and concluded that their levels ofperformance were similar.

Another complication of limited-resource problems is that of capacityconstraints with significant set-up times. This is the more generalizedinterpretation of a capacitated system, in which the act of ordering itselfconsumes capacity. Work on the single-level problem given this condition datesback to 1958 and Mannes formulation[44], which drops the familiar binaryordering variables in capacity insensitive models, replacing them with binaryvariables portraying each possible production sequence, given the sequence is

of the Wagner-Whitin style and feasible to the capacitated problem. Whilesolving the original formulation is computationally prohibitive, Manne's modelproduces a minimal number of fractional values for the binary variables whensolved as a linear program. While rounding these variables to 0 or 1 does notguarantee an optimal (or even feasible) solution, the simulation tests ofDzielinski et al .[45] demonstrated that doing just that provided reliableapproximate solutions to single-level capacitated problems. Efforts to furtherreduce the computational burden of Mannes model include Dzielinski andGomory[46] (application of Dantzig-Wolfe decomposition), Lasdon and

Terjung[47] (application of the Column Generation method) and Newson andKliendorfer[48] (application of Lagrangian relaxation). Further computationalsavings were achieved by the later heuristic procedures of Bahl[49] and Bahl

and Ritzman[50]. Few heuristics addressing this problem are more generalizedthan those that owe their motivation to Manne's model, the heuristic of Arasand Swanson[51] (addressing simultaneous lot-sizing and sequencingdecisions) being a notable exception. More recent is the finite branch-and-boundalgorithm of Erenguc and Mercan[52], which will find an optimal solution to amultiple-family item-scheduling problem, where each item and each family mayrequire significant set-up time.

Historically, research on MSPP-LR systems has lagged behind the effortsinvested in uncapacitated systems. One of the earliest mathematicalprogramming models, proposed by Haehling von Lanzenhauer[53], appeared inthe literature in 1970. Despite the omission of set-up/ordering costs, acomputationally feasible solution procedure is not available. Similar todevelopments in the uncapacitated literature, much of the early work on

capacitated systems was devoted to optimization techniques but relied heavilyon simplifying assumptions. Typical of this is the algorithm put forth byLambrecht and Vandervecken[40] which considers a system that produces onlyone end item with a strictly serial product structure, and only the workstation


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corresponding to the production of the end item is subject to capacityconstraints. The optimization techniques of Gabbay[54], Zahorik et al.[55] andBillington et al.[56] are likewise hobbled by similar assumptions or unwieldy

computational burdens for all but very small problems.Early investigation of multiple-stage lot sizing with capacity constraints

often relied on simulation experiments. Typical of this is Biggs et al.[57], whoevaluated the cost and service level performance of several single-item,uncapacitated lot-sizing rules in a simulated three-stage manufacturingenvironment. One interesting observation was a strong positive correlationbetween high frequency of stockouts and high inventory levels. Single-levelinventory models encourage the intuition that high levels of inventory reducethe frequency of stockouts. In the simulated multiple-level environment,however, higher frequency of stockouts left more work-in-process stranded inthe manufacturing system, inflating overall inventory levels. Similar simulationexperiments comparing single-item, uncapacitated heuristics in MSPP-LRenvironments include Berry[58], Biggs et al.[15], and Collier[59]. An

investigation by Biggs[60] evaluated priority sequencing rules in a multiple-stage system, an alternative to the single-item algorithm approach to the limitedcapacity planning problem. The findings of this study suggest that the samescheduling rule may not be best for all levels of demand versus capacity.

Raturi and Hill[61] also applied a simplistic single-level heuristic (the periodicorder quantity, or POQ) to a simulated MSPP-LR environment. In thisexperiment, however, the issue of capacity was incorporated into the procedureby modifying the set-up costs to reflect the shadow price, or opportunity costof a work-centre set-up, instead of simply the fixed accounting costs. Two suchSP-MSLMP methods were proposed, based on a Lagrangian approach to theoriginal problem formulation. The performance of the two shadow price-basedheuristics were discussed relative to each other and relative to fixed accounting

cost, capacity nave applications of the POQ to the system. The earlierheuristics of Harl and Ritzman[62] represent a similar effort to make single-itemuncapacitated methods capacity sensitive through parameter modification.

Given the complexity of the problem, another approach to simplificationwithout loss of real life applicability is to shift focus from lot sizedetermination to reorder interval setting policies. Jacksonet al.[63] explore thisapproach, building on the earlier, uncapacitated reorder interval model ofMaxwell and Muckstadt[64]. Solutions considered are confined to those inwhich a stage produces 1,2,4,8,...,2ktimes per the reorder interval of itssuccessor stage. While Roundy[33] and Maxwell and Muckstadt[64] showedthat restricting attention to policies of this type increases total cost by no morethan 6 per cent above optimum, it highlights the principle disadvantage to thisapproach: reliance on constant or near-constant end-item demand.

Future research on multiple-stage, capacitated systems may rely on newmathematical frameworks. Recently, Sum and Hill[65] have developed IMPICT,or iterative manufacturing planning in continuous time. In addition to explicitconsideration of capacity constraints, IMPICT includes several modelling


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features the authors feel have been neglected in the literature. Such featuresinclude set-up and processing times, allowing on set-up to run multiple periods,and minimization of combined set-up, holding, and tardiness costs. Once the

problem is modelled on the IMPICT framework, improvement is sought in thetotal cost of the initial schedule by repeated application of one of the threeproposed heuristics, the procedure terminating when total cost stopsimproving. Each heuristic was applied to 324 test problems, schedulingmultiple items over 80 work days. In this experiment, the items consisted of, atmost, two components, and only one work centre was restricted by capacityconstraints.

Supply-chain management

The similarities between the planning frameworks of multiple-stagemanufacturing systems and multiple-stage distribution systems suggests thatmany algorithms devised for the manufacturing model can be applied to

distribution planning or some combination of the two[27,64,66,67]. Recently,increasing globalization of production activity has motivated some researchersand practitioners to reconsider the traditional distinction between the multiplestage manufacturing and the multiple-stage distribution problem. Unificationof these two systems yields a supply chain, defined by Billington[68] as:

a network of facilities that procures raw materials, transforms them into intermediatesubassemblies and final products and then delivers the products to customers through adistribution system.

In the context of MSPP research, the supply-chain model can be thought of asthe most generalized of all the multiple-stage planning problems discussed,incorporating common components, more generalized joint costs, multiplelimited resources and multiple family memberships per item. Unfortunately,

most of the heuristic methods surveyed in this article do not address one ormore of these complications, suggesting that opportunities exist for thedevelopment of techniques which do. Early work has given this problem astochastic framework[69-71], but descriptive articles such as Billington[68] andLee and Billington[72] stress the need to manage inventory throughout acomplex and highly interdependent system as a central element of supply-chainmanagement. This does not exclude the deterministic models examined in thisinvestigation, especially at the tactical level of planning within the chain.

ConclusionsThis brief review of the sizeable body of MSPP literature strongly suggests thata paradigm has evolved from the direction of past research, one which dictates

multiple-stage production planning is an extension of single-item productionplanning. Beginning with Schussels iterative adaptation of the single-itemERLS model and extending through to present day, efforts to develop betterMSPP heuristic techniques can be typified as efforts to render established


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single-item techniques appropriate for multiple-stage environments. Thisobservation and others suggest the following MSPP research opportunities:

This survey uncovered no MSPP heuristics which could be classified SP-

OML or MP-OML. This suggests that an opportunity to develop superiorMSPP heuristics may exist, if the researcher resists the temptation tobegin with a single-item technique and adapt it. One possibility is theinvestigation of structured neighbourhood search techniques.

Evaluating past MSPP heuristic studies with respect to the taxonomypresented in this article highlights the lack of comparative studiesbetween modified, or improved, heuristics. This research opportunitywas noted in the survey by Bahl et al.[36] in 1987, and remains as a gapin the literature to this day.

Many MSPP heuristic studies available neglect the use of an optimalsolution as the benchmark by which to evaluate heuristics. Due tocomputational burden, optimal solutions are rarely secured to MSPP-WO problems over five items and 12 planning periods in size. Withoutthese benchmarks, the researcher must evaluate heuristic techniquesrelative to other heuristic techniques with no sense of the true cost ofany of those techniques, when applied to that particular experiment.Much insight may be lost in these cases, particularly if the researcherwishes to further relate the response of the heuristic to factor levels suchas the depth or complexity of the structure. If securing optimal solutionsis not practical, the use of lower bounds on the optimal solution asperformance benchmarks should be considered, such as the benchmarksutilized in Roundy[34].

The development of efficient methods of generating tight lower boundson the optimal solutions to MSPP-LR problems could greatly enhance

future research into MSPP-LR procedures. Clearly, work in this area hasonly begun.

Most of the MSPP-JC heuristics in the literature are developedspecifically for the case of common components within a productstructure. Adopting a broader definition of joint costs may allow theapplication of MSPP research to a broader spectrum of problems,including the recent supply-chain management merger of MRP andDRP models.

The MSPP problem has inspired considerable research effort over the past threedecades. By adopting more generalized models and holding heuristictechniques to higher standards in the future, the MSPP problem may yieldmany rewarding insights to researchers for many more years to come.

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