Int. J. Production Economics 127 (2010) 85–94

Contents lists available at ScienceDirect

Int. J. Production Economics

0925-52

doi:10.1

n Corr

E-m

p-robin1 Te

journal homepage: www.elsevier.com/locate/ijpe

Evaluation of joint replenishment lot-sizing procedures in rolling horizonplanning systems

Arunachalam Narayanan a,n, Powell Robinson b,1

a Department of Engineering Technology and Industrial Distribution, Texas A&M University, College Station, TX 77843-3367, USAb Department of Information and Operations Management, Mays Business School, Texas A&M University, College Station, TX 77843-4217, USA

a r t i c l e i n f o

Article history:

Received 10 June 2009

Accepted 27 April 2010Available online 7 May 2010

Keywords:

Joint replenishment

Lot sizing

Rolling horizon

73/$ - see front matter & 2010 Elsevier B.V. A

016/j.ijpe.2010.04.038

esponding author. Tel.: +1 979 845 1462; fax

ail addresses: chalam@tamu.edu (A. Narayana

son@mays.tamu.edu (P. Robinson).

l.: +1 979 845 1616; fax: +1 979 845 5653.

a b s t r a c t

Joint replenishment problems are commonly encountered in purchasing, manufacturing, and

transportation planning. Literature evaluates various algorithmic approaches for solving the joint

replenishment problem in a static environment, but their relative performance in a dynamic rolling

horizon system is unknown. This research experimentally evaluates nine joint replenishment lot-sizing

heuristics and policy design variables when implemented in a dynamic rolling schedule environment.

The findings indicate that a single algorithm does excel on both dimensions of schedule cost and

stability. Hence, management must trade off these two performance metrics when choosing the best

approach for their specific problem. Generally, metaheuristics provide the best cost replenishment

schedule, but forward pass based heuristics yield the most stable schedules. The results also indicate

that the choice of lot-sizing heuristic is the major cost performance driver in rolling planning systems,

with policy design variables (frozen interval and planning horizon length) having little impact. While

the simulated annealing heuristic of Robinson et al. (2007a) is the most effective solution procedure for

the static joint replenishment problem, the perturbation metaheuristic of Boctor et al. (2004) produces

lower schedule costs and greater stability in rolling schedule environments.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

Joint replenishment problems (JRPs) determine the time-phasedreplenishment schedule that minimizes sum of ordering andinventory costs for a product family. A joint ordering cost is incurredeach time one or more items in the product family are replenishedand an item order cost is charged for each line item replenished.Silver (1979), Stowers and Palekar (1997), and Robinson andLawrence (2004) provide examples of joint replenishment problemsin production, procurement, and transportation operations. Anillustrative problem is procurement operations of a grocery chainwhere a family of items is sourced from a single vendor. Eachreplenishment order is accessed a fixed delivery charge regardless ofquantity shipped, while each line item incurs a fixed cost forinventory maintenance, receipt, inspection, and put-away. Theobjective is to minimize sum of delivery charge, line item costs,and inventory costs for the product line recognizing that itemreplenishment costs are jointly linked through the shared deliverycost.

ll rights reserved.

: +1 979 845 4980.

n),

This research investigates the JRP in a rolling horizon planningsystem with stochastic demand that is forecast. Moving closer intime, the forecast is replaced with booked customer orders aseach time period enters the replenishment planning horizon. Theresulting problem is a short-term, or ‘‘static’’, JRP with determi-nistic, dynamic demand. After solving this static problem,replenishment decisions within the frozen order interval areimplemented rolling through time. After a pre-specifiedre-planning periodicity, a new short-term plan is constructedusing the updated demand received since the last planning cycle.When constructing the new replenishment plan, orders from theprior planning cycle that lay outside the frozen order interval maybe rescheduled if doing so results in an improved solution for thenew planning cycle. In this manner, solutions to a series of linkedstatic planning problems are implemented rolling through time(Blackburn and Millen, 1982). Fig. 1 depicts two planning cyclesand illustrates the basic definitions and concepts used in rollinghorizon planning systems.

Prior research indicates that effectiveness of rolling horizonplanning systems may be determined by planning horizoninterval, frozen order interval, re-planning periodicity, choice oflot-sizing procedure for solving the short-term planning problem,and cost and demand factors defining the planning environment.

While several researchers study performance of heuristic andoptimization-based approaches for solving the static JRP, their

Free Interval

Free Interval

New demand

data

Frozen Interval

Frozen Interval

Planning Horizon Length

Planning Horizon Length

Total Replenishment Horizon

Re-planning

periodicity

Cycle 1

Cycle 2

Fig. 1. Illustration of rolling horizon planning environment.

A. Narayanan, P. Robinson / Int. J. Production Economics 127 (2010) 85–9486

relative performance in rolling horizon systems is not addressed.This is a major shortcoming of the literature since the bestperforming lot-size procedure in a static environment may not bethe best performer in a rolling horizon environment. This is due tothe end-of-horizon effect, where not knowing the demand beyondthe planning horizon may lead to relatively poor quality short-term schedules. Even with optimal solutions to the staticproblems, their implementation in a rolling horizon environmentprovides at best a heuristic solution (Simpson, 1999). Priorresearch on single-item and multi-level rolling horizon problemsfinds that the problem’s demand pattern, cost structure, choice oflot-sizing procedure, and policy variables governing rollingschedule implementation play a significant role in determiningschedule cost and stability. Hence, studying the joint replenish-ment problem in rolling horizon systems is well justified.

This study conducts extensive computational experimentsevaluating impact of rolling horizon policy variables(e.g., planning horizon and frozen interval lengths) and jointreplenishment lot-sizing procedures on system cost and schedulestability. The findings indicate a tradeoff between schedule costand stability when selecting a particular lot-sizing procedure forimplementation. A single algorithm does not excel on bothperformance dimensions. The results provide managerialguidelines for implementing joint replenishment lot-sizingprocedures in rolling horizon planning systems.

The following section reviews the relevant literature on the JRPand rolling horizon planning systems. Next, the experimentaldesign and computer simulation model is described. Researchfindings are discussed in Section 5 and the final section providesresearch conclusions and implications.

2. Literature review

Numerous researchers study algorithms for solving the JRP in astatic setting. Goyal and Satir (1989) and Khouja and Goyal (2008)survey the broad JRP literature. Robinson and Lawrence (2004)and Robinson et al. (2009) study approaches for solving thedeterministic, dynamic demand JRP. Heuristics for the JRP areoften classified as specialized heuristics, metaheuristics, andmathematical programming-based approaches. Specializedheuristics include dynamic programming based procedures(Fogarty and Barringer, 1987; Silver, 1979), forward passalgorithms (Atkins and Iyogun, 1988; Iyogun, 1991; Robinson

et al., 2007a), and construction based heuristics (Robinson et al.,2007a). The metaheuristics approaches include perturbationmetaheuristic (Boctor et al., 2004) and simulated annealingmetaheuristic (Robinson et al., 2007a). Mathematical program-ming-based heuristics include branch and bound based partitionheuristic (Federgruen and Tzur, 1994) and dual ascent heuristic(Robinson and Gao, 1996). Boctor et al. (2004) and Robinson et al.(2007a) provide extensive evaluation of alternative JRP heuristicsfor solving static planning problems.

There is also an extensive literature addressing rolling horizonplanning systems. Instead of duplicating these reviews, theliterature most closely related to this study is summarized. SeeYeung et al. (1998) and Robinson et al. (2007b) for surveys ofproblem parameters and policy design variables that affectperformance of rolling horizon planning systems.

Baker (1977) investigates single-item rolling schedules, findingthat ideal planning horizon length is an integer multiple of naturaltime between orders (TBO). Sridharan et al. (1987, 1988) studysingle-item single-level systems and introduce the basic tradeoffbetween schedule cost and stability when setting rolling scheduledesign parameters. Their results indicate that longer freezinghorizons promote system stability while shorter freezing horizonsyield lower cost schedules; order-based freezing methods aresuperior to period-based methods; and shorter planning horizonsare favored for schedule stability.

Blackburn and Millen (1980) find that due to end of horizoneffects simple heuristics may perform better than optimizationprocedures in rolling schedule environments. Stadtler (2000)adjusts replenishment fixed costs in later time periods to lessenthe end of horizon effects and improve performance of optimiza-tion-based single-level lot-sizing rules. Heuvel and Wagelmans(2005) show that the ending inventory valuation (EIV, Fisher et al.,2001) heuristic, an extended Wagner Whitin algorithm, andStadtler’s (2000) lot-sizing procedures have similar cost perfor-mance in a rolling schedule environment.

Subsequent researchers (Blackburn and Millen, 1982; Zhao andLee, 1996; Zhao and Lam, 1997; Zhao et al., 2001; Xie et al., 2003;Simpson, 1999, 2001) consider policy design variables andimplementation in different settings, such as multi-level andcapacitated systems. Findings show longer planning horizons reduceschedule cost but increase schedule instability; shorter frozenintervals yield lower system cost; schedule instability is exception-ally high when frozen interval length is less than 50% of planninghorizon length; re-planning at the end of frozen interval reduces

A. Narayanan, P. Robinson / Int. J. Production Economics 127 (2010) 85–94 87

both system cost and nervousness; and choice of lot-sizing heuristichas a major impact on system cost and schedule instability.

Zhao and Lam (1997) examine seven lot-sizing rules inmulti-level uncapacitated systems. Cost-modified versions of theSilver–Meal and part-period balancing yield the most stable andlowest cost schedules. Simpson (1999) evaluates nine lot-sizingheuristics and alternate product structures in a multi-level rollinghorizon environment. The non-sequential incremental part periodalgorithm (NIPPA) provides the lowest cost schedule over avariety of product structures.

Zhao et al. (2001) study the impact of lot-sizing rules andpolicy design variables in multi-item single-level capacitatedsystems. They find that the heuristic of Dogramaci et al. (1981)provides the least cost rolling schedules and choice of lot-sizingrule does not significantly influence the best settings of policydesign variables under capacity constraints. In contrast, Xie et al.(2003) find that policy design variables have a significant impacton schedule cost and instability in multi-item, capacitatedsystems with uncertain demand.

Conclusions from the literature survey indicate that whilevarious algorithms are proposed for the JRP, their implementationand effectiveness in a rolling horizon is not addressed. Thisresearch takes the initial steps in filling this gap.

3. Experimental design

Due to the absence of prior work addressing the JRP in rollinghorizon environments, the experimental design is guided by theJRP and rolling horizon planning literatures.

3.1. Environmental factors

As in Robinson et al. (2007a) and Federgruen et al. (2007), thisstudy considers four environmental factors that prior researchidentifies as drivers of lot-sizing algorithm performance.These are the number of items, demand density or lumpiness,economic time between orders for the product family(TBOFamily), and economic time between orders for items(TBOItem).

The number of items in the product family is taken from theset IA{4, 8, 12}. Demand, dit, is generated from a normaldistribution and varies by item i and time period t. Evennumbered items have a mean demand per time period of 50units with a standard deviation of 20 units. Odd numbered itemshave a mean demand of 100 units with a standard deviation of 20units. Demand density (DD) is tested at three levels,DDA{0.50, 0.75, 1.0}, where DD represents the portion of timeperiods that an item experiences demand. At DD¼0.50 only 50%of time periods for each item have demand, while at DD¼1.0every time period has demand. In order to retain the same totalaverage demand across all DD levels, mean of the demanddistribution is adjusted prior to randomly drawing demandvalues. For example, at DD¼0.75, mean demands of the normaldistribution for odd and even numbered items are set at 67 and133, thereby maintaining the 50 and 100 average unit demand pertime period, respectively.

The TBOs are calculated as in Maes and Wassenhove (1988)and Federgruen et al. (2007). TBOFamily ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2St0=hD

p� �, where St’ is

the joint replenishment fixed cost, D the average demand for theproduct family, and h the holding cost per unit per timeperiod per item, where h¼$1.00 in all test problems. For eachproblem set, values of TBOFamily and D are randomly generatedand then St’ is set to coincide with experimental value of TBOFamily.Similarly, each item’s replenishment fixed cost sit’ is calculatedusing TBOItem ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2sit0=hd

p� �, where d represents average

demand for the item and TBOItem is a randomly generatedexperimental parameter. TBOFamily and TBOItem are evaluated atthree levels, {low, medium, high}, whose values are taken from auniform distribution on the intervals [1, 3], [2, 6], and [5, 10],respectively.

3.2. Rolling horizon policy variables

Policy design variables that define implementation strategiesfor rolling horizon procedures include planning horizon length,frozen interval length, and re-planning frequency (Zhao et al.,2001). Baker (1977) and Lundin and Morton (1975) conclude thatthe best planning horizon length is an integer multiple ofeconomic order cycle length. Hence, planning horizon length(PH) is set as an integer multiple, KA{2, 4, 8} of theeconomic natural order cycle, N, i.e. PH¼KN. As in Ballou(1988), the natural order cycle length, N, for the JRP is set at

N¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðSt0 þSI

i ¼ 1sit0 Þ=hDq

.

Frozen interval length, n, is modeled as a fraction F of planninghorizon length, where n¼F � PH. The frozen interval is evaluatedat four levels, FA{0.25, 0.5, 0.75, 1}. Sridharan and Berry (1990),Zhao and Lee (1996), and Zhao and Lam (1997) find thatre-planning at the end of the frozen interval yields both lowerschedule cost and stability. Hence, the research design imple-ments re-planning at the end of the frozen interval and does notinclude re-planning frequency as an experimental factor.

3.3. Lot-sizing heuristics for the joint replenishment problem

Boctor et al. (2004), Narayanan and Robinson (2006), andRobinson et al. (2007a) compare relative performance of JRPheuristics when solving a static problem. During preliminarytesting, the researchers evaluated the performance of theseheuristics in a rolling horizon environment and selected the topnine performers for inclusion in this research. Due to computa-tional difficulty of finding optimal solutions for the NP-hard JRPand large size of test problems, optimization procedures are notconsidered. The following lot-sizing procedures are considered inthis study.

Forward pass heuristics:

FP-E: forward-pass heuristic using a modified Eisenhutdecision criterion (Eisenhut, 1975; Robinson et al., 2007a),FP-E-RS: FP-E with a right-shift subroutine (Robinson et al.,2007a),FP-LV: forward-pass heuristic using a modified LV decisioncriterion (Lambrecht and Vanderveken, 1979; Robinson et al.,2007a),FP-LV-RS: FP-LV with a right-shift subroutine (Robinson et al.,2007a).

Fogarty–Barringer (FB) based heuristics:

FB: Fogarty–Barringer heuristic (Fogarty and Barringer, 1987),FB–SK: Fogarty–Barringer heuristic with the Silver–Kelleprocedure (Silver and Kelle, 1988; Boctor et al., 2004),PM: perturbation metaheuristic initialized by FB–SK (Boctoret al., 2004).

Two-phase heuristics:

TP: two-phase heuristic (Robinson et al., 2007a),SAM: simulated annealing metaheuristic initialized by TP(Robinson et al., 2007a).

A. Narayanan, P. Robinson / Int. J. Production Economics 127 (2010) 85–9488

3.4. Simulation procedures

The research design uses a full factorial design in theexperimental analysis. The environmental factors include threelevels for number of items, demand density, TBOFamily, andTBOItem. The rolling horizon policy factors include three levelsfor length of planning horizon and four levels for frozen intervallength. The full factorial design results in 972 combinations ofenvironmental and rolling horizon policy design factors. For eachof the 972 factor combinations, we randomly generate tenproblem instances. Each problem instance is solved in a rollinghorizon environment using each of the nine joint replenishmentlot-sizing heuristics, resulting in a total of 87,480 data points foranalysis. This experimental design facilitates evaluating theperformance of the heuristics, while at the same time discerningthe superiority of a specific algorithm in any environment orpolicy factor.

The simulation procedures are coded in C++ and implementedon a computer laptop running a Pentiums M processor at1.7 GHz. Blackburn et al. (1986) and Sridharan et al. (1987) showthat experimental run length of 300 time periods eliminates bothinitialization and termination effects; this experiment uses a runlength of 400 time periods.

3.5. Illustration of a joint replenishment rolling horizon problem

The following example illustrates the differences between staticand rolling horizon problem scenarios and implementation ofplanning horizon and frozen interval policy variables. Consider a12-period, four-item problem where the joint replenishment fixedcost (St’) in each period is $2000, individual item setup costs (sit0) are$200, $500, $250, $300 for items 1–4, respectively, and holding costis $1 per item per time period. The static planning problem assumesthat demand for all items in all time periods is known with certainty.Fig. 2 provides an example of 12 period static planning problem andreplenishment schedule found using the SAM. As illustrated, itemsare replenished in periods 1, 5, and 10 with a schedule cost of$16,230. This is the optimal schedule for this problem.

In a rolling horizon problem, demand is not known for the fullhorizon length, but reveals itself over time as customer orders arefirmed up. Hence, replenishment schedules are determined usingthe available demand information and a subset of the earliestreplenishment decisions are implemented. When it is time for thenext planning cycle, the analyst incorporates recently updateddemand and solves the replenishment schedule again. In thismanner, replenishment schedules are updated rolling through time.

Fig. 3 illustrates two cases of the rolling horizon procedureusing the demand data given in Fig. 2, the SAM, and a 6 periodplanning horizon (i.e. demand is known for only six periods intothe future). In the first case, replenishment schedules are frozenfor 6 periods and re-planning occurs at the end of the frozen

Static planning problem: Assume known demand for all 12 periods

Demand

Time 1 2 3 4 5 6 7 8 9 10 11 12 Item 1 109 115 113 0 102 96 0 136 108 0 0 0

Item 2 254 248 210 189 192 186 192 205 0 197 235 192

Item 3 133 119 0 107 85 83 116 103 0 81 129 0

Item 4 223 234 0 181 229 0 0 0 0 0 0 197

Cost of this replenishment schedule is $16,230

Fig. 2. Illustration of a joint replenishment sta

horizon. As seen in Fig. 3a, the first planning cycle schedulesreplenishments in periods 1 and 4, while the second cycleschedules replenishments in periods 7 and 10. The total cost forthe 12 periods is $17,002. In the second case, replenishmentschedules are frozen for 3 periods with re-planning at theend of the frozen period. The first planning cycle schedulesreplenishments in periods 1 and 4, but the replenishments in onlyperiod 1 are implemented. Next, the planning process rollsforward to period 4 and a new replenishment schedule iscomputed for periods 4–9 using the newly received demanddata for periods 7–9. Planning cycle 2 schedules replenishmentsin period 4, which cover the demand in periods 4–9. Similarly, thethird planning cycle begins in period 7 and incorporates demandthrough period 12. In this example, replenishments are scheduledin periods 1, 4, and 10, for a cost of $16,632.

As illustrated in Figs. 2 and 3, the rolling schedulingprocedures provide a heuristic solution to the overall planningproblem. The quality of the heuristic solution is influenced by thespecific lot-sizing heuristic and decision policy variables govern-ing implementation of the lot-sizing procedures (in this examplefrozen interval length). Rolling schedule procedures provide amechanism for dealing with incomplete demand data. Thedifference between optimal solution for the static problem, whichassumes demand is known for the entire problem duration, andthe rolling schedule solution provides a measure for the cost oflimited demand information.

3.6. Performance metrics

As in Simpson (1999, 2001), this research uses cost error toevaluate economic performance of the heuristics, which is[(C1�C2)/C2]�100, where C1 is the total cost of rolling schedulefor the test heuristic and C2 the lowest schedule cost found by anyof the nine lot-sizing heuristics. A more accurate metric wouldhave been to use optimal cost of the static 400-time-periodproblem. However, this is impractical due to the large problemsize and complexity of the joint replenishment problem. Hence, asin Simpson (1999), this research uses the lowest known schedulecost to compute cost error.

Schedule instability is evaluated using three metrics to capturethe portion of items, item setups, and product family setups thatare rescheduled. Following Sridharan et al. (1988), the Type 1

instability metric assesses number of units rescheduled as apercentage of total number of units in all planning cycles. TheType 2 instability metric tracks changes in timing of itemreplenishment orders as a percentage of number of item ordersthat are executed in the simulation run (see Sahin et al., 2008).The Type 3 instability metric tracks changes in timing of joint(product family) setups as a percentage of number of joint setupsthat are executed in a simulation run. Mathematically, the

Replenishment Schedule

1 2 3 4 5 6 7 8 9 10 11 12

337 0 0 0 442 0 0 0 0 0 0 0

901 0 0 0 775 0 0 0 0 624 0 0

359 0 0 0 387 0 0 0 0 210 0 0638 0 0 0 229 0 0 0 0 197 0 0

tic planning problem using SAM heuristic.

Planning Horizon = 6 periods, Frozen Interval = 6 periods

Planning Cycle 1: Demand

Replenishment Schedule

1 2 3 4 5 6

109 115 113 0 102 96

254 248 210 189 192 186

133 119 0 107 85 83 223 234 0 181 229 0

1 2 3 4 5 6

337 0 0 198 0 0

712 0 0 567 0 0

252 0 0 275 0 0

457 0 0 410 0 0 Planning Cycle 2:

Demand Replenishment Schedule

Cost of this replenishment schedule is $17,002

7 8 9 10 11 12

0 136 108 0 0 0

192 205 0 197 235 192

116 103 0 81 129 0

0 0 0 0 0 197

7 8 9 10 11 12

244 0 0 0 0 0

397 0 0 624 0 0

219 0 0 210 0 0

0 0 0 197 0 0

Planning Horizon = 6 periods, Frozen Interval = 3 periods

Planning Cycle 1 Demand

Replenishment Schedule

1 2 3 4 5 6 109 115 113 0 102 96 254 248 210 189 192 186133 119 0 107 85 83 223 234 0 181 229 0

1 2 3 4 5 6 337 0 0 198 0 0 712 0 0 567 0 0 252 0 0 275 0 0 457 0 0 410 0 0

Planning Cycle 2 Demand

Replenishment Schedule

4 5 6 7 8 9 0 102 96 0 136 108 189 192 186 192 205 0 107 85 83 116 103 0 181 229 0 0 0 0

4 5 6 7 8 9 442 0 0 0 0 0 964 0 0 0 0 0 494 0 0 0 0 0 410 0 0 0 0 0

Planning Cycle 3 Demand

Replenishment Schedule

Cost of this replenishment schedule is $16,632

7 8 9 10 11 12 0 0 0 0 0 0 0 0 0 197 235 192 0 0 0 81 129 0 0 0 0 0 0 197

7 8 9 10 11 12 0 0 0 0 0 0 0 0 0 624 0 0 0 0 0 210 0 0 0 0 0 197 0 0

Fig. 3. (a) Illustration of a joint replenishment rolling horizon problem using SAM heuristic. (b) Illustration of a coordinated rolling horizon problem using SAM heuristic.

A. Narayanan, P. Robinson / Int. J. Production Economics 127 (2010) 85–94 89

instability metrics are

Type 1¼ 100XI

i ¼ 1

Xj41

XMjþPH�1

t ¼ Mj

Qjit�Qj�1

it

������

0@

1A=TQ

Type 2¼ 100XI

i ¼ 1

Xj41

XMjþPH�1

t ¼ Mj

Yjit�Yj�1

it

������

0@

1A=U

Type 3¼ 100P

j41

XMjþPH�1

t ¼ Mj

Zjt�Zj�1

t

������

0@

1A=V

where i is the index of item, j the index of planning cycle, t thetime period, Mj the starting period of planning cycle j, Qj

it thereplenishment order scheduled for item i in time period t duringplanning cycle j, Yj

it ¼ 1 if an order is scheduled for item i in time t

during planning cycle j, Zjt ¼ 1 if a joint setup is scheduled in

period t during planning cycle j, TQ is the total number of units inall planning cycles, U the total number of item replenishmentorders executed over the simulation run, and V the total numberof joint setups or family replenishments that are triggered in asimulation run.

4. Experimental results

This section provides experimental results for cost error andschedule instability.

4.1. Impact of experimental factors on cost error performance

Table 1 summarizes results for 9720 test problems. The topfour performing heuristics based on average cost error are PM(0.37%), SAM (0.67%), FB–SK (1.25%), and FB (1.51%). The bestheuristics based on number of best found solutions are PM (5972),FB–SK (4261), FB (3183), and SAM (2421). However, the highvalues of standard deviation and maximum cost error for the FBand FB–SK heuristics for the DD¼50% problem sets indicate theirpoor performance in lumpy demand environments. Hence the PMand SAM procedures are recognized as the overall best JRPalgorithms when implemented in rolling horizon environments.

As illustrated in Table 1, the PM and SAM metaheuristicssignificantly improve the performance of their foundationalheuristics while the right-shift routine improves the performanceof both forward pass heuristics. Among the constructionheuristics, FB–SK performs better than the FB and TP heuristics.However, the performances of FB and FB–SK are very sensitive todemand lumpiness. Performance of the forward pass andFB-based heuristics improves at higher levels of demand density,whereas the opposite holds for the TP-based heuristics.

4.1.1. Forward pass heuristics

Table 2 indicates that the right shift (RS) subroutineconsistently improves performance of the forward pass heuristicfor all experimental factors. Performance of forward passheuristics improves with higher values of TBOFamily. This

Table 1Cost error: summary results by demand density.

Overall average

cost error (%)

Average cost error (%) Std. dev. of cost error (%) Max. cost error (%) No. of best solutions

DD (%) 50 75 100 50 75 100 50 75 100 50 75 100

Forward pass heuristics

FP-E 4.37 5.03 4.35 3.72 5.15 5.33 6.13 23.74 28.09 39.53 260 326 272

FP-E-RS 3.07 4.00 2.92 2.30 3.61 3.72 5.22 21.18 28.09 39.53 322 425 380

FP-LV 6.57 9.81 6.24 3.67 7.68 6.43 4.96 34.20 30.63 32.63 48 30 116

FP-LV-RS 5.43 9.35 4.94 1.99 6.97 4.96 4.18 32.92 30.60 32.60 48 117 675

FB based heuristics

FB 1.51 3.49 0.63 0.41 5.21 0.99 0.95 35.86 7.95 7.31 524 991 1668

FB–SK 1.25 2.90 0.48 0.36 4.88 0.88 0.87 31.13 5.67 6.27 892 1429 1940

PM 0.37 0.44 0.32 0.36 0.85 0.73 0.87 5.91 5.67 6.26 1790 2109 2073

Two-phase based heuristics

TP 1.87 1.56 1.89 2.15 0.85 0.86 1.00 5.41 6.12 7.72 132 114 135

SAM 0.67 0.56 0.70 0.74 0.76 0.77 0.85 5.40 5.67 6.26 1167 728 526

Overall average 4.13 2.50 1.74

Table 2Cost error: detailed summary results by experimental factors.

Experimental

factors

Overall

average (%)

FP-E (%) FP-E-RS (%) FP-LV (%) FP-LV-RS (%) FB (%) FB–SK (%) PM (%) TP (%) SAM (%)

Forward pass Heuristics FB-based heuristics TP based heuristics

I 4 3.14 3.98 3.60 6.03 5.65 3.30 2.99 0.46 1.59 0.70

8 2.65 4.51 2.96 6.49 5.12 0.85 0.59 0.51 2.05 0.76

12 2.58 4.60 2.66 7.21 5.52 0.37 0.17 0.15 1.97 0.54

TBOFamily Low 3.53 6.95 4.00 9.32 6.89 1.33 0.78 0.47 1.55 0.44

Med. 2.56 3.62 2.72 6.11 5.22 1.24 1.08 0.35 1.95 0.72

High 2.29 2.52 2.50 4.29 4.18 1.96 1.88 0.29 2.11 0.85

TBOItem Low 1.72 2.21 2.13 3.02 2.80 1.22 1.17 0.37 1.81 0.77

Med. 2.59 3.98 2.92 5.81 4.84 1.56 1.30 0.42 1.84 0.66

High 4.05 6.90 4.17 10.89 8.65 1.74 1.27 0.33 1.95 0.58

PH 2 3.06 4.39 3.27 6.79 5.75 2.20 1.86 0.82 1.75 0.73

4 2.66 4.40 3.03 6.47 5.26 1.25 1.04 0.21 1.91 0.35

8 2.65 4.30 2.92 6.46 5.28 1.06 0.84 0.09 1.95 0.92

F 0.25 2.33 3.48 1.89 5.54 4.08 1.42 1.23 0.49 2.03 0.81

0.50 2.30 3.34 1.88 5.40 4.10 1.45 1.26 0.50 2.00 0.81

0.75 2.28 3.14 1.91 5.40 4.35 1.53 1.26 0.42 1.82 0.70

1.00 4.24 7.50 6.60 9.97 9.19 1.63 1.23 0.08 1.62 0.36

A. Narayanan, P. Robinson / Int. J. Production Economics 127 (2010) 85–9490

contradicts the findings for forward pass heuristics whenimplemented to solve static planning problems (Robinson et al.,2007a). This result is understandable considering that only thefirst few replenishment decisions are implemented beforere-planning and due to the myopic nature of forward passheuristics early schedule decisions are typically of high qualitysince they are not adversely impacted by end-of-horizon effects.In contrast, cost performance degrades with increases in TBOItem

(e.g., high item fixed costs). This is due to the mechanics offorward pass heuristics, where cost-saving calculations maypermit high item setup costs to dominate joint setup cost.

A counterintuitive result is seen for impact of planning horizonlength on cost error performance for the forward pass heuristics.While longer planning horizon lengths are anticipated to result inlower schedule cost, this holds only for the highest density(DD¼1.0) problem sets. There is no significant impact on other

demand density problems. Hence the researchers do not see anydiscernible effect of planning horizon length in Table 2 for theforward pass heuristics. The impact of frozen interval length onperformance is as expected, where low cost schedules areassociated with shorter frozen intervals.

4.1.2. FB based heuristics

The relative performance among the FB based heuristics isconsistent with results for solving static coordinated lot-sizingproblems (Narayanan and Robinson, 2006), where PM performsbetter than FB–SK, which in turn performs better than FB acrossall experimental factors. The most important driver of cost error isdemand density and its interaction with other design factors. PM’scost error is low across all experimental factors, with its highestcost error of 0.82% being associated with the shortest planninghorizon. Low number of items yields the worst performance for

Table 3Cost error: ANOVA results.a

Dependent variablesb Deg. freedom F-statistic Significance

I 2 689.815 0.000

TBOFamily 2 1651.627 0.000

TBOItem 2 4449.004 0.000

DD 2 12,829.268 0.000

PH 2 225.121 0.000

F 3 1810.950 0.000

Heuristic 8 18,684.879 0.000

I� TBOFamily 4 941.719 0.000

I� TBOItem 4 119.717 0.000

I�DD 4 857.744 0.000

I� PH 4 326.077 0.000

I� F 6 170.839 0.000

I�Heuristic 16 1177.287 0.000

TBOFamily� TBOItem 4 902.389 0.000

TBOFamily�DD 4 21.680 0.000

TBOFamily� PH 4 23.358 0.000

TBOFamily� F 6 61.645 0.000

TBOFamily�Heuristic 16 618.308 0.000

TBOItem�DD 4 130.011 0.000

TBOItem� PH 4 31.990 0.000

TBOItem� F 6 13.427 0.000

TBOItem�Heuristic 16 617.277 0.000

DD� PH 4 225.609 0.000

DD� F 6 69.889 0.000

DD�Heuristic 16 1639.401 0.000

PH� F 6 59.400 0.000

PH�Heuristic 16 1073.145 0.000

F�Heuristic 24 1139.178 0.000

I� TBOFamily� TBOItem 8 49.974 0.000

I� TBOFamily� PH 8 47.134 0.000

I� TBOFamily�Heuristic 32 83.749 0.000

I�DD� PH 8 155.934 0.000

I�DD�Heuristic 32 323.193 0.000

I� PH� F 12 54.308 0.000

I� PH�Heuristic 32 171.215 0.000

I� F�Heuristic 48 129.136 0.000

TBOFamily� TBOItem�DD 8 61.166 0.000

TBOFamily� TBOItem�Heuristic 32 115.539 0.000

TBOFamily�DD�Heuristic 32 82.263 0.000

TBOItem�DD�Heuristic 32 50.807 0.000

DD� PH� F 12 49.999 0.000

DD� F�Heuristic 48 94.678 0.000

PH� F�Heuristic 48 191.084 0.000

Adjusted R2¼0.811, intercept was included in the model.

a Based on Yeo and Johnson (2000) transformation.b Only 3-way interaction terms with F445 are included in the table.

A. Narayanan, P. Robinson / Int. J. Production Economics 127 (2010) 85–94 91

the FB and FB–SK heuristics. PM improves the solution of FB–SKby an average of 92% for such cases. Cost error increases withTBOFamily and TBOItem for the FB and FB–SK methods. This result iscounterintuitive since FB based procedures are anchored on theassumption that every item is setup in each joint replenishmentperiod, which is more characteristic of optimal solutions forproblems with higher TBOFamily values. Similar counterintuitiveresults are not seen in PM, where performance is rather invariant,especially across the TBOItem values.

As anticipated, the longer planning horizon produces the leasterror measure. However, the effect of frozen interval length isunexpected, where F¼1.0 yields the lowest cost schedules inFB–SK and PM.

4.1.3. TP based heuristics

As illustrated in Table 2, the findings for the TP based heuristicin the rolling horizon environment closely mirror earlier resultsfor the static problem. Specifically, cost error increases withdemand density and TBOFamily. The only surprising result is theeffect of number of items on cost performance. In static problemscost error decreases with increasing number of items while thisresult does not hold for rolling schedules.

The planning horizon and frozen interval lengths havevery little impact on performance of TP heuristic. However, acounterintuitive result occurs for SAM, where lower cost errorsare associated with longer frozen intervals.

4.1.4. Tests for significance

As confirmed by a t-test2 (p-value 0.001), a major finding of theresearch is the superior cost performance of PM over SAM acrossexperimental factors. The only setting in which SAM outperformsPM is for low TBOFamily values, which is also observed in staticproblem experiments (Robinson et al., 2007a).

The research applies ANOVA procedures to identify signifi-cance of main and interaction effects on cost performance. Basedon residual analysis, the researchers apply 1/x2 transformation onthe dependent variable as suggested by Yeo and Johnson (2000).

As illustrated in Table 3, the top three drivers of costperformance are the lot-sizing heuristic, DD, and TBOItem. Thiseffect is also seen in Tables 1 and 2, where average cost errorsvary significantly according to the lot-sizing heuristic, DD, andTBOItem. To a lesser extent, F, DD�Heuristic, TBOFamily, I�Heuristic,F�Heuristic, PH�Heuristic, I� TBOFamily, TBOFamily� TBOItem, andI�DD impact cost performance. Almost all of the 2-way and3-way interactions are significant, but the F-ratio values for manyof the interactions are low, indicating little impact on costperformance. An important finding from the ANOVA analysis isthat while environmental factors impact schedule efficiency,choice of lot-sizing heuristic is the most critical systemperformance driver.

Simpson (1999) points out one disadvantage of using a relativecost measure in cost error calculation. If all the lot-sizingheuristics included in the study react the same way to a particularfactor level, then the effect would go unnoticed in the analysis.From Table 2, the researchers observe that the lot-sizingheuristics react differently to environmental factors (TBOFamily,TBOItem, DD, and I), but there is no such evidence for policy designvariables. This could be the reason for some of the unexpectedresults associated with planning horizon and frozen intervallength. Hence researchers need to find a computationally feasible

2 H0: mSAM�mPMr0; Ha: mSAM�mPM40. Using the averages and standard

deviations (ySAM ¼ 0:6678, std. dev.SAM¼0.799; yPM ¼ 0:3727, std. dev.PM¼0.824;

sample size¼9720), the value of t’ for the t-test is greater than t (p-value: 0.001);

hence H0 is rejected.

benchmark to unambiguously identify the effect of policy designvariables on cost error performance for the coordinated lot-sizingheuristics. However, relative ranking among the heuristics willnot be affected by the change in cost performance.

4.2. Impact of experimental factors on schedule instability

Table 4 summarizes the schedule instability findings. Asindicated there is a greater tendency to reschedule productunits (Type 1) versus altering timing of either the item (Type 2) orproduct family setups (Type 3). As expected, due to their relativeisolation from the end-of-horizon effects, the forward passheuristics produce the most stable schedules. Themetaheuristics provide the highest schedule instability,especially SAM, which has over one third of all items beingrescheduled. A noteworthy result for the FP-E and FP-LVheuristics is that the product family setup is not rescheduled inany of the test problems (see Type 3 results). In business situationswhere the product family cannot be rescheduled in the short termonce a time slot is assigned, such as when reserving container shipcapacity, this is a highly desirable performance characteristic. In

TT

A. Narayanan, P. Robinson / Int. J. Production Economics 127 (2010) 85–9492

such scenarios, the forward pass heuristics may be the mosteffective rolling schedule procedures. Problems with higherdemand density have more opportunities to lower costs through

Table 4Schedule instability: summary results by demand density.

DD (%) Type 1 error a (%) Type 2 error b (%) Type 3 error c (%)

50 75 100 50 75 100 50 75 100

Forward pass heuristics

FP-E 16.89 18.67 20.54 0.70 0.41 0.13 0.00 0.00 0.00

FP-E-RS 21.25 27.93 31.48 1.53 2.29 2.27 1.02 2.08 2.29

FP-LV 13.37 16.81 19.68 0.87 0.51 0.21 0.00 0.00 0.00

FP-LV-RS 15.47 25.65 33.50 1.35 2.46 2.87 0.58 2.16 2.82

FB based heuristics

FB 25.26 26.89 24.24 2.66 3.20 2.98 2.58 3.19 2.98

FB–SK 25.33 26.56 24.24 2.63 3.09 2.95 2.62 3.12 2.96

PM 26.60 27.18 24.68 2.88 3.21 3.02 2.97 3.26 3.04

Two-phase based heuristics

TP 23.10 25.16 27.43 2.18 2.46 2.71 2.17 2.46 2.71

SAM 33.56 37.81 40.87 3.72 4.47 5.06 4.02 4.60 5.12

a Based on demand/unit change.b Based on item/minor setup change.c Based on family/joint setup change.

Table 5Correlation coefficients for the schedule instability measures.

SI Type 1 Type 2 Type 3

Type 1 1 0.855n 0.843n

Type 2 1 0.992n

Type 3 1

n Correlation is significant at the 0.01 level (2-tailed), sample size¼87,480.

able 6ype 1 schedule instability: detailed summary results by experimental factors.

Experimental factors Overall average (%) FP-E (%) FP-E-RS (%)

Forward pass Heuristics

I 4 22.48 24.91 26.34

8 25.95 18.48 27.83

12 27.14 12.71 26.49

TBOFamily Low 25.58 17.60 35.35

Med. 25.05 19.00 25.52

High 24.95 19.49 19.79

TBOItem Low 22.47 17.62 17.83

Med. 24.89 18.65 24.78

High 28.21 19.83 38.04

PH 2 17.58 26.09 26.82

4 27.47 19.73 30.05

8 30.52 10.28 23.79

F 0.25 61.81 33.80 58.87

0.50 28.32 24.06 30.80

0.75 10.64 16.94 17.87

1.00 0.00 0.00 0.00

rescheduling and consequently experience more scheduleinstability; this is clearly seen in forward pass and TP basedheuristics, but FB based heuristics do not reveal any discernablepattern.

Performance of the three schedule instability measures ishighly correlated (see Table 5). Hence, this section presentsresults for only the Type 1 instability measure.

4.2.1. Forward pass heuristics

Table 6 indicates that the FP-E and FP-LV procedures result inmore stable schedules than their counterparts with the right shiftprocedure. FP-E and FP-LV have higher schedule instability withfewer items. For all other forward pass heuristics the results aremixed for number of items. For right shift procedures, higherTBOFamily and lower TBOItem values yield lower scheduleinstability. Results are inconclusive for the FP-E and FP-LVheuristics. Similar to earlier findings for single-item andmulti-level lot-sizing planning environments, schedule stabilityincreases with length of frozen interval. Finally, schedule stabilityis enhanced with longer planning horizons for the FP-E and FP-LVheuristics, which contradicts findings in the literature formulti-item planning systems (Zhao et al., 2001).

4.2.2. FB based heuristics

For the FB based heuristics, schedule instability increasesdramatically with more items and shorter frozen intervals and toa lesser extent with higher TBOFamily values. Schedules also tend tobe more stable with shorter planning horizons. Results fordemand density and TBOItem do not reveal any discernablepattern.

4.2.3. TP based heuristics

Results for the TP based heuristics are similar to those for theFB based heuristics. Schedule instability increases dramaticallywith more items and shorter frozen intervals while the increase isless pronounced with higher TBOFamily values. SAM experiencesgreater schedule instability with higher TBOItem and TBOFamily

values, longer planning horizons, and shorter frozen intervals. The

FP-LV (%) FP-LV-RS (%) FB (%) FB–SK (%) PM (%) TP (%) SAM (%)

FB-based heuristics TP based heuristics

22.09 23.91 19.62 19.83 20.29 20.23 25.10

14.75 23.92 27.01 26.87 28.04 27.88 38.79

13.03 26.80 29.76 29.43 30.12 27.58 48.35

15.80 31.57 23.91 23.72 25.29 23.93 33.02

16.91 24.47 25.51 25.38 25.87 25.32 37.48

17.15 18.59 26.97 27.04 27.30 26.44 41.74

16.58 17.28 24.21 24.18 24.51 24.90 35.13

16.92 23.27 25.72 25.56 26.46 25.22 37.40

16.37 34.07 26.45 26.40 27.49 25.56 39.70

25.31 26.74 10.49 10.30 8.77 14.40 9.31

16.45 26.50 27.33 27.21 28.28 33.75 37.94

8.12 21.38 38.57 38.63 41.40 27.53 64.99

30.78 55.81 66.50 66.70 70.12 66.07 107.60

21.06 27.92 29.20 28.91 29.15 27.81 35.95

14.66 15.76 6.15 5.90 5.34 7.04 6.10

0.00 0.00 0.00 0.00 0.00 0.00 0.00

Table 7Type 1 schedule instability: ANOVA results.a

Dependent variablesb Deg. freedom F-statistic Significance level

I 2 1872.890 0.000

TBOFamily 2 84.215 0.000

TBOItem 2 405.879 0.000

DD 2 428.160 0.000

PH 2 13,795.694 0.000

F 3 251,020.854 0.000

Heuristic 8 488.609 0.000

I� TBOFamily 4 131.224 0.000

I� TBOItem 4 20.718 0.000

I�DD 4 59.730 0.000

I� PH 4 1298.916 0.000

I� F 6 1101.467 0.000

I�Heuristic 16 1227.562 0.000

TBOFamily�Minor 4 118.302 0.000

TBOFamily�DD 4 1.505 0.198

TBOFamily� PH 4 5.660 0.000

TBOFamily� F 6 64.764 0.000

TBOFamily�Heuristic 16 46.516 0.000

TBOItem�DD 4 85.349 0.000

TBOItem� PH 4 1.850 0.116

TBOItem� F 6 202.957 0.000

TBOItem�Heuristic 16 75.260 0.000

DD� PH 4 1082.156 0.000

DD� F 6 124.332 0.000

DD�Heuristic 16 323.898 0.000

PH� F 6 2477.719 0.000

PH�Heuristic 16 3859.388 0.000

F�Heuristic 24 1213.359 0.000

I� TBOFamily�DD 8 83.933 0.000

I� TBOFamily� PH 8 56.613 0.000

I� TBOFamily�Heuristic 32 49.793 0.000

I� TBOItem� PH 8 78.329 0.000

I�DD�Heuristic 32 83.655 0.000

I� PH� FH 12 1164.071 0.000

I� PH�Heuristic 32 360.497 0.000

I� FH�Heuristic 48 150.904 0.000

TBOFamily� TBOItem� FH 12 46.455 0.000

DD� PH� FH 12 158.892 0.000

DD� PH�Heuristic 32 74.517 0.000

PH� FH�Heuristic 48 490.795 0.000

Adjusted R2¼0.922, intercept was included in the model.

a Based on Yeo and Johnson (2000) ln(x+1) transformation.b Only 3-way interaction terms with F445 are presented.

FB

FB-SK TP

SAM

PM F

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

40.00%

0.00%Co

Inst

abili

ty (T

ype

1) M

easu

re

1.00% 2.00% 3.00%

Fig. 4. Cost error versus

A. Narayanan, P. Robinson / Int. J. Production Economics 127 (2010) 85–94 93

TP heuristic’s results are inconclusive for TBOItem and planninghorizon length.

4.2.4. Tests for significance

Tests of significance for the Type 1 instability metric areconducted using ANOVA procedures with the ln(x+1) transforma-tion of data as suggested by Yeo and Johnson (2000) to satisfy theresidual error assumptions of ANOVA. As presented in Table 7,the main effects and most interaction terms are significant at the0.001 level. The F-statistic indicates that major drivers of scheduleinstability are policy variables and their interaction effects. Thetop three in order are F, PH, and PH�Heuristic. The main effectsF and PH are visible in Table 6, where the schedule instabilitymetric varies from 0% to 61.81% with F and 17.58% to 30.52% withPH. The interaction effect of PH�Heuristic is also seen in Table 6,where instability increases with length of PH for FB and TP basedheuristics and decreases with length of PH for forward passheuristics. Environmental factors and their interactions have littleimpact on schedule instability. These results contrast with thedrivers of cost performance, where choice of heuristic andenvironmental factor dominates.

5. Conclusions and implications

The research findings and conclusions provide severalimplications for both researchers and managers. Fig. 4 indicatesthe tradeoff between cost error and schedule instability for the JRPheuristics. FP-LV produces the most stable schedule, but has thelargest cost error. In contrast, PM has the lowest cost error but at aconsiderably increase in schedule instability over FP-LV. SAM, whichranks a close second on cost error, has the greatest scheduleinstability of all the lot-sizing heuristics. Six of the heuristics havesimilar levels of schedule instability, but cost error varies greatly.Fig. 4 enables managers to identify the cost error and schedulestability tradeoff in order to select the heuristic that most closelymeets their needs. If cost error is the primary criterion, then the PMis the best choice. However in situations where rescheduling is notdesirable, the FP-E and FP-LV heuristics proved to be particularlystable in terms of product family replenishment timing.

FP-LV-WR

FP-LV

P-E-WR

FP-E

st Error4.00% 5.00% 6.00% 7.00%

schedule instability.

A. Narayanan, P. Robinson / Int. J. Production Economics 127 (2010) 85–9494

The research findings provide additional insights on cost errorand schedule stability performance. The ANOVA results in Table 3indicate that the major driver of cost performance is choice of JRPheuristic followed by the problem’s demand density and TBOItem.The policy variables (frozen interval and planning horizon length)have relatively little impact on cost performance. However,Table 7 indicates that driving factors for schedule stability arereversed. Here, the frozen interval and planning horizon lengthdominate, while the JRP heuristic, demand density, and TBOItem

have a lesser impact on schedule stability.This research also provides new insights on applications of the

JRP heuristics in static versus rolling schedule environments. WhileRobinson et al. (2007a, 2007b) suggest that the SAM is the mosteffective solution procedure for the static JRP this does not hold inrolling schedule environments. Instead, PM produces lower schedulecosts, and at substantially greater schedule stability. Consideringthat JRP heuristics are commonly applied in rolling schedules andthe importance of maintaining stable schedules, these findings areimportant for both researchers and practitioners.

The effect of policy design variables on cost error alsocontradicts the results in the rolling horizon literature for multi-item systems. Shorter planning horizon and longer frozen intervallength produce the least cost error. As discussed in Simpson(1999) the use of relative cost measure may explain some of thisunexpected results.

Suggested research extensions include development of moreefficient optimization approaches for the JRP and their testing inboth static and rolling schedule environments. This researchshould also test modified fixed cost approaches as suggested byStadtler (2000). It is also worthwhile to consider extending theresearch to include capacitated variant of the JRP. The capacitatedproblem is frequently encountered in systems with limitedshipping and/or production resources. Finally, the researchfindings highlight the tradeoff between schedule cost andstability. Extending the analysis to explicitly consider reschedul-ing cost in successive planning iterations would provide anintegrated approach to evaluating both schedule cost andschedule stability. This more comprehensive treatment of theproblem is well justified, but not addressed in the literature.

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