multiperiod remanufacturing planning with uncertain quality of inputs

394 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 57, NO. 3, AUGUST 2010

Multiperiod Remanufacturing Planning WithUncertain Quality of InputsMeltem Denizel, Mark Ferguson, and Gilvan “Gil” C. Souza

Abstract—In this paper, we consider production planning wheninputs have different and uncertain quality levels, and there arecapacity constraints. This situation is typical of most remanufac-turing environments, where inputs are product returns (also calledcores). Production (remanufacturing) cost increases as the qualitylevel decreases, and any unused cores may be salvaged at a valuethat increases with their quality level. Decision variables include,for each period and under a certain probabilistic scenario, theamount of cores to grade, the amount to remanufacture for eachquality level, and the amount of inventory to carry over for futureperiods for ungraded cores, graded cores, and finished remanu-factured products. Our model is grounded with data collected at amajor original equipment manufacturer that also remanufactures.We formulate the problem as a stochastic program; although it isa large linear program, it can be solved easily using Cplex. Weprovide a numeric study to generate insights into the nature of thesolution.

Index Terms—Production planning, remanufacturing, stochas-tic programming.

I. INTRODUCTION

IN THIS paper, we consider a production planning prob-lem where the inputs have different and uncertain quality

levels, and there are capacity constraints; a situation typicalof remanufacturing operations, where inputs are product re-turns. Remanufacturing is the process of restoring used products(returns or cores) to a common aesthetic and operating standard.Similar to the planning of production of new products, develop-ing a production plan to remanufacture product returns typicallyinvolves determining the quantity of units to process during eachperiod under time-varying demand and a capacity constraint onthe number of units that can be processed during each period.Despite these benefits and similarities, developing a remanu-facturing operation often poses significant challenges to firmswho are more accustomed to planning production of new prod-ucts in the forward chain because inputs are variable; this paperaddresses some of these challenges.

Manuscript received July 8, 2008; revised October 11, 2008 and January 17,2009; accepted February 3, 2009. Date of publication August 25, 2009; date ofcurrent version July 21, 2010. This work was supported in part by the NationalScience Foundation (NSF) under Civil, Mechanical, and Manufacturing Inno-vation Grant 0522557. The work of M. Denizel was supported by the UnitedStates Department of State, Bureau of Educational and Cultural Affairs under aFulbright Scholarship. Review of this manuscript was arranged by DepartmentEditor J. Sarkis.

M. Denizel is with the Faculty of Management, Sabanci University, Istanbul34956, Turkey (e-mail: [email protected]).

M. Ferguson is with the College of Management, Georgia Tech, Atlanta, GA30332 USA (e-mail: [email protected]).

G. C. Souza is with the Kelley School of Business, Indiana University, Bloom-ington, IN 47405 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TEM.2009.2024506

The most common method used to develop production plans(for both new and remanufactured products) is to treat the de-mand as deterministic and solve for the plan using a linearprogramming formulation. Uncertainty in the demand streamis typically accounted for by solving the production plan on arolling horizon, where the demand forecasts are frequently up-dated, the linear program is resolved, and each solution onlyimpacts the current period’s production quantity as a new planwill be generated with updated information before the quanti-ties in the future periods are produced. While this method hasworked reasonably well for many years and for many firmswho only produce new products, there are particular character-istics of the remanufacturing industry that create problems whenthe traditional production planning methods are used to plan aremanufacturing process. Two of the main characteristics thatdifferentiate remanufacturing from new product production areuncertainty in the number of returned products and uncertaintyin the quality of the returns.

When producing new products, as long as proper planninghas been done to procure the components that go into the newproducts, any quantity (up to the capacity of the facility) canbe produced in any period. In addition, there is typically littlevariation in the amount of capacity required to produce one newunit from another of the same make and model. Now considera remanufacturing operation. The number of units remanufac-tured during each period cannot exceed the number of returnedcores available, even if there is capacity at the facility to pro-duce more. In addition, a returned shipment of used cores willtypically vary in the quality of the cores, with some cores re-quiring significantly more capacity to bring the unit up to arequired quality standard than others. As an example, IBM’s re-manufacturing facility in Raleigh, NC may receive a shipmentof end-of-lease laptops from one of their customers. Each lap-top must be remanufactured to a predetermined configurationand quality level before being offered for sale on IBM’s website. A random draw of two laptops from the incoming ship-ment may result in one of the laptops simply requiring testing,cleaning, formatting of the harddrive, and the loading of thestandard software configuration; jobs requiring approximately15 min of production capacity at this facility. The second lap-top, however, may have a broken latch, a worn keyboard, anda nonstandard memory configuration. This laptop will require45 min of production capacity before it matches the qualityand configuration standards, a difference of 300% in the ca-pacity required of the first laptop. There are very few produc-tion environments for new products that have this much vari-ation in the processing times of individual units of the sameproduct.

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DENIZEL et al.: MULTIPERIOD REMANUFACTURING PLANNING WITH UNCERTAIN QUALITY OF INPUTS 395

Several firms that are actively involved in remanufacturingrecover their used cores by either leasing their new products(ensuring the used units will be returned at the end of the lease)or by offering trade-in credits on the return of an old unit whena new product is purchased. Product leasing results in the mostpredictable return stream (from a volume perspective) and is apopular option for firms that also remanufacture. In fact, productleasing corresponded to 28%, or $199 billion, of all capitalexpenditures by American businesses in the United States in2004 [9]. Examples of firms that remanufacture products off-lease include IBM and Pitney Bowes [10]. Firms that chooseto recover their cores through a trade-in program are also ableto accurately forecast their return stream, as the quantity ofreturned cores is highly correlated with the sales forecast forthe firm’s new products. Both sets of firms (firms who lease andfirms who offer trade-in credits), however, still struggle with thevariation in the quality of the cores once they arrive, and this isthe focus of this paper. As the preceding example demonstrates,this issue makes the production planning process much morecomplex compared to production planning for new products.

To address this problem, we formulate a stochastic programfor a multiperiod setting where the quality levels of a lot ofreturned cores may take on any configuration of bad to goodquality levels based on pre-established probabilities; we elabo-rate on a desirable probabilistic definition of this problem later.Stochastic programming (SP) is an attractive methodology forthis problem because (unlike a regular linear programming for-mulation) it guarantees a feasible solution under all possiblequality-level realizations. Regular linear programming (LP) for-mulations, in contrast, must use the expected value for the qual-ity levels and the subsequent capacity requirements, and thus,do-not always produce an implementable solution. Similar toregular LP, SP allows for a capacity constraint and simply re-quires the solution of a (large) LP. We demonstrate how SP canbe applied to the remanufacturing planning problem, and testthe method in an extensive numerical analysis. By analyzingthe results of our numerical analysis, we perform a sensitivityanalysis to determine the parameter values that have the largestimpact on the solution. The results of this sensitivity analysisprovide guidance for firms seeking to make investments to im-prove their remanufacturing operations. The SP formulation hassome history in the new product production planning problembut is new to the remanufactured product planning problem, asdiscussed in the next section.

II. LITERATURE REVIEW

Our paper is positioned among two main streams of literature.The first concerns the consideration of uncertainty in planningproduction for “regular” (or forward) manufacturing, and thesecond is related to planning production in a remanufacturingenvironment. Regarding the first stream, the literature recog-nizes the pitfalls of not considering uncertainty when planningproduction; the main sources of uncertainty considered are de-mand and environmental (which may impact multiple parame-ters). For a comprehensive review, see Mula et al. [26]; we focushere on work most relevant to this paper, namely, research that

uses SP in multiperiod settings. SP (see Birge [3] for a review)is an attractive methodology in production planning problemsunder uncertainty because of its ability to incorporate multipleconstraints (e.g., capacity) that are common in most productionenvironments. Production planning problems are typically for-mulated as two-stage SP problems where a random outcome(e.g., random demand, state of the economy, etc.) occurs onlyonce, at the beginning of the second stage. Examples includeGupta and Maranas [20] who model demand uncertainty; Eppenet al. [8], and Leung et al. [25], who model environmental uncer-tainty (which impacts demand and cost parameters). In all threeexamples, decisions are made in the first stage—production de-cisions in the first, plant selection in the latter two—before therandom outcome is realized. A multistage (MS) SP approachhas been proposed for the more strategic problem of capacityplanning for semiconductor manufacturing over a several-yearplanning horizon [6] under different demand and technologyscenarios. MS SP approaches to (tactical) production planningproblems are not commonly found in the literature, perhaps dueto their size and computational complexity. We propose an SPmodel to solve for a multiperiod production plan in a remanufac-turing environment, which has the unique (to remanufacturing)characteristic of nonuniformity of inputs (cores). The modelis intuitive, allows for uncertainty in the quality levels of thecores, and is computationally manageable (we solved realisticscenarios in an average of 79 s on a shared Unix server).

There is a growing body of literature addressing productionplanning for remanufacturing. For comprehensive overviews,see Inderfurth et al. [21], Inderfurth and Teunter [22], Guide[19], and Fleischmann et al. [13]. Many papers consider a single-period problem where there are only two categories of cores,good and bad; good units are all remanufactured at the samecost whereas bad units are salvaged (e.g., [2], [11], and [31]).Other papers acknowledge different remanufacturing costs forcores of different quality, but still under a single-period setting(e.g., [1], [14], and [17]). In contrast, we consider a multiperiodproblem where cores have different qualities, and there are dis-posal and inventory carrying options across periods. Multiperiodproduction planning problems have been addressed by Fergusonet al. [10], Guide et al. [18], and Golany et al. [15]. All of thepapers mentioned earlier, however, assume a deterministic qual-ity of cores, or rather, consider the expected values of cores of agiven quality level at each period to formulate the optimizationproblem.

To our knowledge, ours is the first paper to consider stochasticquality of cores, with its associated capacity and cost implica-tions, for production planning in a multiperiod setting. We dothis by using a stochastic programming formulation. Althoughthere is a significant stream of research in stochastic inventorymodels that include remanufacturing options (e.g., [7], [12],[23], [27], and [28]), the Markov decision process approachused in these papers prevents modeling complexities, such ascapacity constraints and multiple quality grades. Although ourapproach does not yield closed-form solutions, it always resultsin implementable solutions, as the model formulation guaranteesfeasibility for each random outcome in the planning horizon.This desirable property does not hold when either 1) expected


values are used in other mathematical programming formula-tions or 2) capacity constraints and/or the existence of multiplequality grades are ignored. Regarding 2), consider the fact thatcores of worse quality require more production resources thanthose of better quality; given that the mix changes in each pe-riod (depending on the outcome of the grading process), thencapacity is also impacted.

III. MODEL

Consider a remanufacturing environment where end-of-leasereturned products (cores) are graded and grouped into I differentquality levels. We consider a single product type, so our prob-lem is an aggregate production planning problem; the modelcan be extended to multiple product types although computa-tional complexity would increase significantly. Graded cores arethen remanufactured to meet demand for remanufactured prod-ucts. The total amount of cores is known in advance; however,their quality levels are uncertain. The demand for remanufac-tured products is forecasted over a planning horizon of T peri-ods. Assume that T is considerably smaller than the product’slife cycle, such that the parameters of the problem (e.g., costs,grading outcomes, and probabilities) can be considered to beconstant during T . Remanufactured products are identical andindependent of the quality of the original cores used; however,remanufacturing costs differ across quality grades. Cores canbe kept in stock to be graded in the future and graded corescan be kept in stock to be remanufactured in the future. Theholding cost for a remanufactured product is higher than fora core. Graded cores can also be salvaged at any time; higherquality cores have higher salvage values corresponding to theopportunity to harvest higher quality cores for spare parts. Theproblem is to determine how many of the available cores tograde, how many of the graded cores to be remanufactured, andhow many of them to be salvaged each period. Due to capacityconstraints, the firm may not be able to meet the nonstationarydemand, as lower quality cores consume more capacity thanhigher quality cores. Therefore, backlogs are allowed. The ob-jective is to find a plan that maximizes total expected profit. Ourmodel is a tactical production planning model, where forecastsfor the demand for remanufactured products are given. To the ex-tent that there is any interaction (cannibalization) between newand remanufactured products in the marketplace, the given de-mand forecasts already reflect this interaction. Our model doesnot apply to hybrid remanufacturing systems—the case wherenew and remanufactured products are perfect substitutes (e.g.,Kodak’s single-use camera), as a single inventory pile is usedfor both types of products.

We formulate our problem as a stochastic program. Therandom variable in each period is the outcome (r(j,t)

i , as de-fined next) of the grading process. For example, suppose thereare two quality grades (good and bad), and two possible gradingoutcomes. In outcome 1, which happens with probability p1 , theinspection process yields 70% of good quality cores, whereas inoutcome 2, the inspection process yields 40% of good qualitycores. We define scenarios for each time period t as follows.Each time period t has J(t) scenarios, each corresponding to

Fig. 1. Construction of scenarios.

a realization of all grading outcomes in periods 1, . . . , t. So,for example, period 2 has four possible scenarios (J(2) = 4)pertaining to the percentage of good quality cores during eachperiod ((0.7, 0.7), (0.7, 0.4), (0.4, 0.7), (0.4, 0.4)), whereas pe-riod 4 has 16 scenarios ((0.7, 0.7, 0.7, 0.7), . . . .(0.4, 0.4, 0.4,0.4)). Period 0 (the period prior to the start of the planning hori-zon) has only one scenario, since none of the random variableshave been observed prior to the planning horizon. This is shownin Fig. 1. Note the “nested” structure of the scenarios: scenario1 of period 1 is nested in scenarios 1 and 2 of period 2, whichare nested in scenarios 1–4 of period 3. We denote scenario jin period t as scenario (j, t). For a given time period t, define(pred(j), t − 1) as the scenario, in period t − 1, i.e., the pre-decessor of scenario (j, t). In the earlier example, for period 2,pred(1, 2) = pred(2, 2) = (1, 1), pred(3, 2) = pred(4, 2) = (2,1), and so forth.

In our model, there is no correlation in the quality of returns(outcome of the grading process) across periods, because weconsider the situation where there are more returns than de-mand (we discuss this assumption in more detail later), andtherefore, there is little incentive for a firm to carry over a batchof ungraded returns in inventory. In fact, in our numerical study(Section IV), it is optimal to not carry any inventory of ungradedreturns in the majority of experiments. Ungraded cores can besalvaged although it would be a very rare event when it wouldbe profit increasing for the firm to do so. Without first grad-ing the cores, the firm would salvage good, medium, and badquality cores equally. For most scenarios that we have encoun-tered, however, all good quality cores are remanufactured. Infact, in our numerical examples, 99% of the good quality coresare remanufactured. Our experience with remanufacturing firmsindicates that it is standard policy to grade all incoming coresbefore determining the number to salvage.

Outcomes in our model should be thought of from an im-plementation perspective, in terms of the traditional stochasticprogramming idea of higher level events (e.g., optimistic, av-erage, and pessimistic), rather than more granular probabilitydistributions for quality of returns. Using granular probabilitydistributions for quality of returns (e.g., multivariate normal)would yield very large problems, with (most likely) infeasible


TABLE INOTATION

computational times; this is a topic for future research. Our nota-tion is defined in Table I. We note that subscripts or superscriptsof a variable are dropped if that variable is constant in a certainindex.

In the description of our model given later, we assume demandto be deterministic. Stochastic demand can be incorporated intothe model, at least from a theoretical standpoint, by redefininga random outcome in a period as a combination of the ran-dom realization of the grading process and the random demandrealization; we expand upon this point later. From a model for-mulation perspective, this is not a difficult extension. From apractical implementation standpoint, however, it presents sig-nificant challenges because the number of variables increasesexponentially with the number of random outcomes in a pe-riod. A model with both random quality of cores and randomdemand can feasibly be solved only for a limited number of sce-

narios and time periods. Stochastic demand has been exploredin the previous production planning literature (see Section II),and since the novelty of our paper resides in the peculiarity ofthe remanufacturing environment with its nonuniform input, wemaintain in our numerical analysis the assumption of determin-istic demand (i.e., no forecast error). Further, the impact of thedeterministic demand assumption can be mitigated by solvingthe model using a rolling horizon.

In our model, the first- and second-stage decisions take placeat the same period. First, the amount of cores to be graded xis determined; based on that decision, the amounts of cores ineach quality grade are revealed. These determine the scenariosin each period. Then, for each scenario, the amounts of coresto remanufacture z, salvage, and hold in inventory from eachquality grade are decided upon. Fig. 2 depicts this sequence ofevents and the notation used in our model.


Fig. 2. Notation and sequence of events in the model (two outcomes and threeperiods).

Our problem (assuming zero initial inventories) can thus beformulated as

max Π =T∑

t=1

J (t)∑j=1

p(j,t)

{I∑

i=1

((Pr − ci)z

(j,t)i + siv

(j,t)i

− hiu(j,t)i

)− hry

(j,t)+ − πy(j,t)−

}

−T∑

t=1

J (t−1)∑j=1

p(j,t)(gx(j,t−1)t + hb

(j,t−1)t

)(1)

subject to

y(pred(j ),t−1)+ − y(pred(j ),t−1)− − y(j,t)+ + y(j,t)−

+I∑

i=1

z(j,t)i = Dt ∀(j, t) (2)

b(j,t−1)t + x

(j,t−1)t − b

(pred(j ),t−2)t−1 = Bt ∀t,

j = 1, . . . , J(t − 1) (3)

z(j,t)i +u

(j,t)i −u

(pred(j ),t−1)i + v

(j,t)i − r

(j,t)i x

(pred(j ),t−1)t = 0

∀(j, t) (4)

I∑i=1

aiz(j,t)i ≤ Ct ∀(j, t) (5)

y(j,T ) = 0, j = 1, . . . , J(T ) (6)

y(j,t)+ , y(j,t)−, z(j,t)i , v

(j,t)i , u

(j,t)i , x

(j,t−1)t , b

(j,t−1)t ≥ 0

∀i, (j, t). (7)

The objective function (1) maximizes total profit: revenuefrom remanufactured products plus salvage revenue minuscosts—holding cost for graded cores, holding cost for remanu-factured products, backlogging cost, grading cost, and holdingcost for ungraded cores. Constraint (2) is the inventory balanc-ing constraint for remanufactured products under each scenario(j, t): for a given scenario (j, t), one can meet demand from thecurrent period’s production and starting inventory of remanufac-tured products (all scenarios (j, t) with a common predecessorshare the same starting inventory y(pred(j ),t−1)+ ), or demand canbe backlogged. Constraints (3) and (4) are the inventory balanceconstraints for ungraded and graded cores, respectively; notethat in (4), all scenarios (j, t) that share the same predecessorhave the same amount of graded cores x

(pred(j ),t−1)t . Constraint

(5) regards the capacity constraint for period t, which shouldbe met for any scenario. Finally, constraint (6) ensures that thefirm does not produce in excess of demand over the planninghorizon.

If demand is stochastic, we redefine an outcome in any pe-riod to be the joint realization of the random grading processand random demand in that period. For example, if there arethree possible grading realizations (good, medium, bad) andtwo possible demand realizations (high, low), then each periodhas 3 × 2 = 6 outcomes. The only change in the aforesaid for-mulation is that Dt is replaced by D(j,t) in (2). Although thechange in the formulation is simple, it implies a much large num-ber of variables, as they increase exponentially in the number ofscenarios.

Thus, the stochastic programming approach generates a solu-tion for each scenario in the grading process; it is only feasibleif there is enough capacity in the system to guarantee feasibilityunder all scenarios, including the worst case scenario (a suc-cession of “bad” outcomes–outcomes with a high percentage ofbad quality cores, which occurs with a positive probability).

A. Computational Considerations

The formulation we present for grading and remanufactur-ing planning is an MS stochastic linear program (MS-SLP) thatmay become difficult to solve for the size of problems foundin practice. The size of the problem grows exponentially withrespect to the number of quality grades and number of periods.The structure of the problem, however, can be exploited to de-velop algorithms in case a general-purpose linear programmingcode fails to provide solutions in a reasonable computationaltime. One such approach can be the use of decomposition meth-ods that use the block-angular structure of the MS-SLP for-mulations. The L-shaped method, first suggested by Van Slykeand Wett [29] for two-stage problems and then generalized to


multiple stages as a nested decomposition method [4], shouldprovide better computational results.

In this paper, however, we were able to solve problems ofrealistic sizes in very reasonable computation times based ondata grounded in a real remanufacturing environment, withoutthe need to resort to such a special algorithm. We consider fiveoutcomes for the quality grades and six planning periods (nextsection). This choice of the parameter values also leads to an eas-ier interpretation of the results. The solution provides a courseof action for each outcome in each period. When the number ofpossible outcomes grows very large, tracing the solution paths(which differ from each other with very small probabilities)for implementation may be impractical. The same applies tothe number of periods in the planning horizon. Furthermore,we recommend that the problem be solved on a rolling hori-zon basis, and a six-period planning horizon serves the purposeof preventing myopic decisions. In our computational experi-ments, the solution times observed (about 79 s on average) witha general-purpose linear programming software were very rea-sonable and should enable the problem to be solved on a rollinghorizon basis. It seems that the sparse factorization schemes em-bedded in most general-purpose software take into account thestructure of the problem to a certain extent. In the next section,we provide details about our numerical study, and interpret theresults.

IV. NUMERICAL ANALYSIS

The objective of this section is to investigate how the variousproblem parameters (costs, demand pattern, capacity) impactthe firm’s optimal solution (profit, amount of cores graded, re-manufactured, inventoried, and salvaged). This is done througha large-scale numerical study.

A. Experimental Design

The choice of parameter values for our numerical study islargely taken from existing research (see [10]), which is basedon actual data provided by Pitney Bowes; we provide justifi-cation for additional parameters for the purpose of sensitivityanalysis when necessary. Pitney Bowes is a company based inStamford, CT, which produces large-scale mailing equipmentthat matches customized documents to envelopes, weighs theparcel, prints the postage, and sorts mail by zip code. The major-ity of Pitney Bowes revenues originates from leasing of its newproducts; the length of a typical leasing contract is four years.At the end of the lease, if the customer upgrades to a newertechnology equipment (when available), end-of-lease productsare returned from customers to a central facility, where they canbe remanufactured. Remanufactured products are then sold tocustomers at a discount compared to the price of purchasing anew unit of the same model (the new model is typically of anewer technological generation).

Costs are normalized by choosing the remanufactured prod-uct’s selling price to be Pr = 100. We choose a common pro-duction planning horizon of six months, so T = 6. We considerthree demand patterns, as shown in Table II; these demand pat-terns are slightly modified from the 12-month data of Ferguson

TABLE IIDEMAND DATA FOR NUMERICAL STUDY

Fig. 3. Curves for c(q) with α0 = 60, α1 = 25.

et al. [10] to fit a six-period planning horizon. The variabilityof demand type 1 is evenly distributed throughout the planninghorizon, while demand type 2 has two months with significantlyhigher than average demand. Finally, demand type 3 is constantduring the first three months, but ramps up to about double thislevel in the last two months of the planning horizon.

For Pitney Bowes, returns typically exceed demand for re-manufactured products; this also holds for other industries, suchas imaging supplies and cell phones [14], PCs, printers, andpower tools [16], except during the beginning of the product’slife cycle. Thus, we set the number of incoming cores, acrossall quality grades, at 150% of the average demand per period;this corresponds to 540 incoming cores per period. The qualityof an incoming core q is the realization of a random variableQ between 0 and 1, where q = 0 represents the worst possiblecore and q = 1 the best possible core. For simplicity, we assumethat Q ∼ U [0, 1]. The cost to remanufacture a core of quality qis given by c(q) = α0 + (α1 − α0)qβ , where α0 , α1 , and β areparameters; this functional form represents considerable diver-sity of curve shapes depending on the chosen parameter values.The parameter α0 is the remanufacturing cost for the worstpossible quality core; likewise, α1 is the remanufacturing costfor the best possible quality core. Similarly to Ferguson et al.,we use α0 = 60, α1 = 25 (recall that Pr = 100), and considerβ ∈ {1/3, 1.0, 3.0}; this results in the curves shown in Fig. 3.


Under this setup, if there are I quality grades distributed uni-formly over [0, 1], then ci = I

∫ 1−(i−1)/I

1−i/I c(q)dq, which re-

sults in ci = α0 + (α1 − α0)/((1 + β)Iβ )((I − i + 1)1+β −(I − i)1+β ).

The salvage value for a core comprises its value in reusableparts and material; the higher the quality of a core, the higher itssalvage value. Because remanufacturing is a value-added oper-ation, the salvage value can be viewed as a fraction 0 ≤ θ < 1of the net remanufacturing profit margin, i.e., si = θ(Pr − ci),where in our design, we allow θ ∈ {0.10, 0.40, 0.70}. (If θ > 1,then remanufacturing is not profitable, and the optimal solutionis trivial: salvage all units.) The value θ = 0.10 represents thecase where cores are primarily salvaged for materials recovery,whereas the value θ = 0.70 represent a case where a significantvalue can be recovered from the harvesting of spare parts. Thethree levels for unit holding cost for cores, which is the same forall quality categories (hi = hj∀i, j), are hi ∈ {1, 2, 3}. Thesevalues correspond to 12%, 24%, and 36% of the remanufacturedproduct’s price per year; they represent typical holding cost val-ues, from low time-sensitivity to high time-sensitivity products(we account for market price depreciation in the product’s hold-ing cost value). Further, we set h = 0.5hi and hr = 1.5hi , sothe holding cost for an ungraded core and for a remanufacturedproduct are 50% less and more, respectively, than the holdingcost for a graded return. We set unit grading cost to be 0%, 10%,or 30% of the remanufacturing cost for the best possible qualitycore, i.e., g ∈ {0, 0.1α1 , 0.3α1}.

We consider unit backlogging cost of π ∈ {10, 20, 40}. Giventhat hr varies between 1.5 and 4.5, the ratio π/hr varies between2.2 and 26.7; numbers of this magnitude have been used in pre-vious research (e.g., [5]). Further, π can be interpreted as apercent discount off the remanufactured product’s price if thefirm postpones the due date for meeting demand by one month.For example, π = 10 means that the firm gives the customer a10% discount off the list price if delivery is postponed by onemonth. We normalize resource usage for the best quality gradeto a1 = 1. Resource usage for the worst quality grade core isaI = 1 + A, where A ∈ {0.25, 0.50, 0.75}; resource usage forintermediate quality grades is a linear interpolation of these twopoints such that ai = 1 + A(i − 1)/(I − 1). Capacity availabil-ity is set at 20%, 60%, and 100% of excess capacity over averagedemand if the firm only remanufactures high-quality cores; thus,we consider Ct/µD ∈ {1.2, 1.6, 2.0}.

We complete our experimental design by describing thegrading outcome. We assume three quality grades (good,medium, bad), and five outcomes for the grading process (worst,worse, average, better, best). The probabilities are described inTable III. As an example, the “best” case scenario resulting fromthe grading process would result in 66.7% good-quality cores,33.3% medium-quality cores, and 0% bad-quality cores.

Our experimental design is summarized in Table IV that fol-lows. In total, there are 38 = 6,561 experiments. For each cell,we solve the resulting linear program in Cplex 9.0, using theCplex callable library. Each cell results in a linear program with222,647 variables and 101,556 constraints. Run time for onecell ranges between 31 and 488 s, averaging 79 s on a shared

TABLE IIIGRADING OUTCOME AND ASSOCIATED PROBABILITIES

TABLE IVEXPERIMENTAL DESIGN FOR NUMERICAL STUDY

(Pr = 100, α0 = 60, α1 = 25)

Sun V480 computing environment. For each optimal solution,we also compute the associated values of the decision variables.Thus, our metrics include the optimal objective function value(Π), and the expected values per period for each of the problem’sdecision variables (optimal solution values weighted by the re-spective probabilities in each scenario). That is, we compute theexpected number of graded cores (denoted in short by xt/T ),expected number of remanufactured zt

i/T and salvaged unitsvt

i/T for each quality category, expected inventory for ungradedcores bt/T , graded cores ut

i/T , remanufactured units yt+/T ,and expected backlogs yt−/T .

B. Numerical Results

In Table V, we report descriptive statistics for each of the 14metrics of interest. Recall that average demand per period is 360units; average arriving cores are 540 units per period or 180 ofeach of the three quality grades, on average. On average, thefirm remanufactures 178, 143, and 39 units, and it salvages 2,37, and 141 units of good, average, and bad quality cores perperiod, respectively. Thus, about 99% of all good quality coresare remanufactured (this corresponds to about half the overalldemand), with the remainder half of demand being met mostlyby average, and to a less extent, bad-quality cores. In contrast,for incoming bad-quality cores, 22% are remanufactured, and78% are salvaged, on average.

The firm does not carry much inventory of graded cores, asdemonstrated by median values for ut

i/T of 13, 4, and 0 unitsfor good-, average-, and bad-quality cores, respectively. Thefirm only carries inventory of remanufactured products becauseof capacity constraints coupled with a nonstationary demandpattern; the median value is 20 units, with a mean of 43 units.Similarly, the median value of backlogged units is 1 unit but the


TABLE VDESCRIPTIVE STATISTICS FOR EXPERIMENTAL DESIGN

TABLE VIMULTIPLE REGRESSION RESULTS FOR PROFIT AS DEPENDENT VARIABLE

(p < 0.00001 FOR ALL COEFFICIENTS)

average is 13 units; backlogs only occur if there is insufficientcapacity, as is the case when A = 0.75 and Ct/µD = 1.2. Inmost cases (96% of cases), the firm grades all 540 cores in allperiods and scenarios, and as a result, it does not carry anyinventory of ungraded cores (median value is 0 unit).

C. Impact of Parameters on Profitability

Of particular importance to a decision maker is the impact ofproblem parameters on profitability. Rather than using a one-parameter-at-a-time sensitivity analysis, we perform a multipleregression analysis, where the dependent variable is the objec-tive function Π, the independent variables are the experimentalfactors, and there are 6,561 observations corresponding to eachexperimental cell. This is a form of global sensitivity analysis,where all parameters are varied at the same time [30]. The re-sults are shown in Table VI, where the dummy variable di takesthe value of 1 if demand pattern i is used and 0 otherwise. Thevalue of adjusted R2 is quite high at 0.943, and all coefficientsare highly significant at p < 0.00001. We also note that separatesimple regressions (one for each factor) provide almost identicalresults; a simple plot of the relationship between each parameterand profit (shown later) provide further evidence of the linearityapproximation. Thus, the regression analysis provides a reliabletool for a decision maker when assessing the impact of a fac-tor on profitability, considering the range studied in the design,which is comprehensive and based on industry values.

In the table, the absolute values of the t-values for each coef-ficient provide an assessment of the influence of that factor inexplaining the variance of profit in our experimental design; a

larger t-value indicates a more influential parameter [30]. Thus,the factors β, θ, and g/a1 have the largest influence on profitwithin the range of values in the experimental design, witht-values of −251, 189, and −96, respectively.

The coefficients (slopes) provide a direct measure of themonetary impact of that parameter on profitability. As theremanufacturing cost curve c(q) goes from linear (β = 1) toconcave (β = 3), profit decreases by 2*15,561 = $31,112, or22%, on average (average profit is Π = 143, 604). This factor,however—which describes how remanufacturing cost decreaseswith quality—is one where production managers are likely tohave little control over; as it is determined primarily by the typeof product. As the relative salvage value θ increases from low(0.1) to medium (0.4), profit increases by 0.3*54,253 = $16,276,or 11%, on average. This is a significant increase, and such anincrease in salvage values can be possible; actions, such as useof salvaged returns for spare parts (as opposed to material recy-cling), increases salvage values. Finally, a decrease in gradingcost from high (0.3) to low (0) increases profit by 0.3*54,000 =$16,200 or 11%; again, this is a significant increase and pos-sible through the use of design and grading technologies, suchas electronic data logs that record product usage, as illustratedby the classical examples of Bosch power tools [24], and HPprinters [16]. On the other hand, other cost parameters, suchas holding cost hi and backlog cost π, do not impact profit assignificantly, mitigating the need for actions toward reducingthese costs; similarly for the demand patterns di .

D. Sensitivity Analysis: Optimal Solution

In this section, we provide an analysis about which factorshave the largest impact on a given performance metric of theoptimal solution. This is done in two steps. First, we performsimple regressions, one for each metric and experimental factor.In each regression, the dependent variable is the metric of in-terest (e.g., zt

2/T ), the independent variable is the experimentalfactor (e.g., β), and there are 6,561 observations, correspondingto the experimental cells. The value of R2 for each regressionprovides a measure of the impact of the respective factor valueon the respective metric [30]. The R2 values are reported in theAppendix for the interested reader. In the second step, we thenselect the factors with relatively high R2 values—the factors ofmost impact on the optimal solution—and plot that relationshipas follows. We compute, for each factor level (e.g., θ = 0.1),the average value of the metric of interest across all respectiveexperimental cells, and normalize it by subtracting the mean(across all cells) and dividing by the standard deviation. Thisnormalization is done so that all metrics are measured on the


Fig. 4. Normalized metric value versus relative capacity Ct/µD .

Fig. 5. Normalized metric value versus relative salvage value.

Fig. 6. Normalized metric value versus relative grading cost.

same scale, i.e., relative to its mean value. The results are shownin Figs. 4–7. In all charts, the scale in the vertical axis is thesame; we also use the same marker and line pattern for eachmetric across charts.

First, consider in Fig. 4 the factor with the highest impact inmost metrics, capacity relative to demand. With excess capacity,the firm can afford to salvage good and average quality cores(thus the increase in v1 and v2 and decrease in z2) to boostrevenue and use the excess capacity for capacity-intensive bad

Fig. 7. Normalized metric value versus extra capacity usage for worst quality.

cores (increasing z3 , and consequently, decreasing v3). Morecapacity also reduces the need to carry inventory of remanufac-tured products y+ , and reduces the amount of backlogs (y−).The largest impact on the metrics occurs as Ct/µD increasesfrom 1.2 to 1.6; it can be shown that Ct/µD = 1.6 providessufficient capacity to meet demand with expected (average) mixof good, average, and bad cores. Nonetheless, there is still someimprovement in the metrics as Ct/µD increases from 1.6 to 2.0because of the stochastic nature of the grading process.

Now, consider in Figs. 5 and 6, the twin impacts of relativesalvage value and relative grading cost. Typically, as discussedbefore, the firm grades all cores (and remanufactures all gradedhigh-quality cores); the exception is when salvage values arelow (Fig. 5) and grading cost is high (Fig. 6). Low salvage val-ues, relative to grading costs, cause the amount of grading x todecrease (which increases the inventory of ungraded cores b);the decreased availability of good-quality cores decreases theabsolute number of good-quality cores remanufactured z1 , atthe same rate (since the firm remanufactures almost all good-quality cores it grades). However, a lower salvage value foraverage quality cores decreases their amount salvaged v2 , andconsequently, increases their amount remanufactured z2 . Every-thing else being equal, a higher salvage value or a lower gradingcost per unit increases expected profit, and the relationship isapproximately linear.

Fig. 7 shows the impact of extra capacity usage A on theoptimal solution. Essentially, a higher value of A means lessbad-quality cores z3 are remanufactured, with more of them sal-vaged v3 ; the firm remanufactures more of the average qualitycores z2 (with less of them salvaged), and some of the demandis backlogged because of insufficient capacity. Good-qualitycores are not impacted by A because the firm remanufacturesmost of them. Regarding the impact of the demand pattern, wehighlight that demand pattern 2, which has the “spike in themiddle,” requires more carrying of inventory u1 and y+ fromearlier periods; we omit the chart here for brevity. Backloggingcost π has a considerably smaller impact on the optimal solu-tion. In general, the firm only carries inventory due to capacityconstraints and nonstationary demand; backlogging only occursdue to insufficient capacity as explained earlier.


V. CONCLUSION

In this paper, we consider a production planning problemwhere the inputs (returned units) have different and uncertainquality levels, and there are capacity constraints; a situationtypical of remanufacturing operations. From a production plan-ning perspective, this scenario has several implications: the firmneeds to grade returned units (cores) at a certain cost; the grad-ing operation yields an uncertain outcome with respect to themakeup of core quality, with each quality grade requiring aspecific cost and processing time, and graded cores can be sal-vaged at potentially different salvage values depending on theirquality. We propose a stochastic programming formulation tosolve the remanufacturing production planning problem Ourmodel determines the quantity of cores of each quality gradethat should be remanufactured, held in inventory for a futureperiod, or salvaged during each period in a finite planning hori-zon. Unlike a deterministic production planning model that useslinear programming with inputs based on the expected valuesof the returned units’ quality levels, our SP approach guaran-tees an implementable solution for all realizations of the qualitylevels.

Through an extensive numerical study based on parametervalues observed in practice, we measured the relative impactof each parameter on the firm’s profit (our objective function)as well as each of the corresponding decision variables. Fromthe results of this test, we conclude several interesting insightson the remanufacturing production planning problem. First, thefirm’s profit is most heavily influenced by the shape of the pro-duction cost curve as it relates to the quality of the core (convexincreasing, linear, or concave increasing in the lower quality ofthe core), the salvage value of an unprocessed core, and the costof grading. Thus, firms should take extra care in estimating thesevalues, and take actions toward reducing grading costs (throughgrading technology, such as electronic data logs), and increas-ing salvage values for nonremanufactured items (through usefor spare parts, for example). The per unit holding cost of thecores, as well as unit backlogging cost, on the other hand, havea smaller impact on profitability. This finding is encouragingas firms often have a difficult time estimating these parametervalues. It is often optimal for a firm to remanufacture all of thehighest quality cores.

While our model formulation and numerical study were basedon the assumptions of deterministic demand (no forecast error)and a known return stream of cores, the model can be modifiedto accommodate uncertainties in both of these inputs. For in-stance, the set of scenarios evaluated in the stochastic programcould be nested to include a given quality set for a given returnamount and a given demand realization. The tradeoff, of course,is an exponential increase in the number of outcomes, and conse-quently, on the computational requirement of solving the linearprogram. Future research could provide guidance on the bestway to handle demand and supply uncertainties in the planningof a remanufacturing process, particularly through specializedalgorithms for solving the stochastic program. The alternativesare to include the uncertainties in the model formulation withthe subsequent increase in computational times or to solve the

model frequently, on a rolling horizon, with forecast updates forboth the demand and the return stream of cores.

APPENDIX

TABLE VIIR2 (×100) VALUES FOR SIMPLE LINEAR REGRESSIONS WHERE THE

DEPENDENT VARIABLE IS METRIC AND THE INDEPENDENT

VARIABLE IS EXPERIMENTAL FACTOR

ACKNOWLEDGMENT

The authors would like to thank J. Sarkis and two anony-mous referees for valuable suggestions in an earlier draft of thispaper. A significant part of this research was carried out whenG. C. Souza was on sabbatical at Georgia Tech’s College ofManagement.

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Meltem Denizel received the B.Sc. and M.Sc.degrees in industrial engineering from Middle EastTechnical University, Ankara, Turkey, and the Ph.D.degree from the Department of Decision and Infor-mation Science, Warrington School of Business, Uni-versity of Florida, Gainesville.

She is currently a Professor of operations manage-ment in the Faculty of Management, Sabanci Univer-sity, Istanbul, Turkey, where she is teaching M.B.A.,E.M.B.A., Ph.D., and Executive Development Pro-grams. She has been involved in interdisciplinary re-

search in organization studies and technology management, and marketing.Her research has been in the areas of flexible manufacturing, mathematicalmodeling, and production lot sizing. She is the author or coauthor of paperspublished in Operations Research, Naval Research Logistics, Production andOperations Management, Operations Research Letters, Computers and OR,European Journal of Operations Research, and the Journal of the OperationalResearch Society, the Journal of International Research in Marketing, and ORSpectrum. Her current research interests include product recovery systems andsustainable operations.

Mark Ferguson received the B.S. degree in mechan-ical engineering from Virginia Tech, Blacksburg, theM.S. degree in industrial engineering from GeorgiaTech, Atlanta, and the Ph.D. degree in business ad-ministration, with specialization in operations man-agement, from Duke University, Durham, NC, in2001.

He had five years of experience as a Manufactur-ing Engineer and Inventory Manager with IBM. He iscurrently the Steven A. Denning Professor of technol-ogy and management and the John and Wendi Wells

Associate Professor of operations management in the College of Management,Georgia Tech. Several of his research projects have been funded by the NationalScience Foundation. His current research interests include many areas of sup-ply chain management including supply chain design for sustainable operations,contracts that improve overall supply chain efficiency, pricing and revenue man-agement, and the management of perishable products. He is the Coordinator forthe focused research area on dynamic pricing and revenue management.

Prof. Ferguson was the recipient of Best Paper Awards for two of his papersfrom the Production and Operations Management Society.

Gilvan “Gil” C. Souza received the B.S. degree inaeronautical engineering from Instituto Tecnologicode Aeronautica (ITA), Saao Jose dos Campos,Brazil, the M.B.A. degree from Clemson University,Clemson, SC, and the Ph.D. degree in operationsmanagement from the University of North Carolina,Chapel Hill. Prior to entering academia, he was aProduct Development Engineer at Volkswagen inBrazil for several years. He is currently an AssociateProfessor of operations management at the KelleySchool of Business, Indiana University, Blooming-

ton. Prior to Indiana, he was an Associate Professor at the Smith School ofBusiness, University of Maryland. He is the author or coauthor of several re-search papers published in California Management Review, European Journalof Operational Research, Management Science, Manufacturing and ServiceOperations Management, and Production and Operations Management. Hiscurrent research interests lie in supply chain management, including productionplanning, remanufacturing, and sustainable operations.

Dr. Gil was the recipient of the Wickham Skinner Early-Career Research Ac-complishments Award from Production and Operations Management Society(POMS) in 2004.

multiperiod remanufacturing planning with uncertain quality of inputs

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