production rate control for stochastic remanufacturing systems
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Int. J. Production Economics 112 (2008) 37–47
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Production rate control for stochastic remanufacturing systems
A. Gharbi�, R. Pellerin, J. Sadr
Production Systems Design and Control Laboratory, Department of automated production engineering, University of Quebec,
Ecole de Technologie Superieure, 1100 Notre Dame-Ouest Montreal, Que., Canada H3C 1K3
Received 21 September 2005; accepted 28 August 2006
Available online 10 April 2007
Abstract
We consider the control of a manufacturing system executing single-product remanufacturing operations in a close-loop
system. In this paper, the remanufacturing resources respond to planned demand executed at the end of the expected life of
each individual equipment and unplanned demand triggered by a major equipment failure. Remanufacturing operations
for planned demands can be executed at different rates, which referred to different component replacement and repair
strategies. The system objective is to define the remanufacturing rate as a function of the serviceable equipment inventory
level that minimizes the repair and inventory/shortage cost over an infinite horizon. The proposed policy, called multiple
hedging point policy (MHPP), is described by two thresholds related to two accelerated repair rates. To determine the
parameters of the control policy, and hence, to achieve a close approximation of the optimal repair policy, an experimental
approach based on design of experiment, simulation modeling and response surface methodology is used. Our results show
that the optimal cost incurred under the developed control policy is lower than that incurred under the classical hedging
point policy (HPP).
r 2007 Elsevier B.V. All rights reserved.
Keywords: Remanufacturing; Maintenance; Control policy; Simulation; Experimental design
1. Introduction
Over the last decade, equipment-driven organiza-tions such as airlines companies and armed forceshave been forced to become more and more flexibleand efficient within existing budgets, and withoutcompromising their operations. Those firms havealso faced continuous pressures to become moreenvironmentally responsible. In that context, many
front matter r 2007 Elsevier B.V. All rights reserved
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sses: [email protected] (A. Gharbi),
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of these firms have implemented a material recoverysystem in order to increase their equipment productlife through remanufacturing activities (Guide et al.,2000). Remanufacturing, also referred to majoroverhaul, is here defined as the process in whichworn-out equipments are restored to like-newcondition at a lower cost than new products (Coxet al., 1995). Remanufacturing systems offer poten-tial advantages, including increased profitabilitythrough reduced material requirements, reducedacquisition cost, and improved market share basedon environmental image (Guide et al., 2000).
Organizations involved in product remanufactur-ing can perform a wide range of repair tasks varyingfrom minor repairs to complex technical upgrades
.
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programs (Pellerin, 1997). In contrast to manufac-turing operations, remanufacturing tasks are subjectto a great deal of variability and uncertainty due tothe unique condition of the items being repaired(Pellerin, 1997; Gharbi et al., 1997). A typicalremanufacturing task starts with a disassemblyactivity and an inspection phase where the produc-tion content is defined. The next step, the mostimportant in terms of workload, includes thereplacement and/or repair of a large number ofcomponents. At the same time, the main structureof the equipment can be overhauled and modified toaccept new components and the repaired ones. Theremanufacturing process then ends with the assem-bly and a final inspection.
Researchers have proposed a number of modeladdressing different issues related to productremanufacturing operations, including supply chainplanning and management (Guide et al., 2000;Fleischmann et al., 1997) inventory control (Maha-devan et al., 2003; Van der laan, 1997), andoperations scheduling (Gharbi et al., 1997, 1999;Guide et al., 1997). To this point, the executioncontrol of this unique type of operations has notbeen considered. The difficulty of controllingremanufacturing operations resides in the variablenature of each operation. The amount of workdepends on the specific condition of each compo-nent being repaired and on the selected repairprocess. For each component, managers or techni-cians must decide on either repairing or replacingthe failed component. While repairing a componentusually cost less than replacing it, it does howevertake more time. When serviceable equipmentinventory exceeds the actual demand or servicelevel, remanufacturing organizations will typicallychoose to repair most of the components in order toreduce costs. When the number of serviceableequipments moves near the required service level,it is possible to accelerate the remanufacturingprocess by replacing components and sub-assemblyinstead of repairing them. While more costly, thisaccelerated process allows to quickly add moreserviceable equipment to stock and therefore redu-cing the risk of not meeting the required servicelevel. Between the repair and the full replace-ment remanufacturing control policies, it is possibleto adopt an intermediate repair rate that consistsof replacing only critical parts and repairing allother components. In defense organizations, thistype of intermediate remanufacturing process isformalized and called inspect and repair program
(Gharbi et al., 1997). It is important to note thatwhen the number of serviceable equipment isinsufficient, the operating firm engages significantcosts by having to cancel operations or by rentingequipments.
The purpose of this paper is to find a suboptimalsolution for the remanufacturing control problem ofone product subject to uncertain demand. Theproposed repair rate control policy takes advantageof the possibility of executing remanufacturing tasksat different repair rate with the objective ofminimizing total incurred costs. Before presentingthe proposed approach, we present the remanufac-turing problem and the related repair rate controlpolicy in Section 2. In Section 3, the description ofresolution approach is presented. The description ofthe model, based on discrete and continuoussimulation, is presented in Section 4. Section 5outlines the experimental design and responsesurface methodology. Results obtained in theprevious sections are thereafter used to comparethe proposed control policy with the classic hedgingpoint policy (HPP) in Section 6. The paper isconcluded in Section 7.
2. Problem statement
We consider a close-loop remanufacturingsystem repairing a single-equipment type withconstant demand. The system is said to be closeloop since the operating firm controls both service-able and non-serviceable stocks. Equipment dis-posal is not allowed and the equipment fleet,which corresponds to the total number of service-able and non-serviceable equipments, remainsconstant. In this paper, the remanufacturing re-sources respond to two types of demand: Planneddemand executed at the end of the expectedlife of each individual equipment and unplanneddemand triggered by a major equipment failure.Unplanned demands are here described by astochastic variable. Major equipment repairs areprocessed in priority in order to put the equipmentback in service as quickly as possible. Remanufac-turing operations for planned demands can beexecuted at different rates. The system objective isto define the remanufacturing rate as a function ofthe serviceable equipment inventory level thatminimizes the remanufacturing and inventory/shortage cost over an infinite horizon. The structureof the remanufacturing system under study ispresented in Fig. 1.
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Fig. 1. Structure of the remanufacturing system under study.
A. Gharbi et al. / Int. J. Production Economics 112 (2008) 37–47 39
The following notations will be used to describethe proposed control model:
x(t) equipment surplus at time t: A positivevalue of x(t) represents inventory, while anegative value represents backlog;
u(t) control remanufacturing rate at time t;SL service level (i.e. the number of serviceable
equipment required to support operationsof the organization);
d demand rate;c� cost to be paid per equipment per unit
time for failing to meet the servicelevel;
c+ cost to be paid per equipment per unit timefor keeping inventory at a higher level thanthe service level;
u1 minimum repair rate (repair policy);u2 intermediate repair rate (inspect and repair
policy);u3 maximum repair rate (replace policy);c1 cost per unit time when remanufacturing at
the minimum rate;c2 cost per unit time when remanufacturing at
the intermediate rate;c3 cost per unit time when remanufacturing at
the maximum rate;l12 unplanned demand arrival rate;l21 unplanned demand processing rate;J(x, a) long-run average cost function; and
a reparation mode when the system is onfunctional mode.
The system capacity is initially designed toproduce at the expected demand rate (u1 ¼ d). Asunplanned demand occurs, the remanufacturingresources are pre-empted until this demand issatisfied. Planned demands are therefore not satis-fied during that period and the number of service-able equipment x(t) decreases. The managers of theremanufacturing system have the possibility ofexecuting the planned demand at three differentrates (u1, u2, u3), where u1ou2ou3. Remanufactur-ing cost is rate related and increases as the repairrate is accelerated (c1oc2oc3).
The inventory/backlog level of the remanufactur-ing system, x(t), evolves according to the followingdifferential equation:
_xðtÞ ¼ uaðxðtÞÞ � d; xð0Þ ¼ x, (1)
where x is the initial stock levels and uaðxðtÞÞ is therepair rate control in mode a.
The repair rate takes its value on a convex set Oa
in the closed interval ½0; ua�, where ua represents themaximum repair rate allowed by the capacityconstraints of the manufacturing system when it isin mode a. The state of remanufacturing system canbe classified as ‘‘producing planned demand’’,denoted by 1, or ‘‘producing unplanned demand’’,denoted by 2. This process could be modeled as a
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�12 �21
1
2
Fig. 2. Two-state Markovian representation of planned and
unplanned demand.
A. Gharbi et al. / Int. J. Production Economics 112 (2008) 37–4740
continuous time Markov chain when the transitionfrom state to another is constant. The irreduciblecontinuous time Markovian two-state transitionprocess of producing planned demand and produ-cing unplanned demand is shown in Fig. 2.
This problem has many similarities with theproduction control problem of one product manu-factured by one unreliable machine having twostates (up and down). Akella and Kumar (1986)found an exact solution to that problem. Bieleckiand Kumar (1988) solved the same problem usingan approach that considers the long-run averageexpected inventory/shortage cost. Sharifnia (1988)solved the same problem considering multiplemachine states. The objective in this latter workwas to find the optimum production rate ua, foreach machine state a and surplus level x(t), so as tominimize the average surplus cost per unit timepresented in below
J� ¼ limT!1
1
TE
Z T
0
gðxÞdt ðx0; a0Þ��� �
, (2)
where E½�jx0; a0�is the expectation operator condi-tional on initial condition, gðxÞ ¼ cþxþðtÞ þc�x�ðtÞ,and xþ ¼ maxð0;xÞ and x� ¼ maxð0;�xÞare, re-spectively, the positive and negative part of theinventory/backlog surplus at time t.
The optimal solution of the control problemconsidering a cost function as described in(2) has never been found because of the complexityof stochastic Hamilton–Jacobi–Bellman (HJB)coupled equations. To simplify the analyticalcalculation, a near optimal HPP is considered. Inthis paper, the control policy is obtained byminimizing Eq. (2), adapted from the multiplehedging point policy (MHPP) proposed in Sharifnia(1988) and in Kenne et al. (2003) in the case ofclassical unreliable manufacturing system. Such
policy suggests that when the machine is up, oneshould produce at the maximum possible rate (ua)when inventory level is less than a threshold (za),production should be set at the demand rate if theinventory level is exactly equal to the threshold, andno production is required if the inventory levelexceeds the threshold.
Using the same structure of the MHPP model andaccording to the remanufacturing system describedearlier, the cost of maintaining the inventory/short-age x(t) for one time unit, denoted g(x), takes thefollowing form:
gðxÞ ¼ cþxþðtÞ þ c�x�ðtÞ þX3a¼1
capaua, (3)
where in (3), pa represents stationary probability ofremanufacturing system state (i.e. pa gives thepercentage of the time for which the system produceat rate ua in long run).
The optimal solution of this problem is thereforedetermined by minimizing the total cost of theremanufacturing processing rate and serviceableequipment inventory/backlog. Our proposed sub-optimal control policy is characterized by the pair ofthresholds (z1, z2), where (z2oz1). A problem withcost function similar to (3) (penalty on repair rate)with minimizing discounted cost function has beenshown numerically in Kenne et al. (2003) to be amultiple hedging points levels. Based on Sarifnia(1988) and Kenne et al. (2003), the repair ratecontrol policy is proposed as follows:
uðxÞ ¼
u1 if xðtÞ ¼ z1;
u2 if z2oxðtÞoz1;
u3 if xðtÞoz2:
8><>: (4)
Consequently, the remanufacturing rate is set atthe expected demand rate (u1 ¼ d) when x(t) is athedging point level z1. As x(t) drops below z1, theremanufacturing process is accelerated at rate u2.If x(t) drops below z2, the remanufacturing pro-cess is further accelerated at rate u3 to preventthe inventory level to decrease under the servicelevel SL.
3. Resolution approach
In the sphere of control theory, results fromtraditional production and maintenance planningmethods are not sufficient to reach a comfortablelevel of desired performances. Optimal solutions areoften difficult to calculate or obtained under strict
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conditions which limit the use for real cases. Toimprove these methods, we propose to combine thedescriptive capacities of conventional simulationmodels with analytical models, experimental design,and response surface methodology techniques (seeGharbi and Kenne, 2000, 2003). A synoptic schemeof the resolution approach is depicted in Fig. 3.
The structure of the proposed approach consistsof the following sequential steps:
1.
The control problem statement of the remanufac-turing system, as described in Section 2, consistsof the representation of the repair executionproblem of the remanufacturing system througha stochastic optimal control model based oncontrol theory. Hence, the problem of the optimalflow control for the remanufacturing systemconsidered is described in this first step, whichcontains a specification of the objective of thestudy. That objective is to find the controlvariables (ui) called the repair rates in order toimprove the related output (i.e., the incurred cost).2.
The structure of near optimal control policy (3) isobtained, based on the work of Akella andKumar(1986), Bielecki and Kumar (1988) andSharifnia (1988).3.
The control factors zi, i ¼ 1,y, n for repairplanning of the remanufacturing system, describethe control policy obtained.Fig. 3. Proposed reso
4.
lutio
The simulation model uses the near optimalcontrol policy defined in the previous step asinput for conducting experiments in order toevaluate the performances of the remanufactur-ing system. Hence, for given values of the controlfactors, the cost incurred is obtained from thesimulation model presented in Section 4.
5.
The experimental design approach defines howthe control factors can be varied in order todetermine the effects of the main factors andtheir interactions (i.e., analysis of variance orANOVA) on the cost through a minimal set ofsimulation experiments.6.
The response surfacemethodology is then used toobtain the relationship between the incurred costand significant main factors and interactionsgiven in the previous step. The obtained regres-sion model is then optimized in order todetermine best values of factors called here z�ifor repair.7.
The near-optimal control policy u(zj*) is thenan improved HPP to be applied to the remanu-facturing system. The application of the pro-posed control approach gives the repairrates described by Eq. (3) for best values offactors z�i .
The simulation model used in our approach isfurther described in the next section.
n approach.
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4. Simulation model
A simulation model that combines discrete–con-tinuous changes describing the dynamics of thesystem presented in Fig. 1 was developed using theVisual SLAM language (Pritsker and O’Reilly,1999). This model consists of several networks anduser routines, each of which describes a specific taskin the system (i.e., demand generation, controlpolicy, states of the system, inventory control,etc.). The diagram of the simulation model is shownin Fig. 4 with the following block descriptions:
1.
The initialization block initializes the variables(current surplus, repair rates, incurred cost, etc.).2.
The Planned demand arrival block performs thearrival of a planned demand for equipments to beoverhauled at each 1/d unit of time. Verificationis then performed on the inventory value ofequipments surplus, and the inventory or thebacklog level is updated.3.
The control policy segment block is defined in theprevious section (see Eq. (3) for the system repairrates). The control policy is defined by the outputof the FLAG block. This block is used topermanently verify the variation in the surpluslevel x(t). If x(t)4z1, then the repair rate is set toa zero value; otherwise the repair rate is either setto the demand rate (u1 ¼ d) if x(t) ¼ z1, to theFig. 4. Diagram of simul
intermediate repair rate u2 if z2px(t)oz1, or tothe maximum repair rate u3 if x(t)oz2.
4.
The remanufacturing system block performs theremanufacturing activities of equipments accord-ing the repair rate as set by the control policy.5.
The Unplanned demand arrival block performstwo functions : It defines the time betweenunplanned demand arrival ðl�112 Þ, and time totreat this unplanned demand ðl�121 Þ, using prob-ability distributions. Recall that when an un-planned demand arrives, the remanufacturingresources are pre-empted until this demand issatisfied. Planned demands are therefore notsatisfied during that period.6.
The Update Inventory block traces the variationof the surplus level as equipment get remanufac-tured or as demand arrival occurs (i.e. remanu-factured equipments either increase surplus ifthere is no backorder or it satisfies the cumulativedemand, and hence decreases the backorderlevel). In Fig. 5 , we present the equipmentsurplus trajectory for remanufacturing systemdefined by the following numerical values:d ¼ 20; u2 ¼ 25; u3 ¼ 40; l12 ¼ 4; l21 ¼ 10;z1 ¼ 15.02, z2 ¼ 4.14 and SL ¼ 0. Note that l�112and l�121 follow exponential distributions. It isinteresting to note that: (i) The surplus levelincreases to z1 and remains at this value; (ii) thesurplus level decreases during unplanned demand
ation model.
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Z1=15.02
-10
-5
0
5
10
15
20
0 20 40 60 80 100 120
Time
Inventory
Z2=4.14
SL
Fig. 5. Equipments surplus trajectory (z1 ¼ 15.02, z2 ¼ 4.14 and SL ¼ 0).
Table 1
Level of independent variables
Factor Low level Center High level
A. Gharbi et al. / Int. J. Production Economics 112 (2008) 37–47 43
treatment period; (iii) the surplus level ofequipments depends on remanufacturing rates(u1, u2 or u3 ) and unplanned demand character-istics (l12 and l21).
z1 1 10.5 20
7. A 0.05 0.5 0.95The Updates incurred cost block calculates in areal time the costs of, inventory, backlogs andrepair which depend on the remanufacturingrates (u1, u2 or u3).
5. Experimental design and response surface
methodology
In this study, we collect and analyze steady statecost data. Given that an optimal solution of thestochastic control problem described in Section 2exists and the convexity property of the costfunction, we need three levels for each factor z1and z2 to obtain a convex estimated cost function.These observations give rise to a complete 32
experimental design (Montgomery, 2001).
5.1. Numerical example
The following numerical values were used aspresented previously: d ¼ u1 ¼ 20; u2 ¼ 25;u3 ¼ 40; l12 ¼ 4; l21 ¼ 10; c+ ¼ 10; c� ¼ 100;c1 ¼ 20; c2 ¼ 40; c3 ¼ 100. We also define a newvariable A (0pAp1 and z2 ¼A*z1) such as toensure that the constraint (z2oz1) is respected.Based on off-line simulation runs, where theminimum and the maximum values of the factors
were observed, the independent variable levels werechosen as presented in Table 1.
Two replications were conducted for each combi-nation of the factors, and therefore, 18 (32� 2)simulation runs were made. To unsure that thesteady state of the cost is achieved the simulationmodel is run during 100,000 time units for eachreplication. To reduce the number of replications,we used a variance reduction technique calledcommon random numbers (Law and Kelton, 2000).We conducted some preliminary simulation experi-ments using two replications, and noticed that thevariability allows the effects to be distinguished.
5.2. Result analysis
The statistical analysis of the simulation dataconsists of the multifactor analysis of variance(ANOVA). This is done using a statistical softwareapplication (STATGRAPHICS) to provide theeffects of the two independent variables (z1, andA) on the dependant variable (incurred cost). TheANOVA table corresponding to the generated datais illustrated in Table 2. All p-values are less than
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Table 2
ANOVA table
Sum of squares d.f. Mean square F-ratio P-value
z1 38761.7 1 38761.7 129.18 0.0000
A 40436.0 1 40436.0 134.76 0.0000
z21 25505.6 1 25505.6 85.00 0.0000
z1A 7930.37 1 7930.37 26.43 0.0003
A2 94827.2 1 94827.2 30.43 0.0002
Blocks 251.119 1 251.119 0.84 0.3799
Total error 3300.61 11 300.055
Total (corr) 125315.0 17 R2¼ 0.9736
Z1
A
Cost
0 2 4 6 8 10 12 14 16 18 2000.20.40.60.81
1200
1250
1300
1350
1400
1450
1500
Fig. 6. Estimated response surface.
A. Gharbi et al. / Int. J. Production Economics 112 (2008) 37–4744
5%. Consequently, we conclude that the mainfactors z1, and A, their quadratic effects, as wellas their interactions are significant at the 0.05level. The R-squared value of 0.9736 from theANOVA table states that about 97.4% of the totalvariability is explained by the model (Montgomery,2001).
The residual analysis was used to verify theadequacy of the model. A residual versus predicted
value plot and normal probability plot were used totest the homogeneity of the variances and theresidual normality, respectively. We conclude thatthe model is satisfactory, and there is no need forthe transformation of response variables or foradditional replications for the simulation model.The second order model is then given by
Cost ¼ 1445:94� 28:2456 � z1 � 184:254 � A
þ 0:884791 � z21 þ 7:36488 � z�1Aþ 235:92 � A2:ð5Þ
The near-optimal control policy to be applied tothe manufacturing system considered is defined bythe minimum of the cost function (4) located atz�1 ¼ 15:33, A� ¼ 0:1513ðz�2 ¼ 3:31Þ, as shown inFig. 6. z�1 and z�2 define the best MHPP to beapplied to the remanufacturing system considered.A cost value of 1215.48 results from this controlpolicy. With the aforementioned optimal values ofthe independent factors or input parameters, thecost is minimized and the corresponding controlpolicy is the best approximation of the optimalcontrol one. In practice, the calculated controlpolicy would be applied to the manufacturingsystem as follows:
�
if surplus level of equipments is greater than15.33, then the repair rate is set to zero value;�
if surplus level of equipments is equal 15.33, then,then produce at the demand rate (20 pieces ofequipment/unit of time); � if surplus level of equipments is between 2.31 and15.33, then produce at the rate u2 (25 pieces ofequipment/unit of time); and
� if surplus level of equipments is less than 2.31,then produce at the maximum rate u3 (40 piecesof equipment/unit of time).
5.3. Sensitivity analysis
A set of numerical examples are considered tomeasure the sensitivity of the obtained controlpolicy with respect to inventory, backlog and repaircosts (i.e., c+, c�, c1, c2 and c3). The followingvariations, illustrated in Table 3, are explored andcompared to the basic case (experience no. 1):
�
Decreasing c+: This must result in a tendency toincrease the stock level in order to avoid furtherbacklog costs (z�2 increases)—(experience no. 2). � Increasing c+: This must result in a tendency todecrease the stock level in order to avoid furtherinventory costs (z�2 decreases)—(experience no. 3).
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Table 4
Table 3
Sensitivity analysis table for multiple hedging point policy (MHPP)
No. c+ c� c1 c2 c3 z�1 A* (z�2) C*MHPP
a Remark
1 10 100 20 40 100 15.33 0.15 (2.31) 1215.48 Basic case
2 5 100 20 40 100 15.76 0.283 (4.46) 1190.28 Z�1 increases
Z�2 increases
3 15 100 20 40 100 15.34 0.05 (0.76) 1231.76 Z�1 stable
Z�2 decreases
4 10 80 20 40 100 15.69 0.082 (1.28) 1206.06 Z�1 increases
Z�2 decreases
5 10 150 20 40 100 15.02 0.275 (4.14) 1232.41 Z�1 decreases
Z�2 increases
6 10 100 20 40 80 13.89 0.197 (2.73) 1106.42 Z�1 decreases
Z�2 increases
7 10 100 20 40 120 16.08 0.137 (2.2) 1322.71 Z�1 increases
Z�2 decreases
aThe optimal incurred cost under MHPP.
A. Gharbi et al. / Int. J. Production Economics 112 (2008) 37–47 45
�
Sensitivity analysis table for hedging point policy (HPP)No. c+ c� c1 c3 z* C*HPP
a Remark
Decreasing c�: This must result in a tendency todecrease the stock level in order to avoid furtherinventory costs (z�2 decreases)—(experience no. 4).
�1 10 100 20 100 6.40 1390.15 Basic case
2 5 100 20 100 8.31 1361.94 z* increases*
Increasing c�: This must result in a tendency toincrease the stock level in order to avoid furtherbacklog costs (z�2 increases)—(experience no. 5).
3 15 100 20 100 5.11 1410.95 z decreases*
�4 10 80 20 100 5.50 1383.83 z decreases
5 10 150 20 100 7.60 1401.78 z* increases
6 10 100 20 80 6.40 1161.44 z* stable
7 10 100 20 120 6.40 1618.86 z* stable
Decreasing c3 (repair cost): This must result in atendency to increase the stock level in order toavoid further backlog costs with less increasingof repair cost (z�2 increases)—(experience no. 6);
a
� The optimal incurred cost under HPP.Increasing c3 (repair cost): This must result in atendency to decrease the stock level in order toavoid further repair cost (z�2 decreases)—(experi-ence no. 7).
The above analysis demonstrates that the resultsobtained are coherent and that the proposedapproach is robust.
6. Comparison of MHPP to classic HPP
In this section, we will show that the controlpolicy presented in this paper (MHPP), whichconsiders two thresholds related to the two acceler-ated repair rates, gives better results than the classicone from the literature (Akella and Kumar, 1986;Bielecki and Kumar, 1988), which considers onethreshold related to a unique accelerated repair rate.
The fitting of a second-order polynomial model todescribe the relationship between the cost and thethreshold (Z) for the basic case (Table 4) is thengiven by
Cost ¼ 1441:14� 15:9448 � zþ 1:24658 � z2. (6)
We present in Table 4 the incurred optimal costsfor the same sensitivity analysis input, conductedwith the HPP. It is important to note that the resultspresented in Table 4 were obtained under the sameconditions (simulation, experimental design andRSM), and following the same approach underwhich the sensitivity analysis was conducted for theMHPP (Table 3). Through this analysis, it isinteresting to note that z* does not depend on c3,a result already mentioned in the literature. Thisresult clearly demonstrates that the model iscoherent.
If we compare the optimal cost values obtainedby MHPP and HPP (Tables 3 and 4), we notice thatin all the cases C�MHPPoC�HPP. To confirm thenumerical observation and hence the advantage ofthe proposed MHPP policy compared to that of theclassic HPP, a student test has been conducted inorder to compare the performance of the twopolicies. The confidence interval of C�HCP � C�MHCP
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Table 5
Incurred cost difference confidence interval (95%) for cases 1–7
Case 1 2 3 4 5 6 7
Lower bound 156.04 162.04 146.52 153.95 157.84 45.16 269.16
Upper bound 157.07 162.84 147.82 155.01 158.93 46.00 270.51
A. Gharbi et al. / Int. J. Production Economics 112 (2008) 37–4746
is given by
C�
HPP � C�
MHPP � ta=2;n�1 � s:e:ðC�
HPP � C�
MHPPÞ
pC�HPP � C�MHPPpC�
HPP � C�
MHPP þ ta=2;n�1�
s:e:ðC�
HPP � C�
MHPPÞ, ð7Þ
where ta=2;n�1 is the student coefficient function of n
and a, with n the number of replications (set at 10)and (1�a), the confidence level (set at 95%);
s:e:ðC�
HPP � C�
MHPPÞ ¼ SD=ffiffiffinp
; standard error,
S2D ¼ð1=ðn�1ÞÞð
Pni¼1ðC
�HPPi �C�MHPPiÞ
2� nðC
�
HPP�
C�
MHPPÞ2Þ; C
�
HCP the average optimal cost incurred
under HPP; and C�
MHCP the average optimal cost
incurred under MHPP.The two configurations under study (HPP and
MHPP) were simulated with their optimal designparameters. The incurred cost difference ðC�HPP �
C�MHPPÞ confidence interval obtained is presented inTable 5.
It has been shown that in all cases, it can beconcluded that C�HCP � C�MHCP40 at the 95%confidence level. Consequently, the MHPP givesthe lower optimal cost, and furthermore, it appearsthat the MHPP is better than the HPP, and can beused to better approximate the optimal controlpolicy of the remanufacturing system.
7. Conclusion
In this paper, we studied the multiple repair ratecontrol problem of remanufacturing systems andsolved the problem in the case one product type.Based on control theory, a near-optimal controlpolicy was proposed. This policy, called multiplehedging point policy (MHPP), is described by twothresholds related to the two accelerated repairrates. To determine the parameters of the controlpolicy, and hence, to achieve a close approximationof the optimal repair policy, an experimentalapproach based on design of experiment, simulationmodeling and response surface methodology hasbeen presented. The proposed approach shows thatthe optimal cost incurred under the developed
control policy is lower than that incurred underthe classical hedging point policy (HPP) andshould be used for the control of remanufacturingsystems.
Acknowledgment
This research was supported by Grant NSERC-121739-01, Canada.
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