inventory control for a markovian remanufacturing system with stochastic decomposition process
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Int. J. Production Economics 108 (2007) 416–425
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Inventory control for a MARKOVIAN remanufacturing systemwith stochastic decomposition process
Katsuhiko Takahashi�, Katsumi Morikawa, Myreshka, Daisuke Takeda,Akihiko Mizuno
Department of Artificial Complex Systems Engineering, Graduate School of Engineering,
Hiroshima University, 1-4-1, Kagamiyama, Higashi-Hiroshima 739-8527, Japan
Available online 25 January 2007
Abstract
In this paper, we consider a remanufacturing system with reproduction and disposal. In the system, a decomposition
process where recovered products are decomposed and classified into wastes to be disposed and materials and parts to be
used in the processes for producing parts and products, is considered. For the remanufacturing system, two control policies
are proposed. The performance of the proposed policies is analyzed by Markov analysis.
r 2007 Elsevier B.V. All rights reserved.
Keywords: Remanufacturing system; Stochastic decomposition; Inventory controls; Markov analysis
1. Introduction
Recently, remanufacturing systems that recycleused resources have been popular for saving limitedresources. For the remanufacturing systems, researchhas been done from various view points, andFleischmann et al. (1997) reviewed the literature oninventory control and production planning in reverselogistics and described the problems. In the literatureon remanufacturing systems, Inderfurth and van delLaan (2001) dealt with the model as shown in Fig. 1.In the model, demands from customers can besatisfied not only by brand-new products but alsorecovered products. But recovered products had onlytwo possibilities, one of that was disposed, and
e front matter r 2007 Elsevier B.V. All rights reserved
e.2006.12.023
ng author. Tel.:+8182 424 7705;
024.
ss: [email protected] (K. Takahashi).
another was stocked with dedicated inventory. Uenoet al. (2000), Kiesmuller and van der Laan (2001), andMahadeven et al. (2003) used a similar model in theirresearch. Ueno et al. (2000) proposed a control policybased on pull control, and Mahadeven et al. (2003)proposed push inventory policies for the remanufac-turing system. Kiesmuller and van der Laan (2001)considered dependent product demands and returnsin the remanufacturing system. On the other hand,Kleber et al. (2002) dealt with a different model ofremanufacturing system as shown in Fig. 2. Also,Inderfurth (2004) studied inventory control systemsfor a product recovery system. As shown in Fig. 2, therecovered products are stocked only for reusing themfor multiple options. In addition, demands for eachproduct are satisfied by production of new items orby remanufacturing returned products. However, inthe paper, the process of decomposing the recoveredproducts was not considered.
.
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Stock ofpart
Manufacturing Stock ofproduct
RemanufacturingRecover
Dispose
Fig. 1. The model of remanufacturing system shown by Inderfurth and van der Laan (2001).
Recoverable
Dispose
Recover
RemanufacturingPart
ManufacturingPart
Demand
Demand
Demand
Fig. 2. The model of remanufacturing system shown by Kleber et al. (2002).
K. Takahashi et al. / Int. J. Production Economics 108 (2007) 416–425 417
Based on this background, this paper considers adecomposition process as shown in Fig. 3 whererecovered products are decomposed and classifiedinto wastes to be disposed and materials and partsto be used in the processes for producing parts andproducts, respectively. Usually, recovered productsand the decomposition process are uncertain, andthis paper considers the decomposition process as astochastic process. In the stochastic decompositionprocess, recovered products are decomposed, andwhether the decomposed items are used as materialand/or part or should be disposed is determinedstochastically. Then, a demand of the productproduced by using used and new materials, usedand new parts is supposed to be the same in thispaper. For the remanufacturing system, this paperproposes policies for controlling inventories ofparts, produced products, and recovered products.The performance of the proposed policies isanalyzed by Markov analysis, and the optimal
policy is obtained. Then, the characteristics of theoptimal policy are investigated.
This paper is organized as follows: after theintroduction, the remanufacturing system consid-ered in this paper is defined and the control policiesare proposed in Section 2. For the system, aMarkov chain model is developed and the flowbalance equations of the system are formulated inSection 3. After solving the flow balance equationsand calculating the performance measures, theperformance of the proposed policies is analyzedunder various conditions in Section 4. Finally, thefindings obtained in this paper are summarized asconcluding remarks in Section 5.
2. Production system
In this section, the assumptions of the remanu-facturing system considered in this paper are
ARTICLE IN PRESS
Stock ofproduct
Producingproduct
Stock ofpart
Producingpart
RecoverDecomposition
Disposal
Stock ofmaterial
Demand
Purchasematerials
Fig. 3. The model of remanufacturing system in this paper.
K. Takahashi et al. / Int. J. Production Economics 108 (2007) 416–425418
defined, and the control policies for the remanu-facturing system are proposed.
2.1. Assumptions
In this paper, the following assumptions aredefined:
�
A single standard product that can be stocked isproduced. � The production system consists of three pro-cesses: producing products from parts, producingparts from materials, and decomposing recoveredproducts.
� A recovered product is decomposed into materi-als, parts, and others at the decompositionprocess.
� By decomposing a recovered product, only apart, only a material, or a part and a material areobtained stochastically.
� The arrived demand is lost when there is no stockof product.
� The parts obtained from recovered products arestored as the same parts as produced from thenew materials.
� The materials obtained from recovered productsare stored as the same materials as purchased.
� Demands, recoveries, productions of products,and productions of parts occur according to aPoisson process with rate ld, lr, Mp, and Mpa,respectively.
�
The decomposed products are disposed with rateldis, and they are reused as parts, materials, orboth parts and materials with rates lpa, lm, orlpam (lr ¼ ldis+lpa +lm+lpam), respectively. � Materials are purchased without the lead time forpurchase when the stock of parts runs out, andthe lot size is assumed as e.
2.2. Control policies
In this paper, the following policies are proposedfor the remanufacturing system.
2.2.1. Policy 1
�
Producing products: products are producedunless the stock of products reaches the upperlimit g, or the stock of parts runs out. � Producing parts: parts are produced unless thestock of parts reaches the upper limit b or thestock of materials runs out.
� Purchasing materials: materials are purchasedwith the lot size e when the stock of materialsruns out.
� Disposing materials and/or parts: decomposedmaterials and/or parts are disposed when thestock of materials or the stock of parts reachesthe upper limit, a or b, respectively.
In this policy, it can be expected that the numberof disposals increases because independent parts
ARTICLE IN PRESSK. Takahashi et al. / Int. J. Production Economics 108 (2007) 416–425 419
production and decomposition lead to an increasein the stock of parts, and the stock is apt to reachthe upper limit. Then, we propose policy 2 forimproving the control of parts production by usinga threshold.
2.2.2. Policy 2
In this policy, controls of producing products,purchasing materials, and disposing materials and/or parts are the same as in policy 1. The onlydifference is control of producing parts as follows:
�
MaterialInventory
(α,0,0)
(α,0,γ) (α,δ,γ) (α,β,γ)
(α,δ,0) (α,β,0)
Producing parts: parts are produced unless thestock of parts reaches the threshold d or the stockof materials runs out.
By introducing a threshold different from and lessthan the upper limit to dispose parts, the productionof parts stops before the number of stocked partsreaches the upper limit and the number of stockedparts can be suppressed.
ProductInventory
PartInventory
Produce parts Don’t Produce parts
(1,0,0) (1,δ,0) (1,β,0)
(1,0,γ) (1,δ,γ) (1,β,γ)
Fig. 4. Markov chain model of the proposed policies.
3. Markov analysis
The performance of the proposed policies for theremanufacturing system is analyzed by Markovanalysis in this section. A Markov chain model isdeveloped, the flow balance equations are formu-lated, and the performance measures are presented.
(1,0,γ) (1,1,γ)
(2,1,γ)
(ε,1,γ)
�d
Mpa
�pam
Mp
�pa
(1,1,γ−
(2,0,γ)
(1,0,γ −1)
Mpa
�m
Fig. 5. Part of Markov chain m
3.1. Markov chain model
Let x be the number of stocked materials, y thenumber of stocked parts, and z the number ofstocked products. Then, the state of the productionsystem considered can be expressed as (x, y, z), andall of the transitions from or to each state can beexpressed as a continuous time Markov chain bydefining Poisson arrivals of demand and recovery,and an exponential processing time at each process.In the proposed control policies, the production ofmaterials, parts, and products stop when thenumbers of stocked materials, parts, and productsreach the upper limits, a, b, and g, respectively.
Demand Arrival
PurchasingMaterials
Obtaining a Materialfrom Recovered Products
Obtaining a Partfrom Recovered Products
Obtaining a Part andMaterial fromRecovered Products
Producing a Product
1)
Producing a Part
odel around state (1, 0, g).
ARTICLE IN PRESSK. Takahashi et al. / Int. J. Production Economics 108 (2007) 416–425420
Also, the numbers of stocked items are not negative.Thus 0oxpa, 0pypb, 0pzpg. Fig. 4 shows theMarkov chain model of the proposed policies, andFig. 5 shows a part of the Markov chain modelaround state (1, 0, g).
When a demand arrives at production system andthe stock of products is not zero, the arriveddemand is satisfied by the stocked product, andthe number of stocked products decreases by oneunit. And state (x, y, z) changes to (x, y, z�1) by thedemand arrival of rate ld at this time. When thestock of parts is not zero and the stock of productsis less than g, a part is produced. When a unit ofproduct is produced with rate Mp, the number ofstocked parts decreases one unit and the number ofstocked products increases one unit, simultaneously.Then, at this time, state (x, y, z) changes to (x, y�1,z) with rate Mp. When the stock of materials isnot zero and the stock of parts is less than d, apart is produced. When a unit of part is producedwith rate Mpa, the number of stocked materialsdecreases one unit and the number of stockedparts increases one unit. Therefore, at this time,state (x, y, z) changes to (x�1, y, z) with rate Mpa.If a part is produced when the stock of materialsis equal to 1, the stock of materials runs out,and materials are purchased at the same time.Then, when a unit of part is produced with rateMpa, the number of stocked materials decreasesone unit, then it will be e at the moment, andthe number of stocked parts increases one unit.Therefore, at this time, state (1, y, z) changes to(e, y+1, z) with rate Mpa. When a unit of productis recovered, it is decomposed to a part withrate lpa, a material with rate lm, a part and amaterial with rate lpam, or disposed with rateldis. Therefore, at this time, state (x, y, z) changesto (x, y+1, z) with rate lpa, (x, y, z+1) with ratelm, (x, y+1, z+1) with rate lpam, or state doesnot change with rate ldis. In policy 1, there is nothreshold for stopping the production of parts, andit means that d in policy 2 is equal to b.
3.2. Steady-state flow balance equations
Let Px,y,z be the steady-state probability of state(x, y, z). Each transition of producing products,producing parts, purchasing materials, decompos-ing recovered products, and disposing materialsand/or parts is the same regardless of policies orstates. Therefore, the steady-state flow balanceequations for probability Px,y,z can be formulated
as follows:
I1Mp þ I2Mpa þ I3ld þ I4lm þ I5lpa þ I6lpam�þI13Mpa
�Px;y;z ¼ I7MpPx;yþ1;z�1
þ I8MpaPxþ1;y�1;z þ I9ldPx;y;zþ1 þ I10lmPx�1;y;z
þ I11lpaPx;y�1;z þ I12lpamPx�1;y�1;z
þ I14MpaP1;y�1;z. ð1Þ
In the equations, Ii (i ¼ 1, 2, y, 14) are thevariables which take only the values of 0 or 1, andthe values differ with policies as follows:
3.2.1. Policy 1
Producing products:
I1 ¼1 ðy40 and zogÞ
0 ðotherwiseÞ
(I7 ¼
1 ðyob and z40Þ
0 ðotherwiseÞ
(
Producing parts:
I2 ¼1 ðx41 and yobÞ
0 ðotherwiseÞ
(I8 ¼
1 ðxoa and y40Þ
0 ðotherwiseÞ
(
Demand arrival:
I3 ¼1 ðz40Þ
0 ðotherwiseÞ
(I9 ¼
1 ðzogÞ
0 ðotherwiseÞ
(
Obtaining materials from recovered products:
I4 ¼1 ðxoaÞ
0 ðotherwiseÞ
(I10 ¼
1 ðx41Þ
0 ðotherwiseÞ
(
Obtaining parts from recovered products:
I5 ¼1 ðyobÞ
0 ðotherwiseÞ
(I11 ¼
1 ðy40Þ
0 ðotherwiseÞ
(
Obtaining parts and materials from recoveredproducts:
I6 ¼1 ðxoa and yobÞ
0 ðotherwiseÞ
(I12 ¼
1 ðx40 and y40Þ
0 ðotherwiseÞ
(
Purchasing materials:
I13 ¼1 ðx ¼ 1 and yobÞ
0 ðotherwiseÞI14 ¼
1 ðx ¼ � and y40Þ
0 ðotherwiseÞ
((
3.2.2. Policy 2
As defined before, only the difference betweenpolicies 1 and 2 is control of producing parts, andthe values related to the control in policy 2 are
ARTICLE IN PRESSK. Takahashi et al. / Int. J. Production Economics 108 (2007) 416–425 421
as follows:Producing parts:
I2 ¼1 ðx41 and yodÞ
0 ðotherwizeÞ
(I8 ¼
1 ðxoa and 0oypdÞ
0 ðotherwiseÞ
(
In addition, the sum of the steady-state prob-ability of all states is 1, and the following balanceequation is added to the equations:
Xax¼1
Xby¼0
Xgz¼0
Px;y;z ¼ 1. (2)
As the Markov chain model formulated as shownin Figs. 4 and 5 is ergodic, that is, the model isirreducible and all the states are positive regenera-tive, the steady-state probability of each state canbe obtained by solving the simultaneous equations(for more details, see Bolch et al., 1998), and theprobability is used for evaluating the performanceof the proposed policies.
3.3. Performance measures
As performance measures, the following expecta-tions are calculated from the steady-state probabil-ity:
Expected number of stocked materials:
Ex ¼Xax¼1
Xby¼0
Xgz¼0
xPx;y;z. (3)
Expected number of stocked parts:
Ey ¼Xax¼1
Xby¼0
Xgz¼0
yPx;y;z. (4)
Expected number of stocked products:
Ez ¼Xax¼1
Xby¼0
Xgz¼0
zPx;y;z. (5)
When the stock of products runs out, the arriveddemand is lost, and the expected number ofshortages is calculated from the product sum ofthe demand arrival rate and the steady-stateprobability at z ¼ 0.
Expected number of shortages:
Er ¼ ldXax¼1
Xby¼0
Px;y;0. (6)
When the number of stocked materials is at theupper limit a or the number of stocked parts is at
the upper limit b, the materials and/or partsobtained from the recovered products are disposed.Also, the recovered products are disposed withdisposal rate ldis. Then, the expected number ofdisposals is calculated from the product sum ofthe recovery rate of products and the steady-state probabilities at x ¼ a and y ¼ b and thedisposal rate.
Expected number of disposals:
Ed ¼ lrXax¼1
Xgz¼0
Px;b;zþ
( Xby¼0
Xgz¼0
Pa;y;z
)þ ldis. (7)
The expected number of part productions, orproduct productions is calculated from the partsproduction rate or products production rate and thesteady-state probability as follows:
Expected number of part productions:
Epa ¼Mpa
Xax¼1
Xd�1y¼0
Xgz¼0
Px;y;z. (8)
Expected number of product productions:
Ep ¼Mp
Xax¼1
Xby¼1
Xgz¼0
Px;y;z. (9)
The expected number of purchased materials iscalculated from the product sum of the partsproduction rate, the lot size of purchasing materials,and the steady-state probability at x ¼ 1.
Expected number of purchased materials:
Eb ¼ �Mpa
Xby¼0
Xlz¼0
P1;y;z. (10)
Then, the expected total cost per time unit (in steadystate) is calculated as follows:
C ¼ hxEx þ hyEy þ hzEz þ crEr þ cdEd
þ cpaEpa þ cpEp þ hbEb. ð11Þ
Here, hx, hy, hz, cr, cd, cpa, cp, and cb are unit costsfor carrying the stock of materials, the stock ofparts, the stock of products, running out of pro-ducts, disposing recovered products, producingparts, producing products, and purchasing mate-rials, respectively.
4. Numerical calculations
In this section, the performance of the proposedcontrol policies is calculated numerically undervarious conditions.
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4.1. Effect of Rates
As a result of numerical calculations, Fig. 6 showsthe effects of each rate upon the expected total costand each cost in both control policies. Also, in thefigure, the optimal upper limits and the optimalthreshold are shown. Under the calculations, thecost parameters are fixed as hx ¼ 1, hy ¼ 1, hz ¼ 1,cr ¼ 20, cd ¼ 20, cp ¼ 10, cpa ¼ 5, cb ¼ 5, and the
Policy 1
96
6
4
2
Policy 1
9
6
9
4
3
52
Demand
Production rate
4 6
160
140
120
100
80
60
40
20
0
160
180
140
120
100
80
60
40
20
0
Material Purchase
Material Inventory
Part Production
Part Inventory
Product Production
Product Inventory
Disposal
Shortage
Material Purchase
Material Inventory
Part Production
Part Inventory
Product Production
Product Inventory
Disposal
Shortage
α∗
β∗
γ∗
ε∗
δ∗
α∗
β∗
γ∗
ε∗
δ∗
9 96 6
6 8
4 4
2 1 2
6 9
6 6
9 9
3 4
2 1 2
Fig. 6. Effects of each rate upon the expected tota
rates ld, lr, Mp, and Mpa are varied as shown inFig. 6 from the basic setting as ld ¼ 6, lr ¼ 6,Mp ¼ 5 and Mpa ¼ 4, where lpa ¼ 2, lm ¼ 2,lpam ¼ 1, ldis ¼ 1 (lr ¼ 6), lpa ¼ 2, lm ¼ 2,lpam ¼ 2, ldis ¼ 1 (lr ¼ 7), and lpa ¼ 2, lm ¼ 2,lpam ¼ 2, ldis ¼ 2 (lr ¼ 8).
In Fig. 6, it can be seen that the expected totalcost of policy 2 is less than that of policy 1 althoughin some cases the total cost and the optimal upper
1 2 2
9 9 96 8 9
9 9 9
4 4 4
1 2 2
9 6 7
6 6 6
9 9 9
4 3 3
418
rate Recover rate
Production rate of partof product
88 6 7
9 9 6 9 9 96 6 6 6 6 6
9 9 9 9 9 9
4 4 3 4 4 4
2 2 3 3 4
7
1 2 1 2 1
6 9 7 9 6 9
6 6 6 6 6 6
9
4
9 9 9 9 9
3 4 3 4 3
3 2 3 3 2
1 2 1 2 1
l cost and each cost in both control policies.
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limits have not so much difference between controlpolicies. Based on the results, we can claim thatpolicy 2 is superior to policy 1. This is obviousbecause policy 1 is contained in policy 2, and theoptimal total cost of policy 1 can never be lowerthan that of policy 2. As the effects of each rate, thefollowings can be seen in the figure. As the demandrate increases, the expected total cost in bothpolicies increases since the increased demand rateleads to an increase in producing products and partsand running out of products. However, in policy 2,the increase in running out of products can besuppressed much more than that in policy 1.Although the disposal cost increases by controllingproduction of products in policy 2, the productioncosts of parts and products can be suppressed,and the total cost in policy 2 is lower than that inpolicy 1.
As the recovery rate increases, the expected totalcost in both policies increases since more and morematerials and parts come to be obtained and theproduction of parts and products increases. Inpolicy 2, since parts production is controlled byintroducing the threshold d, the increase in partsproduction and disposal can be suppressed, and theexpected total cost becomes much less than that inpolicy 1.
As the production rate of products increases, thespeed to produce products increases and theexpected number of shortages decreases. Then,the expected total cost decreases in policies 1 and2. In policy 2, the production costs of parts andproducts are suppressed by introducing the thresh-old d, and the expected total cost becomes lowerthan that in policy 1.
As the production rate of parts increases,the expected total cost decreases in policies 1 and2 because the expected shortage decreases. Inpolicy 2, the production costs of parts and pro-ducts are suppressed by introducing the threshold d,and the expected total cost becomes lower thanthat in policy 1. The difference increases as theproduction rate of parts increases and the lowerstock of parts leads to a decrease in the produc-tion cost of parts without increasing the shortagecost.
4.2. Effect of cost parameters
Next, as another result of numerical calculations,Fig. 7 shows the effects of each cost parameter uponthe total cost and each cost in both control policies.
Also, in the figure, the optimal upper limits and theoptimal threshold are shown. Under the calcula-tions, the rates and the following cost parametersare fixed as Mp ¼ 6, Mpa ¼ 6, ld ¼ 7, and lr ¼ 7(lpa ¼ 2, lm ¼ 2, lpam ¼ 2, ldis ¼ 1), hx ¼ 1 hy ¼ 1,hz ¼ 1, and cb ¼ 10. The cost parameters cp, cpa, cr,and cd are varied as shown in Fig. 7 from the basicsetting as cp ¼ 5, cpa ¼ 5, cr ¼ 6, and cd ¼ 7.
In Fig. 7 as in Fig. 6, it can be seen that theexpected total cost of policy 2 is less than that ofpolicy 1, although in some cases the total cost andthe optimal upper limits have not so muchdifference between control policies. Based on theresults, we can claim that policy 2 is superior topolicy 1. As the effects of each cost parameter, thefollowing can be seen in the figure: as the unitproduction cost for products increases, the expectedtotal cost increases in policies 1 and 2. In policy 2,the expected cost for products production can besuppressed much more than that in policy 1 inincreasing the unit production cost of products, andthe difference of the expected total cost betweencontrol policies increases as the unit production costincreases.
As the unit production cost for parts increases,the expected total cost increases in policies 1 and 2.In policy 2, the increase in the expected cost forparts production can be suppressed much more thanthat in policy 1 in increasing the unit productioncost of parts, and the difference of the expected totalcost between control policies increases as the unitproduction cost increases.
As the unit disposal cost increases, the expectedtotal cost increases in policies 1 and 2 because theexpected cost for disposal increases according to theincrease in the unit cost. The main difference ofthe expected total cost between control policies isthe expected cost for materials purchasing. As thedisposal of materials can be suppressed, theexpected cost for material purchasing in policy 2 isless than that in policy 1. However, even if the unitdisposal cost increases, the difference of theexpected total cost between control policies doesnot increase so much.
As the unit shortage cost increases, the expectedtotal cost increases in policies 1 and 2 because theexpected shortage cost increases according to theincrease in the unit cost. The contents ofthe expected total cost differ between controlpolicies. The expected costs for materials purcha-sing, products production, and carrying products inpolicy 1 are much more than those in policy 2, and
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2 5 8 2 5 8
9 9 7 66 6 6 6
9 9 9 9
4 4 3 3
4
9 9 6 9 6 9 9 96 6 6 6 6 6 6 6
9 7 9 6 9 9 9 9
4 4 3 4 3 4 4 4
2 2 4 3 2
9 9 6 66 6 6 6
9 9 9 9
4 4 3 3
2
6 9 9 9 9 9 9 96 6 6 6 6 6 6 6
9 9 9 9 9 9 9 9
3 4 4 4 4 4 4 4
3 3 2 2 4
α∗
β∗
γ∗
ε∗
δ∗
α∗
β∗
γ∗
ε∗
δ∗
Material Purchase
Material Inventory
Part Production
Part Inventory
Product Production
Product Inventory
Disposal
Shortage
Material Purchase
Material Inventory
Part Production
Part Inventory
Product Production
Product Inventory
Disposal
Shortage
240
200
160
120
80
40
0
Policy
Policy
1 1 2 2
4 86 8 6 7
2 1 2 1 2 1 2 1
1 1 2 22 1 2 1 2 1 2 1
Production cost of partProduction cost of product
Disposal cost Shortage cost
160
180
140
120
100
80
60
40
20
0
Fig. 7. Effects of cost parameters upon the expected total cost and each cost in both control policies.
K. Takahashi et al. / Int. J. Production Economics 108 (2007) 416–425424
the difference does not increase even if the unitshortage cost increases. However, the expected costfor disposal in policy 1 is much less than that inpolicy 2, and the difference increases as the unitshortage cost increases and the threshold d forpreventing the shortage of products increases.
5. Concluding remarks
In this paper, we considered the remanufacturingsystem which controls stocks by three productionprocesses: producing products, producing parts,and decomposing recovered products. For the
ARTICLE IN PRESSK. Takahashi et al. / Int. J. Production Economics 108 (2007) 416–425 425
remanufacturing systems, two control policies wereproposed, and the performance of the proposedpolicies was analyzed by Markov analysis. Althoughit is based on numerical calculations without proof,the analysis clarified the effects of rates and costparameters upon the performance of the proposedpolicies. Especially in policy 2, the number ofproduced parts and the number of disposed partsare reduced due to the introduced threshold d. As aresult, the expected total cost in policy 2 becomesmuch less than that in policy 1 although in some casesthe differences are not so big.
In this paper, the setup of production was notconsidered. In policy 2, the number of stocked partsis suppressed much less than that in policy 1 by theintroduced threshold d, and this may lead to anincrease in the number of stops and starts of partsproduction, and the number of setups. Consideringthe setup in evaluating control policies and propos-ing control policies that suppress the number ofsetups can be suggested as future research work.
Acknowledgments
The work in this paper was partially supported bythe Grant-in-Aid for scientific research of theJapanese Ministry of Education, Science, Sportsand Culture in 2002–2005.
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