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Northeastern University

Department of Mechanical and IndustrialEngineering

January 01, 2000

A kanban control mechanism for a multi-echelonInventory system with returns

Surendra M. GuptaNortheastern University

Aybek KoruganNortheastern University

This work is available open access, hos ted by Northeastern University.

Recommended CitationGupta, Surendra M. and Korugan, Aybek, "A kanban control mechanism for a multi-echelon Inventory system with returns" (2000). .Paper 6. http://hdl.handle.net/2047/d10013714

http://iris.lib.neu.edu/mech_ind_enghttp://iris.lib.neu.edu/mech_ind_enghttp://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://iris.lib.neu.edu/mech_ind_enghttp://iris.lib.neu.edu/mech_ind_eng

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Bibliographic Information

Korugan, A. and Gupta, S. M., "A Kanban Control Mechanism for a Multi-EchelonInventory System with Returns", Proceedings of the 2000 Annual Meeting of theNortheast Decision Sciences Institute, Atlantic City, New Jersey, March 22-24, pp.105-107, 2000.

Copyright Information

Copyright 2000, Surendra M. Gupta.

Contact Information

Dr. Surendra M. Gupta, P.E.

Professor of Mechanical and Industrial Engineering andDirector of Laboratory for Responsible Manufacturing334 SN, Department of MIENortheastern University360 Huntington AvenueBoston, MA 02115, U.S.A.

(617)-373-4846 Phone(617)-373-2921 Faxgupta@neu.edu e-mail address

http://www.coe.neu.edu/~smgupta/ Home Page

Laboratory for Responsible Manufacturing

L R m

http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714http://hdl.handle.net/2047/d10013714

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A KANBAN CONTROL MECHANISM FOR A MULTI-ECHELON INVENTORY SYSTEM WITH

RETURNS

Aybek Korugan, Northeastern University, Boston, MA 02115, (617)-373-7635, koru@coe.neu.edu

Surendra M. Gupta*, Northeastern University, Boston, MA 02115, (617)-373-4846, gupta@neu.edu

(*Corresponding author)

ABSTRACT

In this paper, we present a two-stage kanban controlmodel to study a two-echelon hybrid inventory systemwith disposals. We then express the expected cost func-tion for the inventory system in terms of the performancemeasures obtained from the analysis of the kanbanmodel. Finally, we present the results of a series of ex-periments that are conducted to measure the impact ofthe kanban sizes on the expected total cost of the system.

INTRODUCTION

Recent regulations aimed towards protecting the envi-ronmental resources have brought the issue of the reuseof products at the forefront of production and planning.Many companies are involved in retrieving used productswhere they repair, refurbish and upgrade the products inorder to sell them for profit. Production systems of thistype use both the returned products as well as new itemsas raw materials for the products they produce. Since tra-ditional inventory models do not take returns into ac-count, they are not suitable to address such systems.Therefore, new inventory models are needed to minimizetotal inventory costs.

Most of the models developed so far modify classicalinventory techniques with control policies such as ,s S

or ,s . In general, the return and demand processesare assumed to be mutually independent. A single eche-lon inventory system with this assumption was first mod-eled using an / / 1/ M M N queue by Heyman [4]. Theobjective was to determine the optimal keep level N thatminimized the total inventory cost. The model did notconsider any lead times. A latter paper by Muckstadt andIsaac [10] considered lead times but ignored the disposalactivity, when exploring a continuous (Q, r) policy in-ventory model. The results obtained from the single-echelon model were then applied to a multi-echelonmodel. Laan et al. [7] added the disposal option to thesingle-echelon model in [10]. Their study included acomparison between inventory policies with and without

disposal which showed that disposal is a necessary actionfor cost minimization. Korugan and Gupta considered atwo-echelon inventory system with disposals using anopen queueing network model with finite buffers [5].Laan et al. [9] showed that the pull control strategy wasmore cost effective than the push control strategy for in-ventory systems with return flows.Laan et al. [6] com-pared several inventory control policies with disposal op-tion and showed that a four-parameter control policy wasoptimal. A further study by Laan and Salomon [8] veri-fied these results while adding the disposal option to the

earlier model. A more detailed overview of such systemsis given by Salomon et al. [12] and Gungor and Gupta[3].

In this paper, we consider a two-echelon supply sys-tem that satisfies customer demands through direct salesand allows returns. The returned items are first collectedby the retailers at their recoverable item inventory (r)and then sent to the warehouse to be remanufactured. Atthe warehouse, the returned items are kept in the re-manufacturable item inventory (rm) until they areremanufactured. After the remanufacturing process, theitems are assumed to be restored into 'as good as new'condition and placed in the serviceable item inventory (s)

to satisfy the demand from customers. The return rate isassumed to be smaller than the demand rate. The differ-ence is produced at the facility.

As both demand and return rates are probabilistic,there is a chance that the recoverable item inventory ex-ceeds the predefined limit. In an effort to control such aninstance, we allow returned items to be disposed off fromthe recoverable item inventory at a fixed disposal cost perdisposed item. For this study, we assume that the dispos-als take place only at the lower echelon. In addition tothe disposals, we also allow a predefined amount of un-satisfied demand to be backordered. When this amount isexceeded, we assume that the unsatisfied demand is lost.

In the next section, we introduce a two-stage kanban

control mechanism with extra buffers for returned prod-ucts and backordered demand in order to model the in-ventory system. In the following section, we give the nec-essary performance measures of the kanban model anddefine an average total cost function with respect tothem. Following that, we measure the effect of the kan-ban sizes on the average total cost of the system for dif-ferent costs and arrival and service rates. Finally, weconclude our work by analyzing the results obtained inthe previous section.

MODEL DESCRIPTION

We consider a two-echelon inventory system with N re-

tailers and one warehouse as described above, where thedemands and returns occur independently of each otherand are Poisson distributed with rates l and

ig , (for re-

tailer i, 1, ,i N= L ), respectively. In order to simplifythe model, we assume, without loss of generality, thatthere exists only one retailer that receives returns withrate 1i ig g== . We further assume that only one itemis transported at a time and the transportation times areexponentially distributed with rate

rm . In addition, we

assume that the remanufacturing and manufacturingtimes are exponentially distributed with rates

rmm and

mm , respectively. With these assumptions, we can model

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2

the problem as a production line given in Figure 1. Here,the servers represent the transportation; remanufacturingand manufacturing activities while the buffers representthe earlier mentioned inventories. In order to analyze thismodel we employ a two-stage kanban control mechanismgiven in Figure 2. Here,

1,

2and

2are the kanban

control parameters. The demand arrives at the buffer, D,of synchronization station,

2with a Poisson arrival rate

l and is accepted when the buffer is not full. Otherwise,it is considered lost at a cost of

lc per unit of demand.

The accepted demand is queued in D for a backorder feeof

bc per unit time, until it is matched with a new or a

remanufactured product at the serviceable item inven-tory, P2. When a product is matched with a demand, thekanban card attached to the product is detached and, de-pending on its type, sent either to the kanban buffer, F2,of the synchronization station, J1, to pull a returnedproduct for remanufacturing or to node

2to trigger a the

manufacturing of a new product. Here, we assume thatthere is sufficient raw material available. A kanban re-ceived at J1 waits at the queue until it is matched with areturned product in P1 and placed in the buffer of S2 forremanufacturing. When this match takes place the kan-ban on the returned product is detached and sent to thekanban buffer, F1, of synchronization station J0, to pull areturned item from

0. When a returned item arrives at

0, it joins the queue at

0to be pulled by a kanban in

1, if

0is not full. Otherwise, it is rejected for disposal

at a cost ofd

c per unit item.

Rr

Rw

Sw

mr mrm

mm

g

l

Rr: Recoverable Item Inventory of the Retailer

Rw

: Recoverable Item Inventory of the WarehouseS

w: Serviceable Item inventory of the Warehouse

Figure 1: Two-echelon inventory problem as a production line

THE COST FUNCTION

The average total cost of this system consists of threeparts, viz. the operational costs, the holding costs and theincurred penalties. A unit operational cost incurs when aunit product is sent to the warehouse for remanufacturingat a cost of

rc , or a unit remanufactured or manufacturedproduct is sent to the serviceable inventory at a cost of

rmc or

mc , respectively. Thus, in order to find the total

expected operational cost, we need the average through-put rates

rT H of the retailer,

rmT H of the remanufac-

turing system andm

of the manufacturing system.Similarly, a unit item accumulates holding costs of

rh ,

rmh , or

sh , per unit time at the recoverable inventory, the

remanufacturable inventory or the serviceable inventory,respectively. Therefore, for the calculation of the ex-pected total inventory carrying cost, it is necessary to findthe average queue lengths

rQ of the retailer inventory,

rmQ of the remanufacturing inventory and

sQ of the

serviceable inventory. Finally, in order to find the ex-pected total penalties, we have to obtain the averagebackorders

b, the lost sales rate

land the disposal

rated

g . Then, we give the expected total cost functionas,

K1K

2

J0 J1

P0 P1 P2

F1 F2D

S1 S2

S'2

J2

K'2

Figure 2: The kanban model of the two-echelon inventory problem

[ [ ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ] ]

0

1lim

.

T

r r rm rm m mT

r r rm rm s s

b b l l d d

E T H t c T H t c T H t cT

h Q t h Q t h Q t

c Q t c c t dt l g

= + +

+ + +

+ + +

(1)

Here,

( ) ( )

( ) ( )

( )

( )

0 1

1 2

2

,

,

,

.

r

rm

s

b

Q Q P Q S

Q Q P Q S

Q Q P

Q Q D

= +

= +

=

=

Also, let B and BL give the capacities of0

P and D ,

respectively. Then,

( )

( )0

, and

.

l

d

D BL

P B

l l

g g

= R =

= R =

The performance measures of interest can be obtainedby analyzing the kanban network as a queueing networkwith synchronization mechanisms as given in [2]. To thisend, we first decompose the network into two sub-networks 1L and 2L , with station J as the output syn-chronization station of 1L and the input synchronizationstation of 2L . Then using a recursive algorithm, we ana-lyze 1L as a closed queueing network in isolation by re-garding the kanbans as customers [1]. When the algo-rithm converges, we obtain the state dependent arrival

and service rates and the steady state probabilities of eachstation in 1L . Then we pass the arrival rate of buffer 1calculated by this algorithm to a similar algorithm for

2L . Once this algorithm converges, we pass the arrivalrate of buffer 2F to the first algorithm. We repeat thesetwo iterations until the arrival rates for 1 and

2F converge. We then calculate the performance meas-ures and the expected total cost function of the systemusing the probabilities obtained from the recursive itera-tions.

The uniqueness of our problem comes from the fact

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8/2/2019 A Kanban Control Mechanism for a Multi-echelon Inventory System w

5/5

3

that there are two sources, manufacturing and remanu-facturing, that satisfy the demand. Thus, in order to re-

tain the

Table 1: Levels of the experimental variables

Levels K rmq rc rmc mc rh rmh sh bc lc dc

Low 3 0.8 0.3 2 1 1 1 1 1 1 1 2 1

Med. 5 1.4 0.6 3 2 2 3 2 2 3 2 3 2

High 10 2 0.9 4 3 3 5 3 3 5 3 4 5

balance of the network, we introduce the routing prob-ability rmq = , for, , for the output of 2 whenwe analyze

2. Let,

2be the throughput of the

closed queueing network2. Then the arrival rate of

kanbans at2is given as

2rmwhile the arrival rate

of kanbans at2

is estimated as2rm

- .

EXPERIMENTATION

In the previous section, we defined the expected totalcost, , of the system as a function of several perform-ance parameters of the kanban system. Since, all of thesemeasures can be written as functions of kanban sizes, wededuce that is a function of the kanban sizes 1K , 2Kand 2 . Here, let 2 2 2= + , then, 2 2rmq= .Thus, 1 2, .

In this section, in order to explore the behavior of thisfunction for different values of 1K and 2K under differ-ent cost and arrival stream scenarios, we designed an ex-periment, 27 , using orthogonal arrays [11]. Or-thogonal arrays enabled us to cover the entire experi-mental region with only 27 experiments for a three leveldesign of 13 independent variables as opposed to a fullfactorial design of 3 1594323= experiments. The lev-els of the independent variables are given in Table 1.Here, K represents all buffer and kanban sizes while

represents all service rates.

CONCLUSIONS

The results of the experimental design and the kanbansizes are plotted in Figure 3. Here, the experimental datais sorted with respect to kanban sizes in order to demon-strate the impact of the kanban sizes on the total costfunction. In addition, the trend of is fitted with apolynomial function of the fifth order. Thus, from these

results we can easily conclude that an increase in thekanban sizes causes an increase in the expected total cost.In order to compare this observation with other cases, thesame analysis is applied to other parameters in the ex-periment. However, none of them had an impact as sig-nificant as the kanban sizes. Thus, kanban sizes are theprimary control parameters in systems similar to the onewe examined.

REFERENCES

[1] Dallery, Y., (1990), Approximate Analysis of Open QueueingNetworks with Restricted Capacity, Performance Evaluation,vol. 11, pp. 209-222.

[2] Di Mascolo, M., Y. Frein, and Y. Dallery, (1996), An Ana-lytical Method for Performance Evaluation of Kanban Con-

trolled Production Systems, Operations Research, vol. 44,pp. 50-64.

[3] Gungor, A. and S. M. Gupta, (1999), Issues in environmen-tally conscious manufacturing and product recovery: a survey,

Computers & Industrial Engineering, vol. 36, pp. 811-853.[4] Heyman, D. P., (1977), Optimal disposal policies for a single-

item inventory system with returns, Naval Logistics Quar-

terly, vol. 24, pp. 385-405.[5] Korugan, A. and S. M. Gupta, (1998), A multi-echelon in-

ventory system with returns, Computers and Industrial Engi-

neering, vol. 35, pp. 145-148.[6] Laan van der, E. A., R. Dekker, and M. Salomon, (1996),

Product remanufacturing and disposal: A numerical compari-

son of alternative control strategies, International Journal of

Production Economics, vol. 45, pp. 489-498.[7] Laan van der, E. A., R. Dekker, M. Salomon, and A. Ridder,

(1996), An (s,Q) inventory model with remanufacturing and

disposal, International Journal of Production Economics,vol. 46-47, pp. 339-350.

[8] Laan van der, E. A. and M. Salomon, (1997), Productionplanning and inventory control with remanufacturing and dis-

posal,European Journal of Operational Research, vol. 102,pp. 264-278.

[9] Laan van der, E. A., M. Salomon, R. Dekker, and L. Wassen-hove, (1996)Production planning and inventory control in hy-brid systems with remanufacturing, Erasmus University, Rot-terdam Management Report Series 272.

[10] Muckstadt, J. A. and M. H. Isaac, (1981), An analysis of sin-

gle item inventory systems with returns, Naval Research Lo-gistics Quarterly, vol. 28, pp. 237-254.

[11] Phadke, M. S., (1989), Quality Engineering Using Robust

Design. New Jersey: PTR Prentice Hall Inc., 1989.[12] Salomon, M., E. A. Laan van der, R. Dekker, M. Thierry, and

A. A. N. Ridder, (1994)Product remanufacturing and its ef-fects on production and inventory control, Erasmus Univer-sity, Rotterdam ERASM Management Report Series 172.

0

30

60

90

1 6 11 16 21 26Experiments

ExpectedTotalC

ost

K TC Poly. (TC

Figure 3: Trend of the total cost while Kincreases.

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