Pergamon Computers ind. Engng Vol. 35, Nos 1-2, pp. 145-148, 1998

© 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain

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A Multi-Echelon Inventory System with Returns

Aybek Korugan and Surendra M. Gupta t

(t corresponding author) Laboratory for Responsible Manufacturing

334 Snell, Department of MIME Northeastern University 360 Huntington Avenue

Boston, MA 02115, U.S.A.

A B S T R A C T

This paper considers a two-echelon inventory system with return flows, where demand and return rates are mutually independent. An open queueing network with finite buffers is used to model this system. The model is analysed using the expansion methodology. © 1998 Elsevier Science Ltd. All rights reserved.

I N T R O D U C T I O N

As a consequence of environmental necessities, reuse of products has recently become a key issue for production and planning. Many companies are involved in retrieving used products where they repair, refurbish and upgrade the products in order to sell them for profit. Production systems of this type use both the returned products as well as new items as the raw material for the products they sell. Since traditional inventory models do not take returns into account, they are not adequate to address such systems. Therefore, new inventory models are needed to minimize total inventory costs.

A system with mutually independent demand and return rates was first modeled using an M/M/1/N queue by Heyman [1]. The objective was to determine the optimal keep level N that minimized the total inventory cost. The model did not consider the lead times. A latter paper by Muckstadt and Isaac [9] considered lead times but ignored the disposal activity, when exploring a continuous (Q, r) policy inventory model. The results obtained on the single-echelon model were applied to a multi-echelon model. Laan et. al. [8] added the disposal option to the single-echelon model in [9]. Their study included a comparison between inventory policies with and without disposal that showed disposal is a necessary action for cost minimization. Laan et. al. [7] compared several inventory control policies with disposal option and showed that a four- parameter control policy was optimal. Laan et. al. [6] showed that the pull control strategy was more cost effective than the push control strategy for inventory systems with return flows. A further study by Laan and Salomon [5] verified these results while adding the disposal option to the earlier model. A more detailed overview of such systems is given by Salomon et. al. [10].

In this paper, we consider a supply system that satisfies customer demands through direct sales and allows returns. The returned items are first collected by the retailers and then sent to the warehouse to be remanufactured. At the warehouse, the returned items are kept in the recoverable item inventory until they are remanufactured. After the remanufacturing process, the items are assumed to be restored into 'as good as new' condition and placed in the serviceable item inventory to satisfy the demand from customers. The return rate is assumed to be smaller than the demand rate. The difference is produced at the facility.

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146 23rd International Conference on Computers and Industrial Engineering

As both demand and return rates are probabilistic, there is a chance that the on hand in- ventories exceed the predefined limits. In an effort to control such instances, we allow returned items to be disposed off from recoverable inventory at a fixed disposal cost per disposed item. For this study, we assume that the disposals take place only at the lower echelon. In addition to the disposals, we also take lost sales into account. Other incurred costs are holding costs, remanufacturing costs, manufacturing costs and transportation costs.

M O D E L D E S C R I P T I O N

We consider a two-echelon inventory system of the type shown in Figure 1 (a). The demand and return are Poisson distributed with rates )~ and %~, (for retailer i, (i = 1 , . . . , N)), respectively. We approach the described problem by modeling it as an open queueing network with finite buffers. To this end we assume, without loss of generality, that there exists only one retailer that has 7c = ~'-~N=I ~ci as return rate. By imposing this assumption, we reduce the problem to the queueing network given in Figure l(b). Here, the buffer of node one is the recoverable item inventory of the retailer and the activity represents the transportation of returned products from the retailer to the warehouse. Hence, transportation times are represented by the service time of node one, which is exponentially distributed with rate Pt. Similarly, the buffer of node two is the recoverable item inventory of the warehouse, while the activity represents the remanufacturing process with exponential rate Prin. The third node models the demand process. The departures are the demands satisfied by the warehouse. Note that the demand cannot be satisfied when the buffer of node 3 is empty. In such a case, the demand is considered lost. Hence, #3 = A and the buffer represents the serviceable item inventory of the warehouse. The sizes of each of the buffers, one, two and three, determine the maximum number of items its corresponding inventory is allowed to hold and are represented as K1, K2 and K3, respectively. The fourth node of the network models the manufacturing of new items. We are only interested in the output of this node, which is defined as the demand rate minus the return rate.

R: Recovm'abk item i . ~ u ~ / S: Set-vteab~ Irma ~v~ttw~ DS: 131~t~al of Items

Pt It~ P3

1 2 a

node 1 : Vat.priSon of ~ o v m ~ I/7/4

n0~3: ~ p~¢~ ~ 4 : p ~ i ~ 0 f~ ~ a

(b)

Figure 1: (a) Echelons of the Problem, (b) The queueing network model

The performance parameters of the inventory control problem are approximated using the parameters of the queueing network. The recoverable item inventory of the retailer Rr(t) is given by the number of items at node 1, that is Rr(t) = Li(t). Similarly, the recoverable item inventory of the warehouse is represented by node 2, Rw(t) = L2(t). The on-hand inventory of the serviceable items at node 3 is O(t) = L3(t). The procurement of new products, P( t ) , is defined by the throughput of node 4 as TH4(t). Also, as we consider transportation and remanufacturing costs, the outputs of transportation process, Zt(t), and remanufacturing process, Zrm(t), are approximated by the throughputs THz (t) and TH2(t) of nodes 1 and 2 respectively. Other processes observed for their impact on the total cost are disposal, DS(t), and lost sales, S(t). Disposal occurs when the recoverable item inventory, the buffer of node 1, is full. That amounts to the rejected portion of the arrival process A(t). Similarly, the lost sales is given by

23rd International Conference on Computers and Industrial Engineering 147

the unfulfilled portion of the demand process D(t), which occurs when the buffer of node three is zero.

Let, hi, (i = 1, 2, 3), l, ct, am, Crm and d be holding, lost sales, transportation, manufacturing, remanufacturing and disposal costs respectively. Then the total expected inventory cost is: 1/T

C = lira [hlRr(t)+h2Rw(t)+h30(t)+dDS(t)+S(t)l+Zt(t)ct+Zrm(t)Crm+P(t)cm]dt (1) T--+oo

Using the queueing network model, C can be approximated as follows: 1// C - - T~oolim ~ [hiLl(t) + h2L2(t) + h3L+(t) + dA(t)pl,K1 + ID(t)p3,0

+THl(t)ct + TH2(t)Crm + TH4(t)cm]dt

Note that Pi,j is the probability that the buffer size at node i is equal to j.

(2)

ANALYSIS A N D E X P E R I M E N T A T I O N

We use the 'expansion methodology' [2, 3, 4] to analyse the queueing network model. In order to observe the effect of the system parameters on the cost function, we developed a computer code that calculates the total expected cost for all the points in a predetermined subspace of {K1, K2, K3}. With the help of this code we can determine the local optimal values {K~, K~, K~} and C*. We have conducted several experiments to test the sensitivity of the optimal values to various system parameters. The experiments and their results are given in Table 1. Note that, while constructing the experimental data, we assumed that ct + Crm < Cm + d [1].

The sensitivity analysis is conducted relative to the first experiment. First the effect of high lost sales cost is analyzed (exp. 2). Then the changes in manufacturing, remanufacturing and transportation costs are tested (exp. 3, 4). In experiment 5 the system is observed for arbitrarily high disposal costs. The next two experiments are conducted to create a bias towards a chosen inventory (exp. 6, 7). Finally, the effects of higher return and demand rates on the total cost function are tested (exp. 8, 9, 10, 11).

(a) {1,1,1}, C=15.51

) )

30

2! 2

(b) {1, 2, 3}, C = 26.49 (c) {9, 1, 3}, C = 27.70 (d) {1, 1, 3}, C = 48.71

Figure 2: Cost function graphs with local optimum points {K~, K~, K~} and C* for four typical cases; (a) Experiment 1, (b) Experiment 2, (c) Experiment 5, (d) Experiment 11

R E S U L T S A N D C O N C L U S I O N S

The results of the experiments show that in the predefined subspace, for most cases, the cost function is monotonically increasing for the inventory at retailer and for the recoverable item inventory of the warehouse. This observation is also valid for the serviceable item inventory of the warehouse in five of the 11 experiments including the first experiment depicted in Fig. 2(a). The rest of the experiments had concave cost functions for K3 and the local minimum value, KJ, was obtained for each experiment (Table 1). In experiment 2, the response of the system for a large lost sales cost (see Fig. 2(b)) was tested and the system tended to increase both

148 23rd International Conference on Computers and Industrial Engineering

exp hi h2 ha ct cr,n cm d l lit prm pS 7c K~ K~ K~ C* 1 2 2 4 2 3 6 3 4 1 1.25 1.5 0.5 1 1 1 15.51 2 2 2 4 2 3 6 3 30 1 1.25 1.5 0.5 1 2 3 26.49 3 2 2 4 5 5 15 5 10 1 1.25 1.5 0.5 1 1 2 31.39 4 2 2 4 5 7 10 1 5 1 1.25 1.5 0.5 1 1 1 23.83 5 2 2 4 2 3 6 50 28 1 1.25 1.5 0.5 9 1 3 27.70 6 10 2 4 2 3 6 3 4 1 1.25 1.5 0.5 1 1 1 18.18 7 2 5 8 2 3 6 3 4 1 1.25 1.5 0.5 1 1 1 18.42 8 2 2 4 2 3 6 3 4 1.25 1.35 1.5 1 1 1 1 14.95 9 2 2 4 2 3 6 3 4 1.25 1.35 1.5 1.2 1 1 1 14.62 10 2 2 4 2 3 6 3 4 1.25 1.35 3 1 1 1 2 26.96 11 2 2 4 2 3 6 3 4 1.25 1.35 6 1.2 1 1 3 48.71

Table 1: Various experiments and their results

recoverable and serviceable inventory levels of the warehouse. In experiment 5, where the impact of an arbitrarily large disposal cost and a correspondingly large lost sales cost were measured, the cost was asymptotical to the retailer inventory within the observed subspace (see Fig. 2(c)).

Moderate changes in holding costs did not effect the decision parameters even though they caused fluctuations in the total cost. A significant increase in manufacturing costs forced a slight increase in the serviceable inventory, while a low disposal cost encouraged the system to hold the minimum possible inventory. When we tested the effect of high demands, the serviceable inventory tended to increase in order to dampen the impact of lost sales. We also observed that higher return rates resulted in lower holding costs. Obviously, this means that remanufacturing of used products may have a positive effect in overall cost reductions for production systems.

R E F E R E N C E S

[1] D. P. Heyman. Optimal disposal policies for a single-item inventory system with returns. Naval Logistics Quarterly, 24:385-405, 1977.

[2] A. Kavusturucu and Gupta S. M. Manufacturing systems with machine vacations, arbitrary topology and finite buffers, to appear in International Journal of Production Economics, 1998.

[3] A. Kavusturucu and Gupta S. M. Tandem manufacturing systems with machine vacations. to appear in Production Planning and Control, 1998.

[4] L. Kerbache and Smith J. MacGregor. Asymptotic behavior of the expansion method for open finite queueing networks. Computers and Operations Research, 15(2):157-169, 1988.

[5] E. A. Laan van der and Salomon M. Production planning and inventory control with reman- ufacturing and disposal. Technical Report, Erasmus University, Rotterdam, 1997.

[6] E. A. Laan van der, Salomon M., Dekker R., and Wassenhove L. Production planning and inventory control in hybrid systems with remanufacturing. Technical Report, Erasmus University, Rotterdam, 1996.

[7] E. A. Laan van der, Dekker R., and Salomon M. Product remanufacturing and disposal: A numerical comparison of alternative control strategies. International Journal of Production Economics, 45:489-498, 1996.

[8] E. A. Laan van der, Dekker R., Salomon M., and Ridder A. An (s, Q) inventory model with remanufacturing and disposal. International Journal of Production Economics, 46-47:339- 350, 1996.

[9] J. A. Muckstadt and Isaac M. H. An analysis of single item inventory systems with returns. Naval Research Logistics Quarterly, 28:385-405, 1981.

[10] M. Salomon, Laan van der E. A., Dekker R., Thierry M., and Ridder A. A. N. Product re- manufacturing and its effects on production and inventory control. Technical Report ERASM Management Report Series, 172, Erasmus University, Rotterdam, 1994.