Model Multi Echelon

8/2/2019 Model Multi Echelon

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Int J Adv Manuf Technol (2009) 41:12081220

DOI 10.1007/s00170-008-1567-5

ORIGINAL ARTICLE

A two-level distribution inventory system with stochasticlead time at the lower echelon

A. Thangam R. Uthayakumar

Received: 28 August 2007 / Accepted: 12 May 2008 / Published online: 14 June 2008 Springer-Verlag London Limited 2008

Abstract We consider a two-level distribution inven-

tory system with a number of identical retailers atthe lower echelon and a single supplier at the upper

echelon. The replenishment policy is continuous review

policy (R, Q) at all installations. We assume indepen-

dent Poisson demands with stochastic lead time for the

retailers and a constant transportation time for replen-

ishing supplier orders from an external warehouse. Un-

satisfied demands are assumed to be lost in the retailers

and unsatisfied retailer orders are backordered in the

supplier. We develop an approximate cost function to

find optimal reorder points for given batch sizes in

all installations, and the related accuracy is assessed

through simulation. We present numerical examples forthe gamma distributed lead time for the retailers orders

at the supplier.

Keywords Multi-echelon Stochastic lead time

Poisson demand Continuous review policy

1 Introduction

Large, multi-echelon inventory systems usually con-

sist of hundreds of thousands of stock keeping units(SKUs). These SKUs can be classified into two main

categories: consumables and repairables. Multi-echelon

A. Thangam (B) R. UthayakumarDepartment of Mathematics,Gandhigram Rural University, Dindigul 624 302,Tamil Nadu, Indiae-mail: [email protected]

inventory systems are important to large corporations

and to the military to support their operations.In large supply networks like Wal-Mart and the

US-Navy, thousands of SKUs are stocked at different

inventory holding points (IHPs). These holding points

might be at different echelons, where the higher ech-

elons supply the lower echelons. Each of these IHPs

might follow different stocking policies resulting in

decentralized control of the supply network. This case

is most likely to occur when each of the locations that

constitute the supply network are owned by different

owners who are not willing to give control of their in-

ventories to external parties. Under this case, each loca-

tion might not take into consideration interactions withthe other locations that might have a significant effect

on the efficiency of the whole supply network, as well

as on each single location. On the other hand, if all of

these locations are owned or managed by a centralized

management system, a single inventory control system

might be implemented. Tremendous improvements are

attainable if a centralized inventory management sys-

tem is considered for the entire supply network. This

motivated building inventory models that consider the

entire supply network and the interactions between

their constituent IHPs. Most of these models have their

own assumptions and characteristics. Some of these

models are built for a special class of supply networks,

such as slow-moving and expensive spare part supply

networks. Other models are built for a particular struc-

ture of a supply network that might not be applicable

to other supply networks. Hence, modeling two-level

distribution inventory systems is still a rich area for

research.

Lead time plays an important role and has been a to-

pic of interest for many authors in two-level distribution


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Int J Adv Manuf Technol (2009) 41:12081220 1209

inventory management. In general, the control parame-

ters (Q and R) depend on both the demand process

and a replenishment lead time. Although many studies

have treated lead time as constant, focusing solely on

demand, a two-stage inventory model with stochastic

lead time could be a building block in supply chain man-

agement. Variability in lead time between successive

stages is often what disturbs supply chain coordination.Our work is motivated by situations where lead time,

while random, can be influenced by the decision maker.

As one example, the inventory manager of a just-in-

time system could invest in decreasing the lead time in a

stochastic order sense. This is the case where (symmet-

rically) a rebate would be given to the customer who

accepts an above-average lead time.

In most of the early literature dealing with two-level

distribution inventory models, either in deterministic or

probabilistic models, lead time for the retailers orders

at the supplier is viewed as a constant. However, in

real inventory systems, it is not so. Moreover, the leadtime for the retailer orders is equivalent to the service

time for the supplier to meet demand from retailer

without considering the transportation time. We know

from queueing theory that the service time can follow

any probabilistic distribution. Hence, we have assumed

that the lead time for the retailer orders at the supplier

follows a general probabilistic distribution.

In this paper, we consider a two-level supply chain

(see Fig. 1) consisting of a sole supplier at the higher

echelon and many identical retailers at the lower

echelon demanding for a consumable item at the

supplier. The replenishment policy is a continuous

review policy (R, Q) at all installations, which means

that, when the inventory position reaches a predeter-

mined value of R, an order size Q is placed. The

demand process is assumed to follow a Poisson distribu-

tion. The use of a Poisson process to model the demand

for both high-value capital goods and spare parts is

reasonable because, in both cases, demands will almost

certainly be for a single unit. The supplier is notified

immediately when a retailer order is placed. The out-

side warehouse is assumed always to have sufficient

stock to meet the supplier orders.

Fig. 1 A two-level distribution inventory system

Unsatisfied demand at a retailer is lost. Technically,

this may mean either that a demand is lost to the system

or that it is expedited (satisfied, but by means external

to the normal replenishment system). Orders on the

supplier are met on a first-come, first-served basis. We

assume throughout that 0 R < Q for all installations,

with the consequence that not more than one order

from any given retailer may be outstanding at any time.The technical reason for making this assumption is that

the continuous review lost sales model with 1 Q < R

has no known analytical solution. Our assumption is

reasonable ifQ is much greater than the mean demand

in lead time at a retailer.

We allow the procurement lead time for the re-

tailer orders to be a random variable with a general

probabilistic distribution. The transportation time for

replenishing the supplier orders from an external ware-

house is assumed to be constant, and we assume that

the unsatisfied retailers orders are backordered in the

supplier and all backordered orders filled according toa first-in-first-out policy. The reorder point and batch

size of the supplier are assumed to be integer multiples

of the retailers identical batch size.

1.1 Literature review

One of the oldest papers in the field of continuous

review multi-echelon inventory systems is a basic and

famous one written by Sherbrooke [16] in 1968. He

assumed (S 1, S) policies in a depot-base system for a

repairable item in the American air force and approxi-

mated the average unit years of inventory and stockout

in the bases. Continuous review models of multi-

echelon inventory systems in the 1980s concentrated

more on repairable items in a depot-base system than

on consumable items. For example, Graves [9] worked

on determining stocking levels in such a system, Moin-

zadeh and Lee [13] considered the issue of determining

the optimal order batch size and stocking levels at the

stocking locations by using a power approximation, Lee

and Moinzadeh [12] generalized previous models on

multi-echelon repairable inventory systems to cover the

cases of batch ordering and batch shipment. On con-

sumable items, Deuermeyer and Schwarz [8] proposed

a simple approximation for a complex, multi-echelon

system (one warehouse and multiple retailers) assum-

ing backordering of unsatisfied demands in all installa-

tions with a batch ordering policy. Svoronos and Zipkin

[18] proposed several refinements on the latter paper

by considering a second-moment information (standard

deviation as well as mean) in their approximations.

In the 1990s, Axster [3] provided a simple recursive

procedure for determining the holding and stockout


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1210 Int J Adv Manuf Technol (2009) 41:12081220

costs of a system consisting of one central warehouse

and multiple retailers with (S 1, S) policy, indepen-

dent Poisson demands in the retailers, and backordered

demand during stockouts in all installations and con-

stant lead times. Axster [4] proposed exact and ap-

proximate methods for evaluating the previous system

for the case of a general batch size in all installations

but with identical retailers. For the case of nonidenticalretailers and a general batch size, Axster [5] proposed

methods for the exact evaluation of a two-retailers

case and approximate evaluation for the case of more

than two retailers. The common assumptions of the

above papers are that demand during stockout in the

retailers is backordered. However, on some situations,

demands may be lost. Anderson and Melchiors [2]

have proposed an approximate method for the case of

lost sales when inventory control policy is (S 1, S) in

all installations (one warehouse and multiple retailers)

and unsatisfied demands are lost in the retailers. Yang

and Wee [21] have developed a single-vendor, multiple-buyers production inventory model for deteriorating

items. They obtained that optimal policy for the inte-

grated system results in an impressive cost reduction

when it is compared with the independent decisions

made by the vendor and the buyers. Axster [6] has

developed an approximate method for a two-level dis-

tribution inventory system with normal approximations

for both the retailer demand and the demand at the

warehouse, i.e., the orders from the retailers. Wee and

Yang [19] have obtained optimal and heuristic solu-

tions for a multi-echelon distribution network using the

principle of strategic partnership. Here, supply chain

integration involves joint decision making among the

producer, the distributors, and the retailers. Axster [7]

has developed a decentralized, two-echelon inventory

control for the paper Axster [6]. Wee et al. [20] have

examined and pointed out two possible flaws in the

total cost function of Yang and Wees [21] model. They

illustrated a proposal to eradicate the flaws. Hill et al.

[11] considered a two-echelon inventory model with

(SQ, (S 1)Q) policy at the warehouse and continuous

review (R, Q) policy with lost sales at the retailers.

Olsson and Hill [15] developed an inventory model with

a number of retailers that operate under base stock

policy and supplier functions as an M/D/1 queue.

2 Preliminaries and notations

The batch replenishment quantity Q is fixed for all

installations, having been determined by packaging and

handling constraints. For example, Q may represent

the number of units packed into a carton or a shrink-

wrapped palletthe carton or pallet representing the

product unit of purchase, storage, handling, and ship-

ment purposes. The objective is to find the optimal

reorder points by minimizing the total holding costs

(of the supplier and retailers) and the stockout costs

of retailers. The results that are needed for solving the

proposed model are presented in Sections 2.1 and 2.2.Let us introduce the following notations.

N Number of retailers

r Demand rate at a retailer

0 Demand rate at the supplier

r Random variable represents the lead time for

deliveries of retailers orders from the supplier

0 Constant transportation time for deliveries of

supplier orders from an external warehouse

Q0 Batch size of the supplier

Qr Batch size of a retailer

R0 Reorder point of the supplier (integer value, mul-tiple ofQr)

Rr Reorder point of a retailer (integer value, since

demand is one at a time)

hr Holding cost per unit per unit time at a retailer

h0 Holding cost per unit per unit time at the supplier

r Penalty cost per unit of lost sale at a retailer

Cr Cost per unit time of a retailer in steady state

C0 Cost per unit time of the supplier in steady state

TC Total cost of the inventory system per unit in

steady state

2.1 Exact solution for the backordering problem

with Poisson demands

Considering a single-echelon inventory system with

continuous review policy, reorder point ofR and batch

size of Q, constant lead time for replenishing orders,

demand generated by a Poisson process, and backo-

rdered demand during a stockout, Hadley and Whitin

[10] developed formulae for the average stock level

D(Q, R) and for the average stockout level B(Q, R).

Assuming linear unit costs of holding and stockout,

they obtained the related annual cost. We briefly reviewtheir results and introduce the parameters that they

have used in their formulae since we will use them later.

B(Q, R) =1

Q

(R) (R + Q)

(1)

(v) =(L)2

2P(v 1; L) (L)vP(v; L)

+v(v + 1)

2P(v + 1; L) (2)


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D(Q, R) =Q + 1

2+ R L + B(Q, R) (3)

P(x; L) =

i=x

eL(L)i/i! x = 0, 1, 2, 3....

where

Q Ordering batch size of continuous review policyR Reorder point of continuous review policy

Demand rate (mean of Poisson demand

distribution)

2.2 Exact solution for lost sales problem with Poisson

demand and stochastic lead time

Consider an inventory system with continuous review

control policy, reorder point of R, and batch size of

Q. Demands are assumed to be Poisson with rate

> 0, and one unit is demanded at a time. Demands

not covered immediately from the inventory are lost.

The replenishment lead times are independent and

distributed as a generic random variable L. Denote

E[L] by and let X be a random variable, which

is distributed as a lead time demand, i.e., X takes

the value x with probability p(x) = E[p(x; L)], where

p(x, t) =

i=xet(t)i

i!,x = 0, 1, 2, ... and t> 0. The in-

ventory cycles are defined as the time period between

two successive replenishment orders, and we assume

that, at most, one order is outstanding at a time (i.e.,

R < Q ). We review the exact solutions for this model

from Johansen and Thorstenson paper [17].When R is the fixed reorder point, the expected

number of lost sales during one cycle

U(R) =

x=R

Pr[X> x] =

R1x=0

Pr[X> x] (4)

The last term of Eq. 4 vanishes ifR = 0. Expected time

during which the system is in out of stock is given by

T=U(R)

, > 0 (5)

Expected number of cycles is

Q+ T(6)

Average number of lost sales incurred per year is

E =

Q + T

T

(7)

Average stock level is

D =Q

Q +U(R)

R +

Q + 1

2 +U(R)

(8)

3 Modeling

In this section, we are going to approximate the demand

process in the higher echelon and retailers lead time in

the lower echelon.

3.1 Approximating the demand process

in the higher echelon

The average number of cycles per unit time in a con-

tinuous review inventory system when demand is lost

during a stockout is Q+T

(Hadley and Whitin [10])

without any special assumption concerning the nature

of stochastic processes generating demands and lead

times in steady state. Formula Eq. 6 is just a special case

of this relation when the stochastic process generating

demands is Poisson and the lead times are stochastic.

Since a batch size of Q is ordered in each cycle, the

mean rate of demand (from this inventory system to a

higher echelon) will be Q+T

in terms of the identical

batch size ofQ.

As Moinzadeh and Lee [13] mentioned, when the

stockout is backordered in the retailers and the demand

process at each retailer is Poisson, the arrival process

of orders at the supplier is a superposition of Narrival

processes in the case of one supplier and N retailers,

each interarrival time is Erlang distributed with shape

parameter Q. When the number of retailers in the

model is large, the arrival process can be well approxi-

mated by a Poisson process with mean rate Ni=1

iQ

(i

is the demand rate at the retailer i and Q is the iden-tical batch size of the retailers). They also stress that

such an approximation has been used or suggested by

Muckstadt [14], Deuermeyer and Schwarz [8], Albin

[1], and Zipkin [22]. Using the spirit of this nice approx-

imation and extending it for the case of lost sales (with

identical retailers) when the number of retailers in the

model is large, the arrival process can be well approx-

imated by a Poisson process with mean rate NrQr+rTr

in

terms of the identical batch size of Qr, noting that Tris the expected length of time per cycle that a retailer

is out of stock. The accuracy of this approximation is

assessed through simulation in the numerical example.

3.2 Retailers lead time approximation

As mentioned before, retailers at the first echelon of

the model experience independent Poisson demand

processes. Demand during a stockout is assumed to be

lost. Each order, i.e., placed at the supplier by each

retailer, will have a minimum lead time, which is equal

to E [Lr] = r. Since some of the retailers orders are


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placed when there is a stockout at the supplier, the

lead time may be more than r. So, the lead time of

the retailer can be approximated by adding an addi-

tional waiting time, which results from a stockout in

the supplier to r. This waiting time does not have

any clear distribution, and we just know that it is zero

when orders do not incur stockouts in the supplier and

has a positive value when they are backordered in thesupplier.

Based on the approximation of demand at the sup-

plier described in Section 3.1, the supplier behaves

just like an inventory system of the type described in

Section 2.1. From Littles formula in queuing theory

(as Anderson and Melchiors [2] used it in their approx-

imation), we can use the expression for the average

stockout level given by Eq. 1 to obtain the average

waiting time of each retailer order as given by Eq. 9.

It should be noticed that formula Eq. 1 is valid when

customer demands occur one at a time. Since each

retailer orders a batch size of Qr, we can still use theformula Eq. 1 if we make the additional assumption

that the batch size and reorder point of the supplier

(Q0, R0) are integer multiples of identical batch size of

the retailer (Qr). The average waiting time of the orders

placed by the identical retailers is

w =B(Q0/Qr, R0/Qr)

0(9)

0 =Nr

Qr+ rTr(10)

In the above formula, w is the average waiting timeof the orders placed by the identical retailers. Based on

our approximation, w is added to r of each retailer

to approximate lead time for the orders, and it can

be used for evaluating the retailers costs (holding and

stockout costs). Formula Eq. 10 follows directly from

the result in Section 3.1. Our experience may show

that the effect of this approximation is negligible on 0.

However, our numerical results will show how accurate

this approximation is.

4 Total cost of the two-echelon inventory system

Based on the results of the previous sections, the total

cost of holding and shortage in the retailers and holding

cost in the supplier are as follows:

TC = C0 + NCr (11)

The supplier cost consists of holding cost as follows:

C0 = h0D(Q0/Qr, R0/Qr)Qr (12)

In the above formula, D(Q0/Qr, R0/Qr) is the average

stock level in the supplier and is obtained as follows,

using formula Eq. 3 in Section 2.1 and noting that Q0should be an integer multiple ofQr:

The average stock level in the supplier is

D(Q0/Qr, R0/Qr) =

1

2Q0Qr + 1

+

R0

Qr 0L0

+B(Q0/Qr, R0/Qr) (13)

In formula Eq. 13, 0 is obtained through formula

Eq. 14 (as was explained in Section 3.1). B(Q0/Qr,

R0/Qr) is the average backorder level in the sup-

plier in terms of Qr and is obtained as in formula

Eq. 15 using formulae Eqs. 1 and 2 from Section 2.1,

noting again that Q0 should be an integer multiple

ofQr:

0 = NrQr+ rTr

(14)

B(Q0/Qr, R0/Qr) =1

Q0/Qr

(R0/Qr)

R0 + Q0

Qr

(15)

where

(v) = (0L0)2

P(v 1; 0L0) (0L0)vP(v; 0L0)

+v(v + 1)

2P(v + 1; 0L0) (16)

Knowing 0 and B(Q0/Qr, R0/Qr), we can determine

w from formula Eq. 17, as explained in Section 3.2,

w =B (Q0/Qr, R0/Qr)

0(17)

An important point in our approximation is that w

is dependent on 0 and 0 itself depends on Tr =

U(Rr)/r as in the formula Eq. 5 in Section 2.2.Each retailer cost consists of shortage cost and hold-

ing cost as follows:

Cr = rEr+ hrDr

In the above formula, Er is the average number of

lost sales incurred per unit time in a retailer and Dris the average stock level in a retailer. Based on the

approximation in Section 3.2, r+ w is approximated


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lead time in a retailer. Hence, using formulae Eqs. 7

and 8 in Section 2.2, we have

Er =r

Qr+ rT

r

rT

r

(18)

Dr =Qr

Qr

+ rT

r

Rr+

Qr+ 1

2 r(r+ w) + rT

r

(19)

T

r = (r+ w) 1

r

Rr1x=0

Pr[X> x] (20)

and

p(x) = Ep(x; Lr+ w)

p(x; t) =et(t)x

x!,x = 0, 1, 2, ....

T

r is the average length of time per cycle for which a

retailer is out of stock when expected lead time is r+ w.

It is clear that we should find the optimal values

of the reorder points of all installations through min-

imizing TC, which is nonlinear function with integer

variables of R0 and Rr. It is necessary to state that

the reorder point of a retailer is bound by zero and

Qr, i.e., 0 Rr < Qr. There should not be more than

one order outstanding in each retailer at any time, and

this constraint satisfies this condition for a continuous

review inventory system with lost sales (Hadley and

Whitin [10]). Furthermore, since there are N retailersin our model and none of them can have more than

one order outstanding, we have R0 NQr . This is

because ifR0 < NQr , then the reorder point is never

reached in the supplier. Summarizing the above, our

optimization problem is

Minimize TC(R0, Rr)

subject to

0 Rr < Qr

R0 NQr (21)

Noting the above explanations, we can easily find the

local optimal of the approximate total cost TC by using

the optimization tool box in Matlab 7.0.

5 Numerical examples

In order to determine the power of our approximation,

we have designed a set of 18 numerical problems. To

the best of our knowledge, in a two-echelon inventory

system with (R, Q) policy in all installations, no work

has been done on the case of lost sales and stochastic

lead times for the retailers orders in the supplier with

the policy of batch ordering in all installations. Since

we did not have any previous numerical problems as

a reference to compare our approximated results with

existing research work, we developed a problem setthat offered a reasonable range of model parameters. It

is necessary to mention that the optimal reorder points

of the supplier and retailers were found by solving the

optimization problem via the optimization tool box in

Matlab 7.0.

We also simulate each numerical problem 20 times,

for the optimal reorder points obtained from the ap-

proximate model, using GPSS/H simulation software.

The simulation time length of each run is 110,000 unit

times as a run inperiod. Different starting random

number seeds were employed for each problem. All

of the results show that this length of time is sufficientfor the system to reach a steady state. This is also clear

from the standard deviation of the total system cost.

The cost error is obtained by the following relation:

Cost error =|simulated total cost - approximated total cost|

simulated total cost

The average stock in transit between supplier and

retailer is not of interest in the backorder model be-

cause all demand is met, and so, this represents a fixed

overhead that does not depend on the control policy. It

is, however, of interest in the lost sales model because

it is directly proportional to service level. We reportthe retailers service levels as their ready rate (the

fraction of time with positive stock on hand). This can

be obtained through Hadley and Whitin [10] as

Service level =1Average number of lost sales per unit time

Average number of demands per unit time

The above relation has been employed in the ap-

proximation, and also in the simulation, model to find

the service levels. The numerical problems are as in

Table 1. The number of retailers is considered 10 and

20 (a large enough number to approximate the demand

distribution as Poisson in the supplier as Moinzadeh

and Lee [13]) to compare the results when the number

of retailers is changed. The holding costs of the supplier

and retailers per unit time are assumed to be 1, h0 =

hr = 1 and L0 = 1.

We assume that the lead time Lrof retailer is gamma

distributed with mean r and variance (r)2/ where

is positive. The parameter affects the shape of the

lead time distribution. For example, = 1, exponential

distribution is obtained and for , the degenerate


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Table 1 Input data

No r Q0 Qr r

1 50 16 8 0.5

2 50 16 8 1.0

3 50 16 8 1.5

4 50 16 16 0.5

5 50 16 16 1.0

6 50 16 16 1.57 50 32 8 0.5

8 50 32 8 1.0

9 50 32 8 1.5

10 50 32 16 0.5

11 50 32 16 1.0

12 50 32 16 1.5

13 50 64 8 0.5

14 50 64 8 1.0

15 50 64 8 1.5

16 50 64 16 0.5

17 50 64 16 1.0

18 50 64 16 1.5

case of the distribution for constant lead time results.

The Erlang distribution is obtained when is an inte-

ger. Since the demands in the retailers are Poisson with

rate r, the lead time demand (X) in the retailer has

a negative binomial distribution. Hence, X takes the

value x with probability

p(x) = + x 1

x

+ rr

rr

+ rrx

,

x = 0, 1, 2, .... and > 0 (22)

The mean lead time demand in the retailer is E[X] =

rr, and variance ofX isrr

1+rr/. Since the geometric

distribution is the special case of the negative binomial

distribution with = 1, exponentially distributed lead

times result in X having a geometric distribution with

mean rr and variancerr

1+rr. The limiting case of

, corresponding to constant lead time, implies

that Xhas the Poisson distribution with mean rr and

variance rr. Using Eq. 22, we can find out U(Rr) as in

formula Eq. 4.

Tables 2, 3, 4, and 5 give the optimum results for

total cost, service level, and average waiting time for

the cases of 10 and 20 retailers. The tables show the

results when = 1, 3, 5, and 10. In order to evaluate

the supply chain structures, we consider the following

parameters (consider problem no. 7 in Table 1): Q0 =

32, Qr = 8, r = 0.5, r = 50, h0 = hr = 1, L0 = 1, and

{1, 3, 5, 10, 100, 1000, 10000}, and numerical results

are obtained in Table 6.

5.1 Comment on the numerical examples

With regards to the obtained results from Tables 2 to 6,

we observe the following phenomena:

1. As can be seen from Tables 2 to 5, the errors in

the approximate total cost, average waiting time,

and service level are small in comparison with thesimulated values.

2. As N increases, the mean error decreases in each

table. This implies that if the number of retailers

increases, then the demand process at the supplier

can be well approximated by Poisson process (it is

also very clear from Moinzadeh and Lee [13]).

3. When r increases, TC increases. It implies that

a higher value of demand rate at the retailers,

resulting in a shorter replenishment time, increases

the optimal total cost of the supply chain.

4. When Q0 increases, TC increases. This implies that,

although the increase in ordering quantity of thesupplier decreases stockout level at a retailer, be-

cause of a relatively high penalty cost (r) com-

pared to unit holding cost, the expected total cost

increases.

5. From Table 6, we observe that, as decreases, TC

increases. This implies that, as expected variations

(2r/ ) in lead time at a retailer increase, the total

cost of the supply chain increases. The reason for

this phenomenon is as follows: When lead time

parameter decreases, it increases the expected

variations in the order processing time (i.e., lead

time) at a retailer. Simultaneously, it increases thestockout level at the lower echelon, and hence, the

total cost increases.

6. Table 6 shows that as increases the service level

increases, but the average waiting time of a retailer

decreases. This implies that shortage cost per unit

lost sale has to be raised at a retailer if the service

level is to be kept constant when the variation in

lead time demand increases due to an increase in

waiting time variation. The reason for this effect

is as follows: When increases, it decreases the

stockout level at the retailers. Decrease in stockout

level at the lower echelon decreases the averagewaiting time of orders, and therefore, service level

increases.

From observations 5 and 6, we see that the varia-

tion in lead time of retailer orders disturbs the supply

chain structures. The following observation is made

intuitively:

7. The lost sales model can give a higher service level

than the backorder level because the loss of sales


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Table 6 Results for differentvalues of the shape parameter of the gamma lead timedistribution at the retailers

AWTaverage waiting time ofa retailer, TCtotal cost of thesupply chain system

N= 10 N= 20

TC Service level (%) AWT TC Service level (%) AWT

1 68.36 95.99 2.3259 192.31 88.53 2.0877

3 64.31 96.25 1.9039 177.55 91.81 1.9155

5 59.96 98.06 1.7351 123.29 91.95 1.6216

10 59.26 98.51 1.6313 122.38 97.59 1.5671

100 57.32 98.76 1.4396 115.73 97.82 1.08191000 54.87 98.81 0.8319 108.38 98.16 0.9156

10000 53.54 98.97 0.7374 98.80 98.30 0.4943

(Equivalent

to )

decreases the effective demand rate and, therefore,

limits further loss of sales. If stockout rarely occur,

then the distinction between lost sales and backo-

rders is not significant.

6 Conclusions and future research

Multi-echelon inventory models with lost sales during

stockout at the lower echelon are most likely to be of

interest to powerful high street spare part companies

in which the demand for important spares during a

stockout is lost rather than backlogged. This paper has

presented a means of studying the steady state behavior

of a single-item, two-level, continuous review inventory

model that, on the basis of simulation results, provides

very accurate answers. We have made a number of

assumptions but most of these are consistent with ourassumed context of the stocking of slow-moving, high-

value capital goods or vital spare parts in the retail

sector. The main point of this paper is to introduce

the concept of stochastic lead time for retailers orders

since most of the existing research in this field has

assumed constant lead times. We have approximated

demand distribution as Poisson in the supplier, and

the average waiting time of the retailers orders due

to stockout in the supplier is obtained using Littles

formula in queueing theory. From the numerical ex-

amples developed for gamma distributed lead times for

the retailer orders, it is observed that the variability inlead time disturbs the supply chain coordination. Com-

paring our analytical results with simulation results for

18 numerical problems, the error levels are consistent

with those reported for similar approximations by other

research in this area.

This work can be extended for the non-identical-

retailers case at the lower echelon. Another interesting

issue to be investigated is the impact of perishable items

on the optimal investment cost and replenishment de-

cisions with order frequency reduction.

Acknowledgement We sincerely express our gratitude to theanonymous referees for their valuable suggestions. This researchwork is fully supported by Senior Research Fellowship (grant no.:09/715(0002)/2006 EMR-I) under the Counsel of Scientific andIndustrial ResearchIndia. We also thank the University GrantsCommission, India, for providing a special assistance program(UGC-SAP).

References

1. Albin S (1982) On Poisson approximation for superpositionarrival processes in queues. Manag Sci 28:126137

2. Andersson J, Melchiors P (2001) A two echelon inventory

model with lost sales. Int J Prod Econ 69:3073153. Axster S (1990) Simple solution procedures fora class of twoechelon inventory problems. Oper Res 38:6469

4. Axster S (1993) Exact and approximate evaluation of batchordering policies for two-level inventory systems. Oper Res41:777785

5. Axster S (1998) Evaluation of installation stock based(R, Q) policies for two-level inventory systems with Poissondemand. Oper Res 46:135145

6. Axster S (2003) Approximate optimization of a two leveldistribution inventory system. Int J Prod Econ 8182:545553

7. Axster S (2005) A simple decision rule for decentralizedtwo-echelon inventory control. Int J Prod Econ 9394:5359

8. Deuermeyer B, Schwarz LB (1981) A model for the analy-sis of system service level in warehouse/retailer distribution

system: the identical retailer case. In: Schwarz L (ed) Multi-level production/inventory control systems (TIMS studies inmanagement science 16). Elsevier, New York, chapter 13

9. Graves SC (1985) A multi-echelon inventory model for arepairable item with one for one replenishment. Manag Sci31:12471256

10. Hadley G, Whitin TM (1963) Analysis of inventory systems.Prentice-Hall, Englewood Cliffs

11. Hill RM, Seifbarghy M, Smith DK (2007) A two-echeloninventory model with lost sales. Eur J Oper Res 181:753766


13/13


12. Lee HL, Moinzadeh K (1987) Operating characteristics of atwo-echelon inventory system for repairable and consumableitems under batch ordering and shipment policy. Nav ResLogist 34:365380

13. Moinzadeh K, Lee HL (1986) Batch size and stocking levelsin multi-echelon repairable systems. Manag Sci 32: 15671581

14. Muckstadt JA (1977) Analysis of a two-echelon inventorysystem in which all locations follow continuous review(s,S)policy. Technical report no. 337, School of OperationsResearch, vol 16, pp 122141

15. Olsson RJ, Hill RM (2007) A two-echelon base stock inven-tory model with Poisson demand and the sequential process-ing of orders at the upper echelon. Eur J Oper Res 177:310324

16. Sherbrooke CC (1968) METRIC: a multi-echelon techniquefor recoverable item control. Oper Res 16:122141

17. Johansen SG, Throstenson A (1993) Optimal and approx-imate (Q, r) inventory policies with lost sales and gammadistributed lead times. Int J Prod Econ 3031:179194

18. Svoronos AP, Zipkin P (1988) Estimating the performance ofmulti-level inventory systems. Oper Res 36:5772

19. Wee HM, Yang PC (2004) The optimal and heuristicsolutions of a distribution network. Eur J Oper Res 158:626632

20. Wee HM, Jong JF, Jiang JC (2007) A note on a single-vendorand multiple-buyers production-inventory policy for a deteri-orating item. Eur J Oper Res 180:11301134

21. Yang PC, Wee HM (2002) A single-vendor and multiple-buyers production inventory policy for a deteriorating item.Eur J Oper Res 143:570581

22. Zipkin PH (1986) Models for design and control of stochastic,multi item batch production systems. Oper Res 34:91104

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