performance modeling and analysis of integrated logistic chains: an analytic framework

European Journal of Operational Research 162 (2005) 83–98

www.elsevier.com/locate/dsw

Performance modeling and analysis of integratedlogistic chains: An analytic framework

Ming Dong a,*, F. Frank Chen b

a TEAMS, Inc., 8882-3C Storch Woods Drive, Savage, MD 20763, USAb Grado Department of Industrial and Systems Engineering, Virginia Polytechnic Institute and State University,

Blacksburg, VA 24061, USA

Received 1 October 2001; accepted 14 October 2003

Available online 22 January 2004

Abstract

This paper is geared toward developing a network of inventory-queue models for the performance modeling and

analysis of an integrated logistic network. An inventory-queue is a queueing model that incorporates an inventory

replenishment policy for a store, which is a basic modeling element for an integrated logistic network. To achieve this

objective, first, this paper presents an analytical modeling framework for integrated logistic chains, in which the in-

terdependencies between model components are captured. Second, a network of inventory-queue models for perfor-

mance analysis of an integrated logistic network with inventory control at all sites is developed. Then this paper extends

the previous work done on the supply network model with base-stock control and service requirements. Instead of one-

for-one base stock policy, batch-ordering policy and lot-sizing problems are considered. In practice, the assumption of

uncapacitated production is often not true, therefore, GIx/G/1 queueing analysis is used to replace the Mx/G/1 queue

based method. To include lot-sizing issue in the analysis of stores, a fixed-batch target-level production authorization

mechanism is employed to explicitly obtain performance measures of the logistic chain queueing model. The validity

of the proposed model is illustrated by comparing the results from the analytical performance evaluation model and

those obtained from the simulation study.

� 2003 Elsevier B.V. All rights reserved.

Keywords: Logistic chains; Integrated framework; Analytic performance analysis; Queueing theory; Lot sizing

1. Introduction

An integrated logistic chain can be viewed as a

network of suppliers, fabrication sites, assembly

sites, distribution centers, and customer locations

* Corresponding author. Tel./fax: +1-240-5689792.

E-mail address: [email protected] (M. Dong).

0377-2217/$ - see front matter � 2003 Elsevier B.V. All rights reserv

doi:10.1016/j.ejor.2003.10.030

through which components and products flow.The inventory in each site is controlled by some

inventory control policy. An important issue in

integrated logistic network design is to control the

inventory at different sites or stores while meeting

end-customer service level requirements, therefore

quantifying the trade-off between inventory in-

vestment and end-customer service levels. The dy-

namic nature of complex logistic chains causes

ed.

mail to: [email protected]

84 M. Dong, F.F. Chen / European Journal of Operational Research 162 (2005) 83–98

that this trade-off changes over time. In turn, theperformance of logistic chains must be reevaluated

continuously. Therefore, an analytic framework

to be used for both performance modeling and

analysis is in need. Furthermore, to answer ‘‘what–

if’’ questions quickly, such an analytic model has

to be computed efficiently.

The arborescent structure of the networks

treated by multi-echelon inventory networks andrelated models are appropriate for distribution

networks. However, in a general logistic chain

network, the model involves not only distribution

but also assembly, in which multiple components

are required for the production of one part. Inte-

grating manufacturing and logistic functions to-

gether is the key determinant for an organization

to obtain the competing advantages in business.Extending multi-echelon inventory networks into

integrated manufacturing logistic chains is not

straightforward. See Ernst and Pyke (1993) and

Cohen and Lee (1988) for research in this direc-

tion.

Lee and Billington (1993) present an analytical

model for a decentralized supply chain. Their

model assumes each site operates under a periodic-review, base-stock inventory policy and the de-

mands for end-products are normally distributed.

A key assumption in the model is that the re-

plenishment lead time for a product at a node

comprises the material lead time, the production

lead time, and the delay time. The network per-

formance is obtained by aggregating the solutions

of multiple single-product, single-site inventoryproblems. One of the advantages of this frame-

work is the ability to analyze the first two moments

of all the performance metrics.

Garg (1999) develops a decentralized supply

chain modeling and analysis tool (SCMAT) on

designing products and processes for supply chain

management. SCMAT contains two sub-models:

queueing network sub-model which is used to at-tain the production lead time for a site and

inventory network sub-model which is used to

calculate the base-stock level for each stock keep-

ing unit (SKU) flowing through every site. The

output (production lead time) of the queueing

network sub-model is one input of the inventory

network sub-model. Given the base-stock level and

the values of its three drivers (demand, replenish-ment lead time, and the service-level), one can

compute various performance measures useful for

obtaining managerial insight. These performance

measures include the mean and variance of on-

hand inventory, backorder level and response

time, and the capacity utilization at each node in

the network. SCMAT explicitly incorporates the

congestion effects due to capacity limitations ateach node, and the interference effect because of

multi-product flows through each node.

Ettl et al. (2000) develop a supply chain model

that takes as input the bill-of-materials, the nom-

inal lead times, the demand and cost data, and

the required customer service levels. In return,

the model generates the base-stock level at each

store––the stocking location for a componentor an end-product. They assume a distributed

inventory control mechanism whereby each site in

the network operates according to a base-stock

control policy. This base-stock policy makes au-

thors avoid the consideration on determining the

lot sizes at each store. They further assume that,

given there are no stock-outs at upstream stores,

the replenishment lead time of any store is inde-pendent of the number of outstanding orders.

In this sense, stores are uncapacitated.

The paper by Srinivasa Raghavan (2001) pres-

ent an analytical method for evaluating the per-

formance of make-to-order supply chains using

general queueing networks (GQNs). Through

some examples, they illustrate the use of multi-

class open generalized queueing networks tocompute the mean and variance of the lead time.

They also investigate the effect of variance reduc-

tion on the average lead time and average WIP

inventory. For supply chain configurations with

both assembly function (convergent structure) and

distribution function (divergent structure), the

authors provide a decomposition–aggregation ap-

proach for the performance analysis with a fork-join queueing network (FJQN). The network is

decomposed into GI/G/1 queues and these queues

can be analyzed independently of each other and

for each queue the performance measures can be

calculated. After composing the network again it is

possible to determine the performance measures

for the entire network. In order to use a general

M. Dong, F.F. Chen / European Journal of Operational Research 162 (2005) 83–98 85

open fork-join queueing network, this paper con-siders a supply chain where all the members work

on a make-to-order basis and assumes that de-

mand arrivals are deterministic, while processing

times at the facilities are normally distributed. In

practice, the demand information flows backward

(from customers to manufacturers to suppliers)

and the demand interdependence between different

stages complicates the analysis of the supply chain.To use the open fork-join queueing network, this

paper assumes an independent demand relation-

ship between different stages and the demand

information flows forward in their models. This

limits the application of their approach. Inventory

control in multi-echelon systems is the core of

integrated supply chain analysis, however, this

paper does not consider this issue.This paper is geared toward developing effective

modeling framework and analyzing methods

for performance evaluation of integrated logistic

chains. In view of the above, the following objec-

tives are pursed: (1) to provide an integrated mod-

eling framework for logistic chains, in which the

interdependencies between model components are

captured; (2) to develop a network of inventory-queue models for performance analysis of an inte-

grated logistic network with inventory control at all

sites; and (3) to extend the previous work developed

for supply network model with base-stock control

and service requirements. Instead of one-for-one

base stock policy, batch-ordering policy and lot-

sizing problems are considered. The assumption of

uncapacitated production is often not true inpractice, therefore, GIx/G/1 queueing analysis is

used to replace the Mx/G/1 queue based method.

The remainder of this paper is organized as

follows. Section 2 provides an analytical model-

ing framework for integrated logistic chains. In

Section 3, performance analysis with GIx/G/1

queue is developed. Through the performance

analysis, different performance measures of astore such as order fill-rate and stock-out prob-

ability in the network are obtained. Section 4

introduces the performance analysis for the entire

logistic chain. The performance measures of other

stores at different stages can be derived by mak-

ing use demand dependence and lead time

dependence. Several numerical examples are used

to illustrate the validity of the proposed meth-odology in Section 5. Section 6 summarizes this

research.

2. An integrated modeling framework for logistic

chains

Logistic chains may differ in the networkstructure (serial, parallel, assembly and arbores-

cent distribution), product structure (levels of Bill-

Of-Materials), transportation modes, and degree

of uncertainty that they face. However, they have

some basic elements in common.

2.1. Sites and stores

A logistic chain network can be viewed as a

network of functional sites connected by different

material flow paths. Generally, there are four types

of sites: (1) Supplier sites: they procure raw mate-

rials from outside suppliers; (2) Fabrication sites:

they transform raw materials into components; (3)

Assembly sites: they assemble the components into

semi-finished products or finished goods; and (4)Distribution sites: they delivery the finished prod-

ucts to warehouses or customers. All sites in the

network are capable of building parts, subassem-

blies or finished goods in either make-to-stock

or make-to-order mode. The part that a site pro-

duces is a single-level BOM.

Typically, there are two types of operations

performed at a site in a logistic chain: material

receiving and production. A material receiving

operation is one that receives input materials from

upstream sites and stocks them as a stockpile to be

used for production. A production operation is

one in which fabrication or assembly activities

occur, transforming or assembling input materials

into output materials. Correspondingly, each site

in the logistic chain has two kinds of stores: inputstores and output stores. Each store stocks a single

SKU. The input stores model the stocking of dif-

ferent types of components received from up-

stream sites, and output stores model the stocking

of finished-products at the site (in Fig. 1, a site is

represented by the dashed box containing input

and output stores).

Fig. 1. An integrated modeling framework for logistic chains.


The sites can be treated as the building blocks

for modeling the whole logistic chain. Therefore,

for the performance analysis of a logistic chainmodel, the performance of each store is analyzed

first, then, the whole logistic chain performance is

analyzed.

2.2. Links

All stores in the logistic chain are connected

together by links that represent supply and de-mand processes. Two types of links are defined:

internal link and external link. Internal links are

used to connect the stores within a site, i.e., they

represent the material flow paths from input stores

to output stores within a site. A link connecting an

output store of one site to an input store of an-

other site is called an external link. This kind of

link represents that the output store providesreplenishments to the specified downstream input

store. In the network topology, we define that a

downstream input store has only one link between

it and its upstream output store (see Fig. 1). In

other words, we are assuming ‘‘single sourcing’’

for the total logistic chain.

2.3. The relationships between stores

Let ST be the collection of stores in a logistic

network and i be a store in ST. The set of directlyupstream supplying stores of store i is denoted as

UPST ðiÞ. The set of directly downstream receiving


stores from store i is denoted as DOWNST ðiÞ. If i isan input store, then UPST ðiÞ is a singleton set, i.e.,

it contains only one upstream supplying store.

That is, each input store can obtain replenishment

from only one supplier. On the other hand,

DOWNST ðiÞ consists of one or more output stores

at the same site. If i is an output store, then

UPST ðiÞ is either empty, in which case i is a source

store (e.g., a supplier), or contains one or morestores, which are input stores at the same site. For

DOWNST ðiÞ, it is either empty, in which case i is anend store, or contains one or more input stores

at its downstream site.

3. Performance analysis of a single-product-type

store

Previous literature on integrated logistic chains

(Chen et al., 2000; Cohen and Lee, 1988; Ettl et al.,

2000; Feigin, 1999; Garg, 1999; Lee and Billington,

1993; Srinivasa Raghavan, 2001) mainly focuses

on strategic aspects of inventory management such

as lead time approximations, fill rates of stores and

impact of variance reduction. The operational as-pects of inventory management in integrated lo-

gistic chains, such as determinations of order sizes

and order-up-to levels at each store, are not

studied enough so far.

Ettl et al. (2000) model each store as an infinite-

server queue operating under a base-stock control

rule. In particular, they use the Mx/G/1 queue,

where demand arrivals follow a Poisson processwith rate k, and each arrival brings a batch of Xorders. The use of one-for-one base-stock policy

avoids the determination of lot size at each store.

This may be suitable for the situation in which the

production set-up cost is negligible. Since the one-

for-one replenishment policy requires the ma-

chines to reset-up for each production of an order,

this policy will result in a higher cost if the set-upcost is not negligible. And in reality, most time this

set-up cost cannot be neglected. Therefore, instead

of using ðS � 1; SÞ base-stock policy, we adopt

ðs; SÞ inventory policy. In addition, the assumption

of uncapacitated production limits the practical

application of their approach. In the following, we

use GIx/G/1 queue operating under ðs; SÞ inven-

tory control rule to analyze the performance of asingle-product store.

The paper by Yao et al. (1984a) gives a simple

and effective approach to derive bounds for bulk

arrival queues by making use of the known bounds

for single arrival queues. The authors focus on

upper and lower bounds of mean waiting times

and mean queue length for the bulk arrival queues

GIx/G/1 and GIx/G/c. They formulate an equiva-lent system GI/G/1 for the original system GIx/G/

1. Instead of individual customer, arrivals and

services are carried out group by group, every

group is treated as a unit. The mean waiting time

in this equivalent system is equal to the waiting

time of the first-serviced customer in the group of

the original system. The results show that for all

traffic intensities their bounds are closer to exactresults than the diffusion approximations.

Yao et al. (1984b) studied the delay in the queue

GIx/G/1 by making use of the �dual� queue GIx/G/

1. They found that the queueing delay of the first-

serviced customer is equal to the delay in the queue

GIx/G/1, where the service time is a sum of a

random number (X ) of i.i.d. random variables. It is

proved in this paper that the GIx/G/1 queueingdelay is the sum of two independent random

variables: the waiting time of the first customer in

the group and the waiting time for the service of

the group-mates who are served before the random

customer under consideration. The first part can

be expressed explicitly in terms of the first two

moments of the relevant distributions of the sys-

tem, while the other is given by the GIx/G/1 queue.Since GIx/G/1 is a special class of the GI/G/1

queues, the well-known results of the queue GI/G/

1 (e.g., approximations) can be readily applied.

To analyze the performance of a make-to-stock

store under ðs; SÞ inventory policy, we adopted the

target-level production authorization (PA) mech-

anism with fixed lot size. This mechanism works as

follows: let S be the order-up-to level of a storeinventory, each item at the store is attached with a

tag. When a demand stream arrives and there are

some available items at the store, this demand is

met and the tag with this demand is removed from

the item. Then this tag is activated into a PA card.

Whenever q or more PA cards are accumulated

at the store, q PA cards are transmitted to the


manufacturing facility. In other words, q is the lotsize. Each of these q PA cards authorizes the ma-

chine to produce one item. And the overlapping

production initiation policy is adopted: as soon as

the manufacturing of all q items of a lot of PA cards

is initiated, the production of items for the sub-

sequent lot of PA cards can be initiated (whenever

machines become available). This is so called ðq; SÞfixed-batch target-level PA mechanism (Buzacott,1989; Buzacott and Shanthikumar, 1993). When

the production of q items is completed and shipped

to the store inventory area, the PA cards with these

items will be converted into tags. Observe that the

maximum number of items in the store will always

be less than or equal to S. Therefore, this mecha-

nism is essentially the same as the traditional

reorder point/order quantity inventory controlpolicy. Unlike in inventory modeling, the fixed-lot

target-level PA mechanism will explicitly model the

process that determines the lead time for replen-

ishment of items at the store.

3.1. Performance analysis with GIx/G/1 queue

Consider a queuing system consisting of awaiting area and a service facility. Customers ar-

rive at the queueing system according to the arrival

process fAn; n ¼ 1; 2; . . .g. The number of orders

brought in by the nth customer is Xn ¼ q,n ¼ 1; 2; . . .. In other words, the lot size is fixed

and equals to q. Here, we suppose that each order

corresponds to one SKU. If the number of orders

in the service facility is less than S, orders from thewaiting area are sent into the service facility one at

a time until there are S customers in the service

facility. Assume that the store is initially full at

time zero. The nth customer arrives at time~An ¼ Anq and brings q orders. The service time of

the nth order is Tn, n ¼ 1; 2; . . . Let Dn be the

departure time of the nth order from this queueing

system. Then eDn ¼ Dnq is the time epoch where thenth lot of q PA cards order is filled. Let NðtÞ be

the number of orders in the batch arrival eGIq/G/1

queueing system with arrival process f~An; n ¼ 1;2; . . .g, service times fTn; n ¼ 1; 2; . . .g and batch

size fXn ¼ q; n ¼ 1; 2; . . . :g. For the batch of PA

cards currently being manufactured, define CðtÞ bethe number of finished SKUs at the manufacturing

facility waiting to be shipped. Since for each batchthere are at most q items that are in manufactur-

ing, we have

CðtÞ ¼ NðtÞq

� �� q� NðtÞ; tP 0: ð1Þ

At each arrival epoch of ~An ¼ Anq, we create a

lot tag (i.e., one tag for every q orders) and destroy

one whenever a lot of q PA cards is returned to

inventory area. Let RðtÞ be the number of orders

arrives at or before time t, but after the last lot tagwas created. Then

RðtÞ ¼ AðtÞ � AðtÞq

� �� q; tP 0; ð2Þ

where AðtÞ is the number of orders that arrived

during ð0; t�.Since CðtÞ þ NðtÞ is the number of orders within

the manufacturing facility and RðtÞ is the number

of orders waiting at outside the manufacturing

facility, the total number of orders during ð0; t� isCðtÞ þ NðtÞ þ RðtÞ. Therefore the number of or-ders backlogged at time t is

BðtÞ ¼ ½CðtÞ þ NðtÞ þ RðtÞ � S�þ

¼ NðtÞq

� �� q

�þ RðtÞ � S

�þ

; tP 0: ð3Þ

Similarly, the inventory of finished SKUs at

inventory area is

IðtÞ ¼ S�

� NðtÞq

� �� q� RðtÞ

�þ

; tP 0: ð4Þ

From Eq. (2), it can be seen that as AðtÞ in-

creases, RðtÞ will uniformly increase from 0 toq� 1, and from q� 1 to 0, then increase from 0 to

q� 1, and so on. Then we have

PfR ¼ ng ¼ 1

q; n ¼ 0; 1; . . . ; q� 1: ð5Þ

Assume that NðtÞ and RðtÞ are independent, andlet pN ðnÞ be the distribution of NðtÞ, from Eqs. (1),

(3) and (4), it can be shown (Buzacott and

Shanthikumar, 1993)

PfC ¼ ng ¼ pNnq

� �� q

�� n

�; n ¼ 1; 2; . . . ;

ð6Þ


PfB ¼ ng ¼ 1

qpN

nq

� �� q

�þ S

�; n ¼ 1; 2; . . . ;

ð7Þ

PfI ¼ ng ¼ 1

qpN S

�� n

q

� �� q�; n ¼ 1; 2; . . . ; S:

ð8ÞNext, some approximations given by Buzacott

and Shanthikumar (1993) are used to derive the

distribution of NðtÞ, i.e., pN ðnÞ.For the renewal customer arrival processes (not

necessarily Poisson), the properties of the GIX /G/1

queue can be modeled by an equivalent GI/ eG/1

queue, where each job is the aggregate of orders

brought in by each customer. The correspondingarrival process is fAn; n ¼ 1; 2; . . .g and service

times can be aggregated as eTn ¼PXn

j¼1 Tnj, n ¼1; 2; . . .. The mean of the service times eT becomes

E½X �E½T � (refer to Wolff (1989) for general back-

ground materials for queues). The squared coeffi-

cient of variation of the service times eT , which

is defined as C2eT ¼ VarðeT Þ=E½eT �2, is then C2X þ ð1=

E½X �ÞC2T .

Define eWGI=G=1ðk; l;C2a ;C

2T Þ be one of the

approximations for the average waiting time inqueue for a GI/G/1 queue with mean interarrival

time 1=k, mean service time 1=l, squared coeffi-

cient of variation of interarrival time C2a , and

squared coefficient of variation of service time C2T .

Then the average waiting time of a customer in the

GI/ eG/1 queue can be approximated by (Buzacott

and Shanthikumar, 1993)

wc ¼ eWGI=G=1ðk; 1=E½X �E½T �;C2a ;C

2X þ ð1=E½X �ÞC2

T Þ:ð9Þ

The average number of customers waiting in the

queue can be obtained from Little�s formula:

Nc ¼ k � wc: ð10ÞSince each customer in queue consists of an

average of E½X � (or q) orders, the average numberof orders in the system corresponding to those

customers waiting in the queue (i.e., orders in

waiting) is

Now ¼ k � E½X � � wc ¼ k � q � wc: ð11Þ

The average number of orders in the systemcorresponding to the customers in service is

Nos ¼E½X 2� þ E½X �

2E½X � : ð12Þ

Because the probability that a customer is inservice is q ¼ kE½X �E½T �, the average number of

orders, E½No�, in the system can be approximated by

E½No� ¼ Now þ Nos � q

¼ k � E½X � � wc þk � ðE½X 2� þ E½X �ÞE½T �

2

¼ k � q � wc þk � ðq2 þ qÞE½T �

2: ð13Þ

The time-average probability of the server beingidle is 1� q, so P ð0Þ ¼ 1� q. Assume that the

probability has a form of P ðnÞ ¼ a � rn�1. Since the

total probability should be one, we have a ¼qð1� rÞ. And this approximated distribution has

a mean qð1� rÞ. Finally, let the approximated

mean number of jobs in the system given by

Buzacott and Shanthikumar (1993) equal to the

mean of the approximated distribution of thenumber of jobs in the system. Then, the distribu-

tion of the number of orders in this GIX /G/1 can

be approximated by

PfNo ¼ ng � ð1� qÞ; n ¼ 0;qð1� rÞrn�1; n ¼ 1; 2; . . . ;

�ð14Þ

where

r ¼ E½No� � qE½No�

ð15Þ

is chosen such that the average of the approxi-

mated distribution is equal to E½No�.Substitute Eq. (14) into Eqs. (7) and (8), we

have

PfB ¼ ng ¼ qqð1� rÞr n

qd e�qþS�1; n ¼ 1; 2; . . . ;

ð16Þ

and

PfI ¼ ng ¼ qqð1� rÞrS� n

qd e�q�1; n ¼ 1; 2; . . . ; S:

ð17Þ


Our next target is to find the means of BðtÞ andIðtÞ, we have

E½B� ¼X1n¼1

n � qqð1� rÞr n

qd e�qþS�1

¼ qqð1� rÞrS�1

X1n¼1

n � r nqd e�q: ð18Þ

The last summation in Eq. (18) can be obtained as

follows:X1n¼1

n � r nqd e�q

¼ ½1þ 2þ � � � þ ðq� 1Þ þ q�rq þ ½ðqþ 1Þ

þ ðqþ 2Þ þ � � � þ 2q�r2q þ � � � þ ½ðmqþ 1Þ

þ ðmqþ 2Þ þ � � � þ ðmqþ qÞ�rðmþ1Þq þ � � �

¼ ð1þ qÞq2

rq þ ð1þ 3qÞq2

r2q þ ð1þ 5qÞq2

r3q

þ � � � þ ½1þ ð2mþ 1Þq�q2

rðmþ1Þq þ � � �

¼ q2rqfð1þ qÞ þ ð1þ 3qÞrq þ ð1þ 5qÞr2q þ � � �

þ ½1þ ð2mþ 1Þq�rmq þ � � �g:

Let

Z ¼ fð1þ qÞ þ ð1þ 3qÞrq þ ð1þ 5qÞr2q

þ � � � þ ½1þ ð2mþ 1Þq�rmqg; ð19Þ

and multiply rq to both sides of Eq. (19), then

rqZ ¼ fð1þ qÞrq þ ð1þ 3qÞr2q

þ ð1þ 5qÞr3q þ � � �þ ½1þ ð2mþ 1Þq�rðmþ1Þqg: ð20Þ

From (19) and (20), we have

ð1� rqÞZ ¼ ð1þ qÞ � ½1þ ð2mþ 1Þq�rðmþ1Þq

þ 2qrq½1þ rq þ r2q þ � � � þ rðm�1Þq�

¼ ð1þ qÞ � ½1þ ð2mþ 1Þq�rðmþ1Þq

þ 2qrq 1� rmq

1� rq

� ; ð21Þ

thus

Z ¼ 1þ q1� rq

� ½1þ ð2mþ 1Þq�rðmþ1Þq

1� rq

þ 2qrq

1� rq� 1� rmq

1� rq

� : ð22Þ

From Eq. (15), we know jrj < 1. So the limit of

Z is

limm!1

Z ¼ 1þ q1� rq

þ 2qrq

1� rq� 1

1� rq

� : ð23Þ

Therefore

E½B� ¼ qqð1� rÞrS�1

X1n¼1

n � r nqd e�q

¼ ð1þ qÞð1� rÞqrSþq�1

2ð1� rqÞ

þ qr2qþS�1qð1� rÞð1� rqÞ2

: ð24Þ

Similarly,

E½I � ¼XS

n¼1

n � qqð1� rÞrS� n

qd e�q�1

¼ qqð1� rÞrS�1

XS

n¼1

n � r� nqd e�q; ð25Þ

and define Sq

l m¼ mþ 1, thenXS

n¼1

n � r� nqd e�q

¼ ½1þ 2þ � � � þ ðq� 1Þ þ q�r�q þ ½ðqþ 1Þþ ðqþ 2Þ þ � � � þ 2q�r�2q þ � � � þ f½ðm� 1Þqþ 1� þ ½ðm� 1Þqþ 2� þ � � � þmqgr�mq

þ ½ðmqþ 1Þ þ ðmqþ 2Þ þ � � � þ S�r�ðmþ1Þq

¼ ð1þ qÞq2

r�q þ ð1þ 3qÞq2

r�2q þ � � �

þ ½1þ ð2m� 1Þq�q2

r�mq

þ ð1þmqþ SÞðS �mqÞ2

r�ðmþ1Þq

¼ q2r�qfð1þ qÞ þ ð1þ 3qÞr�q þ � � �

þ ½1þ ð2m� 1Þq�r�mqg

þ ð1þmqþ SÞðS �mqÞ2

r�ðmþ1Þq:


Let

Z 0 ¼ fð1þ qÞ þ ð1þ 3qÞr�q þ � � �þ ½1þ ð2m� 1Þq�r�mqg; ð26Þ

and multiply r�q to both sides of Eq. (26), then

r�qZ 0 ¼ fð1þ qÞr�q þ ð1þ 3qÞr�2q þ � � �þ ½1þ ð2m� 3Þq�r�mq

þ ½1þ ð2m� 1Þq�r�ðmþ1Þqg: ð27Þ

From (26) and (27), we have

ð1� r�qÞZ 0

¼ ð1þ qÞ � ½1þ ð2m� 1Þq�r�ðmþ1Þq

þ 2qr�q½1þ r�q þ r�2q þ � � � þ r�ðm�1Þq�

¼ ð1þ qÞ � ½1þ ð2m� 1Þq�r�ðmþ1Þq

þ 2qr�q 1� r�mq

1� r�q

� ;

that is

Z 0 ¼ ð1þ qÞð1� r�qÞ �

½1þ ð2m� 1Þq�r�ðmþ1Þq

ð1� r�qÞ

þ 2qr�q 1� r�mq

ð1� r�qÞ2

" #: ð28Þ

Therefore

E½I � ¼ qqð1� rÞrS�1

XS

n¼1

n � r� nqd e�q

¼ q2ð1� rÞrS�q�1Z 0

þ qqð1� rÞr�ðmþ1ÞqþS�1

� ð1þ mqþ SÞðS � mqÞ2

: ð29Þ

3.2. Optimization of inventory positions

Since each arrival customer will bring in E½X �(or q) orders, the effective arrival rate is kE½X �. Ifthe setup cost is k1 per batch, the inventory car-

rying cost is k2 per unit time, and the backlogging

cost is k3 per unit time, then the total cost rate

TCðq; SÞ for this target-level PA mechanism withfixed-lot size is

TCðq;SÞ

¼ k1k �E½X �

q

� �þ k2E½I �þ k3E½B�

¼ k1kþ k2q2ð1

��rÞrS�q�1Z 0

þqqð1�rÞr�ðmþ1ÞqþS�1 ð1þmqþSÞðS�mqÞ

2

þ k3

ð1þqÞð1�rÞqrSþq�1

2ð1�rqÞ

"þqr2qþS�1qð1�rÞ

ð1�rqÞ2

#:

ð30ÞThis result can be used to obtain the optimal

values of q and S that minimize the total cost of

inventory, backlogging as well as machine set-up

costs if the machines are set up for each batch ofPA cards. The solution procedure is differentiating

TCðq; SÞ with respect to q and S, equating to zero,

and solving for 0 < q < 1 and SP 0.

3.3. Performance measures of a store

3.3.1. Stock-out probability of a store

The stock-out probability at a store or theprobability of a customer is backlogged, denoted

by p, is the fraction of time that the on-hand

inventory at the store is zero:

p ¼ P ½I ¼ 0� ¼ P S�

6NðtÞq

� �� qþ RðtÞ

: ð31Þ

Since RðtÞ uniformly increases from 0 to q� 1,

drops to 0, increases from 0 to q� 1, and so on.

The distribution of RðtÞ then is 1=q. Thus, we havethe following expressions:

When RðtÞ ¼ 0,

IðtÞ ¼ S�

� NðtÞq

� �q;

NðtÞ ¼ Sq

� �qþ 1;

Sq

� �qþ 2; . . . : ð32Þ

When RðtÞ ¼ 1,

IðtÞ ¼ ðS�

� 1Þ � NðtÞq

� �q;

NðtÞ ¼ S � 1

q

� �qþ 1;

S � 1

q

� �qþ 2; . . . ð33Þ


When RðtÞ ¼ 2,

IðtÞ ¼ ðS�

� 2Þ � NðtÞq

� �q;

NðtÞ ¼ S � 2

q

� �qþ 1;

S � 2

q

� �qþ 2; . . . ð34Þ

..

.

When RðtÞ ¼ q� 1,

IðtÞ ¼ ðS�

� qþ 1Þ � NðtÞq

� �q;

NðtÞ ¼ S � qþ 1

q

� �qþ 1;

S � qþ 1

q

� �qþ 2; . . .

ð35Þ

Then, by the distribution of NðtÞ in Eq. (14), we

have

p ¼ P ½I ¼ 0�

¼ 1

qqð1� rÞrq S

qb cX1n¼0

rn þ 1

qqð1� rÞrq S�1

qb cX1n¼0

rn

þ � � � þ 1

qqð1� rÞrq S�qþ1

qb cX1n¼0

rn

¼ qqrq r

Sqb c

nþ r

S�1qb c þ � � � þ r

S�qþ1qb c

o: ð36Þ

Let S ¼ mqþ r, where 06 r < q, from Eq. (35)

we have the stock-out probability

p ¼ P ½I ¼ 0�

¼ qqrq½ðr þ 1Þrm þ ðq� r � 1Þrm�1�: ð37Þ

3.3.2. Fill rate of a store

The fill rate at a store, denoted by f , is the

fraction of customer orders that is filled by on-hand inventory. The fill rate is also the fraction of

time that the on-hand inventory at the store is

greater than zero:

f ¼ P ½I > 0� ¼ P S�

>NðtÞq

� �� qþ RðtÞ

¼ 1� p ¼ 1� q

qrq½ðr þ 1Þrm þ ðq� r � 1Þrm�1�:

ð38Þ

Let ~f ¼ 1� f , we have ~f ¼ p.

4. Performance analysis of the entire logistic chain

For stores at different stages, there exist two

types of dependences: demand dependence and lead

time dependence. Demand dependence describes

how demand at upstream stores is derived from

downstream stores. While lead time dependence

describes how inventory availability at upstreamstores influences the lead time at downstream stores.

4.1. Demand dependence in logistic networks

The demand transmission is used to determine

the mean and variance of the demands for SKUs

of upstream stages of end-product stages. The

demand placed on SKUs at a downstream site is

translated into a demand for components at the

current site via the bill of materials. The down-

stream demand, in turn, creates demand at the

supplying site. This is called the demand transmis-

sion process (Garg, 1999).

The derived demand for each SKU at upstream

sites is not stationary, uncorrelated and normally

distributed since the commonality of components

among products will result in positively correlated

demands for common components. However, by

comparing some performance measures (such as

lead times, backorder levels, etc.) of correlateddemand and uncorrelated demand situations,

Feigin (1999) indicates that the difference between

them is insignificant when service level require-

ments for finished goods are high. Therefore, the

derived demand for each SKU at upstream sites

can be approximately assumed to be stationary,

uncorrelated and normally distributed.

Through the bill-of-materials, the demandtransmission process can be represented mathe-

matically as follows:

Let uij ¼ the number of component i needed for

each SKU j;dij¼ the indicator variable,

¼ 1 if component i is used in SKU j;¼ 0 otherwise;

lðj; hÞ ¼ mean demand per unit time period for

SKU j at store h;


mðj; hÞ ¼ variance of the demand per unit time

period for SKU j at store h;fiðg; hÞ ¼ the proportion of the requirement of

component i from store g at its downstream

store h;DOWNST ðg; iÞ ¼ downstream Stores of store gwhich supplies components i;DOWNST ðh; jÞ ¼ downstream Stores of store hwhich supplies components j.

From Fig. 2, the following equations can be ob-

tained (Garg, 1999):

lði; gÞ ¼X

h2DOWNST ðg;iÞ

Xj

dijlijfiðg; hÞlðj; hÞ; ð39Þ

mði; gÞ ¼X

h2DOWNST ðg;iÞ

Xj

dijl2ijf

2i ðg; hÞmðj; hÞ: ð40Þ

4.2. Nominal and actual replenishment lead times of

sites

In the case that there are a unique output store

and a unique input store within a site, e.g., a non-

assembly site, the nominal lead time (NLT) of this

site is defined as its output store�s actual replen-

ishment lead time (ALT) under the conditions that

its input store has a stock-out, but the supplier of

the input store has available components. Thus,NLT of a site equals to the summation of input

store NLT and output store NLT.

The actual lead time of a non-assembly site is

the actual lead time of its output store when its

Fig. 2. The illustration of a demand transmission process.

input store has a stock-out and the supplier of thisinput store also has a stock-out. Thus, ALT of a

site equals to the summation of input store ALT

and output store NLT.

When a site has more than one output stores,

the corresponding NLT and ALT of this site

would be the ones with the maximal values among

several output stores� NLTs and ALTs, respec-

tively, under above conditions.For assembly sites, an order is triggered by an

output store which has a BOM consisting of more

than one type of SKU, the assembly operation will

only proceed when sufficient quantities of all input

SKUs are available. These input materials have

different nominal shipping lead time, and delays

resulting from the availability of materials at the

supplying sites differ.In terms of above definitions, for assembly sites,

we have:

NLT ¼ maxj2OS

maxi2UPST ðjÞ

½NLT ðiÞ��

þ NLT ðjÞ; ð41Þ

and

ALT ¼ maxj2OS

maxi2UPST ðjÞ

½ALT ðiÞ��

þ NLT ðjÞ; ð42Þ

where OS is the set of output stores in a site,

UPST ðjÞ is the set of directly upstream supplying

stores j.If some of the upstream stores are empty or

stock-out, denoted by the set E � UPST ðiÞ, actualreplenishment lead time (ALT) at store i can be

expressed as follows:

ALT ðiÞ ¼ NLT ðiÞ þmaxj2E

ðsjÞ; ð43Þ

where sj denotes the additional delay at store

j 2 E.In general, the quantity sj is quite intractable.

Here, we use an approximation given by Ettl et al.

(2000). At store j, suppose there is zero on-hand

inventory, and Rj orders are being processed. Then

sj can be approximated as (through a birth–deathqueue analysis):

sj ¼ eLjrj; where rj ¼EðBjÞ

pjðRj þ 1Þ ; ð44Þ

pj can be obtained from (37).


5. Numerical examples

In this section, two numerical examples are used

to illustrate the validity of the proposed frame-

work. In particular, the numerical comparison

between results from the analytical performance

evaluation model and those obtained from simu-

lation study is made.

Example 1. Retailer 1 fill rate

E½X �¼ q S E½T � l ¼E½X � E½T � k q ¼ k=l C2a C2

T C2T� wc

8 26 3.50 28 5 0.179 1 2.4 9.3 3.540

Now Nos E½No� r E½B� E½I � p f (Retailer 1 fill rate)

141.599 4.5 142.403 0.9987 17.15 7.87 0.176 0.814


E½X �¼ q S E½T � l ¼E½X �E½T � k q ¼ k=l C2a C2

T C2T� wc

8 20 3.50 28 3 0.107 1 2.4 9.3 3.522


84.530 4.5 85.012 0.9987 10.32 2.82 0.106 0.894


E½X �¼ q S E½T � l ¼E½X �E½T � k q ¼ k=l C2a C2

T C2T� wc

8 32 4.50 36 2 0.056 1 2.4 9.3 4.508


72.135 4.5 72.385 0.9992 8.80 3.65 0.055 0.945

Example 1

R1 fill rate R2 fill rate R3 fill rate

Ave SD Ave SD Ave SD

0.792 0.090 0.929 0.058 0.917 0.062

5.1. Performance analysis results

Fig. 3 gives a hypothetical logistic network with

fourteen stores: two supplier stores, four manu-

facturing stores, three assembly stores, two ware-

house stores and three retailer stores. It can be

seen it is a multi-stage logistic chain. Each product

consists of three type 1 components and two type 2components. The transshipment lead times are

assumed to be normal distributions. In this paper,

we are only concerned with the fill-rates of retail-

ers, which are indicators of end-customer service

level. For stores at other stages of the logistic

chain, their demands and lead times can be derived

through demand dependence and lead-time depen-

dence. The similar performance computation canbe achieved based on their inventory control para-

meters (batch order size q and reorder point level

S), mean and variance of facility service time (1=land r2

s ) and demand transmitted from downstream

stores. The corresponding input parameters aregiven in Table 1 (Dong, 2001).

5.2. Example 1

In terms of the analytical results from previous

sections, the performance of three retailers in the

logistic chain can be calculated as follows:

The simulation model is written in STRO-BOSCOPE, which is a general-purpose discrete-

event simulation language based on activity

scanning and activity cycle diagrams (Martinez,

1998).

The corresponding simulation results are

given as follows (with 500 independent replica-

tions):

In this hypothetical example, the low fill rates at

retailers indicate that the batch ordering size q and

order-up-to level S are not chosen appropriately

Fig. 3. An example of an integrated logistic chain network.

Example 2

R1 fill rate R2 fill rate R3 fill rate

Ave SD Ave SD Ave SD

0.89 0.114 0.934 0.042 0.932 0.053


(e.g., not at their optimal values). This causes some

upstream stores become make-to-order mode, that

is, their on-hand inventory levels become zero atsome time points.

5.3. Example 2

By adjusting the order-up-to points of retailers

and the order size per batch to their optimal val-

ues, the performance of the logistic network can be

calculated as follows (It can be seen that the fill-rate performance of retailers are much better than

the previous one):


E½X � ¼q S E½T � l ¼ E½X �E½T � k11 24 3.50 38.5 5

Now Nos E½No� r E½B�115.870 6 116.338 0.9993 10.37


E½X � ¼q S E½T � l ¼ E½X �E½T � k9 26 3.50 31.5 3

Now Nos E½No� r E½B�94.963 5 95.439 0.9990 10.29


E½X � ¼q S E½T � l ¼ E½X �E½T � k14 23 4.50 63 2

Now Nos E½No� r E½B�189.170 7.5 189.527 0.9997 13.44

Similarly, the corresponding simulation results

are given as follows:

5.4. Comparisons

From the comparisons between results from the

analytical performance evaluation model and

q ¼ k=l C2a C2

T C2T� wc

0.078 1 2.4 9.218 3.511

E½I � p f (Retailer 1 fill rate)2.13 0.077 0.923

q ¼ k=l C2a C2

T C2T� wc

0.095 1 2.4 9.267 3.517

E½I � p f (Retailer 2 fill rate)

3.68 0.094 0.906

q ¼ k=l C2a C2

T C2T� wc

0.048 1 2.4 9.171 4.504

E½I � p f (Retailer 3 fill rate)

0.94 0.047 0.953

Table 1

Data sets used in examples

Input parameters Example 1 Example 2

DDR1 (Demand Distribution for Retailer 1) Exponential[12] Exponential[12]



SDR1 (Service Distribution for Retailer 1) Normal[3.5,0.3] Normal[3.5,0.3]



BOC1 (Batch Order Size for Component 1) 5 5

BOC2 (Batch Order Size for Component 2) 5 5

OLR1 (Order-up-to Level for Retailer 1) 26 24



ROPW (Re-Order Point for Warehouse) 32 32

SDW (Service Distribution for Warehouse) Normal[12,1] Normal[12,1]

ROPFPInv (Re-Order Point for Finished Product Inventory, i.e., store 6) 26 26

ROPCM1Inv (Re-Order Point for Component 1 Inventory, i.e., store 7) 18 18

ROPCM2Inv (Re-Order Point for Component 2 Inventory, i.e., store 8) 10 10

SDA (Service Distribution for Assembly) Normal[2,0.2] Normal[2,0.2]

EOQWR1 (Economic Order Quantity from Warehouse to Retailer 1) 8 11



EOQFPW (Economic Order Quantity from Finished Product Inventory to

Warehouse)

60 60

EPQFPInv (Economic Production Quantity for Finished Product Inventory) 42 42

EOQCM1Inv (Economic Order Quantity from Manufacturing to Component 1

Inventory)

25 25

EOQCM2Inv (Economic Order Quantity from Manufacturing to Component 2

Inventory)

10 10

SDCM1 (Service Distribution for manufacturing of Component 1) Normal[7.8,0.6] Normal[7.8,0.6]

SDCM2 (Service Distribution for manufacturing of Component 2) Normal[30,3.2] Normal[30,3.2]

BSR1 (Batch Size of shipment to customers from Retailer 1) 9 9



WOC (Warehouse Order Cost per unit of product) 10$ 10$

ROC (Retailers� Order Cost per unit of product) 15$ 15$

RMOC (Order Cost per unit Raw Material) 7$ 7$

UVW (Unit Value of the product stored at the Warehouse) 1$ 1$

UVR (Unit Value of the product stored at the Retailers) 5$ 5$

ICC (Inventory Carrying Cost) 2$/$/unit time 2$/$/unit time

BOC (Back Order Cost) 2.9$/unit 2.9$/unit

HCW (Holding Cost at Warehouse) UVW· ICC UVW· ICCHCR (Holding Cost at Retailers) UVR · ICC UVR· ICCSCMCM1 (Set-up Cost for Manufacturing of a batch of Component 1) 10$ 10$

SCMCM2 (Set-up Cost for Manufacturing of a batch of Component 2) 15$ 15$


those obtained from the simulation study, it can beseen that these results are close to each other. The

differences between results are less than 5% (rela-

tive error). Therefore, the simulation results show

that the proposed methodology is effective to lo-

gistic network performance analysis.

6. Conclusions

This paper presents an integrated analytical

framework for logistic chains that can be used to

model the different network topologies such as

serial, parallel, assembly and arborescent struc-

S k Analytical results Simulation results Difference Percentage

Example 1

Retailer 1 fill rate Simulation for Retailer 1 Retailer 1 fill

rate

Retailer 1 fill

rate26 5 0.814 0.792± 0.090 0.022 2.778%


rate

Retailer 2 fill

rate

20 3 0.894 0.929± 0.058 0.035 3.767%


rate

Retailer 3 fill

rate

32 2 0.945 0.917± 0.062 0.028 3.053%

Example 2


rate

Retailer 1 fill

rate

24 5 0.9227 0.89± 0.114 0.0327 3.674%


rate

Retailer 2 fill

rate

26 3 0.9058 0.934± 0.042 0.0282 3.019%


rate

Retailer 3 fill

rate

23 2 0.9526 0.932± 0.053 0.0206 2.210%


tures. Inventory policies are specified throughoutthe network, performance measures such as order

fill-rate at each store in the network, expected

number of back-orders, expected number of orders

waiting in the queue, expected inventory level, and

the stock-out probability of a store (or the prob-

ability of a customer is backlogged) in the network

can be analyzed by using queuing theory and

production authorization card control mechanism.The proposed framework is integrated since it

provides a unified basis to decompose the organi-

zation functionality into detailed sub-processes

and activities. Through this framework, the man-

ufacturing and logistic activities can be logically

integrated to provide a facility for effective design,

analysis and optimization of integrated logistic

chains.Currently, our framework is used for single-

product logistic chains and their performance

analysis only. Two aspects of potential extension

of this framework are to consider multi-product

environments and performance optimization issue.The multiple-class queueing system could be used

to analyze the performance of logistic chains for

cases of multi-product situations. The objective of

the optimization model could be minimizing the

total expected inventory cost throughout the net-

work while satisfying end-customer service-level

requirements.

Acknowledgements

The authors are grateful to anonymous referees

whose valuable comments helped to improve the

content of this paper.

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