Problem Description
Throughout my term paper research, I focus on a distribution system which provides
two classes of service. These two classes of service differ by their respective demand
lead-times, where the term demand lead-time refers to the time elapsed between the
arrival of an order and its fulfillment. My research is mainly based on a paper by Wang,
Cohen and Zheng (2002). This paper is motivated by an actual case study observed in a
semi-conductor equipment manufacturer producing online testers. These testers, in turn,
are used in capital-intensive production environments, where equipment downtime is
critical resulting in substantial costs. Although highly reliable, the testers are subject to
random parts failure. As a result the customers feel obligated to hold on-site inventory to
avoid the costly downtimes. To increase the efficiency of its service system, the firm has
introduced two classes of service: emergency and non-emergency class service. While the
non-emergency class service is slower due to the presence of a positive demand leadtime, it compensates customers with a substantially lower price. Obviously this reduced
price is justified by the fact that the presence of the demand lead-time allows the firm to
reduce its inventory. As a result, the customers have the flexibility to choose between the
two classes of service depending on their cost versus parts requirement trade-off.
Generally speaking, a customer who is ordering to replenish its on-site inventory would
use the cheaper non-emergency service whereas a customer ordering because of a
machine break-down (the customer either does not hold on-site inventory or its on-site
inventory has depleted) would use the emergency service.
The firm uses a two-echelon service logistics network which is composed of a central
repair depot (CD) and multiple regional distribution centers (DC’s) which replenish their
inventory from the central depot. The figure below illustrates the general structure of the
service parts logistic network of the firm.
Depot
Repair Center
0
1
2
Customer
r
arrivals
Depot inventory
center
Distribution Centers
Customer
arrivals
Good part
Defective
N
Customer
arrivals
Figure 1: Material flows in the two-echelon service network
When an emergency customer arrives at the DC, the DC satisfies (or backlogs) the
customer immediately and at the same time places an emergency replenishment order
with the CD. On the other hand, when a non-emergency order arrives at the DC, the DC
does not satisfy (or backlog) the order until a demand lead-time passes. The DC places a
non-emergency replenishment order with the CD and the CD fills this non-emergency
order in a just-in-time manner. What is meant with a just-in-time manner here can be
explained as follows: assume the demand lead-time quoted to a non-emergency customer
is 20 days whereas the transportation lead-time between the CD and DC is 5 days. Then
the CD will send the order on the 15th day so that the order arrives exactly at its due date.
Although the model is based on a repairable service parts distribution network, it is
obvious that the model can also be applied to non-repairables in which case the repair
facility would be replaced by an exogenous supply. In fact, the cases where customer
services is differentiated based on demand lead-time ranges from semi-conductor
industry to online booksellers which makes the problem extremely important. Chen
(2001) provides an excellent motivation on applications of these types of systems. Thus
my research will be based on the service parts network model introduced by Wang,
Cohen and Zheng (2002).In the next section, I will examine the related literature which
deals with similar systems or concepts.
Literature Review
The focus of the model that I investigate is on repairable service parts. Such parts are
very expensive and face a demand that is low and random. Therefore the model assumes
independent Poisson demand arrivals at each DC, a one-for-one replenishment policy at
all locations and i.i.d. replenishment lead-times at the central depot.
A single location service parts system was first considered by Scarf (1958) where
there exists only emergency service class. Scarf, efficiently solved the model by
observing that the replenishment process is equivalent to an M/G/∞ queue. This fact
makes Palm’s theorem applicable which states that in the steady the number of customers
waiting in the queue, which are the outstanding orders in our case, is Poisson distributed
with a mean equal to the arrival rate multiplied by the average service time. Using the
outstanding order distribution and the standard inventory balance equation (on-hand
inventory = base-stock level – outstanding orders), it becomes easy to derive performance
metrics such as on-hand inventory distribution and random customer delay. Later,
Sherbrooke and Feeney (1966) extend this model to allow for compound Poisson arrivals.
Pioneered by the classic METRIC model (Sherbrooke, 1968) many researchers have
studied service parts systems in the context of multi-echelon distribution systems. Some
other references are Graves (1985), Nahmias (1981), Axsater (1993) and Wang et. al.
(2000). However, as a result of the introduction of the non-emergency service class, the
standard inventory balance equation is no longer valid for the model I consider. This fact
is the key difference between the literature described above and the model that I consider.
The notion of demand lead-time was first introduced by Simpson (1958) under the
name “service time” for base-stock, multi-stage production systems. Hariharan and
Zipkin (1995) then gave it the name demand lead-time to describe inventory/distribution
systems where customers do not require immediate delivery of orders but allow for a
fixed delay. The key observation of both papers is that the presence of a demand leadtime results in a reduction of the inventory required for achieving a desired service level.
Obviously this fact also applies to the system I consider but the existence of the two
service classes and their interaction makes the system more complex preventing to adopt
their approach. Moinzadeh and Aggarwal (1997) consider a two echelon system with two
modes of inventory replenishment. Although at first glance the systems seem similar, in
their case the two order classes differ only in their transportation lead-times from the CD
to DC. Otherwise, all orders, regardless of their class, are satisfied on a FCFS basis.
Therefore both classes will stochastically experience the same random delay. On the
other hand in the system I consider, orders are satisfied on a FDFS (first-due-first-serve)
basis. As a result the two classes are likely to experience different random delay and thus
different service levels.
Another way to differentiate customer service is inventory rationing. In these policies
customers are differentiated into priority levels based on measures such as lost sales or
backorder costs, annual sales or profitability. The usual assumption is that when on-hand
inventories drop below a certain level – usually called critical level, rationing level or
threshold level of the associated customer class- the demands of the lower priority classes
are rejected with the expectation of future high priority class customer demands. As seen
this approach is in sharp contrast with the model I described in which customers are
differentiate based on demand lead-time. In other words, the literature of inventory
rationing seems irrelevant at first glance. However inventory rationing is very important
to me because of the specific research question I address. Therefore, I will focus on the
inventory rationing literature highlighting important issues to consider.
Veinott (1965) was the first to consider the problem of several demand classes in
inventory systems. He analyzed a periodic review inventory model with n demand classes
and zero lead-time and introduced the concept of a critical level policy. Topkis (1968)
proved the optimality of this policy both for the case of backordering and for the case of
lost sales. He made the analysis easier by breaking down the period until the next
ordering opportunity into a finite number of subintervals. In any given interval the
optimal rationing policy is such that demand from a given class is satisfied from existing
stock as long as there remains no unsatisfied demand from a higher class and the stock
level does not drop below a certain critical level for that class. The critical levels are
generally decreasing with the remaining time until the next ordering opportunity.
Independent of Topkis, Evans (1968) and Kaplan (1969) essentially derived the same
results but for two demand classes. In his paper Kaplan (1969) suggested to let the critical
level depend on the time until next replenishment. A single period inventory model where
demand occurs at the end of a period is presented by Nahmias and Demmy (1981) for
two demand classes. This work was later generalized by Moon and Kang (1998).
Nahmias and Demmy (1981) generalized their results to a multi-period model with zero
lead-times and an (s,S) inventory policy. Atkins and Katircioglu (1995) analyzed a
periodic review inventory system with several demand classes, backordering and a fixed
lead-time; where for each class a minimum service level was required. For this model
they presented a heuristic rationing policy. Cohen, Kleindorfer and Lee (1998) also
considered the problem of two demand classes, in the setting of a periodic review (s,S)
policy with lost sales. However, they did not use a critical level policy. At the end of each
period the inventory is issued with priority such that stock is used to satisfy high-priority
demand first, followed by low-priority demand.
The first contribution, considering multiple demand classes in a continuous review
inventory model, was made by Nahmias and Demmy (1981). They analyzed an (s,Q)
inventory model, with two demand classes, Poisson demand, backordering, a fixed leadtime and a critical level policy, under the crucial assumption that there is at most one
outstanding order. This assumption implies that whenever a replenishment order is
triggered, the net inventory and the inventory position are identical. Their main
contribution was the derivation of approximate expressions for the fill rates. In their
analysis they used the notion of the hitting time of the critical level, i.e., the time that the
inventory reaches the critical (or rationing) level of the low priority class. Conditioning
on this hitting time, it is possible to derive approximate expressions for the cost and
service levels. Dekker, Klein and De Rooj (1998) considered a lot-for-lot inventory for
item model with the same characteristics, but without the assumption of at most one
outstanding order. They discussed a case study on the inventory control of slow moving
spare parts in a large petrochemical plant, where parts were installed in equipment of
different criticality. Their main result was the derivation of approximate expressions for
the fill rates of the two demand classes. The model of Nahmias and Demmy is analyzed
in a lost sales context by Melchiors, Dekker and Klein (1998).
Ha (1997b) discussed a lot-for-lot model with two demand classes, backordering and
exponentially distributed lead-times and showed that this model can be formulated as a
queuing model. He showed that in this setting a critical level policy is optimal, with the
critical level decreasing in the number of backorders of the low-priority class. Moreover
he proved that it is optimal to increase the stock level upon arrival of a replenishment
order even if there are backorders for low-priority customers if the inventory level is
below the critical level.
A critical level policy for two demand classes where the critical level depends on the
remaining time until the next stock replenishment was discussed by Teunter and Klein
Haneveld (1996). A so-called remaining time policy is characterized by a set of critical
stocking times L1, L2,…; if the remaining time until the next replenishment is at most L1
no items are reserved for the high-priority customers, if the time is between L1+L2 then
one item should be reserved, and so on. They first analyze a model, which is the
continuous equivalent of the periodic review models by Evans (1968) and Kaplan (1969).
Teunter and Klein Haneveld also present a continuous review (s,Q) model with nonnegative deterministic lead-times. Under the assumption that an arriving replenishment
order is large enough to satisfy all outstanding orders for high-priority customers, they
derived a method to find near optimal critical stocking times. They showed such a
remaining time policy outperforms a simple critical level policy where all critical levels
are stationary.
Ha (1997a) considered a single item, make-to-stock production system with n demand
classes, loss sales, Poisson demand and exponential production times. He modeled the
system as an M/M/1/S queuing system and proved that a lot-for-lot production policy and
a critical level rationing policy is optimal. Moreover the optimal policy is stationary. For
two demand classes he presented expressions for the expected inventory level and the
stockout probabilities. To determine the optimal policy he used an exhaustive search, and
made the assumption that the average cost is unimodal in the order-up-to level. Ha (2000)
generalized his policy for Erlang distributed lead times where he stated that the critical
level policy would also provide good results under generally distributed lead-times.
Later on, Dekker, Hill, Kleijn and Teunter (2002) analyzed a similar system, with n
demand classes, lost sales, Poisson demand and general distributed lead-times. They
modeled this system to derive expressions for the average cost and service levels. In
addition the authors have derived efficient algorithms to determine the optimal critical
level, order-up-to level policy, both for systems with and without service level
constraints. Moreover they presented a fast heuristic approach for the model without
service level constraints. In this model the different demand classes are characterized by
different lost sales costs. Dehpande, Cohen and Donohue (2002) consider a rationing
policy for two demand classes differing in delay and shortage penalty costs and demand
arrivals under a continuous review (Q,r) environment. They do not make the assumption
of at most one outstanding order which makes the allocation of arriving orders a major
issue to consider. They defined a so-called threshold clearing mechanism to overcome the
difficulty of allocating arriving orders as well as providing an efficient algorithm for
computing the optimal policy parameters which are defined by (Q,r,K), K being the
threshold level.
Finally, I want to discuss the research paper that I base my analysis on. Wang, Cohen
and Zheng (2002) analyze the two echelon system in order to derive the transient and
steady performance metrics of the system. Beginning with a single location system they
derive expressions for the inventory level distribution and random customer delay. To
derive the inventory level distribution they partition the demands in (0,t) into the
following three groups depending on the way they affect the inventory level at t, I(t). The
first group consists of all class 1 (emergency) customers arriving in (0,t). Only those
demands whose corresponding replenishment orders have not been received by time t has
a net effect on I(t)- each decrease it by unit.
The second group consists of all the class 2 (non-emergency) customers arriving in
(0,t-T) where T denotes the demand lead-time. As seen these are the customers, just as in
the previous case for class 1 customers, whose due date has passed by time t. therefore
only those whose corresponding replenishment order have not been received by time t
has a net impact on I(t)- each decrease it by one unit.
The last group consists of all class 2 customers arriving in (t-T, t). Since these are the
customers whose due date has not come by time t only those whose corresponding
replenishment order have been received by time t has a net impact on I(t)- each increase it
by one unit. As can easily be seen all the arrivals of the mentioned groups constitute
Poisson variables with associated means. As a result the inventory level distribution at
time t, I(t), can be expressed in terms of the base stock level and the three respective
Poisson random variables. Finally by taking the limit of t as it approaches infinity they
derive the steady state inventory level distribution of the single location system.
With a similar but more involved analysis the authors derive the random delay
incurred by each customer class. As a result they come up with the important observation
that the service level of class 2 customers, that is the non-emergency class, is higher than
the class 1 customers, that is the emergency class, as long as there is a positive
probability that the replenishment order corresponding to a non-emergency customer
arrives before its demand due date.
After deriving the steady state performance metrics for the single location system, the
authors extend the model to the two echelon system. By following an approach similar to
the well-known METRIC, they decompose the multi-echelon network into single location
subsystems. After the analysis of the two-echelon setting, they conduct an optimization
study to see the effects of the introduction of a non-emergency service class. As a result it
is seen that the system with two service classes results in significant cost savings in terms
of inventory which obviously comes from the presence of the non-zero demand leadtime. The research question that I am addressing differs from the ones addressed by the
authors although it is related. I will discuss my specific research question in the next
section.
Specific Research Questions and My Further Research
There are some further research issues about the paper of Wang, Cohen and Zheng.
Generalization issues might be the first step to take. Multiple non-emergency service
classes with different demand lead-times can be considered. Besides, the usual
assumptions of the repairable parts system can also be generalized by allowing
condemnations and enforcing a capacity constraint on the repair facility. Also the demand
arrivals are assumed to be i.i.d. which may not be the case especially when the model is
applied to non-repairables. In addition to these, the authors themselves provide two other
research directions. The first one is related to pricing issues, that is, how the benefits of
the introduction of the non-emergency service class will be shared with the customers
given the ability to quantify such benefits. Another interesting issue is stated as
incorporating the price elasticity of demand.
Despite the available extensions of the model, the research I will conduct differs from
all. I will rather try to incorporate inventory rationing with the investigated policy. In
other words, I will try to incorporate the two policies of customer service differentiation
with the hope of obtaining better control of the overall system. The main motivator of this
idea is the fact that when differentiating customer service on the basis of delivery leadtimes you will incur a higher service level for the non-emergency service class given that
there exists a positive probability that the replenishment order of a non-emergency class
customer arrives before the demand lead-time. Although this might in some cases be
what is desired, usually a higher service level would be desired for the emergency or high
priority customers. In addition, it might be the case that a joint policy will work better
than a single policy which differentiates service only based on demand lead-time. That is
rationing inventory might result in further benefits in terms of increased profit by
reserving some part of the inventory to the more profitable customers. Another
consideration is the fact that given the customer is willing to wait you may choose to
ration the inventory so that certain part of the inventory will be reserved for the customer
who cannot wait. Given a certain stock level, this type of a system might increase the
service level of emergency customers while not decreasing the service level of the nonemergency customers who are willing to wait for a certain period of time.
Briefly, the policy will work like this: the supplier will offer two service levels which
are the emergency and non-emergency class service levels where the non-emergency
service level allows a demand lead-time. In addition, the supplier will ration the
inventory whenever the on-hand inventory will drop below a critical level. Here, the high
priority demand class will be the emergency service customers which both result a higher
backorder costs and higher revenues because of the higher price of this service. Demand
not satisfied at the due date (immediately for emergency class and a demand lead-time
after order arrival for non-emergency class customers), is assumed to be backlogged.
Hence, when the inventory level drops below a certain level, the orders of the nonemergency class will be backordered with the expectation of future emergency class
arrivals. The research question that I ask is whether the system performance can be
improved by using a joint policy. In other words, given specified target service levels for
both classes, can these service levels be achieved through inventory rationing and
differentiating customer service on the basis of demand lead-time simultaneously? If this
is the case, how can the optimal policy be defined?
My intuitive answer, which also depends on my prior knowledge about inventory
rationing policies, is that indeed the system performance can be improved by
simultaneously applying both policies. Obviously, introducing a demand lead-time
decreases the replenishment lead-time which in turn decreases the required inventory to
achieve a specified service level. However this approach results in a higher service level
for the non-emergency service level. Therefore, Wang, Cohen and Zheng (2002), when
conducting their optimization study, only constrain the emergency class. But it might be
the case that different combinations of service levels are required, usually a higher
service level being required by the emergency or high priority class. Because of that, I
believe, simultaneously using inventory rationing and differentiating customer service on
the basis of demand lead-time can be a better way of controlling inventory in such cases,
resulting in lower base-stock requirements.
Solution Methodology
I will begin by modeling the single location problem to derive some performance
measures of this simpler system. For the sake of simplicity, I will begin with
deterministic replenishment lead-times and a fixed demand lead-time. In addition, I will
adopt the usual assumptions of the repairable service parts literature, that is, I will assume
Poisson arrivals and a one-for-one replenishment policy. Therefore the problem at the
first step will be to derive the performance metrics of this simplified system. Later on, I
will try to extend the analysis to a two-echelon system and determine the optimal system
parameters.
Obviously, the procedure reminds that of Wang, Cohen and Zheng (2002). However
the addition of the inventory rationing policy complicates the situation. Several other
issues to consider are the type of the rationing policy and the treatment of arriving orders
in terms of allocation between the two classes. Again, for the sake of simplicity, I will
first consider a static rationing policy, that is a rationing policy with static critical levels.
To avoid the consideration of how to allocate arriving orders at first step, I might make
the usual but weak assumption that at most one outstanding order can exist in the system
at any time. However, I must note that by using a dynamic rationing policy, it is most
likely that better results are obtained, since a dynamic policy will work at least as good as
a static one.
Conclusions
Through this paper, I first investigated a two-echelon distribution system with two
classes of service which differ in their demand lead-time. Based on the analysis in a
recent paper, it is seen that such systems outperform traditional systems where no
demand lead-time is allowed. Therefore I raised the research question whether system
performance can further be improved by incorporating inventory rationing in this
practice. Inventory rationing has also proved to outperform traditional inventory systems
incurring multiple classes. Thus it is likely that such a combination of customer
differentiation procedures will result in a better system performance. I believe the
detailed consideration of multiple demand classes is important since such cases exist in
many industries and I hope to obtain results as soon as possible.
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