Value of Information in Capacitated Supply Chains.
We incorporate information flow between a supplier and a retailer in a two-echelon model that captures the capacitated setting of a typical supply chain. We consider three situations: (1) a traditional model where there is no information to the supplier prior to a demand to him except for past data; (2) the supplier knows the (s, S) policy used by the retailer as well as the end-item demand distribution; and (3) the supplier has full information about the state of the retailer. Order up-to policies continue to be optimal for models with information flow for the finite horizon, the infinite horizon discounted and the infinite horizon average cost cases. Study of these three models enables us to understand the relationships between capacity, inventory, and information at the supplier level, as well as how they are affected by the retailer's (S - s) values and end-item demand distribution. We estimate the savings at the supplier due to information flow and study when information is most beneficial.
Introduction
The industrial supplier-customer relations have undergone radical changes in recent years
as the philosophy behind managing manufacturing systems continues to be influenced by
several Japanese manufacturing practices. As more organizations realize that successful in-house
implementation of Just-In-Time alone will have limited effect, they are seeking other
members of their supply chain to change their operations. This has resulted in a certain
level of co-operation, mainly in the areas of supply contracts and information sharing, that
was lacking before. This is especially true when dealing with customized products, and is
most commonly seen between suppliers and their larger customers.
Our primary motivation to develop the models in this paper came from two sources.
One was from the food industry, when North Side Foods - the largest supplier on the East
Coast for McDonald's - was asked to take the lead in integrated supply chain management.
One of the issues was to improve the understanding of the interactions between informa-
tion, inventory and capacity (because it affects customer service and cost) and to provide
insights for the entire McDonald's supply chain, consisting of 160 suppliers. Consequently,
we take the supplier's point of view in this paper. The second (and a broader) motivation
to study the benefits of information arose because of differing reactions to Electronic
Data Interchange (EDI) benefits from industrial sources: while some were very happy with
improved information, others were disappointed at the benefits (see [3] and [20]). Thus,
while information is always beneficial, we would like to know when it is most beneficial
and when it is only marginally useful. In the latter case, some other characteristic of the
system, such as end-item demand variance or supplier capacity may have to be improved
before expecting significant benefits from information. Thus, our computational efforts will
be directed towards understanding some of these issues.
The degree of co-operation varies significantly from one supply chain to another. The
information sharing could range from generic (e.g. type of inventory control policy being
used, type of production scheduling rules being used) to specific (e.g. day-to-day inventory
levels, exact production schedules). This is similar to long term contracts, which may be
loose (e.g. 'customer will purchase a certain cumulative quantity over the next six months')
or tight (e.g. 'customer will purchase a certain quantity every Monday'); see Anupindi and
Akella [1] and Anupindi and Bassok [2]. We develop and analyze new models of these recent
developments in information sharing because traditional inventory models were developed
under demand and informational assumptions that still hold, perhaps, when dealing with
commodity products or at a retail level but no longer universally hold in the manufacturing
sector.
Previous research in this area, i.e. incorporating information flow into inventory control
and supply chains, has mainly followed two approaches. The first approach used by Scarf
[19], Iglehart [12], Azoury [4], and Lovejoy [17] was to use the information to more accurately
forecast the demand distribution using Bayesian updates. The second approach, followed
in this paper, incorporates the information flow by developing new analytical models. This
approach was used by Zheng and Zipkin [22] to analyze the value of information flow in a
two product setting. They showed that using the information about the outstanding orders
of both the products resulted in improvement of system performance. Zipkin [24] extended
this to a multi-item production facility. Hariharan and Zipkin [11] incorporated information
about order arrivals, termed demand leadtimes, into inventory control and studied its effect
on system performance. They observed that demand leadtimes behave in a fashion that
is exactly opposite to supply leadtimes. An increase in demand leadtime improves the
system performance exactly like a reduction in supply leadtime. Chen [6] studied the
benefit of information flow by observing that echelon-stock policies and installation-stock
policies in a multi-echelon environment differ in their informational requirements. He also
studied the effect of parameters such as number of stages, leadtimes, demand variance, and
customer service on the benefit of information flow. Traditional capacitated models have
recently received attention: Federgruen and Zipkin [7] [8] provide the optimal policy for
the stationary demand case; Glasserman and Tayur [9], [10] study multi-echelon systems
operated via a base stock policy; Kapuscinski and Tayur [13] study the periodic demand
case.
Demand
Control
Inventory
Supplier

Figure

1: Model 0 - The traditional supply chain.
In this paper, we study two different levels of information flow in a two-echelon capacitated
supply chain (and also compare them to a base case of no information). We are
interested in policies for a supplier who faces demand from a customer who is in reality a
middleman facing final (end-item) demand. In some sense, we are broadening the 'informa-
tion structure' (see concluding remarks of [11]) by providing differential abilities to better
anticipate the next demand. Model 0 (shown in Fig. 1) represents the traditional view of
supply chains in which the demands are seen as originating from a black box. The supplier
is not aware of the logic behind the demand process, and so there is no 'advance warning' of
the demand, and has to rely on past data. When there is some co-operation in the supply
chain, the producer finds out the real driver behind his demand process. As represented
in Model 1 (shown in Fig. 2) the customer is using an (s; S) policy for inventory control
while facing an i.i.d. (independent and identically distributed) demand process. Here, the
supplier can improve his prediction of the demand by incorporating the value, the
Control
Inventory
i.i.d Demand
Customer
Demand
Supplier

Figure

2: Model 1 - Supply chain with some information flow.
time that has elapsed since the last demand from the customer and the distribution of the
end-item demand. Model 2 (shown in Fig. 3) represents a tighter co-operation in the supply
chain. In every period the supplier also knows the current inventory level at the customer,
either because of an EDI link or because the customer explicitly provides the information
on a periodic basis. This helps the supplier to predict the demand process more reliably,
and comes close to an 'early warning' or an `advance ordering' situation.
Control
Inventory
i.i.d Demand
Customer
Demand
Day to Day Inventory levels
Supplier

Figure

3: Model 2 - Supply chain with full information flow.
Some of our qualitative results are as follows. Information is always beneficial, and as
it becomes more precise it appears to retain its incremental value. However, when the end-
item demand variance is high, or the values of customer are very high or very low,
information is not very beneficial. On the other hand, if the end-item demand variance is
moderate, the values are not extreme and holding costs (at the supplier) are high, the
benefits of information are high. Similarly, information is not as beneficial if the supplier's
capacity is low as compared to when his capacity is high. The benefits of information, as
penalty costs (at the supplier) increase, appears to first increase and then drop off, especially
when the end-item demand variances are moderate to high. Thus, before embarking on an
EDI implementation, the supplier should verify his capacity and the fixed cost imputed onto
his customer (resulting in the customer's while the customer should attempt to
reduce the end-item demand variance. When have reduced to moderate
values, and the supplier has moderate to high capacity, the information link is bound to be
most beneficial. We justify these insights and other results in subsequent sections.
The rest of the paper is organized as follows: In section 2, we develop optimal policies
for both models 1 and 2. We show that modified order up-to policies are optimal for the
finite horizon, infinite horizon with discounted cost, and infinite horizon with average cost.
Since the results (and the proofs) are similar for both models, we only present detailed
proofs for Model 1. In section 3, we present structural properties of the optimal order
up-to levels. In section 4, we first present a fast recursive solution procedure to compute
the optimal parameters for the uncapacitated problem. Then, for the capacitated situation
we develop and validate the Infinitesimal Perturbation Analysis (IPA) solution procedure
to compute the optimal parameters. In section 4, we also provide solution procedures for
Model 0 and Model 2. Using these solution procedures we estimate the savings due to
information flow. A detailed computational study reveals interesting insights with regard
to when information is most beneficial. Section 5 contains these computational results and
insights. As by-products, we also gain some insight on how to manage multiple customers
and obtain a simple approximation to compute the optimal parameters. We conclude in
section 6 by summarizing our findings and discussing future directions of research.
Models
We consider a periodic review inventory control problem (at the supplier level) with linear
holding and penalty costs (h and b per unit), finite capacity (C), and no purchase or salvage
costs. The unit variable purchase cost does not affect our results; see [13]. The customer
is following a (s; S) policy. The (s; S) policy was shown to be optimal for infinite horizon
inventory problem with periodic review, fixed ordering cost, linear holding and penalty costs,
constant leadtime, facing stationary demands and full backlogging by Scarf [18]. This is the
setting of the customer.
Let us first consider Model 1 where the supplier knows that the customer is using a
(s; S) policy and is also aware of the end-item demand distribution. The sequence of events
in every period is as follows. The supplier decides on his production quantity for the period.
The customer realizes the end-item demand for the period and after satisfying the demand,
if her inventory level is below s, she places an order with the supplier to bring her inventory
level to S. If the supplier cannot satisfy the full order of the customer, we assume that the
customer acquires the missing part of the order elsewhere and brings her inventory level
to S. All this happens with no lead time. fixed non-zero delivery lead time does not
change the analysis.) The supplier pays holding cost if there is inventory at his level at the
end of the period or pays penalty cost if a portion of the demand was lost (i.e. satisfied
from elsewhere). The objective is to find a control policy for the supplier that minimizes
the total holding and penalty costs.
At the supplier level, incorporation of this information into inventory models results in
a non-stationary demand process although the end-item demand is stationary. From the
supplier's point of view, every period is in one of k possible `states' (see remark below). State
is defined as the number of periods since the last demand. Let p i be the probability that a
demand to the supplier will be realized in state i. If the demand is realized in state i, it is
from a cumulative distribution function (cdf) \Phi i (:) (and probability distribution function,
pdf, OE i (:)) with mean  i and the next period will be in state 1. If the demand is not realized,
then the next period will be in state i + 1. We further assume that p i  p
and \Phi i (:)  st \Phi i+1 (:) i.e. the chance and quantity of realization of demand increases in
time since last demand ('monotonicity'). If a period reaches state k, a demand will be
realized in that period from distribution \Phi k (:). This monotonicity assumption is satisfied
if the end-item demand to the customer has an Increasing Failure Rate (IFR) distribution
(such as Uniform, Normal or Erlang distributions); see Appendix A for a proof. We further
assume that the end-item demand has a finite mean, and so
Remarks. We have made two assumptions above that simplify our analysis. (1) We assume
a finite k; this is satisfied if the end-item demand is non-zero in every period, bounded below
by some ffl ? 0. Then we can choose
1. Alternatively, we can ignore the states
larger than k
is the end-item demand
and ffl sufficiently small. (2) We allow the customer to obtain all the requirements not met
from this supplier at a higher price from elsewhere (or due to overtime at this supplier),
and pass on the cost difference to this supplier. This cost differential is what b represents.
Both these assumptions are reasonable based on our industry experience. We believe that
without the finite k assumption, the analysis will be more technical, but provide no sharper
insights on benefits of information. Similarly, if the customer at the time of placing her order
could not be guaranteed of the full amount demanded, then she will not follow a simple
stationary (s,S) policy. Again we believe, the analysis will become complicated, without
adding significant insights into the benefits of information flow. Even the assumption of
fixed lead time is reasonable, since transportation can be assumed to be fairly reliable. In
fact, in the McDonald's supply chain, the period may be considered as one week and the
transportation time is less than two days, thus allowing the 'zero lead time' assumption to
be applicable directly.
In Model 2, the state captures the inventory level at the customer exactly. As we shall
see in section 4.2, Model 2 is a straightforward extension of Model 1 from structural point-
of-view, and some of the demand assumptions made above can be relaxed. In what follows,
we will deal with Model 1 and provide a summary of results for Model 2 in section 4.2.
In the base case, Model 0, the supplier faces a demand process that we assume comes
from an independent and identically distributed process, both in inter-arrival time as well as
order quantity, whose distribution is determined from past data. (Thus this model assumes
that the supplier does not know that the customer is using a (s,S) policy.) The policy used
to manage under Model 0 assumptions and the method to compute the specific parameters
are described in section 4.3.
Note that at the supplier-customer interface there is an implicit fixed ordering cost in
our model, and hence the customer uses a (s; S) policy (with optimal parameters based on
her holding and penalty costs). Once the supplier uses the optimal policy, which we derive
for models 1 and 2, the entire supply chain is 'optimal'. This is a pleasing property obtained
due to the simplicity of our setting.
The rest of this section is devoted to showing the structure of optimal policies for Model
1. As is standard practice in inventory theory, we start with the finite horizon case.
2.1 Finite Horizon : Model 1
As defined in [7] [8], a modified order up-to policy with level z is one where if the inventory
level is less than z, we raise it to z; if this level cannot be reached, we exhaust the available
capacity; if the inventory level is above z, we produce nothing.
Property 1 For the one period problem and any starting state i  k, a modified order
up-to policy is optimal.
Proof Suppose we have x units on hand at the beginning of the period. We can place an
order and bring the level to y(x) such that y(x) 2 [x; x+C]. In state i, the one period cost
1 (y) is
where
y
Since the holding and penalty costs are linear, L i (y) is convex, J i
1 (y) is also convex and its
minimum occurs at \Phi i (y i
stands for max(0,x).) The optimal control
policy is as follows:
which is a modified order up-to policy. 2
Consider an n-period problem with initial state i. If the demand is realized in period n,
then we have a (n \Gamma 1)-period problem with initial state 1, otherwise we have a (n \Gamma 1)-period
problem with starting state (i + 1). Let  i be the demand in period i, with cdf \Phi i (:). The
optimal n-period cost V i
We use to denote first and second derivatives respectively. See [13] for a proof of the
existence of the derivatives we use below.
Property 2 For all values of n:
1. J i
n (y) is convex in
2.
(x) is convex in x; and
3. a modified order up-to policy is optimal.
Proof These statements can be proved using induction. From our assumption that
we know that V i
0 (x) is convex in x and V 0 i
\Gammab. Assume that J i
are convex and V 0 i
\Gammab. Notice that J i
n (y) is
y
(y)]:
Using Leibniz's rule, the derivatives of these cost functions,
are:
\Gammab;
0;
The non-negativity of J 00 i
n follows from the convexity of V i
(x) and the fact that
(x) is convex. Based on the fact that if f(t) is convex then
also
n (x) is also convex. Also since V 0 i
(y) is convex in y, it is optimal to order up-to y i
that minimizes J i
(y). If we do not have
enough capacity to reach y i
n it is optimal to go as close to y i
n as possible. 2
n be the quantities
defined above when the capacities are C 1 and C 2 respectively. If C 2  C 1 then
Proof We will first prove (a) and (b) by induction. They are obviously true for n=0.
Assume they are true for
n (x), and using (b) for
easily established that C 1 J
(x). Furthermore from the convexity
of J n and inductional assumption C 1 J
Using the expression for
n (x) from the proof of Property 2, and the observation that if
it is easily established
that
(x). So, by induction parts (a) and (b) of the property are true
for all n. Since both C
n (x) are convex and C 1
n (x), we have
n . This proves part (c). 2
2.2 Infinite Horizon : Model 1
We consider both the discounted cost criterion as well as the average cost criterion.
2.2.1 Discounted Cost
Under this criterion, the costs in the future are discounted by a factor 1. The finite
horizon recursive relation for V i
(x) is modified as below to include fi.
(y)]:
Let us define V i
(x); the objective is to minimize V i
(x).
Property 4 For any finite x, V i
(x) is finite.
Proof Since the single period cost L i (x) is non-negative for all i and x, V i
n (x) is non-decreasing
in n. Consider the policy that is order up-to zero. For this policy, the holding
cost in any period is less than hx and the penalty cost in any period is less than b i . The
total expected cost of this policy is bounded above by
1\Gammafi which is
finite. Since fV i
(x)g is a non-decreasing sequence with a finite upper bound, lim
exists and is finite. Therefore, the expected cost of the optimal policy, if it exists, must be
finite. 2
Property 5 For each state i, the optimal policy is modified order up-to y i
.
(x) is finite, from pp. 210-212 in Bertsekas [5], the optimal policy exists
and it satisfies the following recursion:
(y)]: (1)
Since L i (:) and yh are continuous, V i
(:) is convex as it is a pointwise limit of convex
functions. Therefore the right hand side of eq. (1) is convex and there exists y i
that minimizes it. As y i
n and since lim sup n!1 fy i
n g is finite (see Property 6
must be finite. 2
2.2.2 Average Cost
The following technical result is repeatedly used to show several interesting properties of
our models.
Property 6 lim supfy i
Proof We will establish this result by first showing that lim supf 0 y i
1. We will prove this by contradiction. Assume that there exists a sequence fn t g
such that 0 y i
for all us choose a particular
n t and compare the cost of the optimal policy,
, to the cost of a policy that is order
up-to zero and starts with zero inventory. The cost of these policies is different only for a
few periods early in the horizon when the inventory at the beginning of a period is positive.
As soon as the inventory at the beginning of a period reaches zero, both the policies operate
the same way resulting in equal costs. Let the number of periods over which the costs differ
be l t . Let
be the demands observed when the inventory is positive. Let
be the inter arrival times of these demands. Thus l d demands occurred during
the l t periods and
For ease of presentation, we will assume that the initial state s n t
1. The proof for the
case when the initial state is different from 1 requires minor adjustments. The demands
are i.i.d. and a 1 ; a
are also i.i.d. Let us assume these demands have
mean  and the inter arrival times have mean . These demands satisfy the condition
and
. This implies that
+D l d
l d
Taking expectations, we get
l d
are i.i.d and l d is a stopping time, from Wald's equality, we have
The total inventory over these l t periods is
l d
l d
l d
Noting that D j 0
is independent of a
and l d is a stopping time, the expected
total inventory is
E(l d
l d
l d
Rewriting the terms in the expectation, the total expected inventory is
E(l d
We know that if two random variables X and Y are negatively correlated then E(XY )
E(X)E(Y are negatively correlated, the total expected
inventory is at least
E(l d
Upon simplification, the expected total inventory is at least
E(l d
The total expected inventory cost over these l t periods is greater than or equal to
E(l d
E(l d )E(l
The expected cost of order up-to zero policy with zero initial inventory is at most
+). The difference in the cost of the optimal policy and the cost of the order up-to
zero policy is greater than or equal to
which is quadratic and is positive for large values of 0 y 1
thus contradicting our assumption
of optimality. This proves that lim supf 0 y 1
n g can not go to infinity as n goes to infinity.
1. Since the sequence fy 1
n g is dominated from above by the sequence
(Property 3), we obtain lim supfy 1
1. Using similar arguments and handling
appropriately, we can establish that lim supfy i
For the average cost criterion, the objective is to determine a policy that minimizes
(:). To analyze this situation we will use the discrete version of the
problem (i.e. end-item demand is discrete). This implies that the possible inventory levels
are also discrete. Furthermore since we know that lim sup n!1 fy i
g is finite, the possible
inventory levels are f0; fy i
gg. The complete state
space Ag which is finite.
Property 7 For each state i, the optimal policy is modified order up-to " y i
.
Proof Since the state space is finite, the result follows from pp. 310-313 in [5]. 2
3 Structural Properties for Model 1
Property 8 When the capacity is infinite, 1, the order up-to levels for the finite
horizon are ordered. i.e. y i
Proof From the results of the one period problem, we have y i
Comparing
with
it is easily seen that J 0 i
(x), which in turn implies that V 0 i
By induction, we have 8n,
which implies y i
as the length of the horizon increases, the order up-to level for state i decreases.
Assume y i
. We will show by induction that the same relation holds for
n+1 and y i+1
are solutions to the equations:
(y
((y
We know that L 0
(y). Since  i is stochastically increasing and V 0 1
decreasing function of  i , we have
Based on
similar reasoning to the one used in proving Property 9 below, we obtain y i
n+1 . By
induction this relation holds for all n. 2
Property 9 The infinite horizon cost function satisfies V 0 i
Proof Recall that V i
and
(y)]:
Compare
(x) and J
(y)]:
Using the relations
and
we establish J 0 (k\Gamma1)
(x). This further implies that V 0 (k\Gamma1)
(x). Assume
that
kg. We will show that V 0 i
(x).
From definition
(y)]:
By simple comparison
and from the induction hypothesis we know that h
(y). Since
(y)  J
(x), we
have
and
(y)  J
implying that their linear combination must be greater or equal to J
(x). So, we have
implying that V 0 i
Corollary 1 A direct result of Property 9 is that the order up-to levels for the infinite
horizon (discounted and average cost) are ordered. That is y i
.
Let us assume that the i.i.d. demand that the customer has cdf \Psi(:) and pdf /(:). Let
be defined as minfijp i ? h
g. Let fl := inffyj\Psi(y)  b
g. Then the optimal solution
satisfies the following property.
the optimal order up-to levels satisfy
1.
2. y i
\Delta; and
3. y k
Proof 1. From Property 1, we know that the myopic solution in period i satisfies the
condition
When i is less than i   then we know that p i  h
implying that
the myopic solution is order up-to
zero. Under infinite capacity, the infinite horizon optimal solution is lower than the myopic
solution. Since a negative order up-to level is never optimal the optimal order up-to level
must be zero.
2. Clearly due to infinite capacity
. Observe that
(\Delta)
which is equal to h which is negative since i  i   . So
implying that
\Delta. By induction for all i  i   we have y i
\Delta.
3. When we reach state k, a demand at the supplier will certainly occur and this demand
will be bounded above by random variable from the distribution \Psi(:).
If part of the demand to the supplier is lost, the penalty cost is b per unit. The holding
cost on remaining inventory is at least h. Thus, in period k, we will never want to have
more inventory than the relation
. In other words, the
optimal order up-to level y k
Note It is possible to tighten the upper bound on y k
by estimating the holding cost
over periods adding it to the denominator in the right hand side of the
equation for \Psi(fl).
4 Solution Methods
In this section, we develop solution procedures for our models.
4.1 Model 1
To find the optimal solution analytically in a capacitated setting is hard even for the stationary
newsboy problem in a general setting. Further, even simple non-stationary cases
are analytically intractable. However, for the uncapacitated case, easy solution methods
have been presented by Karlin [14] and Zipkin [23] for the non-stationary problem with
periodic data. In the first part of this subsection, we present a simple procedure to estimate
the optimal order up-to levels for the uncapacitated case. For ease of presentation, in this
section we represent the infinite horizon optimal order up-to level in state i by z i .
4.1.1 Uncapacitated System,
We know from eq. (1) (Property 5) that the derivative of the infinite horizon cost will follow
the recursive relation
(y
From Property 7, we know that the optimal order up-to levels fz 1 are ordered.
Since we have infinite capacity, whenever
dx
Therefore
Similarly, whenever x  z i+1 implies V
(y
Using these two observations, we will develop a procedure that calculates V 0 i
(x) for all
values of i and x. Let W i
(y(x)).
We discretize the demand to evaluate the integral term. jg. We
assume that when a demand occurs, it is strictly positive. Since the demand has been
discretized, the lowest value the demand can take is \Delta. The derivative at every point for
every state can be estimated using the algorithm below.
values of i.
Repeat the following recursion for values of x starting at 1 and increasing by 1
and noting that L 0
x
x
At any point during the recursion, if W i (x) is negative, we set it equal to zero indicating that
x is below the optimal solution and the future cost does not depend on x. The recursion can
be stopped when W 0 k (x) becomes positive. However, if the derivatives at higher values are
also needed for further analysis, the recursion procedure can be continued until x reaches a
predefined limit.
Step 3 For each state i, find the order up-to level z
Remark. For faster execution, we can start with W i is the
solution to the equation \Phi 1 (z 1
4.1.2 Capacitated System
For the capacitated situation, we will develop a solution procedure using Infinitesimal Perturbation
Analysis (IPA). Let I be the inventory level after demand, demand, and
the state of the system in period n. If z i is the order up-to level in state i, the simulation
recursions are:
there is no demand
there is a demand
I
The derivative recursions are
dI n+1
if I n+1 ! z s n+1
or I n+1 ? z s n+1
and
To validate the derivatives for the finite horizon and the infinite horizon discounted, the
approach is similar to Glasserman and Tayur [9],[10]. To show that the IPA derivatives are
valid for the infinite horizon average cost case, using the approach similar to Kapuscinski
and Tayur [13], we need to show that each of the order up-to levels z are reached
infinite number of times. To show this we make the following non-restrictive assumption
on the demand distributions:  i is continuous and \Phi i when the
customer is following an (s; S) policy.) From here on, when we say fz i g is an order up-to
policy, we will assume that z
Definition z i communicates with z j if we can reach inventory level z j in state j while
starting with inventory z i in state i.
Definition An order up-to policy fz i g is regenerative if for all pairs (i; j), z i communicates
with z j .
Property 11 If C ? \Delta, then every order up-to policy fz i g is regenerative.
Proof If probfz communicates with z 1 . This is
true since  k is continuous above \Delta and z
Next we will show that state i communicates with state i + 1. If z i+1  z
it is obviously true. If z there exists a number N ! 1 such that
z
can be reached in state i due to N demands that are less
than C+\Deltain state i + 1 which occur with a non-zero probability.
We have shown that state k communicates state 1 and state i ! k communicates with
the state i + 1. This implies that the order up-to policy fz i g is regenerative. 2
Property 12 If C  \Delta, than an order up-to policy fz i g is regenerative if and only if for
every state defined above.
Proof (() Let us assume that for every state Cg.
Assume that we are currently in state k with inventory z k . If z 1 is less than C, then
a large demand in period k can bring the inventory to zero and z 1 can be reached in
state 1. If z 1 is bigger than C but smaller that (z close to
will drop the inventory to z which can be brought upto z 1 in state 1. So z k
communicates with z 1 . We will establish that i  1 communicates with i + 1. Let us
Cg. For all must be less than or equal
to z indicating that state i communicates with i + 1. Further we know that
implying that
Consider be the biggest that does not communicate with
communicates with some state j where j ? i, then i communicates with k through state j
and state i + 1 can be reached from state k. Thus i communicates with state i + 1. If i
does not communicate directly with any state j ? i, then we will construct a sequence of
events such that state i + 1 can be reached. We are currently in state i with inventory z i .
In step zero set I 0
, the inventory in step will be I n
using full capacity till we reach state k and then observing a demand of \Delta. I n+1
is clearly
greater than I n
and so there will be no production until we reach state i. Since these events
occur with a non-zero probability, I n
will eventually reach a value equal to or greater than
Cg. If the inventory level reaches z can
be reached with inventory z k from which we can reach state
we reach the inventory level z then we can reach inventory level z i+1 in state i + 1.
In both cases state i communicates with state i + 1. We have shown that every state i
communicates with state communicates with state 1 thus implying that
is regenerative.
Assume that fz i g is regenerative. This implies that any state i\Gamma1  0 communicates
with i. If state directly with i then obviously z i  z
indirectly, then state i has to be reached from
some inventory level X after seeing a demand (say ) which implies that z i
But we know that X must be less than z k and   \Delta. This automatically implies that
thus proving that z i  [z C. The property is obviously true for
state 1 by noting that z 1 will never be reached if it is greater than [z
states i, z i must be less than or equal to minfz
Following the reasoning in [9] and [13], the IPA procedure is valid. A simple gradient
search provides us with optimal parameters.
4.2 Model 2
In Model 2, as shown in Fig. 3, the supplier has information about the day-to-day inventory
levels at the customer as well as the (s; S) values and the end-item demand distribution.
The state of the system is no longer captured fully by the number of periods since the last
demand. So, we redefine the state space to incorporate the information about the total
demand seen so far by the customer. Thus, his inventory model corresponding to Model 2
has the following characteristics. State is defined as the total demand seen by the customer
since receiving her last order from the supplier. (Other equivalent definitions are possible.
All of them will capture the exact inventory at the customer at the beginning of every
period.) Every period is in one of possible \Delta states f0;
As before, p i is the probability that a demand to the supplier will be realized in state i. If
the demand to the supplier is realized in state i, it (exactly equal to i plus the end-item
demand at the customer for one period) is from a distribution with cdf \Phi i (:) with mean
i and the next period will be in state 0. If the demand is not realized, then the next
period will be in state j ? i with probability  (j \Gammai) which is exactly the probability that the
end-item demand in a single period is (j \Gamma i). (In Model It is easily verified
that st \Phi i+1 (:). i.e. the chance and quantity of
realization of demand increases in time. Here we do not need the IFR assumption on the
end-item demand distribution for monotonicity, nor a lowerbound on the end-item demand
for finiteness of states.
With minor adjustments the proofs presented for Model 1 apply to this model as well.
Briefly: (1) the cost function is convex and the optimal policy is modified order up-to;
(2) the optimal policy exists and has finite cost for the discounted cost criterion and the
average cost criterion; (3) the infinite horizon order up-to levels are ordered; (4) for the
uncapacitated system, a recursive procedure can be used to find the optimal order up-to
levels; when the capacity is finite, IPA can be used to find
the optimal order up-to levels.
Control
Inventory
i.i.d Demand
Customer
Demand
Day to Day Inventory levels
Uniform[15,25]
(Model
Supplier

Figure

4: The experimental setup.
4.3 Model 0
In Model 0, the supplier has no information about the customer, except what is available
from past demand data. The supplier observes periods of no demand, followed by a period
with demand. We calculate a policy using the following approach, where the supplier is
implicitly making an assumption that the demands are following an i.i.d. process, both in
inter-arrival time and quantity (that is, assuming a 'traditional' model).
1. Calculate the mean and the variance of the non-zero demands observed.
2. Assuming normal distribution for the non-zero demands, calculate an order up-to
using the newsboy formula and the given values for h and b.
3. Set z
z 8j  l for some value of l ? 0. Adjust the order up-to levels for finite
capacity using the following rule: If an order up-to level in period l
then set the order up-to level z
4. Simulate the system for various values of l and choose the value that gives the lowest
cost.
We believe that this is a reasonable 'straw' policy.
Computational Results
In this section we implement the solution procedures developed in earlier sections for all
three models. Our goal is to understand the trade-offs between inventories, capacities and
information. We vary h; b; \Delta and variance of end-item demand. Note that the customer
policy structure and parameters are the same across all three models for any given set of
cost, capacity and demand values.
The experimental design is shown in Fig. 4. The values of b,\Delta and mean demand were
held constant at 14, 50 and 20 respectively. The holding cost h was varied from 1 to 5 in
increments of 1, while the capacity was varied from 25 to 65 in increments of 10. Three
(Uniform, Erlang and Normal) distributions were used for the end item demand seen by the
customer. For each distribution, three values of the parameters were chosen:
Uniform[10,30], and Uniform[15,25]; Exponential(20), Erlang(10,2), and Erlang(5,4) (mean
20, standard deviation of 10); Normal(20,20), Normal(20,10), and Normal(20,5) where the
first number is the mean and the second number is the standard deviation. It should
be noted that while generating demands using the Normal distribution, the non-positive
demands were discarded and so the non-truncated mean was selected so that the modified
demands had a mean of 20. The standard deviations of the observed demands for these
three cases were 13.5, 9.5 and 5 and these values were used while analyzing the results. An
experiment is a combination of the holding cost, the capacity, the demand distribution and
the model type. There were a total of 675 experiments in this computational setup.
5.1 Costs and Savings
We observed that for all the models the total cost increased with increase in holding cost,
decrease in capacity or increase in variance. These results are expected and so we will not
elaborate on them here. We will look more closely at the savings realized due to information
flow and their change due to changes in capacity, holding cost and variance. The percentage
savings from Model 0 to Model 1 were estimated using the formula:
cost
Model
The percentage savings from Model 1 to Model 2 were also calculated using a similar
formula. We want to understand how these savings are affected by parameters such as
capacity, the ratio of holding cost to penalty cost, the variance of the end-item demand (as
seen by the customer) and \Delta. Notice that the cost for Model 2 was always smaller than
the cost for Model 1 which in turn was smaller than the cost for Model 0. This would
imply that information flow will always result in savings. This leads us to conclude that
information is always beneficial.
First, we study the percentage savings realized between Models 0 and 1: the benefits of
some information. The graphs of these savings versus capacity, holding cost and standard
deviation are given in Figures 5, 6 and 7. These savings vary from 3% to 17%. As shown in

Figure

5: Plot of % savings comparing Model 0 to Model 1 versus capacity.

Figure

5, the savings from Model 0 to Model 1 are highest when the capacity was 55; other
experiments show that the information is most beneficial when the ratio C
\Delta is around one.
In

Figures

6 and 7, these savings form a horizontal band of near constant width indicating
that there is no significant pattern with respect to changes in holding cost and variance.
We next study the percentage savings realized between Model 1 to Model 2: the benefits
of increased (and full) information. The plots of these savings with respect to capacity,
holding cost and variance are given in Figures 8, 9 and 10. These savings have a wide
range, varying from 1% to 35%. In general, the savings realized between Models 1 and 2
are at least as large as the savings realized between Models 0 and 1. This would lead us
to believe that more information results in continued large savings in our setting. Details
of the savings due to changes in capacity, holding cost and variance between Model 1 and
Model 2 are described below.
1. Effect of capacity. For every value of capacity and each demand distribution we
averaged the percentage savings over all the holding costs. The results are presented in
Fig. 8. It is easily noticed that as the capacity was increased, the percentage savings
also increased. This would lead us to conclude that between Models 1 and 2, the
benefit of information flow is higher at higher capacities.
2. Effect of penalty cost. Fig. 9 shows the information benefits as function of
Holding
Figure

Plot of % savings comparing Model 0 to Model 1 versus holding cost.30%
Standard Deviation

Figure

7: Plot of % savings comparing Model 0 to Model 1 versus variance.

Figure

8: Plot of % savings comparing Model 1 to Model 2 versus capacity.
- Normal(20,5)30%BenefitPenalty Cost

Figure

9: Plot of % savings comparing Model 1 to Model 2 versus penalty cost.
Standard Deviation

Figure

10: Plot of % savings from Model 1 to Model 2 versus variance.
penalty cost. For small variances, the information value increases as the service level
increases, while for medium to high variances, the benefits decrease. This is because,
at higher variances, there remains significant uncertainty on the total demand quantity
to the supplier (although the timing uncertainty may have reduced). In the extreme
case when the end-item demand has an Exponential distribution, the excess over \Delta
has the same distribution, independent of the state at the end of the previous period.
So we conclude that if the variance of the demand seen by the customer is small (high),
we can expect the benefit of information flow to increase (decrease) with increase in
penalty cost.
3. Effect of end-item demand distribution and variance. Fig. 10 contains the
plot of percentage savings versus standard deviation for each of the demand distri-
butions. For the uniform distribution as the standard deviation was decreased from
11.55 to 5.78 to 2.89 the percentage savings decreased while for the Erlang distributions
as the variance was decreased from 20 to 14.14 to 10 the percentage savings
increased. For the normal distribution as the standard deviation was changed from
13.5 to 9.5 to 5.0 the percentage savings initially increased from 7.55% to 9.37% and
then dropped to 7.37%. The plot seems to indicate a bell shaped relation between
200 250 300

Figure

11: Plot of % savings from Model 1 to Model 2 versus \Delta.
variance and percentage savings 1 . This could be due to the fact that when the variance
is very high, as in item 2 above, the uncertainty in quantity to the supplier is
not reduced significantly. When the variance is very low the system almost acts like a
deterministic system and the information is not really beneficial. Information is most
beneficial at moderate values of variance.
4. Effect of \Delta. The percentage savings between models 1 and 2 as a function of \Delta are
given in Fig. 11 for two distributions, exp(20) and erlang(5,4). For each distribution,
we averaged the savings over four simulations
For both the distributions, as the value of \Delta is
increased these savings increase initially and then start to decrease. This behavior
can be explained as follows. The value of \Delta determines the size of the order. Due to
finite capacity, in anticipation of a large order we start building up inventory over a
horizon which we call "production horizon". When \Delta is large (compared to capacity
and mean end-item demand), the production horizon is so large that (1) holding costs
accrue and (2) Central Limit Theorem applies reducing the coefficient of variation of
quantity demanded. On the other hand, when small (relative to mean end-item
only the right side of this bell curve in his uncapacitated model, as we would if we did
not test Normal distributions but limited ourselves to the Erlang family.
demand), the end-item demand is passed through to the supplier almost every period.
So we conclude that information is less beneficial at extreme values of \Delta.
5.2 Inventory Levels
The optimal order up-to levels for Model 1 in two situations, namely fexponential(20),
are given in Figures 12 and 13. First consider the case of
that these order up-to levels follow the pattern proved in Property 10
for 1. For the case of exp(20) and the optimal order up-to levels are zero for the
two periods immediately following a demand. In period 3(=i   ) the order up-to level jumps
to 55 and from then on slowly climbs to 80. Similarly for the case the
order up-to level is zero for the first period immediately following the demand. In period 2
it jumps to 63 and slowly climbs to 80 from there on.
Next observe the optimal order up-to levels for other values of capacity for these two
situations: the optimal order up-to levels do not change significantly with capacity for
periods that had non-zero order up-to levels when the capacity was (periods 3 and above
in Fig. 12, and Periods 2 and above in Fig. 13) or higher. The major changes occur, if at
all, in periods immediately after the demand to facilitate reaching the required inventory
level for period i   . Using these observations we can draw the following two insights into
how these systems can be managed.
1. Managing multiple customers. This property of order up-to levels can help
manage multiple customers as follows. For example, in the case of exp(20),
we know that if we have just seen the demand, we need not produce this product
for another 2 periods. If there are other products in the system we can schedule this
product 3 periods from now and use the first 2 periods for some other products. If
there is another product already scheduled in period 3 we can compare the costs of
these two products to come up with a schedule for them in periods 2, 3 or 4. This also
suggests that if the same capacity is being shared by two or more products, it may be
better to assign the total capacity in a period to one product thus treating the system
as multiple large capacity channels and breaking ties among products myopically
rather than pre-allocating capacities to products in every period thus creating many
low capacity channels. A parallel paper studies this issue in greater detail.
2. Approximating order up-to levels in the finite capacity case without
IPA. Solve a given finite capacity system as if it had infinite capacity to obtain the
Order
upto
level
Periods since last demand

Figure

12: The optimal order up-to levels for exp(20), Model 1.
Order
upto
level
Periods since last demand

Figure

13: The optimal order up-to levels for Model 1.
order up-to levels fz i g particularly i   and z i   . To obtain the order up-to levels for
finite capacity, set the order up-to levels for i   and states above it for the capacitated
case are same as the ones from the uncapacitated solution. For states below i   define
order up-to levelsfz c
g in such a way so that the order up-to level in i   can be reached
under the capacity restriction by defining production levels as late as possible. That
is z c
6 Conclusions
In this paper, we have incorporated information flow into inventory control models. This
gave rise to interesting non-stationary demand processes. Optimal policy structures are
found to be order up-to policies. We have provided solution procedures to compute the
optimal parameters. An extensive computational study provided us with insights on the
savings and relative benefits due to information flow.
Future work is in three directions: (1) extending this analysis with multiple customers;
(2) studying inventory-information-capacity interactions in a supply chain with more than
two stages; and (3) studying, at a theoretical level, more general demand processes 2 at
the supplier, where the transition from one state to another follows a general probability
matrix, of which the current non-stationary situations are special cases. We conjecture that
if the transition probability matrix is totally positive of order 2 (TP 2 , see Karlin [15]), then
the properties of order up-to levels presented in this paper still apply.
A Monotonicity of Demands at the Supplier: Model 1
Assume that the end-customer sees an i.i.d demand process with pdf /() and cdf \Psi(). Let
be the sequence of demands seen by the customer and D
be the total
demand in i periods. Then the customer places an order for D i in period i if D i  \Delta and
the probability that the customer places an
order at the end of period i, is
These probabilities represent the distribution of the first passage times of a jump process
and from [16] we know that p i  p i+1 .
We thank Professor George Shanthikumar for this suggestion.
Let \Psi n (\Delta) be the nth convolution of \Psi(\Delta) and clearly \Psi st \Psi n (\Delta). \Phi i (y) and
can be expressed as
If we assume that \Psi(\Delta) is a function with increasing failure rate (IFR), then [\Psi(y\Gammax)\Gamma\Psi(\Delta\Gammax)]
is a decreasing function in x and thus \Phi i (y)  \Phi i+1 (y) implying that \Phi i (\Delta)  st \Phi i+1 (\Delta).



--R

"An Inventory Model with Commitments,"
"Analysis of Supply Contracts with Total Minimum Commitment,"
"The impact of supply chain integration on operating performance,"
"Bayes solutions to dynamic inventory models under unknown demand distributions,"
"Dynamic programming: Deterministic and stochastic models,"
"Echelon Reorder Points, Installation Reorder Points, and the Value of Centralized Demand Information,"
"An inventory model with limited production capacity and uncertain demands I : The average-cost criterion,"
"An inventory model with limited production capacity and uncertain demands II : The discounted-cost criterion,"
"The stability of a capacitated, multi-echelon production- inventory system under a base-stock policy,"
"Sensitivity analysis for base-stock levels in multi-echelon production-inventory systems,"
"Customer-order Information, Leadtimes, and Invento- ries,"
"The dynamic inventory problem with unknown demand distributions,"
"A capacitated production-inventory model with periodic demand,"
"Optimal policy for dynamic inventory process with stochastic demands subject to seasonal variations,"
"Total Positivity Vol. I,"
"Uniform Monotonicity of Markov Processes and its Related Properties,"
"Myopic policies for some inventory models with uncertain demand distributions,"
" The Optimality of (s; S) policies in the dynamic inventory problem,"
"Bayes solutions to the statistical inventory problem,"
" Electronic data interchange (EDI): an avenue to better performance and the improvement of trading relationships?,"
"Computing the optimal policy for capacitated inventory models,"
"A Queueing Model to Analyze the Value of Centralized Inventory Information,"
"Critical number polices for inventory models with periodic data,"
"Performance Analysis of a Multi-item Production-inventory System under Alternative Policies,"
--TR

--CTR
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Infinitesimal Perturbation Analysis;Capacitated Production-Inventory Model;information sharing;S Policy;optimal policies