IIE Transactions A periodic-review inventory model in

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A periodic-review inventory model in a fluctuating environment a

S. Papachristos & A. Katsaros

b

a

Department of Business Administration of Food and Agricultural Products , University of Ioannina , 45110, Greece b

Department of Environment and Natural Resources Management , University of Ioannina , 45110, Greece Published online: 17 Jan 2008.

To cite this article: S. Papachristos & A. Katsaros (2008) A periodic-review inventory model in a fluctuating environment, IIE Transactions, 40:3, 356-366, DOI: 10.1080/07408170701488045 To link to this article: http://dx.doi.org/10.1080/07408170701488045

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IIE Transactions (2008) 40, 356–366 C “IIE” Copyright ISSN: 0740-817X print / 1545-8830 online DOI: 10.1080/07408170701488045

A periodic-review inventory model in a fluctuating environment S. PAPACHRISTOS1,∗ and A. KATSAROS2 1

Department of Business Administration of Food and Agricultural Products, University of Ioannina, 45110 Greece E-mail: [email protected] 2 Department of Environment and Natural Resources Management, University of Ioannina, 45110 Greece

Downloaded by [University of Ioannina] at 10:49 29 March 2015

Received February 2006 and accepted October 2006

We study a single-item periodic-review inventory model in a fluctuating environment with a fixed lead time of λ periods. The state of the environment at the beginning of each period is described by a homogeneous Markov chain. Ordering, holding, penalty costs and the distributions of random variables representing the customer’s demand and the supplier’s capacity level are state environment dependent. By using dynamic programming it is proved in the finite-horizon case that the optimal policy is of a base stock type. Its parameters are monotonic in the number of the periods making up the horizon and also in stochastically ordered random variables representing different supplier capacities. Similar results are proved for the infinite-horizon problem. Keywords: Inventory, dynamic programming, fluctuating environment, variable capacity, stochastic ordering

1. Introduction It is generally true that economic decisions are affected by the prevailing business climate at the time that they are made. For example almost all parameters involved in deciding on an inventory management system are affected by the prevailing business environment. Let us consider one of the most critical parameters, namely demand. For high-technology products (e.g., computers), demand is affected by the technology status, whereas for pharmaceutical products it depends on the society’s status. The demand for clothes is affected by weather conditions, whereas demand for luxury products is affected by the extent of consumer wealth. The term fluctuating environment or simply environment is used to represent the randomly changing (inside or outside) conditions and factors that affect some or all of the parameters of an inventory system. A system for which all or some of the parameters are affected by the prevailing environment is said to operate in a fluctuating environment. In the majority of existing stochastic inventory models, random demand is the primary component that is subject to external factors. Iglehart and Karlin (1962) introduced an inventory model in which the distribution of demand in a period depends on the state of the environment and it follows a Markov chain (see also Johnson and Thomson (1975)). Song and Zipkin (1993) considered a model in ∗

Corresponding author

C 2008 “IIE” 0740-817X

which the demand is affected by the environment and they extended this work to consider a model with items subject to obsolescence in Song and Zipkin (1996). Sethi and Cheng (1997) considered a similar model with Markovian demand and fixed charge ordering cost. Another source of uncertainty in many inventory systems is caused by the supplier’s random capacity levels and its inability to deliver all the placed orders. Federgruen and Zipkin (1986) considered a single-item periodic-review inventory model with a limited production capacity. Anupindi and Akella (1993) studied an inventory model for the case where only a percentage p ∈ (0,1) of the placed order is satisfied. Wang and Gerchak (1996) investigated a production planning problem with variable production capacity and random yield, i.e., a random portion of the items processed is defective, due to an imperfect production process. Another line of approach considers the supplier’s production to be a random variable that depends on the environment and thus the lot size sent to the buyer’s warehouse is a random variable. Ozekici and Parlar (1999) and Erdem and Ozekici (2002) considered a similar periodic-review inventory model with a zero lead time. In these models, the demand distribution, the availability of the unreliable supplier and the cost parameters depend, in every period, on the state of the fluctuating environment which is geared by a time-homogeneous Markov chain. The concept of the stochastic ordering of probability distributions has also found application in the area of inventory control. The first use of this approach was in Scarf

357


A periodic-review inventory model in a fluctuating environment (1959). Using a Bayesian approach to update the demand density, he managed to compare optimal ordering levels under different statistical information settings. This work was followed by Karlin (1960) who treated the stochastic ordering in a more systematic way. Other important papers in this area are those by Papachristos (1977), Azoury (1985) and Rajashree and Pakkala (2002). Recently, Gupta and Cooper (2005) studied a model with a random yield and considered the effect of the stochastic ordering defined on the random yield distributions to the optimal base stock levels. An extensive treatment of this concept and its applications is discussed in the book by Shaked and Shanthikumar (1994). In this study, the model proposed by Erdem and Ozekici (2002) is extended by introducing a constant lead time of λ periods. The lead time may occur for reasons such as the time required to produce the lot size ordered, transportation time, inspection process time, etc. It is known, see e.g., Zipkin (2000, pp. 404–408), that many systems with a constant lead time can be transformed into equivalent systems with a zero lead time, provided that the environment is constant, and as such they can be solved using Dynamic Programming. For systems operating in a fluctuating environment this technique is difficult to apply (Zipkin, 2000, p. 416). The difficulty arises from the fact that it is hard to compute the expected value of the demand during the lead time period, since the demand at every period depends on the state of the environment. The approach used here is inspired by ideas given in Arrow et al. (1958) who studied a periodic-review inventory model with a constant lead time of λ periods, a fixed supplier’s capacity and a constant envinroment. In this extented model we further examine the effect of stochastic ordering defined on distributions describing the random supplier’s capacity on the optimal policy parameters. The structure of the paper is as follows. The assumptions of the model and the notation are given in Section 2. In Section 3 the optimal policy for the finite-horizon problem is shown to be a base stock level policy and monotonicity properties for its parameters are established. In Section 4 it is shown that stochastic ordering among the distributions describing the random supplier’s capacity is transferred on ordering in the parameters of the optimal policy. The infinite-horizon problem is studied in Section 5. A summary of the results obtained and proposals for further research are given in Section 6.

2. Assumptions and notation The problem studied in this paper is a single-item periodicreview problem in a randomly fluctuating environment. The evolution of the environment is described by a homogeneous, discrete-time Markov chain {In , n ≥ 0}, on a finite state space E. At the beginning of each period, the state of the random environment is observed. Based on this observation, values are assigned to the model parameters and

these remain constant through the period. Thus, all the model parameters depend on the environmental conditions prevailing at the beginning of the period. The notation and assumptions of the paper are as follows. i Pij

Di gi (d) Gi (d) Ai fi (w) Fi (w) ci hi pi γ n λ

= state of the environment at the beginning of any period; = Pr[In+1 = j|In = i] = probability that the environment will make a transition to state j at the beginning of period n + 1 given that it was at i at the beginning of period n; = random variable describing the period’s demand given that the environment is at i; = density function of Di ; = Pr[Di ≤ d|In = i], the cumulative distribution function (c.d.f.) of Di ; = random variable describing the supplier’s capacity given the environment is at i; = density function of Ai ; = Pr[Ai ≤ w|In = i] the c.d.f. of Ai with F¯ i (w) = 1 − Fi (w); = the per unit purchase cost; = the per unit/unit of time holding cost; = the per unit/unit of time shortage cost; = discount factor; = the planning horizon; = the lead time.

We assume that the replenishment rate is infinite and excess demand is completely backlogged. We suppose that the parameters satisfy the condition: (λ+1) (λ) Pi j pj + γ Pi j pj (C.1) ci − j∈E

+ · · · + γ n−1

j∈E (λ+n−1)

Pi j

pj

< 0,

j∈E (λ)

where Pi j is the λ steps transition probability from state i to j. Note that, in the case of a constant environment and a zero lead time, condition C1 results in c − p(1 + γ + · · · + γ n−1 ) < 0 which is given in the paper by Wang and Gerchak (1996) and for the one-period problem (n = 1) it gives p > c which is an assumption that is usually made in the classical inventory model. If the environment at the beginning of a period is i and the buyer places an order for z units, λ periods later he or she receives a quantity equal to min{z, Ai }. This is owing to the supplier’s random capacity and the λ periods lead time. If Li (x) is the holding and backlogging cost for any period starting with a stock of x units (inventory on hand) and the environment is at i, then: Li (x) = E[hi max{0, x − Dι } + pi max{0, Dι − x}] x ∞ (x − d)dGi (d) + pi (d − x)dGi (d) . = hi 0

x

358

Papachristos and Katsaros or

It is easy to find that: Li (x) = (hi + pi )Gi (x) − pi

and

Li (x) = (hi + pi )gi (x) ≥ 0, (1)

and so Li (x) is convex.


3. The finite-horizon problem: mathematical model and optimal policy The analysis of the model will be performed using a Dynamic Programming (DP) approach. For technical reasons the horizon is counted reversely (backward). Thus, referring to the nth period problem the first period is the nth and the horizon remaining is n periods. Let us suppose that at the beginning of the nth period the environment is at state i, the inventory on hand is x, the units y1 , y2 , . . . , yλ−1 are scheduled to enter the system at the beginning of each of the following (λ − 1) periods and that the decision taken is to order z units. The state of the system at this moment is completely described by the vector (x, y1 , . . . , yλ−1 , i). When an order for z units is placed and the stochastic supplier’s capacity is Ai , the quantity entering the system λ periods later is min{z, Ai }. Thus, at the beginning of the next, i.e., (n − 1)th period, the system transits to the state (x + y1 − Di , y2 , . . . , yλ−1 , min{z, Ai }, j), which is a random variable in the three variables Di , Ai and j. Let us denote by Vni (x, y1 , . . . , yλ−1 ) the optimal expected discounted cost for the n periods problem, starting at state (x, y1 , . . . , yλ−1 , i) at the beginning of period n. Also, let TCn (x, y1 , . . . , yλ−1 , z.Ai |In = i) = be the expected discounted cost for the n-period problem, if at the beginning of period n the state is at (x, y1 , . . . , yλ−1 , i), an order for z units is placed and the supplier’s random capacity is Ai . The definitions of Vni (x, y1 , . . . , yλ−1 ) and TCn (x, y1 , . . . , yλ−1 , z, Ai |In = i) imply that: TCn (x, y1 , . . . , yλ−1 , z, Ai |In = i)  λ j γ c A + L (x) + γ Pij EDi Vn−1 (x + y1  i i i    j∈E    −D , y , . . . , y , A ) , i 2 λ−1 i = j λ  Pij EDi Vn−1 (x + y1 γ ci z + Li (x) + γ     j∈E

  −Di , y2 , . . . , yλ−1 , z) ,

Since the lead time is λ periods, it is obvious that there is no sense in placing orders in the last λ periods because these orders will arrive at the system after the end of the planning horizon. To facilitate the analysis, the planning horizon is extended by considering λ extra periods, namely periods 0, −1, . . . , −(λ − 1), so that the first period subjected to control (complete period) is period 1 (Ehrhardt, 1984; Zipkin, 2000). For these periods no cost control can be exercised and assuming a zero salvage cost we have that: V0i (x, y1 , . . . , yλ−1 ) = Li (x) +

λ−1

γ m Li,m (x+y1 + · · · +ym )

m=1

= Li (x) +

λ−1

γ m Li,m (um ),

(4)

m=1

where um = x + y1 + · · · + ym , u0 = x, m = 1, . . . , λ − 1. The functions Li,m (um ) represent the expected costs for each of the −(m − 1) periods and are calculated from the recursive relation: Pi j EDi Lj,m−1 (um − Di ), m = 1, . . . , λ − 1, Li,m (um ) = j∈E

(5) where Li,0 (u0 ) = Li (u0 ) for any i ∈ E, u ∈ R. The examples given below show that these recursive relations correctly calculate the relevant costs: z > Ai , Pi j EDi Lj,0 (u1 − Di ), Li,1 (u1 ) = j∈E

z ≤ Ai ,

where EDi (.) is the expected value operator. The term γ λ ci z in the above expression is explained by the fact that the purchase cost of the ordered lot size z will be paid when it is received λ periods later and so it must be discounted to the present period by the factor γ λ . Using the principle of optimality, the backward (DP) equation for Vni (x, y1 , . . . , yλ−1 ) is Vni (x, y1 , . . . , yλ−1 ) = min EAi [TCn (x, y1 , . . . , yλ−1 , z, Ai |In = i )], (2) z≥0

Vni (x, y1 , . . . , yλ−1 ) = Li (x)  z  j  Pi j EDi Vn−1 (x + y1 γ λ ci w + γ    0  j∈E        −Di , y2 , . . . , yλ−1 , w) dFi (w) (3) + min z≥0  j  λ  + γ c z + γ P E V (x + y i i j Di n−1 1     j∈E       −Di , y2 , . . . , yλ−1 , z) Fi (z).

Li,2 (u2 ) =

Pi j EDi Lj,1 (u2 − Di )

j∈E

=

Pi j EDi

j∈E

Pjk EDj Lk,0 (u2 − Di − Dj ).

k∈E

It has already been shown that Li (x), i ∈ E are convex and since E is finite the functions Li,1 (u) = j∈E Pi j EDi Lj,0 (u − Di ), i ∈ E, are also convex (Zipkin (2000, p. 435)). Using induction on m it is easily established that: Li,m (u) ≥ 0,

m = 0, 1, . . . , λ − 1,

and so Li,m (u) are convex functions.

(6)

359

A periodic-review inventory model in a fluctuating environment Additionally we can prove (proofs are given in the Appendix) that: (m) (u) = Pi j hj , (7) lim Li,m u→+∞

j∈E

(u) = − lim Li,m

u→−∞

(m)

Pi j pj .

(8)

j∈E

We will now proceed to analyze the problem for n = 1 and n = 2. To obtain V1i (x, y1 , . . . , yλ−1 ) from Equation (3) we need j V0 (x + y1 − Di , y2 , . . . , yλ−1 , w) which from Equation (4) is j


V0 (x + y1 − Di , y2 , . . . , yλ−1 , w) = Lj (x + y1 − Di ) λ−2 γ m Lj,m (x + y1 − Di + · · · + ym ) + m=1 λ−1

+γ

γ m Li,m

m=1

(x + y1 + · · · + ym ) + γ λ−1 Li,λ−1 (x + y1 + · · · + yλ−1 )  z   Pi j EDi Lj,λ−1 (x + y1 − Di γ ci w + γ    0  j∈E      + · · · + y + w) dFi (w)  λ−1  λ−1 min +γ z≥0    + γ ci z + γ Pi j EDi Lj,λ−1 (x + y1 − Di     j∈E       + · · · + yλ−1 + z) Fi (z) = Li (x) +

λ−2 m=1

γ m Li,m (um ) + γ λ−1 Li,λ−1 (uλ−1 )

+ γ λ−1 min b1i (uλ−1 , z) , z≥0

where b1i (uλ−1 , z)

=

z

γ ci w + γ

0

Pi j EDi Lj,λ−1

j∈E

If we set

v1i (uλ−1 ) = Li,λ−1 (uλ−1 ) + min b1i (uλ−1 , z) ,

V1i (x, y1 , . . . , yλ−1 )

R1i (uλ−1 + z )Fi (z), where R1i (uλ−1 + z) = γ ci + γ

(12)

EDi Pi j Lj,λ−1 (uλ−1 − Di + z).

As in Erdem and Ozekici (2002), we assume that Fi (z) > 0, which ensures a positive probability of receiving the full amount ordered in the period. The unconstrained minimizing point, if it exists, will result from the root, in u = uλ−1 + z, of the equation: R1i (u) = 0.

(14)

Taking into account Equations (7) and (8) then Equation (13) gives: (λ) Pi j hj > 0, (15) lim R1i (u) = γ ci + γ u→+∞

j∈E

lim R i (u) u→−∞ 1

= γ ci − γ

(λ)

Pi j pj < 0,

(16)

j∈E

where the last inequality results from condition C1 for n = (u) ≥ 0, the function R1i (u) is non1. Furthermore, since Li,m decreasing. Combining this with Equations (15) and (16) we conclude that Equation (14) has a root (if there are more than one then we keep the smallest), say S1i , that is R1i S1i = 0. (17)

j∈E

Pi j EDi Lj,λ−1 (uλ−1 − Di + z) Fi (z).

then Equation (9) becomes:

At this point it is important to notice that b1i (uλ−1 , z) is a function of uλ−1 + z and not of uλ−1 and z separately. This important observation is the key point in the subsequent analysis. The first derivative of b1i (uλ−1 , z) with respect to z is

+ R1i (uλ−1

j∈E

z≥0

(11)

m=1

To see that this root gives a minimum, we check the secondorder derivative of b1i (uλ−1 , z): (9) 2 i EDi Pi j Lj,λ−1 (uλ−1 − Di + z) F¯ i (z) Dz b1 (uλ−1 , z) = γ

(uλ−1 − Di + w) dFi (w) + γ ci z +γ

γ m Li,m (um ) + γ λ−1 v1i (uλ−1 ).

(13)

Substituting this into Equation (3) and using the recursive relations Equation (5) we obtain: V1i (x, y1 , . . . , yλ−1 ) = Li (x) +

λ−2

j∈E

Lj,λ−1 (x + y1 − Di + · · · + yλ−1 + w).

λ−2

= Li (x) +

(10)

+ z)[−fi (z)].

The first term of the sum above is always positive because Lj,λ−1 > 0, while the second is positive for all z such that u = uλ−1 + z < S1i since then R1i (u) < 0. Furthermore, from Equation (12) we get Dz b1i (uλ−1 , z) < 0 for all z values such that u = uλ−1 + z < S1i . Thus, b1i (uλ−1 , z) is convex and decreasing for all z values such that u = uλ−1 + z < S1i . For z values such that u = uλ−1 + z > S1i , but with u close to S1i , the second term is non-positive but the sum of the two terms is still positive, because R1i (uλ−1 + z) is continuous in u and so b1i (uλ−1 , z) is still convex at some interval left of the point S1i . Furthermore, as z → +∞ the term F¯ i (z) goes to zero and it follows from Equation (15)

360

Papachristos and Katsaros


that limu→∞ R1i (uλ−1 + z ) > 0. Thus, there exists a point z0 where Dz2 b1i (uλ−1 , z) becomes non-positive and remains as such for all z values right to this point. The conclusion is that for all z such that u = uλ−1 + z > S1i , b1i (uλ−1 , z) is a quasi-convex and non-decreasing function. The above discussion ensures that b1i (uλ−1 , z) attains its global minimum at S1i . Let uλ−1 + z1i (uλ−1 ) = S1i . If uλ−1 < S1i then z1i (uλ−1 ) = S1i − uλ−1 > 0 and this is the minimizing point of Equation (10). If uλ−1 ≥ S1i then z1i (uλ−1 ) = S1i − uλ−1 ≤ 0 and the constrained minimum of Equation (10) is obtained at z1i (uλ−1 ) = 0. Thus, the optimal ordering policy is z1i (uλ−1 ) (18) i S1 − uλ−1 for uλ−1 = x + y1 + · · · + yλ−1 < S1i , = 0 for uλ−1 = x + y1 + · · · + yλ−1 ≥ S1i . By substituting this optimum value into Equation (10) we obtain:  z1i (uλ−1 )    L γ ci w + γ (u ) +  i,λ−1 λ−1   0   j∈E      Pi j EDi [Lj,λ−1 (uλ−1 − Di + w)] dFi (w)     i z (u ) + γ Pi j EDi [Lj,λ−1 (uλ−1 + [γ c i i λ−1 1 v1 (uλ−1 ) =  j∈E     −Di + z1i (u))]]Fi z1i (uλ−1 ) for uλ−1 < S1i ,       Pi j EDi [Lj,λ−1 Li,λ−1 (uλ−1 ) + γ    j∈E    (uλ−1 − Di )] for uλ−1 ≥ S1i . (19) From Equation (18), it is obvious that the z1i (·) is a continuous function. Based on this and Equation (19) we can verify that lim v1i (u) = lim v1i (u), u→S1i −

u→S1i +

v1i (u)

is also a continuous function. and so We will now establish the convexity of v1i (uλ−1 ). Its first derivative is  z1i (uλ−1 )    L (u ) + γ Pi j EDi  i,λ−1 λ−1   0  j∈E      [Lj,λ−1 (uλ−1 − Di + w)]dFi (w) v1i (uλ−1 ) = uλ−1 < S, (20) − γ ci Fi (z1i (uλ−1 )),       (uλ−1 ) + γ Pi j EDi Lj,λ−1 Li,λ−1     

j∈E  (uλ−1 − Di ) , uλ−1 ≥ S1i . For uλ−1 < S1i the second derivative, is z1i (uλ−1 ) i v1 (uλ−1 ) = Li,λ−1 (uλ−1 ) + γ Pi j EDi Lj,λ−1

(uλ−1 − Di + w) dFi (w)

0

j∈E

− fi (z1i (uλ−1 ))

γ ci + γ

Pi j EDi Lj,λ−1

j∈E

(uλ−1 − Di + w)

/w=z1i (uλ−1 )

.

(21)

Recalling Equations (13) and (17), the last term is zero because Lj,λ−1 (u − Di + w)/w=z1i (u) is calculated at w = z1i (uλ−1 ) and so Equation (21) becomes:  z1i (uλ−1 )      (u ) + γ Pi j L λ−1  i,λ−1    j∈E    E [L (u 0 − D + w)dF (w), (22) Di j,λ−1 λ−1 i i i v1 (uλ−1 ) =   for uλ−1 < S1i     Li,λ−1 (uλ−1 ) + γ Pi j     j∈E   EDi [Lj,λ−1 (uλ−1 − Di )], for uλ−1 < S1i . This proves that each of the two branches of v1i (·) is a convex function. To establish the convexity of v1i (·) it is sufficient to show that, at the boundary u = S1i , the derivatives satisfy the condition:

i v1i (uλ−1 ) ≤ lim v1 (uλ−1 ) . lim uλ−1 →S1i −

uλ−1 →S1i +

Since z1i (u) is continuous from Equation (20) we have: i lim v1i (uλ−1 ) = Li.λ−1 S1 − γ ci , uλ−1 →S1i −

and lim

uλ−1 →S1I +

i S1 + γ v1i (uλ−1 ) = Li.λ−1 Pi j EDi Lj,λ−1 j∈E

(S1i ) − γ ci , S1i − Di = Li.λ−1 where the second equality results from Equations (13) and (17). Now the case where n = 2 is considered. Substituting V1i (·) from Equation (11) into Equation (3) and simplifying results in V2i (x, y1 , . . . , yλ−1 ) λ−2 γ m Li,m (um ) + γ λ−1 v2i (uλ−1 ), (23) = Li (x) + m=1

where v2i (uλ−1 ) = Li,λ−1 (uλ−1 )  z j   [γ ci w + γ Pi j EDi v1    0 j∈E    (uλ−1 − Di + w)]dFi (w) + min (24) j z≥0   z + γ P E v + [γ c  i i j Di 1    j∈E   (uλ−1 − Di + z)]Fi (z). By using similar arguments as in the case n = 1 the following can be shown. The point S2i , which is the root of (25) R2i (u) = 0, j

i (26) Pi j EDi v1 (u − Di ) , R2 (u) = γ ci + γ j∈E

361

A periodic-review inventory model in a fluctuating environment (recall that u = uλ−1 + z) is a candidate for the minimum in Equation (24). This root exists becauseR2i (u ) is non-decreasing, v1i (u) is convex, and furthermore: (λ) (λ+1) Pi j hj + γ 2 Pi j hj > 0, lim R2i (u) = γ ci + γ u→+∞

lim R i (u) u→−∞ 2

j∈E

= γ ci − γ

j∈E

j∈E

(λ) Pi j pj

−γ

2

(λ+1)

Pi j

pj ≤ 0,

j∈E


(27) as is shown in the Appendix on account of condition C1. The part of Equation (24) within the brackets is a quasi-convex function in z and so S2i gives the global minimum Equation (24). The optimal policy is i S2 − uλ−1 , for uλ−1 < S2i , i (28) z2 (uλ−1 ) = 0, for uλ−1 ≥ S2i . By substituting z2i (uλ−1 ) into Equation (24), the first and second derivatives of v2i (uλ−1 ) are v2i (uλ−1 )  z2i (uλ−1 )    γ Pi j EDi Li,λ−1 (uλ−1 ) +    0  j∈E    [vj (u   1 λ−1 − Di + w) ]dFi (w) = − γ ci Fi (z1i (uλ−1 ))      (u ) + γ Pi j EDi L  λ−1 i,λ−1    j∈E   j  [v1 (uλ−1 − Di ) ]

(29)

vni (uλ−1 ) = Li,λ−1 (uλ−1 )  z  j  w + γ Pi j EDi vn−1 γ c  i   0  j∈E        (uλ−1 − Di + w) dFi (w) + min z≥0  j   + γ ci z + γ Pi j EDi vn−1     j∈E       (uλ−1 − Di + z) Fi (z), (32) j

with v0 (u) = Lj,λ−1 (u), j ∈ E. 2. The optimal ordering policy for the nth period is an environment state-dependent base stock level policy defined by Order the quantity: zni (uλ−1 ) = Sni − uλ−1 if uλ−1 < Sni , if uλ−1 ≥ Sni . Do not order: zni (uλ−1 ) = 0 The number Sni is the root of the equation: j

(33) Pi j EDi vn−1 Sni − Di = 0. ci + j∈E

3. The function vni (uλ−1 ) is continuous, convex and for u
0 and so the minimum in Equation (32), if any exists, results from the root, say Sni , of the equation: Rni Sni = 0. (36) i (u) is convex and Rni (u )is non-decreasing. By induction, vn−1 Furthermore, (see Appendix): (λ) Pi j hj + · · · lim Rni (u) = γ ci + γ u→+∞

+γ

n

j∈E (λ+n−1) Pi j hj

> 0,

j∈E

lim R i (u) u→−∞ n

= γ ci − γ −γ

n

(λ) Pi j pj

< 0.

0

0

j vn−1 (uλ−1

(37)

j vn−2 (uλ−1

j∈E

These facts guarantee the existence of Sni . It is easy to establish that the part of Equation (32) within the brackets is a quasi-convex function in z and so Sni gives the minimum of Equation (32). Thus, part 2 of the theorem is obvious. Substituting zni (uλ−1 ) = Sni − uλ−1 into Equation (32) and taking the derivatives, proves part 3. Finally, by combining Equation (31), Equation (32) and the result in part 2 we are led to the optimal cost function. The next theorem establishes the monotonic character of the sequences Sni and zni (u). Theorem 2. i i∈ 1. The sequence Sni is increasing in n, i.e., Sni ≤ Sn+1 E, n ≥ 1. i (u). 2. For any u ∈ R, i ∈ E, n ≥ 1 we have zni (u) ≤ zn+1

Proof. To prove the first part of the theorem it is sufficient i (uλ−1 ) for any uλ−1 < S1i . The to show that vni (uλ−1 ) < vn−1 proof will be done by induction. Let us consider the interval uλ−1 < S1i . In this interval Equation (20) gives: (uλ−1 ) + v1i (uλ−1 ) = Li,λ−1

z1i (uλ−1 ) 0

γ

Pi j EDi [Lj,λ−1

j∈E i γ ci Fi z1 (uλ−1 )

(uλ−1 − Di + w)] dFi (w) − < Li,λ−1 z1i (uλ−1 ) (uλ−1 ) + (−γ ci ) dFi (w) − γ cι F¯ i z1i (uλ−1 ) 0

< Li,λ−1 (uλ−1 ) = v0i (uλ−1 ) , i.e., v1i (uλ−1 ) < v0i (uλ−1 ) .

j∈E

j vn−1 (uλ−1 − Di + w) dFi (w) − γ ci + γ ci Fi zni (uλ−1 ) i zn (uλ−1 ) < Li,λ−1 (uλ−1 ) + γ Pi j EDi

− ···

j∈E (λ+n−1) Pi j pj

The first inequality step is justified as follows. If uλ−1 < S1i and 0 ≤ w ≤ z1i (uλ−1 ) then uλ−1 + w ≤ uλ−1 + z1i (uλ−1 ) = R1i (uλ−1 + w) < 0 and Equation (13) S1i and subsequently

gives γ j ∈E Pi j EDi [Lj,λ−1 (uλ−1 − Di + w)] < −γ ci . The second is obvious. The result v1i (uλ−1 ) < v0i (uλ−1 ) combined with Equations (13) and (25) gives R2i (u) < R1i (u) and since Rni (·), n ≥ 1 are non-decreasing continuous functions, we obtain S1i ≤ S2i . i we have Let us now suppose that for uλ−1 < Sn−1 i i i i (u) for vn−1 (uλ−1 ) < vn−2 (uλ−1 ) . To prove that vn (u) < vn−1 i i uλ−1 < Sn , we consider only the case Sn−1 < uλ−1 < Sni . i is treated in a similar way. For The case where uλ−1 < Sn−1 i uλ−1 < Sn , Equation (34) gives: zni (uλ−1 ) (uλ−1 ) + γ Pi j EDi vni (uλ−1 ) = Li,λ−1

−

j∈E

Di ) + γ ci Fi zni (uλ−1 ) zni (uλ−1 )

(uλ−1 ) + < Li,λ−1

j∈E

− Di + w) dFi (w) + γ Pi j EDi

(−γ ci )dFi (w) + γ

0

Pi j EDi

j∈E

j i vn−2 (uλ−1 − Di ) + γ ci Fi zni (uλ−1 ) = vn−1 (uλ−1 ) .

The first inequality step is justified as follows. If i i then Rn−1 (uλ−1 ) > 0 and so Equation (35) uλ−1 > Sn−1

j gives −γ ci < γ j∈E Pi j EDi [vn−2 (u − Di ) ]. The second step follows from the fact that, if uλ−1 < Sni and 0 ≤ w ≤ zni (u), then uλ−1 + w ≤ uλ−1 + zni (uλ−1 ) = Sni and Rni (uλ−1 + w ) < 0, so Equation (35) gives

j Combin−γ ci > γ j∈E EDi Pi j [vn−1 (uλ−1 − Di + w) ]. i (uλ−1 ) < vni (uλ−1 ) and Equation (35) leads to ing vn−1 i i (uλ−1 ) < Rni (uλ−1 ), i.e., Sni ≤ Sn+1 . The second result Rn+1 is obvious.

4. Stochastic ordering properties We now explore stochastic ordering properties on the base stock levels Sni and ordering quantities zni (uλ−1 ), corresponding to stochastic ordering on the distributions Fi (w) of the supplier’s random capacity. To do this we need the following definition. Definition 1. Let Ai and A˜i be two random variables with densities fi and f˜i and distribution functions Fi and F˜i . If Fi ≥ F˜i then Ai is said to be stochastically smaller than A˜i and we denote it as Ai ≤st A˜i or alternatively fi ≤st f˜i

363

A periodic-review inventory model in a fluctuating environment Theorem 3. Let Ai and A˜i be two random variables that represent different random capacities of the supplier and Sni and S˜in the corresponding order levels and ordering quantities zni (uλ−1 ) and z˜in (uλ−1 ). If Ai is stochastically smaller than A˜i , i.e. Ai ≤st A˜i , i ∈ E then for n ≥ 1 and i ∈ E we have:


1. Sni ≥ S˜ in . 2. zni (uλ−1 ) ≥ z˜in (uλ−1 ) for x ∈ R, y1 , . . . , y≈−1 ∈ R + .

Proof. For n = 1 we have R˜ 1i (u ) = R1i (u) because the different capacities of the supplier do not affect these functions. Thus, S1i = S˜ 1i and z1i (u) = z˜ 1i (u). i i (u)] ≥ [vn−2 (u)] . This imLet us now suppose that [˜vn−2 ˜ i (·) ≥ R i (·) and Si ≥ S˜ i , zni (·) ≥ z˜ ni (·). To plies R n−1 n−1 n−1 n−1 i i (u)] ≥ [vn−1 (u)] , we must consider the three show that [˜vn−1 i i i i cases, uλ−1 < S˜ n−1 , S˜ n−1 < uλ−1 < Sn−1 and uλ−1 > Sn−1 i separately. For uλ−1 < S˜ n−1 referring to Equation (34) we have: i (uλ−1 ) v˜ n−1

=

Li,λ−1 (uλ−1 )

i z˜ n−1 (uλ−1 )

+ 0

j Pi j EDi v˜ n−2

j∈E

i

(uλ−1 − Di + w) dF˜ i (w) − γ ci 1 − F˜ i z˜ n−1 (uλ−1 ) z˜ n−1 i (uλ−1 ) i ˜ n−1 = Li,λ−1 (uλ−1 ) + [R (uλ−1 + w) − γ ci ] 0 i

× dF˜ i (w) − γ ci 1 − F˜ i z˜ n−1 (uλ−1 ) z˜ n−1 i (uλ−1 ) i ˜ n−1 = Li,λ−1 (uλ−1 ) + (uλ−1 + w) dF˜ i (w) − γ ci R 0 z˜ n−1 i (uλ−1 ) i ˜ ˜ = Li,λ−1 (uλ−1 ) + Rn−1 (uλ−1 + w)Fi (w)

γ

0

i z˜ n−1 (uλ−1 )

−

i ˜ n−1 (uλ−1 + w) − γ ci F˜ i (w)dR z˜ n−1 i (uλ−1 ) i ˜ n−1 ≥ Li,λ−1 (uλ−1 ) + R (uλ−1 + w)Fi (w) 0 z˜ n−1 i (uλ−1 ) i ˜ n−1 − Fi (w)dR (uλ−1 + w) − γ ci 0 z˜ n−1 i (uλ−1 ) i ˜ n−1 = Li,λ−1 (uλ−1 ) + (uλ−1 + w) dFi (w) − γ ci R 0 zn−1 i (uλ−1 ) i ˜ n−1 ≥ Li,λ−1 (uλ−1 ) + (uλ−1 + w) dFi (w) − γ ci R 0 zn−1 i (uλ−1 ) i ≥ Li,λ−1 (uλ−1 ) + Rn−1 (uλ−1 + w) dFi (w) − γ ci 0 zn−1 i (u) j = Li,λ−1 (u) + [γ ci + γ Pi j EDi vn−2 0

0

j∈E

(uλ−1 − Di + w) dFi (w) − γ ci

i i i = vn−1 (uλ−1 ) , i.e., v˜ n−1 (uλ−1 ) ≥ vn−1 (uλ−1 ) .

The equality step results from Equation (35). The first ini equality is justified as follows. Since uλ−1 < S˜ n−1 ,0≤w≤ i i i z˜ n−1 (uλ−1 ), then uλ−1 + w ≤ uλ−1 + z˜ n−1 (uλ−1 ) ≤ S˜ n−1 and i ˜ Equations (35) and (36) give Rn−1 (uλ−1 + w) ≤ 0. Further˜ i (·) is non-decreasing, its more, F˜ i (w) ≤ Fi (w) and as R n−1 derivative is positive. The second inequality is justified from R˜ in−1 (uλ−1 + w) ≤ 0 and by the induction assumption that i (uλ−1 ). Finally, the last step results from z˜in−1 (uλ−1 ) ≤ zn−1 ˜ i (uλ−1 + w) ≥ R i (uλ−1 + w), which is true by the inR n−1 n−1 i duction assumption. Proofs for the cases S˜ n−1 < uλ−1 < i i Sn−1 and uλ−1 > Sn−1 are quite similar.

5. The infinite-horizon problem Let us denote by V i (x, y1 , . . . , yλ−1 ) the optimal expected discounted cost for the infinite-horizon problem, starting in state (x, y1 , . . . , yλ−1 , i) at the beginning of the horizon. Based on Bellman’s principle of optimality, V i (·) satisfies the following DP equation: V i (x, y1 , . . . , yλ−1 ) = Li (x)  z  λ  c w + γ Pi j EDi V j γ i    0  j ∈E       (x + y1 − Di , y2 , . . . , yλ−1 , w) dFi (w) + min z≥0   Pi j EDi V j + γ λ ci z + γ     j∈E       (x + y1 − Di , y2 , . . . , yλ−1 , z) Fi (z). (38) λ

For any real-valued function υ : E × R → R, we consider the transformation T defined by T[(υ i (x, y1 , . . . , yλ−1 )] = Li (x) (39)  z   γ λ ci w + γ Pi j EDi υ j    0  j∈E       (x + y1 − Di , y2 , . . . , yλ−1 , w) dFi (w) + min z≥0   Pi j EDi υ j + γ λ ci z + γ     j∈E       (x + y1 − Di , y2 , . . . , yλ−1 , z) Fi (z). This transformation is a contraction mapping and it is known that under the positivity assumption, which requires the expected cost per period to be non-negative, we have limn→∞ Vni (x, y1 , . . . , yλ−1 ) = V i (x, y1 , . . . , yλ−1 ) (Bertsekas, 1997, Proposition 14). However, since the oneperiod cost may not be bounded, the optimal cost function may not be the unique solution of Equation (38). If this is the case we take as the optimal solution the minimal fixed point of Equation (39). This optimal solution satisfies limk→∞ T k [υ0i (x, y1 , . . . , yλ−1 )] = V i (x, y1 , . . . , yλ−1 ), with

364


υ0i (x, y1 , . . . , yλ−1 ) = 0. Moreover, it is known that there exists a stationary policy which minimizes the infinite period total cost. For the infinite-horizon model we have the following.

remainder of the proof can be established using arguments quite similar to those used in Theorem 1. Results similar to those presented in Theorem 3, i.e., Si ≥ S˜ i if Fi ≥ F˜ i , can be easily proved.

Theorem 4.


1. The optimal ordering policy is an environment dependent base stock level policy defined by Order the quantity: zni (uλ−1 ) = Si − (uλ−1 ) if uλ−1 < Si Do not order: zni (uλ−1 ) = 0 if uλ−1 ≥ Si ,

where Sι satisfies the equation γ ci + j∈E Pij EDι [vj (Si − Di )] = 0 and vi (uλ−1 )  zi (uλ−1 )   L γ ci w + γ (u) + Pi j EDi  i,λ−1   0  j∈E     j   [v (uλ−1 − Di + w)] dFi (w)       i z (u ) + γ Pi j EDi [vj (u − Di + γ c i λ−1 =  j∈E     i i  + z (uλ−1 ))] Fi (z (uλ−1 )), for uλ−1 < Si ,        Pi j EDi Li,λ−1 (uλ−1 ) + γ    j∈E   [vj (uλ−1 − Di )], for uλ−1 ≥ Si . 2. The optimal cost function resulting from this policy is λ−1 γ m Li,m (um ) V i (x, y1 , . . . , yλ−1 ) = Li (x) + m=1  zι (u ) λ−1    Pi j EDi vj γ ci w + γ    0  j ∈E      (uλ−1 − Di + w) dFi (w)         ι Pi j EDi vj λ−1 + γ ci z (uλ−1 ) + γ +γ j∈E     ι   (u − D + z (u )) Fi (zι (uλ−1 )), i λ−1       for uλ−1 < Si       Pi j EDi vj (uλ−1 − Di )] for uλ−1 ≥ Si .  γ j∈E

Proof. Recalling Equation (31) and since limn→∞ Vni (x, y1 , . . . , yλ−1 ) = V i (x, y1 , . . . , yλ−1 ), the limn→∞ vni (·) also exists and is a convex function since vni (.) is convex. Let us set limn→∞ vni (·) = vi (·). The function corresponding to Equation (35), that is EDi Pi j [vj (u − Di ) ], (40) R i (u ) = γ ci + γ j∈E

is non-decreasing. Since from Equation (37) limu→+∞ R i (u) > 0 and limu→−∞ R i (u ) < 0 it is concluded that the equation R i (u ) = 0 has a root, say Si i.e., R i (Si ) = 0. The

6. Conclusions In this article the model of Erdem and Ozekici (2002) is extended by introducing a lead time of λ periods and further by considering the effect of stochastically ordered distributions describing the supplier’s capacity. For the finitehorizon problem it was found that the optimal policy is of the base stock type characterized by the levels Sni and the ordering quantities zni (uλ−1 ) and the sequences Sni and zni (·) are monotonic in n. Furthermore, it is proved that the stochastic ordering on random variables representing the supplier’s random capacity is transferred to an ordering on the base stock levels Sni and the quantities zni (·). The same results are established for the infinite-horizon case. A promising topic for further research is to consider stochastic lead times. Some good ideas on these can be found in Chapter 9 of Zipkin (2000). Another extension indicated by one of the referees, is to suppose that the supplier’s random capacity depends on the random environment prevailing at every period in the lead time following the ordering period and not only from the environment prevailing in the ordering period.

Acknowledgements We thank all three anonymous referees for their constructive comments which improved the paper significantly. The work of the first author was co-financed by a grand from the European Union (European Social fund 75%) and the ministry of National Educational and Religion Affairs, in the framework of the Operational Program II for Educational and Initial Vocational Training Archimedes (25%).

References Anupindi, R. and Akella, R. (1993) Diversification under supply uncertainty. Management Science, 43, 944–963. Arrow, K.J., Karlin, S. and Scarf, H. (1958) Studies in the Mathematical Theory of Inventory and Production, Stanford University Press, Stanford, CA. Azoury, K.S. (1985) Bayes solutions to the dynamic inventory models under unknown demand distribution. Management Science, 31, 1150– 1160. Bertsekas, D.P. (1997) Dynamic Programming: Deterministic and Stochastic Models, Prentice Hall, NJ. Ehrhardt, R. (1984) (s, S) policies for a dynamic inventory model with stochastic leadtimes. Operations Research, 32, 121–132. Erdem, A. and Ozekici, S. (2002). Inventory models with random yield in a random environment. International Journal Production Economics, 78, 239–253.

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A periodic-review inventory model in a fluctuating environment Federgruen, A. and Zipkin, P. (1986) An inventory model with limited production capacity and uncertain demands II. The discounted-cost criterion. Mathematics of Operations Resarch, 11, 208–215. Gupta, D. and Cooper, W.L. (2005) Stochastic comparisons in production yield management. Operations Research, 53, 377–384. Iglehart, D. and Karlin, S. (1962) Optimal policy for dynamic inventory process with nonstationary stochastic demands, in Studies in the Mathematical Theory of Inventory and Production, Arrow, K.J., Karlin, S. and Scarf, H. (eds.), Stanford University Press, Stanford, CA, Ch. 8. Johnson, G. and Thomson, H. (1975) Optimality of myopic inventory policies for certain dependent demand processes. Management Science, 21, 1303–1307. Karlin, S. (1960) Dynamic inventory policy with varying stochastic demands. Management Science, 6, 231–258. Ozekici, S. and Parlar, M. (1999) Inventory models with unreliable suppliers in a random environment. Annals of Operations Research, 91, 123–136. Papachristos, S. (1977) Adaptive dynamic programming and inventory control. Ph.D Thesis, Faculty of Economics, University of Manchester, Manchester, UK. Rajashree, K. and Pakkala, T.P.M. (2002) A Bayesian approach to a dynamic inventory model under an unknown demand distribution. Computers and Operations Research, 29, 403–422. Scarf, H. (1959) Bayes solutions of the statistical inventory problem. Annals of Mathematics Statistics, 30, 490–508. Sethi, S. and Cheng, F. (1997) Optimality of policies in inventory models with Markovian demand. Operations Research, 45, 931–939. Shaked, M. and Shanthikumar, G.J. (1994) Stochastic Orders and their Applications, Academic Press, New York, NY. Song, J.S. and Zipkin, P. (1993) Inventory control in a fluctuating demand environment. Operations Research, 41, 351–370. Song, J.S. and Zipkin, P. (1996) Managing inventory with the prospect of obsolescence. Operations Research, 44, 215–222. Wang, Y. and Gerchak, Y. (1996) Periodic review production models with variable capacity, random yield, and uncertain demand. Management Science, 42, 130–137. Zipkin, P. (2000) Foundations of Inventory Management, McGraw-Hill, New York, NY.

=

Pi k [ EDi lim Lk,m−1 (u − Di )] u→−∞

k ∈E

=−

Pi k

j ∈E

k ∈E

=−

(m−1)

Proof of Equations (7) and (8). We use induction to prove Equation (8). Equation (5) for m = 1 is written as lim Li,1 (u) = limu→−∞ Pi j EDi Lj (u − Di ). (A1) u→−∞

(u) = − lim Li,1

u→−∞

=−

Pi j [limu→−∞ EDi Lj (u − Di )]

j∈E

Pi j [EDi limu→−∞ Lj (u − Di )] = −

j∈E

Pi j pj .

j∈E

(u) = − We now suppose that limu→−∞ Li,m−1 (m−1)

Pi j

In the last step the Chapman-Kolmogorov equations were used. Equation (7) is proved in a similar way.

Proof of Equation (37). We again use induction to prove the first part of Equation (37). In order to find the limits in Equation (37), Equation (35) indicates that we must find j the limit: limu→+∞ vn−1 (u − Di ) , n = 1, 2, . . . . Combining Equation (13) with Equation (7) gives: i lim R1 (u ) = lim γ ci + γ Pi k EDi Lk,λ−1 (u − Di ) u→+∞

u→+∞

= γ ci + γ

Pi k

= γ ci + γ

u→−∞

=

k ∈E

k ∈E

Pi k limu→−∞ EDi Lk,m−1 (u − Di )

k∈E (λ−1)

Pk j

hj

j∈E

k∈E

(λ−1)

Pi k Pk j

hj = γ ci + γ

j∈E k∈E

(λ)

Pk j hj .

j∈E

Since we search for the limit as u → +∞ we suppose that u > Sni , n ≥ 1. i ∈ E. By using Equation (7), Equation (20) for n = 2 and u > Sni becomes: lim vi (u) u→+∞ 1 = lim Li,λ−1 (u) + γ Pi k EDi Lk,λ−1 (u − Di ) u→+∞

=

(λ−1) Pi j hj

+γ

k∈E (λ) Pi j hj .

j∈E

Using this result in Equation (25) we obtain: k

i Pi k EDi v1 (u − Di ) lim R2 (u ) = lim γ ci + γ

u→+∞

= γ ci + γ = γ ci + γ

u→+∞

Pi k

k∈E (λ−1) Pk j hj

j∈E (λ) Pi j hj

+γ

2

j∈E

+γ

(λ) Pk j hj

j∈E (λ+1) Pi j hj .

j∈E

Taking the induction step, we suppose now that: (λ−1) (λ) i (u) = Pi j hj + γ Pi j hj lim vn−1 u→+∞

j∈E

+ ··· + γ

pj . Using this relation, Equation (5) gives: (u) = limu→−∞ Pi k EDi Lk,m−1 (u − Di ) lim Li,m

pj

(m)

k∈E

= −pi , so Equa-

j ∈E k ∈E

(m−1)

Pi k Pk j

j∈E

j∈E

From Equation (1) we have limu→−∞ Li (u) tion (A1 ) gives:

Pi j pj .

j∈E

Appendix

pj = −

Pk j

n−1

j∈E (λ+n−2) Pi j hj .

(A2)

j∈E

j∈E

Thus, Equation (35) gives: lim Rni (u) = γ ci + γ

u→+∞

j∈E

+ ··· + γn

(λ)

Pi j hj + γ 2

j∈E

j∈E

(λ+n−1)

Pi j

hj .

(λ+1)

Pi j

hj

366


From the second branch of Equation (34) and the induction hypothesis, Equation (A2), we have: lim vni (u) k

= lim Li,λ−1 (u) + γ Pi k EDi vn−1 (u − Di )

The second part of Equation (38) can be proved in a similar way, but since u → −∞ we have to consider the first branch of Equation (34).

u→+∞

u→+∞

=

(λ−1)

Pi j

hj + γ

j∈E

=

j∈E

+γ


=

k∈E k Pi k [EDi lim [vn−1 (u − Di ) ]] u→+∞

k∈E

(λ−1) Pi j hj

+γ

Pi k

j∈E (λ−1) Pi j hj j∈E

+ ··· + γ

+γ

(λ−1)

Pk j

hj

j∈E

k∈E (λ) Pk j hj

n−1

(λ+n−2) Pk j hj

j∈E (λ) Pi j hj

+ ··· + γn

j∈E

(λ+n−1)

Pi j

j∈E

i.e., i (u) = γ ci + γ lim Rn+1

u→+∞

+ · · · + γ n+1

j∈E

j∈E

(λ+n)

Pi j

Biographies

hj .

(λ)

Pi j hj + γ 2

j∈E

(λ+1)

Pi j

hj

hj .

S. Papachristos holds a B.Sc. in Mathematics from the University of Thesaloniki Greece, a M.Sc. in Probability and Statistics from the University of Sheffield, England and a Ph.D. in Operations Research from the University of Manchester, England. For many years he worked in the Department of Mathematics of the University of Ioannina, Greece and is currently an Associate Professor in the Department of Agribusiness Administration and Management, at the same university. He has published articles in Management Science, Operations Research, Naval Research Logistics, International Journal of Production Economics, O.R. Letters, Journal of the Operations Research Society, Optimal Control and Applications Methods and others. His current research interests include lot sizing, reverse logistics, SCM and dynamic programming. A. Katsaros holds a B.Sc. in Mathematics from the University of Athens, Greece and a Ph.D in Operations Research from the Department of Mathematics, University of Ioannina, Greece. He is currently a Lecturer in the Department of Environment and Natural Recourses Management. He has published articles in Queueing Systems, Journal of the Operations Research Society of Japan and Stochastic Models. His research interests are queuing systems and stochastic inventory systems.