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A production-inventory model with randomly changing environmental conditions
European Journal of Operational Research 174 (2006) 539–552 www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
A production-inventory model with randomly changing environmental conditions Esmail Mohebbi
*
Department of Industrial and Management Systems Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588-0518, USA Received 4 January 2004; accepted 20 December 2004 Available online 14 March 2005
Abstract We consider a limited-capacity production-inventory system with linear production rate and compound Poisson demand in which both production and demand processes are subject to independently and randomly changing environmental conditions. Assuming that these conditions are represented by two continuous-time homogeneous Markov chains and shortages are lost, we derive the limiting distribution of the inventory level and present some numerical results in terms of the ensuing performance measures. 2005 Elsevier B.V. All rights reserved. Keywords: Inventory; Random environments; Markov-modulated processes; Level crossings
1. Introduction Effective management of production and inventory control operations in dynamic and stochastic environments is closely tied with the extent of incorporating the randomly changing ‘‘internal’’ and ‘‘external’’ conditions into the decision making process. These conditions often play defining roles in characterizing the supply and demand processes and as such, their random fluctuations can significantly influence the performance of production-inventory systems. Changes in internal conditions may occur due to a variety of factors (such as equipment conditions and breakdowns, production yields, raw material/subcomponents quality, production mix, workforce level and labor skills, etc.) at random points in time and affect the production (or equivalently, the supply) rate of the system. Similarly, random changes in external conditions
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Fax: +1 402 472 1384. E-mail address: [email protected]
0377-2217/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2004.12.014
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may be caused by sudden changes in economic and market conditions which mainly influence the demand process. The importance of incorporating the random nature of environmental conditions that may impact the status or distributions of supply and/or demand in production/inventory models has been recognized in the literature. Much of the relevant work within the context of inventory control literature has been focused on randomly fluctuating demand environments represented by a Markov-modulated demand process (e.g., [3,5,25,26]). The fundamental feature of these studies is that the demand distribution (or its parameters) may change in time in accordance with state transitions of a discrete- or continuous-time Markov chain which represents the environment. Accordingly, focusing on a fluctuating supply environment, inventory models with random supply disruptions have received considerable attention in the literature (e.g., [1,10,16,20,21]). The underlying scenario in these models is that the status of a supplier switches intermittently between two possible states (available and unavailable) due to random changes in the environment (e.g., strike, embargo, etc.) thereby disrupting the supply process. The same concept has been addressed in production-inventory settings where a failure-prone production facility which switches randomly between on and off states feeds into a storage system which faces the demand (e.g., [2,14,15,22]). Queueing models with Markov-modulated arrival and service rates have been discussed in the queueing theory literature (e.g., [17,24]). A number of related papers in queueing context have studied fluid systems with deterministic or stochastic input and output flows that are modulated by a two-state environment which alternates randomly between up and down states (e.g., [6,11,12,23]). The case of a fluid model with deterministic and continuous flows modulated by a multi-state continuous-time Markov chain has also been discussed in the queueing literature [13]. Another related line of research encompasses a special class of random yield models where the yield structure depends on the state of a randomly changing environment. Ozekici and Parlar [19] considered a periodic-review inventory model with backorders and zero lead time where the distribution of demand, the availability (i.e., up or down status) of an unreliable supplier and the cost parameters in every period depend on an environmental process that is represented by a time-homogeneous Markov chain. More specifically, their model allowed for an all or nothing replenishment scenario where the order placed at the beginning of a period is delivered immediately in full quantity, if the supplier is up, or else, nothing is delivered at all (zero replenishment). Erdem and Ozekici [8] considered a similar setting, but allowed for a different yield structure which is related to the random capacity of the supplying vendor; that is, an order placed at the beginning of a period is immediately received in full quantity if the order size is less than the vendors capacity, otherwise, the quantity received is equal to the available capacity which is determined by the state of the time-homogeneous Markov chain representing the environment. Finally, we note in passing that the implications of random supplier capacity in zero-lead-time inventory systems with/without random yield have received some attention in recent years (e.g., [9,27]); however, these works do not provide an explicit account of the impact of randomly changing environmental conditions on the supply process. Motivated by Berg et al. [2], this paper considers a production-inventory system under randomly changing environmental conditions. In [2], the authors studied a single-item production-inventory system consisting of an arbitrary number of identical parallel machines and a storage facility with limited capacity that faces a compound Poisson demand process. Each machine is subject to a stochastic failure and repair process in which the time to failure and the repair time are independent and exponentially distributed. There is an ample repair capacity so that the repair of a failed machine starts immediately. Each machine can produce the product continuously and uniformly over time at a fixed production rate. All operable machines produce the product simultaneously as along as the inventory level (i.e., stock-on-hand, throughout) remains below the storage capacity, and shortages are lost. Production on all operable machines is stopped whenever the inventory level reaches the maximum storage capacity and is started again as soon as the inventory falls below the storage limit. Clearly, the system considered in [2] can be viewed as a production-inventory system in a randomly changing (internal) environment where the number of operable
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machines at any point in time determines the overall production rate capability of the facility. However, given that each machine can only have two possible states, namely, up or down (under repair), during a production run, changes in the overall production rate of the system (while producing) can only occur as the result of transition between adjacent states; i.e., from m operating machines to m + 1 or m 1 machines. Furthermore, the random operating environment has no impact on the demand process. In contrast, our work is focused on a more general production-inventory setting in which the production and the demand processes are subject to two independent and randomly changing multi-state environmental processes (i.e., internal and external). The environmental processes are represented by two continuous-time homogeneous Markov chains which allow for one-step transitions from any given state to all other states. In this paper we consider a production-inventory system with limited storage capacity in which demand for a single item arises according to a compound Poisson stream whose occurrence rate and random batch sizes at any point in time depend on the state of a random (external) environment, and production occurs continuously and uniformly over time at a rate determined by the state of another independent and randomly changing (internal) environment. As mentioned earlier, we assume that each of the two randomly changing environments is represented by a continuous-time homogeneous Markov chain with a discrete state space. The facility starts production as soon as the inventory level drops below the storage limit and the production continues until the inventory level reaches the storage full capacity. Assuming that the shortages are lost, we derive the steady-state distribution of the inventory level which is used to formulate some measures of common interest regarding the performance of such systems. We also formulate an expected-net-revenue maximization problem and provide some numerical results. The organization of the paper is as follows: The model is described in Section 2, and the solution procedure is presented in Section 3. Section 4 outlines performance measures and presents some graphical and numerical results, followed by Section 5 which contains a summary and direction for further extension of this work.
2. Model formulation We consider a single-item production-inventory system with the following characteristics: (1) Demand is realized against the inventory level of a storage facility which has a limited storage capacity of R units. (2) The random demand process is characterized by a Markov-modulated compound Poisson stream. We assume that the occurrence rate for Poisson demand at any given time can take on a value of k|, | = 1, . . . , nE, according to the state of a continuous-time stochastic process E ¼ fEðtÞ; t P 0g with discrete state space S E ¼ f1; . . . ; nE g, and the demand sizes are conditionally (given the state of E) independent and exponentially-distributed random variables with mean /1 | . The exponential assumption pertaining to demand sizes is intended for obtaining closed form analytical solutions, albeit the modeling process in terms of formulating the system equations can accommodate more general compounding distributions. The modulating process E is described by a continuous-time homogeneous Markov chain in which the sojourn time in each state | 2 S E is exponentially distributed with 0 mean g1 | , and the transition probabilities at any transition epoch are denoted by q||0 for |(5| ) and E E 0 | 2 S with q|| ¼ 0 for 8| 2 S . (3) The storage facility is supplied by a production facility which can produce the item continuously and uniformly at the rate wı , ı ¼ 1, . . . , nI, depending on the state of a continuous-time homogeneous Markov chain I ¼ fIðtÞ; t P 0g with discrete state space S I ¼ f1; . . . ; nI g and transition probabilities pıı0 for ıð6¼ ı0 Þ and ı0 2 S I , and pıı ¼ 0 for 8ı 2 S I at any transition epoch. The sojourn time of I in each state ı 2 S I is exponentially distributed with mean m1 ı .
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(4) The production is halted whenever the inventory level in the storage facility reaches its capacity R and is restarted as soon as the inventory level drops below R ( i.e., at the next demand occurrence epoch). Clearly, such production control policy is mainly prevalent in production systems where setup times/ costs are negligible (in comparison with production times/costs) or nonexistent. As such, the policy is more likely to be realized in process industries such as chemical and pharmaceutical plants where continuous production of standardized goods by highly specialized and automated equipments over a long horizon rationalizes the overlooking of setups. A more general control policy may include a provision for setup time/cost by restarting production only when the inventory level drops to or below a threshold level r (r < R) at a demand occurrence epoch; however, to facilitate the analytical process in this paper, we assume that the setup time/cost for each production run is insignificant. (5) Both E and I are assumed to be ergodic. Moreover, transitions in E and I over time occur independent of each other and of the inventory level of the storage facility; that is, transitions in E and I occur independently irrespective of whether or not the production facility is idle or producing. (6) All stockouts, including the excess demand when a batch size is larger than the inventory level, are considered to be lost. It is clear from the above that the inventory level in this system has a continuous state space which ranges from 0 to R over time. We let X(t) 2 U [0, R] denote the inventory level at time t(P0), and borrowing a term from [2], define W(t) = R X(t) with W(t) 2 U as the ‘‘slack storage capacity’’ available at time t. We note that under the set of assumptions stated above, X(t), and hence W(t), can be characterized as a regenerative process. For example, assuming that at time t = 0 the system is full and E and I are in states ı and |, respectively, then the regenerative points can be defined at epochs at which the production is halted (i.e., the system reaches its full storage capacity) and the internal and external environmental processes are in states ı and | accordingly. We, furthermore, assume that X(t), and naturally W(t), posses limiting distributions [7]. Similar to [2], we focus on the steady-state situation throughout and apply a system-point method of level crossings [4] to develop a model whose solution yields the limiting distribution of the W(t) process. In this regard, we let F(Æ) and f(Æ) denote the limiting cumulative distribution and density functions of the W(t) process, respectively. By level-crossing theory, the three-dimensional stochastic process fW ı| ðtÞ, t P 0} {W(t), IðtÞ ¼ ı, EðtÞ ¼ |, t P 0}, ı 2 S I and | 2 S E , is the system-point process for our system, and corresponds to those portions of sample path tracings of W(t) over time t during which E and I are in states ı and |, respectively. A typical realization of the W ı| ðtÞ process over the state space U fı|g U \ fIðtÞ ¼ ıg \ fEðtÞ ¼ |g is depicted in Fig. 1. (Observe that W ı| ðtÞ is a jump process with right-continuous sample paths.) Under the steadystate conditions, we write W ı| ¼ limt!1 W ı| ðtÞ in distribution, and let F ı| ðwÞ ¼ PrðW ı| 6 wÞ denote its cumu-
Fig. 1. Typical tracings of the W ı| ðtÞ process (ı 2 S I , | 2 S E ).
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lative probability distribution function; i.e., the long-run joint probability of the slack storage capacity not exceeding w while the environmental processesR I and E are in states ı and |, respectively. It is evident that w under our set of assumptions, F ı| ðwÞ ¼ fı|0 þ a¼0 fı| ðaÞda, where fı| ðwÞ is the density function associated 0 with W ı| , and fı| ¼ dF ı| ð0Þ denotes the probability mass at Wı| = 0. Note that fı|0 corresponds to the long-run fraction of time that the inventory system is full (i.e., zero slack storage capacity) and production is halted while I and E, are in states ı and |, respectively. Accordingly, given that the sample paths of the W(t) process can be viewed as the aggregation those of the W ı| ðtÞ processes, it can be readily verified that PI P PnIofP nE nE 0 0 f ðwÞ ¼ nı¼1 f ðwÞ and f ¼ dF ð0Þ ¼ |¼1 ı| ı¼1 |¼1 fı| . We now utilize level-crossing theory to establish the model equations for deriving the partial densities fı| (.), ı 2 S I and | 2 S E , by studying the sample paths of the system-point process W ı| ðtÞ 2 U fı|g . The equations are based on the main principle of level crossings which asserts that the long-run rates at which the sample path tracings of the system-point process enter and exit a set of states are equal. Hence, let us consider the state interval [0, w) with w 2 (0, R) in U fı|g , ı 2 S I and | 2 S E . The long-run average entrance and exit rates of the system-point sample path associated with this set of states are as follows: (i) The long-run average number of downcrossings of the sample path into the interval [0, w) in U fı|g per unit time by level-crossing theory is wı fı| ðwÞ, ı 2 S I and | 2 S E . (ii) The infinitesimal transition generators of the environmental Markov chains I and E assert that the state interval [0, w) in U fı|g can be reached from the same intervals in U fk|g , k 2 S I , k 6¼ ı, and in U fı‘g , ‘ 2 S E , ‘ 6¼ |. Therefore, the long-run average entrance rates into this interval due to such transitions are mk pkı F k| ðwÞ and g‘ q‘| F ı‘ ðwÞ, respectively. (iii) The long-run average exit rate of the sample path from [0, w) in U fı|g due to infinitesimal transition generators of I and E is ðmı þ g| ÞF ı| ðwÞ, ı 2 S I and | 2 S E . (iv) When the sample path is in state a (0 6 a < w) in U fı|g , a demand arrival whose size exceeds w a results in an upward jump which takes the sample path out of the interval [0, w) in U fı|g . By PASTA [28], R w the long-run average exit rate of the sample path from this interval due to such event is k| a¼0 e/| ðwaÞ dF ı| ðaÞ. Considering the above and following level-crossing theory, for any interval [0, w), w 2 (0,R), in U fı|g , ı 2 S I and | 2 S E , we obtain: Z w nI nE X X mk pkı F k| ðwÞ þ g‘ q‘| F ı‘ ðwÞ ¼ ðmı þ g| ÞF ı| ðwÞ þ k| e/| ðwaÞ dF ı| ðaÞ; ð1Þ wı fı| ðwÞ þ k¼1
‘¼1
a¼0
where the right- and left-hand sides of Eq. (1), respectively, represent the long-run average total entrance and exit rates of the system-point sample path with respect to the designated interval. Following the same line of reasoning for the state (level) w = 0 in U fı|g , ı 2 S I and | 2 S E , results in the following boundary condition: wı fı| ð0Þ þ
nI X
mk pki fk|0 þ
k¼1
nE X
g‘ q‘| fı‘0 ¼ ðmı þ g| þ k| Þfı|0 ;
ð2Þ
‘¼1
where wı fı| ð0Þ is the long-run average number of down-crossings of level zero in U fı|g per unit time by the system-point sample path. Finally, the normalizing condition Z R nI X nE X 0 fı| þ fı| ðwÞdw ¼ 1; ð3Þ ı¼1
|¼1
w¼0
completes the model formulation.
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3. Model solution We invoke the differential operator hDi = hd/dwi and let gı| ðwÞ ¼ Df ı| ðwÞ for ı 2 S I and | 2 S E , wherever the derivative exists. Applying hDihD þ /| i to Eq. (1) results in nI nI X X wı Dgı| ðwÞ þ ðwı /| mı g| k| Þgı| ðwÞ ðmı þ g| Þ/j fı| ðwÞ þ mk pkı gk| ðwÞ þ /| mk pkı fk| ðwÞ k¼1
þ
nE X
g‘ q‘| gı‘ ðwÞ þ /|
‘¼1
nE X
k¼1
g‘ q‘| fı‘ ðwÞ ¼ 0:
ð4Þ
‘¼1
Let H denote a square matrix of size n(n = nI · nE) that represents the infinitesimal generator for the two-dimensional {I(t), E(t), t P 0} Markovian process over the state space fðı; |Þ; ı ¼ 1; . . . ; nI ; | ¼ 1; . . . ; nE g; i.e., 2
ðm1 þ g1 Þ 6 gq 2 21 6 6 .. 6 6 . 6 6 g q 6 nE nE 1 6 6 m2 p21 6 H¼6 0 6 6 .. 6 6 . 6 6 0 6 6 .. 6 4 . 0
g1 q12 ðm1 þ g2 Þ .. . gnE qnE 2 0 m2 p21 .. . 0 .. . 0
g1 q1nE g2 q2nE .. .. . . ðm1 þ gnE Þ 0 0 .. .. . . m2 p21 .. . mnI pnI 1
m1 p12 0 .. . 0 ðm2 þ g1 Þ g2 q21 .. . gnE qnE 1 .. . 0
0 m1 p12 .. . 0 g1 q12 ðm2 þ g2 Þ .. . gnE qnE 2 .. . 0
0 0 .. .. . . m1 p12 g1q1nE g2 q2nE .. .. . . ðm2 þ gnE Þ .. . mnI pnI 2
m1 p1nI 0 .. . 0 m2 p2nI 0 .. . 0 .. . gnE qnE 1
0 m1 p1nI .. . 0 0 m2 p2nI .. . 0 .. . gnE qnE 2
.. . .. .
3 0 7 0 7 7 .. 7 7 . 7 7 m1 p1nI 7 7 0 7 7 7: 0 7 7 .. 7 7 . 7 7 m2 p2nI 7 7 .. 7 5 . ðmnI þ gnE Þ
b denote two n dimensional square matrices whose rth rows, Mr. and N b r: , r = 1, 2, . . . , n, Next, let M and N respectively, are characterized as Mr ¼
/rbnr cnE E
wbnr cþ1
HTr ;
E
b r ¼ N
1 HT ; wbnr cþ1 r E
where is the rth row of HT (i.e., the transpose of H) and bsc marks the largest integer smaller than s. Also, let D = [Drr], r = 1, . . . , n, denote a diagonal matrix of size n with HTr
Drr ¼
wbnr cþ1 /rbnr cnE þ krbnr cnE E
E
wbnr cþ1
E
:
E
Then, the system of first-order linear differential equations in Eq. (4) can be expressed in the matrix form of DZðwÞ ¼ AZðwÞ;
ð5Þ T
where ZðwÞ ¼ ½f11 ðwÞ; . . . ; fnI nE ðwÞ; g11 ðwÞ; . . . ; gnI nE ðwÞ is a column vector of size 2n, and " # € 0 €I A¼ M N b þ D and €I and €0, respectively, denoting the n · n identity and null is a square matrix of size 2n with N ¼ N matrices.
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The general solution to the system of matrix differential equations in Eq. (5) is of the form Z(w) = eAwZ(0) [18]. Assuming that A is diagonalizable, let the column vector V.r denote the rth eigenvector of A corresponding to its eigenvalue cr, r = 1, . . . , 2n. Then, the general solution can be conveniently expressed as ZðwÞ ¼ VeCw b:
ð6Þ
Note that in Eq. (6), V = [V.1, . . . , V.2n], C represents a diagonal matrix whose diagonal elements are c1, . . . , c2n, and b = [b1, . . . , b2n]T is a column vector of constants coefficients. Accordingly, it can be readily verified that fðwÞ ¼ !eCw b;
ð7Þ
T where fðwÞ ¼ ½f11 ðwÞ; . . . ; fnI nE ðwÞ is a column vector of size n, and ! = YV with Y ¼ €I €0 . In order to fully characterize the limiting distribution of the W(t) process, we now need to find the values of 3n unknowns; i.e., b1, . . . , b2n and f110 ; . . . ; fn0I nE . In this regard, Let W = [Wrr], K = [Krr] and U = [Urr] denote three diagonal matrices of size n with Wrr ¼ wbr=nE cþ1 , Krr ¼ krbr=nE cnE and Urr ¼ /rbr=nE cnE , r = 1,2, . . . , n. Then, Eq. (1) can be represented in the matrix form as Z w WfðwÞ þ HT FðwÞ ¼ K eUðwaÞ dFðaÞ; ð8Þ a¼0
where FðwÞ ¼ ½F 11 ðwÞ; . . . ; F nI nE ðwÞT . Substituting the general solution, Eq. (7), in Eq. (8) and comparing the coefficient of common exponential terms, e/| w , | 2 S E , gives b € ¼ 0; ð9Þ L1 I f0 where 0 is the null column vector of size 3n, f 0 ¼ ½f110 ; . . . ; fn0I nE T , and L1 = [L1kr], k = 1, . . . , n and r = 1, . . . , 2n, with L1 ¼
kr ; cr þ /kbnk cnE E
and kr marking the (k, r)th element of !. Similarly, expressed in matrix form, Eq. (2) can be written as fð0Þ ¼ L2f 0 ;
ð10Þ
b rr , r = 1, . . . , n, denotes a diagonal matrix of size n with bþN b ¼ ½D b is an n · n matrix, and D where L2 ¼ D b rr ¼ D
krbnr cnE E
wbnr cþ1
:
E
Substituting Eq. (7) for w = 0 in Eq. (10) yields b ½ ! L2 0 ¼ 0: f
ð11Þ
The third set of relations is resulted from the overall transition rate balance equations for W ı| ðtÞ, ı 2 S I and | 2 S E , which can be expressed as HT FðRÞ ¼ 0:
ð12Þ
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Again, substituting Eq. (7) in Eq. (12) provides
L3
H
T
b f0
¼ 0;
ð13Þ
where L3=HT!J1, and J1 = [J1rr], r = 1, . . . , 2n, is a diagonal matrix with J 1rr ¼ more, using Eq. (7) in the normalizing condition depicted in Eq. (3), we obtain b be 0 ¼ 1; f
RR w¼0
ecr w dw. Further-
ð14Þ
where be ¼ e !J1 €I and e = [1, 1, . . . , 1] denote two row vectors of sizes 3n and n, respectively. Considering the above, substituting an arbitrary row in 2 3 €I L1 6 7 b 4 ! L2 5 0 ¼ 0 f L3 QT
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
fX(x)
fW(w)
with Eq. (14) provides a sufficient system of non-homogeneous linear equations to complete the model solution. Fig. 2 depicts sample forms of the limiting density and cumulative distribution functions for the W(t) and X(t) processes.
0.3
0.2
0.1
0.1
0
0
5
10 w
15
0
20
1
0
5
10 x
15
20
5
10 x
15
20
1 0.8
0.8 0.6
fX(x)
fW(w)
0.3
0.2
0.6 0.4
0.4 0.2 0.2 0
5
10 w
15
20
0
0
Fig. 2. Typical plots of the limiting density and distribution functions for the W(t) and X(t) processes for CVw = 0.9, CVk = 0.5 and R = 20.
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4. Performance measures and numerical results Explicit knowledge of the limiting distribution of W(t) (and hence X(t)) enables us to formulate a variety of long-run performance measures for the production-inventory system considered here. Some of the most common measures of interest noted in the literature (e.g., [2]) include: Expected inventory level: Z R lX ¼ R lW ¼ R wdF ðwÞ ¼ R e!J2b; ð15Þ w¼0
where J2 = [J2rr], r = 1, . . . , 2n, is a diagonal matrix with J 2rr ¼ Expected utilization ratio of the production facility: q¼
RR w¼0
wecr w dw.
Expected effective production rate w ¼ PnI e ; Expected potential production rate ı¼0 wı p ı
ð16Þ
where
we ¼ eW FðRÞ f 0 ¼ e!J1b;
ð17Þ
I
and pı ¼ limt!1 PrfIðtÞ ¼ ıg, ı 2 S . Expected rate of lost sales: Z R Z 1 uUeUðRwþuÞ dudFðwÞ ¼ eKðJ3b þ J4f 0 Þ; b ¼ eK w¼0
ð18Þ
u¼0
where J3 = [J3kr], k = 1, . . . , n and r = 1, . . . , 2n, is an n · 2n matrix with J 3kr ¼
/ R kr k ðecr R e kbnE cnE Þ; /kbnk cnE ðcr þ /kbnk cnE Þ E
E
and J4 = [J4kk], k = 1, . . . , n, is a diagonal matrix with /
k
R
e kbnE cnE : J 4kk ¼ /kbnk cnE E
Expected fill rate (fraction of demand satisfied per unit time): fr ¼ 1 PnE
b
k| |¼1 /|
q|
;
ð19Þ
where q| ¼ limt!1 PrfEðtÞ ¼ |g, | 2 S E . For expository purposes, these performance measures were computed for various combinations of the parameter settings displayed in Table 1 with S I ¼ f1; 2; 3g, S E ¼ f1; 2g, p12 = p13 = p21 = p23 = p31 = p32 = 0.5, q12 = q21 = 1, m1 = m2 = m3 = g1 = g2 = 1, /1 = /2 = 0.5, and R = 20. We focused on the impact of variability in production and demand rates caused by fluctuating random environments on the system performance. Hence, the values of wı , ı = 1, 2, 3, and k| , | = 1, 2, in Table 1 were chosen to represent various levels of variability as measured by their coefficients of variation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P3 P3 2 2 P3 and CVk ¼ CVw and CVk, respectively. Clearly, CVw ¼ ı¼1 wı p ı ð ı¼1 wı p ı Þ = ı¼1 wı p ı , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P2 P2 2 2 P2 |¼1 k| q| ð |¼1 k| q| Þ = |¼1 k| q| . The results are shown Fig. 3. Observe that for any chosen value of CVw, the expected utilization of the production facility decreases as CVk increases. This decreasing trend intensifies as CVw decreases. This is primarily due to the fact that the variability of demand rate has direct
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Table 1 Parameter settings for numerical experiments w2
w3
CVw
Production rates: wı, ı 2 S ¼ f1; 2; 3g 35 35 40 35 46 30 51 28 56 25 60 20 66 15 70 15 75 25 80 20 85 15 90 10
35 30 29 26 24 25 24 20 5 5 5 5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
w1 I
k1
k2
CVk
15 13 12 10 9 7 6 4 3 1.5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
E
Demand rates: k|, | 2 S ¼ f1; 2g 15 17 18 20 21 23 24 26 27 28.5
impacts on the product outflow from the storage facility and the rate at which the production process is halted. The increasing variability of the production rate, however, appears to mitigate the impact of demand variability. It should also be noted that for any given value of CVk, the expected utilization ratio decreases as CVw increases. A similar pattern is displayed by the expected fill rate for the reasons made clear from observing the behavior of the expected lost sales rate against changes in CVk and CVw. The expected inventory level shows a decreasing trend for any chosen value of CVk as CVw increases. The domain of such decrease, however, diminishes as CVk increases. This appears to have been caused by a complex interaction between the demand and the production rate variability. In this regard, we note that on one hand, when there is no variability in the production rate of the facility, the expected inventory level decreases (and the expected lost sales rate increases rapidly) as the variability in demand rate increases. On the other hand, increasing the variability of production rate appear to have a compensating effect by dampening the growth of lost sales rate at the expense of creating some excess inventory as CVk increases. In addition to the above, Eqs. (15)–(19) can be readily used as functional components to formulate objective functions for optimization purposes. For illustration, let us assume that production generates revenue of cw per unit, and there is an inventory holding cost of ch per unit per unit time and a penalty cost of cp per unit of lost demand. The long-run expected net revenue accumulation rate for a preset value of the storage capacity, R, can then be characterized as XðRÞ ¼ cw we ch lX cp b:
ð20Þ
Assuming that the decision maker is interested in finding the optimum capacity for the storage facility (R ) so that the expected net revenue accumulated per unit time is maximized, the optimization process can be practically carried out through standard search techniques embedded in a commercially available opti-
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14
1 CVψ= 0
Expected Fill Rate
Expected Inventory
13 12 11 10
CVψ= 1.1
9 8
0.9
CVψ= 0
0.8 0.7 0.6 CVψ= 1.1
0
0.2
0.4
0.6
0.8
0.5
1
0
0.2
0.4
CVλ Expected Utilization of the Facility
Expected Lost Sales Rate
CVψ= 1.1
12 10 8 6
CVψ= 0
4 2 0
0.2
0.4
0.6
0.8
1
CVλ
14
0
549
0.6
0.8
1
0.9 0.8 CVψ= 0
0.7 0.6 0.5 CVψ= 1.1
0.4
0
0.2
0.4
CVλ
0.6
0.8
1
CVλ
Fig. 3. Plots of performance measure for parameter sets in Table 1.
mization package. (It should be pointed out that establishing the concavity or unimodality of X(R) in an analytical sense due to its complex form appears to be impractical). Tables 2 and 3 contain the optimality results for the parameter settings described earlier in this section with cw = 5, ch = 1, and cp = 7. These Table 2 Optimal values R for parameter sets in Table 1 CVw 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
CVk 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
25.59 26.29 28.05 30.96 35.08 39.50 47.37 52.61 61.89 68.72 76.50 84.83
26.39 27.08 28.83 31.68 35.71 40.04 47.78 52.93 62.20 68.94 76.66 84.94
27.41 28.09 29.81 32.61 36.53 40.74 48.32 53.36 62.59 69.24 76.86 85.08
30.80 31.42 33.04 35.64 39.25 43.10 50.16 54.87 63.90 70.24 77.57 85.59
33.12 33.70 35.25 37.72 41.14 44.79 51.50 56.00 64.84 70.99 78.12 85.98
38.77 39.26 40.64 42.83 45.88 49.09 55.05 59.13 67.32 73.04 79.69 87.13
42.00 42.45 43.74 45.79 48.65 51.65 57.24 61.11 68.88 74.38 80.76 87.93
49.09 49.46 50.56 52.34 54.84 57.47 62.38 65.85 72.63 77.70 83.53 90.10
52.88 53.22 54.23 55.88 58.22 60.68 65.27 68.57 74.83 79.70 85.27 91.52
58.82 59.11 60.00 61.47 63.57 65.79 69.97 73.02 78.54 83.12 88.34 94.16
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Table 3 Optimal values X(R ) for parameter sets in Table 1 CVw
CVk 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
122.41 121.23 118.69 114.64 109.27 103.44 93.17 87.33 73.06 66.25 58.01 48.70
121.01 119.84 117.30 113.30 108.03 102.33 92.26 86.51 72.40 65.64 57.47 48.23
119.31 118.16 115.62 111.68 106.52 100.98 91.14 85.49 71.57 64.89 56.80 47.63
114.22 113.12 110.62 106.84 101.97 96.83 87.63 82.27 68.97 62.50 54.66 45.72
110.98 109.91 107.45 103.78 99.06 94.13 85.30 80.10 67.22 60.87 53.19 44.41
103.41 102.44 100.11 96.66 92.26 87.73 79.64 74.79 62.86 56.79 49.49 41.11
99.20 98.27 96.03 92.71 88.46 84.12 76.38 71.69 60.28 54.36 47.27 39.11
90.12 89.29 87.24 84.18 80.24 76.24 69.14 64.75 54.43 48.79 42.10 34.44
85.32 84.54 82.58 79.65 75.87 72.04 65.22 60.97 51.17 45.66 39.17 31.75
77.87 77.15 75.34 72.60 69.05 65.44 59.02 54.96 45.90 40.58 34.36 27.28
results indicate that the larger the variability in the production and demand rates, the higher the required storage capacity in the system. Furthermore, the expected net revenue accumulated by the system per unit time when operating with the optimal storage capacity decreases as the production and demand rates endure higher variability.
5. Summary and extension Motivated by an existing model with randomly varying production rates [2], in this paper we presented a production-inventory model in which the production and demand rates are modulated by two independent continuous-time homogeneous Markov chains. For stochastic piecewise linear production and compound Poisson demands, we applied level-crossing theory to derive the limiting distribution of the inventory level in a limited capacity production-storage system with lost sales. A number of important performance measures including an expected revenue accumulation rate function were formulated and some numerical results in terms of sensitivity analysis and optimality of the system parameters were presented. These results clearly demonstrate the critical impact of the variability induced by randomly changing environments on the system performance. The importance of accounting for the random nature of the surrounding environments has been recognized by researchers in recent years. While most production-inventory or fluid models with finite production (replenishment) or inflow rates in the literature limit their characterization of the state environments to having only two states (i.e., up/down or on/off), this work proposes an adaptation of level-crossing theory which allows for both demand and production processes to depend on multi-state random environments. The study with its findings motivates continuing research in this area. In this regard, an imminent extension of the model proposed in this paper involves the introduction of the two-parameter (r, R) production control policy to address the issue of setup time/cost of the production facility cited earlier. The level-crossing approach adapted here appears to provide a viable mean for modeling this generalized case where once production is halted after the system reaches its maximum storage capacity R, it is only restarted after the inventory level downcrosses the critical level r at a demand occurrence epoch. It should be noted, however, that unlike the present scenario where the sample path tracings of the inventory level consist of both continuous and jump crossings of state levels within the entire state space, the sample path analysis under the new policy must account for pure jump crossings of state levels within the subspace r < X(t) 6 R when pro-
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duction is halted. This is expected to have an increasing effect on the number of balance equations and hence the complexity of the solution process. It is also worth mentioning that although we assumed in this paper that demand and production are modulated by two independent Markovian environmental (i.e, external and internal) processes, it is equally conceivable to be faced with a scenario where there are interdependencies between internal and external conditions; e.g., a high production rate may be correlated with a high demand rate, etc. Yet, assuming that such interdependencies in environmental processes can be captured through redefining the environmental states and structuring a single ergodic Markov chain B ¼ fBðtÞ; t P 0g with discrete state space S B ¼ f1; . . . ; ng, the present model formulation can be readily revised address this scenario as a special case. Pn Pto n More specifically, letting f ðwÞ ¼ ı¼1 fı ðwÞ and f 0 ¼ dF ð0Þ ¼ ı¼1 fı0 , Eq. (1) can be simplified as Z w n X fk okı F k ðwÞ ¼ oı F ı ðwÞ þ kı e/ı ðwaÞ dF ı ðaÞ; wı fı ðwÞ þ k¼1
a¼0
f1 ı 0
represent the mean sojourn time in state ı, and oıı0 denotes the transition probability from state ı where to state ı ð6¼ ıÞ in S B . The rest of the formulation follows accordingly.
Acknowledgement The author is thankful for insightful comments by an anonymous referee which have improved the presentation of this paper.
References [1] A. Arreola-Risa, G.A. DeCroix, Inventory management under random supply disruptions and partial backorders, Naval Research Logistics 46 (1998) 687–703. [2] M. Berg, M.J.M. Posner, H. Zhao, Production-inventory systems with unreliable machines, Operations Research 42 (1994) 111– 118. [3] P.H. Brill, B.A. Chaouch, An EOQ model with random variations in demand, Management Science 41 (1995) 927–936. [4] P.H. Brill, M.J.M. Posner, The system-point method in exponential queues: A level crossing approach, Mathematics of Operations Research 6 (1981) 31–49. [5] F. Chen, J.S. Song, Optimal policies for multi-echelon problems with Markov-modulated demand, Operations Research 49 (2001) 26–234. [6] H. Chen, D.D. Yao, A fluid model for systems with random disruptions, Operations Research 40 (1992) S239–S247. [7] E. C ¸ inlar, Introduction to Stochastic Processes, Prentice-Hall, Englewood Cliffs, New Jersey, 1975. [8] A.S. Erdem, S. Ozekici, Inventory models with random yield in a random environment, International Journal of Production Economics 78 (2002) 239–253. [9] R. Gullu, Base stock policies for production/inventory problems with uncertain capacity levels, European Journal of Operational Research 105 (1998) 43–51. [10] U. Gurler, M. Parlar, An inventory problem with two randomly available suppliers, Operations Research 45 (1997) 904–918. [11] H. Kaspi, O. Kella, D. Perry, Dam processes with state dependent batch sizes and intermittent production processes with state dependent rates, Queueing Systems 24 (1996) 37–57. [12] O. Kella, W. Whitt, A storage model with a two-state random environment, Operations Research 40 (1992) S257–S262. [13] O. Kella, Markov-modulated feedforward fluid networks, Queueing Systems 37 (2001) 141–161. [14] R.R. Meyer, M.H. Rothkopf, S.A. Smith, Reliability and inventory in a production-storage system, Management Science 25 (1997) 799–807. [15] K. Moinzadeh, P. Aggarwal, Analysis of a production/inventory system subject to random disruptions, Management Science 43 (1997) 1577–1588. [16] E. Mohebbi, A replenishment model for the supply uncertainty problem, International Journal of Production Economics 87 (2004) 25–37. [17] M.F. Neuts, The M/M/1 queue with randomly varying arrival and service rates, Opsearch 15 (1978) 139–157.
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[18] B. Noble, Applied Linear Algebra, Prentice-Hall, Englewood Cliffs, NJ, 1969. [19] S. Ozekici, M. Parlar, Inventory models with unreliable suppliers in a random environment, Annals of Operations Research 91 (1999) 123–136. [20] M. Parlar, Probabilistic analysis of renewal cycles: An application to a non-Markovian inventory problem with multiple objectives, Operations Research 48 (2000) 243–255. [21] M. Parlar, D. Perry, Optimal (Q, r, T) policies in deterministic and random yield models with uncertain future supply, European Journal of Operational Research 84 (1995) 431–443. [22] P.R. Parthasarathy, M. Sharafali, On the unutilized capacity of a production-storage system, Operations Research Letters 6 (1987) 233–238. [23] D. Perry, M.J.M. Posner, A mountain process with state dependent input and output and a correlated dam, Operations Research Letters 30 (2002) 245–251. [24] N.U. Parbhu, Y. Zhu, Markov-modulated queueing systems, Queueing Systems 5 (1989) 215–246. [25] S. Sethi, F. Cheng, Optimality of (s, S) policies in inventory models with Markovian demand, Operations Research 45 (1997) 931–939. [26] J.S. Song, P. Zipkin, Inventory control in a fluctuating demand environment, Operations Research 41 (1993) 351–370. [27] Y. Wang, Y. Gerchak, Periodic review production models with variable capacity, random yield, and uncertain demand, Management Science 42 (1996) 130–137. [28] R.W. Wolff, Poisson arrivals see time averages, Operations Research 30 (1982) 223–231.
Production, Manufacturing and Logistics
A production-inventory model with randomly changing environmental conditions Esmail Mohebbi
*
Department of Industrial and Management Systems Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588-0518, USA Received 4 January 2004; accepted 20 December 2004 Available online 14 March 2005
Abstract We consider a limited-capacity production-inventory system with linear production rate and compound Poisson demand in which both production and demand processes are subject to independently and randomly changing environmental conditions. Assuming that these conditions are represented by two continuous-time homogeneous Markov chains and shortages are lost, we derive the limiting distribution of the inventory level and present some numerical results in terms of the ensuing performance measures. 2005 Elsevier B.V. All rights reserved. Keywords: Inventory; Random environments; Markov-modulated processes; Level crossings
1. Introduction Effective management of production and inventory control operations in dynamic and stochastic environments is closely tied with the extent of incorporating the randomly changing ‘‘internal’’ and ‘‘external’’ conditions into the decision making process. These conditions often play defining roles in characterizing the supply and demand processes and as such, their random fluctuations can significantly influence the performance of production-inventory systems. Changes in internal conditions may occur due to a variety of factors (such as equipment conditions and breakdowns, production yields, raw material/subcomponents quality, production mix, workforce level and labor skills, etc.) at random points in time and affect the production (or equivalently, the supply) rate of the system. Similarly, random changes in external conditions
*
Fax: +1 402 472 1384. E-mail address: [email protected]
0377-2217/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2004.12.014
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may be caused by sudden changes in economic and market conditions which mainly influence the demand process. The importance of incorporating the random nature of environmental conditions that may impact the status or distributions of supply and/or demand in production/inventory models has been recognized in the literature. Much of the relevant work within the context of inventory control literature has been focused on randomly fluctuating demand environments represented by a Markov-modulated demand process (e.g., [3,5,25,26]). The fundamental feature of these studies is that the demand distribution (or its parameters) may change in time in accordance with state transitions of a discrete- or continuous-time Markov chain which represents the environment. Accordingly, focusing on a fluctuating supply environment, inventory models with random supply disruptions have received considerable attention in the literature (e.g., [1,10,16,20,21]). The underlying scenario in these models is that the status of a supplier switches intermittently between two possible states (available and unavailable) due to random changes in the environment (e.g., strike, embargo, etc.) thereby disrupting the supply process. The same concept has been addressed in production-inventory settings where a failure-prone production facility which switches randomly between on and off states feeds into a storage system which faces the demand (e.g., [2,14,15,22]). Queueing models with Markov-modulated arrival and service rates have been discussed in the queueing theory literature (e.g., [17,24]). A number of related papers in queueing context have studied fluid systems with deterministic or stochastic input and output flows that are modulated by a two-state environment which alternates randomly between up and down states (e.g., [6,11,12,23]). The case of a fluid model with deterministic and continuous flows modulated by a multi-state continuous-time Markov chain has also been discussed in the queueing literature [13]. Another related line of research encompasses a special class of random yield models where the yield structure depends on the state of a randomly changing environment. Ozekici and Parlar [19] considered a periodic-review inventory model with backorders and zero lead time where the distribution of demand, the availability (i.e., up or down status) of an unreliable supplier and the cost parameters in every period depend on an environmental process that is represented by a time-homogeneous Markov chain. More specifically, their model allowed for an all or nothing replenishment scenario where the order placed at the beginning of a period is delivered immediately in full quantity, if the supplier is up, or else, nothing is delivered at all (zero replenishment). Erdem and Ozekici [8] considered a similar setting, but allowed for a different yield structure which is related to the random capacity of the supplying vendor; that is, an order placed at the beginning of a period is immediately received in full quantity if the order size is less than the vendors capacity, otherwise, the quantity received is equal to the available capacity which is determined by the state of the time-homogeneous Markov chain representing the environment. Finally, we note in passing that the implications of random supplier capacity in zero-lead-time inventory systems with/without random yield have received some attention in recent years (e.g., [9,27]); however, these works do not provide an explicit account of the impact of randomly changing environmental conditions on the supply process. Motivated by Berg et al. [2], this paper considers a production-inventory system under randomly changing environmental conditions. In [2], the authors studied a single-item production-inventory system consisting of an arbitrary number of identical parallel machines and a storage facility with limited capacity that faces a compound Poisson demand process. Each machine is subject to a stochastic failure and repair process in which the time to failure and the repair time are independent and exponentially distributed. There is an ample repair capacity so that the repair of a failed machine starts immediately. Each machine can produce the product continuously and uniformly over time at a fixed production rate. All operable machines produce the product simultaneously as along as the inventory level (i.e., stock-on-hand, throughout) remains below the storage capacity, and shortages are lost. Production on all operable machines is stopped whenever the inventory level reaches the maximum storage capacity and is started again as soon as the inventory falls below the storage limit. Clearly, the system considered in [2] can be viewed as a production-inventory system in a randomly changing (internal) environment where the number of operable
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machines at any point in time determines the overall production rate capability of the facility. However, given that each machine can only have two possible states, namely, up or down (under repair), during a production run, changes in the overall production rate of the system (while producing) can only occur as the result of transition between adjacent states; i.e., from m operating machines to m + 1 or m 1 machines. Furthermore, the random operating environment has no impact on the demand process. In contrast, our work is focused on a more general production-inventory setting in which the production and the demand processes are subject to two independent and randomly changing multi-state environmental processes (i.e., internal and external). The environmental processes are represented by two continuous-time homogeneous Markov chains which allow for one-step transitions from any given state to all other states. In this paper we consider a production-inventory system with limited storage capacity in which demand for a single item arises according to a compound Poisson stream whose occurrence rate and random batch sizes at any point in time depend on the state of a random (external) environment, and production occurs continuously and uniformly over time at a rate determined by the state of another independent and randomly changing (internal) environment. As mentioned earlier, we assume that each of the two randomly changing environments is represented by a continuous-time homogeneous Markov chain with a discrete state space. The facility starts production as soon as the inventory level drops below the storage limit and the production continues until the inventory level reaches the storage full capacity. Assuming that the shortages are lost, we derive the steady-state distribution of the inventory level which is used to formulate some measures of common interest regarding the performance of such systems. We also formulate an expected-net-revenue maximization problem and provide some numerical results. The organization of the paper is as follows: The model is described in Section 2, and the solution procedure is presented in Section 3. Section 4 outlines performance measures and presents some graphical and numerical results, followed by Section 5 which contains a summary and direction for further extension of this work.
2. Model formulation We consider a single-item production-inventory system with the following characteristics: (1) Demand is realized against the inventory level of a storage facility which has a limited storage capacity of R units. (2) The random demand process is characterized by a Markov-modulated compound Poisson stream. We assume that the occurrence rate for Poisson demand at any given time can take on a value of k|, | = 1, . . . , nE, according to the state of a continuous-time stochastic process E ¼ fEðtÞ; t P 0g with discrete state space S E ¼ f1; . . . ; nE g, and the demand sizes are conditionally (given the state of E) independent and exponentially-distributed random variables with mean /1 | . The exponential assumption pertaining to demand sizes is intended for obtaining closed form analytical solutions, albeit the modeling process in terms of formulating the system equations can accommodate more general compounding distributions. The modulating process E is described by a continuous-time homogeneous Markov chain in which the sojourn time in each state | 2 S E is exponentially distributed with 0 mean g1 | , and the transition probabilities at any transition epoch are denoted by q||0 for |(5| ) and E E 0 | 2 S with q|| ¼ 0 for 8| 2 S . (3) The storage facility is supplied by a production facility which can produce the item continuously and uniformly at the rate wı , ı ¼ 1, . . . , nI, depending on the state of a continuous-time homogeneous Markov chain I ¼ fIðtÞ; t P 0g with discrete state space S I ¼ f1; . . . ; nI g and transition probabilities pıı0 for ıð6¼ ı0 Þ and ı0 2 S I , and pıı ¼ 0 for 8ı 2 S I at any transition epoch. The sojourn time of I in each state ı 2 S I is exponentially distributed with mean m1 ı .
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(4) The production is halted whenever the inventory level in the storage facility reaches its capacity R and is restarted as soon as the inventory level drops below R ( i.e., at the next demand occurrence epoch). Clearly, such production control policy is mainly prevalent in production systems where setup times/ costs are negligible (in comparison with production times/costs) or nonexistent. As such, the policy is more likely to be realized in process industries such as chemical and pharmaceutical plants where continuous production of standardized goods by highly specialized and automated equipments over a long horizon rationalizes the overlooking of setups. A more general control policy may include a provision for setup time/cost by restarting production only when the inventory level drops to or below a threshold level r (r < R) at a demand occurrence epoch; however, to facilitate the analytical process in this paper, we assume that the setup time/cost for each production run is insignificant. (5) Both E and I are assumed to be ergodic. Moreover, transitions in E and I over time occur independent of each other and of the inventory level of the storage facility; that is, transitions in E and I occur independently irrespective of whether or not the production facility is idle or producing. (6) All stockouts, including the excess demand when a batch size is larger than the inventory level, are considered to be lost. It is clear from the above that the inventory level in this system has a continuous state space which ranges from 0 to R over time. We let X(t) 2 U [0, R] denote the inventory level at time t(P0), and borrowing a term from [2], define W(t) = R X(t) with W(t) 2 U as the ‘‘slack storage capacity’’ available at time t. We note that under the set of assumptions stated above, X(t), and hence W(t), can be characterized as a regenerative process. For example, assuming that at time t = 0 the system is full and E and I are in states ı and |, respectively, then the regenerative points can be defined at epochs at which the production is halted (i.e., the system reaches its full storage capacity) and the internal and external environmental processes are in states ı and | accordingly. We, furthermore, assume that X(t), and naturally W(t), posses limiting distributions [7]. Similar to [2], we focus on the steady-state situation throughout and apply a system-point method of level crossings [4] to develop a model whose solution yields the limiting distribution of the W(t) process. In this regard, we let F(Æ) and f(Æ) denote the limiting cumulative distribution and density functions of the W(t) process, respectively. By level-crossing theory, the three-dimensional stochastic process fW ı| ðtÞ, t P 0} {W(t), IðtÞ ¼ ı, EðtÞ ¼ |, t P 0}, ı 2 S I and | 2 S E , is the system-point process for our system, and corresponds to those portions of sample path tracings of W(t) over time t during which E and I are in states ı and |, respectively. A typical realization of the W ı| ðtÞ process over the state space U fı|g U \ fIðtÞ ¼ ıg \ fEðtÞ ¼ |g is depicted in Fig. 1. (Observe that W ı| ðtÞ is a jump process with right-continuous sample paths.) Under the steadystate conditions, we write W ı| ¼ limt!1 W ı| ðtÞ in distribution, and let F ı| ðwÞ ¼ PrðW ı| 6 wÞ denote its cumu-
Fig. 1. Typical tracings of the W ı| ðtÞ process (ı 2 S I , | 2 S E ).
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lative probability distribution function; i.e., the long-run joint probability of the slack storage capacity not exceeding w while the environmental processesR I and E are in states ı and |, respectively. It is evident that w under our set of assumptions, F ı| ðwÞ ¼ fı|0 þ a¼0 fı| ðaÞda, where fı| ðwÞ is the density function associated 0 with W ı| , and fı| ¼ dF ı| ð0Þ denotes the probability mass at Wı| = 0. Note that fı|0 corresponds to the long-run fraction of time that the inventory system is full (i.e., zero slack storage capacity) and production is halted while I and E, are in states ı and |, respectively. Accordingly, given that the sample paths of the W(t) process can be viewed as the aggregation those of the W ı| ðtÞ processes, it can be readily verified that PI P PnIofP nE nE 0 0 f ðwÞ ¼ nı¼1 f ðwÞ and f ¼ dF ð0Þ ¼ |¼1 ı| ı¼1 |¼1 fı| . We now utilize level-crossing theory to establish the model equations for deriving the partial densities fı| (.), ı 2 S I and | 2 S E , by studying the sample paths of the system-point process W ı| ðtÞ 2 U fı|g . The equations are based on the main principle of level crossings which asserts that the long-run rates at which the sample path tracings of the system-point process enter and exit a set of states are equal. Hence, let us consider the state interval [0, w) with w 2 (0, R) in U fı|g , ı 2 S I and | 2 S E . The long-run average entrance and exit rates of the system-point sample path associated with this set of states are as follows: (i) The long-run average number of downcrossings of the sample path into the interval [0, w) in U fı|g per unit time by level-crossing theory is wı fı| ðwÞ, ı 2 S I and | 2 S E . (ii) The infinitesimal transition generators of the environmental Markov chains I and E assert that the state interval [0, w) in U fı|g can be reached from the same intervals in U fk|g , k 2 S I , k 6¼ ı, and in U fı‘g , ‘ 2 S E , ‘ 6¼ |. Therefore, the long-run average entrance rates into this interval due to such transitions are mk pkı F k| ðwÞ and g‘ q‘| F ı‘ ðwÞ, respectively. (iii) The long-run average exit rate of the sample path from [0, w) in U fı|g due to infinitesimal transition generators of I and E is ðmı þ g| ÞF ı| ðwÞ, ı 2 S I and | 2 S E . (iv) When the sample path is in state a (0 6 a < w) in U fı|g , a demand arrival whose size exceeds w a results in an upward jump which takes the sample path out of the interval [0, w) in U fı|g . By PASTA [28], R w the long-run average exit rate of the sample path from this interval due to such event is k| a¼0 e/| ðwaÞ dF ı| ðaÞ. Considering the above and following level-crossing theory, for any interval [0, w), w 2 (0,R), in U fı|g , ı 2 S I and | 2 S E , we obtain: Z w nI nE X X mk pkı F k| ðwÞ þ g‘ q‘| F ı‘ ðwÞ ¼ ðmı þ g| ÞF ı| ðwÞ þ k| e/| ðwaÞ dF ı| ðaÞ; ð1Þ wı fı| ðwÞ þ k¼1
‘¼1
a¼0
where the right- and left-hand sides of Eq. (1), respectively, represent the long-run average total entrance and exit rates of the system-point sample path with respect to the designated interval. Following the same line of reasoning for the state (level) w = 0 in U fı|g , ı 2 S I and | 2 S E , results in the following boundary condition: wı fı| ð0Þ þ
nI X
mk pki fk|0 þ
k¼1
nE X
g‘ q‘| fı‘0 ¼ ðmı þ g| þ k| Þfı|0 ;
ð2Þ
‘¼1
where wı fı| ð0Þ is the long-run average number of down-crossings of level zero in U fı|g per unit time by the system-point sample path. Finally, the normalizing condition Z R nI X nE X 0 fı| þ fı| ðwÞdw ¼ 1; ð3Þ ı¼1
|¼1
w¼0
completes the model formulation.
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3. Model solution We invoke the differential operator hDi = hd/dwi and let gı| ðwÞ ¼ Df ı| ðwÞ for ı 2 S I and | 2 S E , wherever the derivative exists. Applying hDihD þ /| i to Eq. (1) results in nI nI X X wı Dgı| ðwÞ þ ðwı /| mı g| k| Þgı| ðwÞ ðmı þ g| Þ/j fı| ðwÞ þ mk pkı gk| ðwÞ þ /| mk pkı fk| ðwÞ k¼1
þ
nE X
g‘ q‘| gı‘ ðwÞ þ /|
‘¼1
nE X
k¼1
g‘ q‘| fı‘ ðwÞ ¼ 0:
ð4Þ
‘¼1
Let H denote a square matrix of size n(n = nI · nE) that represents the infinitesimal generator for the two-dimensional {I(t), E(t), t P 0} Markovian process over the state space fðı; |Þ; ı ¼ 1; . . . ; nI ; | ¼ 1; . . . ; nE g; i.e., 2
ðm1 þ g1 Þ 6 gq 2 21 6 6 .. 6 6 . 6 6 g q 6 nE nE 1 6 6 m2 p21 6 H¼6 0 6 6 .. 6 6 . 6 6 0 6 6 .. 6 4 . 0
g1 q12 ðm1 þ g2 Þ .. . gnE qnE 2 0 m2 p21 .. . 0 .. . 0
g1 q1nE g2 q2nE .. .. . . ðm1 þ gnE Þ 0 0 .. .. . . m2 p21 .. . mnI pnI 1
m1 p12 0 .. . 0 ðm2 þ g1 Þ g2 q21 .. . gnE qnE 1 .. . 0
0 m1 p12 .. . 0 g1 q12 ðm2 þ g2 Þ .. . gnE qnE 2 .. . 0
0 0 .. .. . . m1 p12 g1q1nE g2 q2nE .. .. . . ðm2 þ gnE Þ .. . mnI pnI 2
m1 p1nI 0 .. . 0 m2 p2nI 0 .. . 0 .. . gnE qnE 1
0 m1 p1nI .. . 0 0 m2 p2nI .. . 0 .. . gnE qnE 2
.. . .. .
3 0 7 0 7 7 .. 7 7 . 7 7 m1 p1nI 7 7 0 7 7 7: 0 7 7 .. 7 7 . 7 7 m2 p2nI 7 7 .. 7 5 . ðmnI þ gnE Þ
b denote two n dimensional square matrices whose rth rows, Mr. and N b r: , r = 1, 2, . . . , n, Next, let M and N respectively, are characterized as Mr ¼
/rbnr cnE E
wbnr cþ1
HTr ;
E
b r ¼ N
1 HT ; wbnr cþ1 r E
where is the rth row of HT (i.e., the transpose of H) and bsc marks the largest integer smaller than s. Also, let D = [Drr], r = 1, . . . , n, denote a diagonal matrix of size n with HTr
Drr ¼
wbnr cþ1 /rbnr cnE þ krbnr cnE E
E
wbnr cþ1
E
:
E
Then, the system of first-order linear differential equations in Eq. (4) can be expressed in the matrix form of DZðwÞ ¼ AZðwÞ;
ð5Þ T
where ZðwÞ ¼ ½f11 ðwÞ; . . . ; fnI nE ðwÞ; g11 ðwÞ; . . . ; gnI nE ðwÞ is a column vector of size 2n, and " # € 0 €I A¼ M N b þ D and €I and €0, respectively, denoting the n · n identity and null is a square matrix of size 2n with N ¼ N matrices.
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The general solution to the system of matrix differential equations in Eq. (5) is of the form Z(w) = eAwZ(0) [18]. Assuming that A is diagonalizable, let the column vector V.r denote the rth eigenvector of A corresponding to its eigenvalue cr, r = 1, . . . , 2n. Then, the general solution can be conveniently expressed as ZðwÞ ¼ VeCw b:
ð6Þ
Note that in Eq. (6), V = [V.1, . . . , V.2n], C represents a diagonal matrix whose diagonal elements are c1, . . . , c2n, and b = [b1, . . . , b2n]T is a column vector of constants coefficients. Accordingly, it can be readily verified that fðwÞ ¼ !eCw b;
ð7Þ
T where fðwÞ ¼ ½f11 ðwÞ; . . . ; fnI nE ðwÞ is a column vector of size n, and ! = YV with Y ¼ €I €0 . In order to fully characterize the limiting distribution of the W(t) process, we now need to find the values of 3n unknowns; i.e., b1, . . . , b2n and f110 ; . . . ; fn0I nE . In this regard, Let W = [Wrr], K = [Krr] and U = [Urr] denote three diagonal matrices of size n with Wrr ¼ wbr=nE cþ1 , Krr ¼ krbr=nE cnE and Urr ¼ /rbr=nE cnE , r = 1,2, . . . , n. Then, Eq. (1) can be represented in the matrix form as Z w WfðwÞ þ HT FðwÞ ¼ K eUðwaÞ dFðaÞ; ð8Þ a¼0
where FðwÞ ¼ ½F 11 ðwÞ; . . . ; F nI nE ðwÞT . Substituting the general solution, Eq. (7), in Eq. (8) and comparing the coefficient of common exponential terms, e/| w , | 2 S E , gives b € ¼ 0; ð9Þ L1 I f0 where 0 is the null column vector of size 3n, f 0 ¼ ½f110 ; . . . ; fn0I nE T , and L1 = [L1kr], k = 1, . . . , n and r = 1, . . . , 2n, with L1 ¼
kr ; cr þ /kbnk cnE E
and kr marking the (k, r)th element of !. Similarly, expressed in matrix form, Eq. (2) can be written as fð0Þ ¼ L2f 0 ;
ð10Þ
b rr , r = 1, . . . , n, denotes a diagonal matrix of size n with bþN b ¼ ½D b is an n · n matrix, and D where L2 ¼ D b rr ¼ D
krbnr cnE E
wbnr cþ1
:
E
Substituting Eq. (7) for w = 0 in Eq. (10) yields b ½ ! L2 0 ¼ 0: f
ð11Þ
The third set of relations is resulted from the overall transition rate balance equations for W ı| ðtÞ, ı 2 S I and | 2 S E , which can be expressed as HT FðRÞ ¼ 0:
ð12Þ
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E. Mohebbi / European Journal of Operational Research 174 (2006) 539–552
Again, substituting Eq. (7) in Eq. (12) provides
L3
H
T
b f0
¼ 0;
ð13Þ
where L3=HT!J1, and J1 = [J1rr], r = 1, . . . , 2n, is a diagonal matrix with J 1rr ¼ more, using Eq. (7) in the normalizing condition depicted in Eq. (3), we obtain b be 0 ¼ 1; f
RR w¼0
ecr w dw. Further-
ð14Þ
where be ¼ e !J1 €I and e = [1, 1, . . . , 1] denote two row vectors of sizes 3n and n, respectively. Considering the above, substituting an arbitrary row in 2 3 €I L1 6 7 b 4 ! L2 5 0 ¼ 0 f L3 QT
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
fX(x)
fW(w)
with Eq. (14) provides a sufficient system of non-homogeneous linear equations to complete the model solution. Fig. 2 depicts sample forms of the limiting density and cumulative distribution functions for the W(t) and X(t) processes.
0.3
0.2
0.1
0.1
0
0
5
10 w
15
0
20
1
0
5
10 x
15
20
5
10 x
15
20
1 0.8
0.8 0.6
fX(x)
fW(w)
0.3
0.2
0.6 0.4
0.4 0.2 0.2 0
5
10 w
15
20
0
0
Fig. 2. Typical plots of the limiting density and distribution functions for the W(t) and X(t) processes for CVw = 0.9, CVk = 0.5 and R = 20.
E. Mohebbi / European Journal of Operational Research 174 (2006) 539–552
547
4. Performance measures and numerical results Explicit knowledge of the limiting distribution of W(t) (and hence X(t)) enables us to formulate a variety of long-run performance measures for the production-inventory system considered here. Some of the most common measures of interest noted in the literature (e.g., [2]) include: Expected inventory level: Z R lX ¼ R lW ¼ R wdF ðwÞ ¼ R e!J2b; ð15Þ w¼0
where J2 = [J2rr], r = 1, . . . , 2n, is a diagonal matrix with J 2rr ¼ Expected utilization ratio of the production facility: q¼
RR w¼0
wecr w dw.
Expected effective production rate w ¼ PnI e ; Expected potential production rate ı¼0 wı p ı
ð16Þ
where
we ¼ eW FðRÞ f 0 ¼ e!J1b;
ð17Þ
I
and pı ¼ limt!1 PrfIðtÞ ¼ ıg, ı 2 S . Expected rate of lost sales: Z R Z 1 uUeUðRwþuÞ dudFðwÞ ¼ eKðJ3b þ J4f 0 Þ; b ¼ eK w¼0
ð18Þ
u¼0
where J3 = [J3kr], k = 1, . . . , n and r = 1, . . . , 2n, is an n · 2n matrix with J 3kr ¼
/ R kr k ðecr R e kbnE cnE Þ; /kbnk cnE ðcr þ /kbnk cnE Þ E
E
and J4 = [J4kk], k = 1, . . . , n, is a diagonal matrix with /
k
R
e kbnE cnE : J 4kk ¼ /kbnk cnE E
Expected fill rate (fraction of demand satisfied per unit time): fr ¼ 1 PnE
b
k| |¼1 /|
q|
;
ð19Þ
where q| ¼ limt!1 PrfEðtÞ ¼ |g, | 2 S E . For expository purposes, these performance measures were computed for various combinations of the parameter settings displayed in Table 1 with S I ¼ f1; 2; 3g, S E ¼ f1; 2g, p12 = p13 = p21 = p23 = p31 = p32 = 0.5, q12 = q21 = 1, m1 = m2 = m3 = g1 = g2 = 1, /1 = /2 = 0.5, and R = 20. We focused on the impact of variability in production and demand rates caused by fluctuating random environments on the system performance. Hence, the values of wı , ı = 1, 2, 3, and k| , | = 1, 2, in Table 1 were chosen to represent various levels of variability as measured by their coefficients of variation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P3 P3 2 2 P3 and CVk ¼ CVw and CVk, respectively. Clearly, CVw ¼ ı¼1 wı p ı ð ı¼1 wı p ı Þ = ı¼1 wı p ı , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P2 P2 2 2 P2 |¼1 k| q| ð |¼1 k| q| Þ = |¼1 k| q| . The results are shown Fig. 3. Observe that for any chosen value of CVw, the expected utilization of the production facility decreases as CVk increases. This decreasing trend intensifies as CVw decreases. This is primarily due to the fact that the variability of demand rate has direct
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E. Mohebbi / European Journal of Operational Research 174 (2006) 539–552
Table 1 Parameter settings for numerical experiments w2
w3
CVw
Production rates: wı, ı 2 S ¼ f1; 2; 3g 35 35 40 35 46 30 51 28 56 25 60 20 66 15 70 15 75 25 80 20 85 15 90 10
35 30 29 26 24 25 24 20 5 5 5 5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
w1 I
k1
k2
CVk
15 13 12 10 9 7 6 4 3 1.5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
E
Demand rates: k|, | 2 S ¼ f1; 2g 15 17 18 20 21 23 24 26 27 28.5
impacts on the product outflow from the storage facility and the rate at which the production process is halted. The increasing variability of the production rate, however, appears to mitigate the impact of demand variability. It should also be noted that for any given value of CVk, the expected utilization ratio decreases as CVw increases. A similar pattern is displayed by the expected fill rate for the reasons made clear from observing the behavior of the expected lost sales rate against changes in CVk and CVw. The expected inventory level shows a decreasing trend for any chosen value of CVk as CVw increases. The domain of such decrease, however, diminishes as CVk increases. This appears to have been caused by a complex interaction between the demand and the production rate variability. In this regard, we note that on one hand, when there is no variability in the production rate of the facility, the expected inventory level decreases (and the expected lost sales rate increases rapidly) as the variability in demand rate increases. On the other hand, increasing the variability of production rate appear to have a compensating effect by dampening the growth of lost sales rate at the expense of creating some excess inventory as CVk increases. In addition to the above, Eqs. (15)–(19) can be readily used as functional components to formulate objective functions for optimization purposes. For illustration, let us assume that production generates revenue of cw per unit, and there is an inventory holding cost of ch per unit per unit time and a penalty cost of cp per unit of lost demand. The long-run expected net revenue accumulation rate for a preset value of the storage capacity, R, can then be characterized as XðRÞ ¼ cw we ch lX cp b:
ð20Þ
Assuming that the decision maker is interested in finding the optimum capacity for the storage facility (R ) so that the expected net revenue accumulated per unit time is maximized, the optimization process can be practically carried out through standard search techniques embedded in a commercially available opti-
E. Mohebbi / European Journal of Operational Research 174 (2006) 539–552
14
1 CVψ= 0
Expected Fill Rate
Expected Inventory
13 12 11 10
CVψ= 1.1
9 8
0.9
CVψ= 0
0.8 0.7 0.6 CVψ= 1.1
0
0.2
0.4
0.6
0.8
0.5
1
0
0.2
0.4
CVλ Expected Utilization of the Facility
Expected Lost Sales Rate
CVψ= 1.1
12 10 8 6
CVψ= 0
4 2 0
0.2
0.4
0.6
0.8
1
CVλ
14
0
549
0.6
0.8
1
0.9 0.8 CVψ= 0
0.7 0.6 0.5 CVψ= 1.1
0.4
0
0.2
0.4
CVλ
0.6
0.8
1
CVλ
Fig. 3. Plots of performance measure for parameter sets in Table 1.
mization package. (It should be pointed out that establishing the concavity or unimodality of X(R) in an analytical sense due to its complex form appears to be impractical). Tables 2 and 3 contain the optimality results for the parameter settings described earlier in this section with cw = 5, ch = 1, and cp = 7. These Table 2 Optimal values R for parameter sets in Table 1 CVw 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
CVk 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
25.59 26.29 28.05 30.96 35.08 39.50 47.37 52.61 61.89 68.72 76.50 84.83
26.39 27.08 28.83 31.68 35.71 40.04 47.78 52.93 62.20 68.94 76.66 84.94
27.41 28.09 29.81 32.61 36.53 40.74 48.32 53.36 62.59 69.24 76.86 85.08
30.80 31.42 33.04 35.64 39.25 43.10 50.16 54.87 63.90 70.24 77.57 85.59
33.12 33.70 35.25 37.72 41.14 44.79 51.50 56.00 64.84 70.99 78.12 85.98
38.77 39.26 40.64 42.83 45.88 49.09 55.05 59.13 67.32 73.04 79.69 87.13
42.00 42.45 43.74 45.79 48.65 51.65 57.24 61.11 68.88 74.38 80.76 87.93
49.09 49.46 50.56 52.34 54.84 57.47 62.38 65.85 72.63 77.70 83.53 90.10
52.88 53.22 54.23 55.88 58.22 60.68 65.27 68.57 74.83 79.70 85.27 91.52
58.82 59.11 60.00 61.47 63.57 65.79 69.97 73.02 78.54 83.12 88.34 94.16
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Table 3 Optimal values X(R ) for parameter sets in Table 1 CVw
CVk 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
122.41 121.23 118.69 114.64 109.27 103.44 93.17 87.33 73.06 66.25 58.01 48.70
121.01 119.84 117.30 113.30 108.03 102.33 92.26 86.51 72.40 65.64 57.47 48.23
119.31 118.16 115.62 111.68 106.52 100.98 91.14 85.49 71.57 64.89 56.80 47.63
114.22 113.12 110.62 106.84 101.97 96.83 87.63 82.27 68.97 62.50 54.66 45.72
110.98 109.91 107.45 103.78 99.06 94.13 85.30 80.10 67.22 60.87 53.19 44.41
103.41 102.44 100.11 96.66 92.26 87.73 79.64 74.79 62.86 56.79 49.49 41.11
99.20 98.27 96.03 92.71 88.46 84.12 76.38 71.69 60.28 54.36 47.27 39.11
90.12 89.29 87.24 84.18 80.24 76.24 69.14 64.75 54.43 48.79 42.10 34.44
85.32 84.54 82.58 79.65 75.87 72.04 65.22 60.97 51.17 45.66 39.17 31.75
77.87 77.15 75.34 72.60 69.05 65.44 59.02 54.96 45.90 40.58 34.36 27.28
results indicate that the larger the variability in the production and demand rates, the higher the required storage capacity in the system. Furthermore, the expected net revenue accumulated by the system per unit time when operating with the optimal storage capacity decreases as the production and demand rates endure higher variability.
5. Summary and extension Motivated by an existing model with randomly varying production rates [2], in this paper we presented a production-inventory model in which the production and demand rates are modulated by two independent continuous-time homogeneous Markov chains. For stochastic piecewise linear production and compound Poisson demands, we applied level-crossing theory to derive the limiting distribution of the inventory level in a limited capacity production-storage system with lost sales. A number of important performance measures including an expected revenue accumulation rate function were formulated and some numerical results in terms of sensitivity analysis and optimality of the system parameters were presented. These results clearly demonstrate the critical impact of the variability induced by randomly changing environments on the system performance. The importance of accounting for the random nature of the surrounding environments has been recognized by researchers in recent years. While most production-inventory or fluid models with finite production (replenishment) or inflow rates in the literature limit their characterization of the state environments to having only two states (i.e., up/down or on/off), this work proposes an adaptation of level-crossing theory which allows for both demand and production processes to depend on multi-state random environments. The study with its findings motivates continuing research in this area. In this regard, an imminent extension of the model proposed in this paper involves the introduction of the two-parameter (r, R) production control policy to address the issue of setup time/cost of the production facility cited earlier. The level-crossing approach adapted here appears to provide a viable mean for modeling this generalized case where once production is halted after the system reaches its maximum storage capacity R, it is only restarted after the inventory level downcrosses the critical level r at a demand occurrence epoch. It should be noted, however, that unlike the present scenario where the sample path tracings of the inventory level consist of both continuous and jump crossings of state levels within the entire state space, the sample path analysis under the new policy must account for pure jump crossings of state levels within the subspace r < X(t) 6 R when pro-
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duction is halted. This is expected to have an increasing effect on the number of balance equations and hence the complexity of the solution process. It is also worth mentioning that although we assumed in this paper that demand and production are modulated by two independent Markovian environmental (i.e, external and internal) processes, it is equally conceivable to be faced with a scenario where there are interdependencies between internal and external conditions; e.g., a high production rate may be correlated with a high demand rate, etc. Yet, assuming that such interdependencies in environmental processes can be captured through redefining the environmental states and structuring a single ergodic Markov chain B ¼ fBðtÞ; t P 0g with discrete state space S B ¼ f1; . . . ; ng, the present model formulation can be readily revised address this scenario as a special case. Pn Pto n More specifically, letting f ðwÞ ¼ ı¼1 fı ðwÞ and f 0 ¼ dF ð0Þ ¼ ı¼1 fı0 , Eq. (1) can be simplified as Z w n X fk okı F k ðwÞ ¼ oı F ı ðwÞ þ kı e/ı ðwaÞ dF ı ðaÞ; wı fı ðwÞ þ k¼1
a¼0
f1 ı 0
represent the mean sojourn time in state ı, and oıı0 denotes the transition probability from state ı where to state ı ð6¼ ıÞ in S B . The rest of the formulation follows accordingly.
Acknowledgement The author is thankful for insightful comments by an anonymous referee which have improved the presentation of this paper.
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