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A note on a production control model for a facility with limited storage capacity in a random environment
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European Journal of Operational Research 190 (2008) 562–570 www.elsevier.com/locate/ejor
Short Communication
A note on a production control model for a facility with limited storage capacity in a random environment E. Mohebbi
*
Department of Management/MIS, College of Business, University of West Florida, Pensacola, FL 32514, USA Received 17 January 2006; accepted 17 June 2007 Available online 30 June 2007
Abstract This note presents an extension of our earlier paper on a production–inventory system with limited storage capacity in a multi-state random environment [i.e., E. Mohebbi, A production–inventory model with randomly changing environmental conditions, Eur. J Res. 174 (2006) 539–552]. More specifically, it assumes that the production rate as well as the occurrence rate and batch sizes of a compound Poisson demand stream are influenced by the state of a continuous-time homogeneous Markov chain representing the random environment. We present an adaptation of the system-point method of level crossings which yields the limiting distribution of inventory level (stock-on-hand) for a two-parameter production control policy when stockouts are lost. 2007 Elsevier B.V. All rights reserved. Keywords: Inventory; Production; Limited capacity; Random environments; Markov-modulated processes; Level crossings
1. Introduction Manufacturing systems are known for operating in random environments. A random environment is a common term used in the production control literature to highlight the random nature of operating conditions that play defining roles in characterizing the production and demand processes and as such, can significantly impact the performance of production–inventory systems. Such random changes in operating conditions may be attributed to a variety of factors ranging from equipment condition and breakdowns, production yield, raw material/subcomponent quality, production mix, workforce level and labor skills to economic and market conditions. The globalization of economy and liberalization of marketplace at an increasingly rapid pace has intensified the need for incorporating the resulting operational uncertainties and financial risks into the firms’ production and inventory control decisions. To that end, increased product variety and customization, frequent price
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discounts and promotions, shortened product life cycles, advancement in technology, geopolitical crises and environmental conditions, to name a few, are among the many factors that contribute to such uncertainties in today’s manufacturing world. (The reader is referred to [10,13,14], among others, for typical manifestations of environmental uncertainties in wafer manufacturing, agricultural and mining industries, respectively.) In addition, randomly occurring disruptive events such as equipment breakdowns, capacity limitations, material shortages, or strikes in most supply chains are known to have significant negative impacts on the financial bottom lines of the affected firms. Apple, Motorola and Sony reported problems with meeting their demands due to material shortages caused by unforeseen circumstances in 1995, 1999 and 2000, respectively [24]. A recent study indicates that while much of the emphasis on the increasingly popular supply chain management strategies seems to be on efficiency or reducing costs, overemphasis on efficiency can make the supply chain brittle and more susceptible to the risk of disruptions [11,12]. The study documents that based on a sample of 885 glitches reported by various companies, firms that experience disruptions face on average 6.92% lower sales growth, 10.66% growth in cost, and 13.88% growth in inventories. A large segment of the existing models in this area corresponds to inventory systems in randomly fluctuating demand environments where demand is characterized as a Markov-modulated process (e.g., [6,23,25]); i.e., the rate or distribution of demand is subject to change at random epochs in accordance with state transitions of a discrete-or continuous-time Markov chain which represents the environment. Another line of work within this context has focused on production control in production–inventory settings where a failure-prone production facility, which switches randomly between states available and unavailable, feeds into a storage system which faces the demand (e.g., [1,8,9,17,18,21]). Accordingly, the relationship between fluid queueing systems with deterministic or stochastic input and/or output flows modulated by two-state (up and down) environments, and production-storage systems in random environments has been explored in the queueing literature (e.g., [7,15,16,22]). The economic order quantity (EOQ) model in random environment has also received some attention in the literature [4]. This note presents an exact mathematical model for deriving the limiting (steady state) distribution of the inventory level (i.e., stock-on-hand) of a single item in a production–inventory system with limited storage capacity. We assume that the demand for this item arises according to a compound Poisson stream, and the production occurs continuously and uniformly over time until the storage facility reaches its maximum capacity upon which the production is halted. The production is restarted once the inventory level drops to or below a critical threshold. All shortages are lost. Furthermore, it is assumed that both demand and production processes are modulated by a multi-state random environment which is represented by a continuous-time homogeneous Markov chain. The model is an extension of our earlier work [19] in that it attempts to address a limitation embedded in the previous model by utilizing a two-parameter control policy. More specifically, the control policy introduced above is much broader than the single-parameter policy considered in [19] where the production of the item is halted whenever the inventory level reaches the maximum storage capacity and is immediately restarted at the next demand epoch. It should be clear that while the latter policy is only prevalent in circumstance where the production set-up cost/time is negligible, the former is a more realistic representation of most manufacturing environments where set-ups cannot be ignored. A special case of this problem includes a scenario involving several independent and failure-prone machines producing a single item to meet uncertain demand. Each machine must undergo repair independently upon failure to restore operation. The production on all machines is also halted due to storage capacity limitations, and is restarted once the inventory level crosses a threshold. We note that while the form of the optimal control policy for this type of production–inventory systems in general remains an open research question, the main contribution of this work in reference to our earlier paper [19] lies in its unique conflation of two main characteristics of such systems in an analytical framework which is largely unexplored: namely, a ‘‘multi-state’’ randomly changing environment which affects both production and demand, and a ‘‘two-parameter’’ control policy which accommodates the possibility of incurring a set-up time/cost for each production run. This note is organized as follows: Section 2 outlines our main assumptions; Section 3 contains the model formulation; Section 4 presents the solution process in detail, and is followed by concluding remarks in Section 5.
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2. System characteristics 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 > 0 units. (2) The storage facility is supplied by a production facility which can produce the item continuously and uniformly at the rate of wi ; i ¼ 1; . . . ; n, depending on the state of a continuous-time homogeneous Markov chain E fEðtÞ; t P 0g with discrete state space S E ¼ f1; . . . ; ng and transition probabilities pii0 for 0 i(5i ) and i0 2 S E , and pii = 0 for 8i 2 S E at any transition epoch. The sojourn time of E in each state i 2 S E is exponentially distributed with mean m1 i . (3) The random demand process is characterized by a Markov-modulated compound Poisson stream with an average intensity of ki and independent exponentially distributed batch sizes with mean /1 i , depending on the state of E at any point in time. (4) The production is halted whenever the inventory level in the storage facility reaches its maximum capacity R and is restarted as soon as the inventory level drops to or below a threshold level s(0 6 s < R) at a demand occurrence epoch. Letting r = R s, it should be clear that the two-parameter (s, R), or equivalently, (r, R), control policy considered here is a generalization of the single-parameter policy considered in [19] where production is stopped as soon as the inventory level reaches R and is resumed immediately after the next demand occurs. (5) The Markov process E is assumed to be ergodic. Moreover, transitions of E over time are independent of the inventory level of the storage facility; that is, transitions of E are independent 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. 3. Model formulation Let fX ðtÞ; t P 0g denote the inventory level at time t. Clearly, X ðtÞ 2 ½0; R. We define fW ðtÞ; t P 0g ¼ R X ðtÞ, as the slack storage capacity available at time t with W(t) 2 [0, R]. Note that X(t) under our set of assumptions can be readily characterized as a regenerative process possessing a limiting distribution. Accordingly, W(t) is a stationary process with W = limt!1W(t) in distribution. We utilize a level-crossing approach [2,5] to model the steady-state behavior of the production–inventory system and derive the limiting distribution of W(t), and hence, X(t). Knowledge of such distribution can then be conveniently used to formulate a variety of performance measures (e.g., average inventory level, average total cost/revenue per unit time, fill rate, etc.) for evaluation purposes [19]. Let F(Æ) and f(Æ), respectively, denote the limiting cumulative probability distribution and density functions of W(t). By level-crossing theory and ‘‘method of stages’’, observe that the evolution of the sample-path of W(t) over time can be viewed as the confluent of the sample-paths of Markovian processes Wi1(t) and Wi0(t), where fW i1 ðtÞ; t P 0g ffW ðtÞ; t P 0g \ fEðtÞ ¼ i; i 2 S E ; t P 0g \ fThe production facility is activegg, and fW i0 ðtÞ; t P 0g ffW ðtÞ; t P 0g \ fEðtÞ ¼ i; i 2 S E ; t P 0g \ fThe production facility is idlegg. Moreover, given the structure of the control policy, it follows that W i1 ðtÞ 2 ð0; R and Wi0(t) 2 [0, r), i 2 S E . Fig. 1 depicts typical sample-paths of these processes. Let fi j(Æ) and Fi j(Æ) denote the limiting density and cumulative distribution functions of W ij ðtÞ; i 2 S E and j ¼ 0; 1, respectively. Accordingly, 8 n P 1 P > > > fij ðwÞ w 2 ð0; rÞ; < i¼1 j¼0 f ðwÞ ¼ n > P > > w 2 ½r; R; : fi1 ðwÞ i¼1
Rw and F ðwÞ ¼ i¼1 fi00 þ a¼0 f ðaÞda, where fi00 ¼ dF i0 ð0Þ denotes the long-run point mass probability of having zero slack capacity (i.e., the storage facility is full and the production facility is idle) while E is in state i, i 2 S E . Pn
E. Mohebbi / European Journal of Operational Research 190 (2008) 562–570
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Wι1(t) Lost sales R
φi-1
ψι r
0
t
Wι 0(t)
r
φi-1 0
t
Fig. 1. Sample-path tracings of W ij ðtÞ, i 2 S E , j ¼ 0; 1.
Our level-crossing approach toward formulating the sample-path tracings of the stochastic processes identified above is based on the conservation law of level crossings [3] which asserts the existence of an equilibrium condition between the long-run total rates at which these tracings enter and exit any chosen subset of states. Following this principle over a number of judiciously chosen state intervals and utilizing the PASTA [26] property lead to a series of equations which balance the long-run entrance and exit rates of sample-path tracings of W ij ðtÞ; i 2 S E , j ¼ 0; 1, processes in reference to the designated state intervals. We first consider state-space interval ½0; wÞ; w 2 ð0; rÞ, for W i0 ðtÞ 2 ½0; rÞ and obtain: Z w n X mj pji F j0 ðwÞ þ wi fi1 ð0þ Þ ¼ ki e/i ðwaÞ dF i0 ðaÞ þ mi F i0 ðwÞ: ð1Þ a¼0
j¼1
The first term on the left-hand side (LHS) represents the total rate of sample-path entering ½0; wÞ due to transitions of E, and the second term shows the entrance rate into ½0; wÞ due to downcrossing of level 0 upon system reaching its maximum storage capacity. On the right-hand side (RHS), the first term depicts the samplepath exit rate from ½0; wÞ due to a demand of sufficiently large size while the production facility is idle, and the second term captures its exit rate from the same interval due to transitions of E. Similarly, for W i0 ðtÞ ¼ 0; i 2 S E , in steady state, we have wi fi1 ð0þ Þ þ
n X
mj pji fj00 ¼ ðki þ mi Þfi00 ;
ð2Þ
j¼1
where the first LHS term illustrates the sample-path downcrossing rate of level 0 (while E is in state i) at points of continuity due to a continuous and linear production process (slope = wi) which depletes the slack storage capacity to the point where the storage facility reaches its full capacity and production is halted. The second LHS term expresses the total rate of sample-path transitions from level 0, with E being in state j 2 S E ðj 6¼ iÞ, to the designated level 0 where E is in state i due to environmental transitions. The RHS term renders the longrun rate of sample-path exit from this level caused by a demand occurrence (rate ki) or a change in the environmental state (rate mi). Considering W i1 ðtÞ 2 ð0; R, i 2 S E , the balance equations for state intervals ðw; rÞ, where w 2 ð0; rÞ, and ½r; wÞ with w 2 ½r; R, are established as Z w n X / ðwaÞ ki e i e/i ðraÞ fi1 ðaÞda þ wi fi1 ðrÞ þ mj pji ðF j1 ðrÞ F j1 ðwÞÞ a¼0
¼ mi ðF i1 ðrÞ F i1 ðwÞÞ þ ki
Z
j¼1 r
a¼w
e/i ðraÞ fi1 ðaÞda þ wi fi1 ðwÞ;
ð3Þ
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and wi fi1 ðwÞ þ þ ki
Z
n X
mj pji ðF j1 ðwÞ F j1 ðrÞÞ
j¼1 r
e
/i ðraÞ
e
/i ðwaÞ
a¼0
¼ mi ðF i1 ðwÞ F i1 ðrÞÞ þ ki
dF i0 ðaÞ þ ki
Z
r
e/i ðraÞ e/i ðwaÞ fi1 ðaÞda
a¼0
Z
w
e/i ðwaÞ fi1 ðaÞda þ wi fi1 ðrÞ;
ð4Þ
a¼r
respectively. The first and second LHS terms of Eq. (3), respectively, outline the rates at which the sample-path enters ðw; rÞ due to a demand of sufficiently large size and downcrossing of level r while the production facility is active and E is in state i. The third LHS term describes the total entrance rate into (w, r) due to transitions of E. By the same token, the first, second, and third RHS terms represent the sample-path exit rates from (w, r) due to a transition of E, a demand of sufficiently large size, and downcrossing of level w while the production facility is active, respectively. A similar line of reasoning can be applied to justify the structure of Eq. (4). The limiting distribution of W(t) is obtained by solving Eqs. (1)–(4). 4. Model solution Introducing hDi = hd/dwi and applying hDihD + /ii to Eq. (1), we obtain: ððki þ mi ÞhDi þ mi /i Þfi0 ðwÞ
n X
mj pji hD þ /i ifj0 ðwÞ ¼ 0;
ð5Þ
j¼1
or equivalently, hDif 0 ðwÞ ¼ Af 0 ðwÞ;
ð6Þ T
in the matrix form where f 0 ðwÞ ¼ ½f10 ðwÞ; . . . ; fn0 ðwÞ is an n-dimensional column vector, and 1
A ¼ ½K QT ½/i QTi 8i is a square matrix of size 2 m1 m1 p12 6 m2 p 6 21 m2 Q¼6 .. 6 .. 4 . . mn pn1
mn pn2
n with .. .
3 m1 p1n m2 p2n 7 7 7 .. 7 . 5
mn
representing the infinitesimal generator for E, and QTi denoting the ith row of QT, i ¼ 1; . . . ; n. Similarly, applying hDihD + /ii to Eqs. (3) and (4) results in the following differential equation over their corresponding state intervals:
n X wi hD2 i þ ðwi /i ki mi ÞhDi mi /i fi1 ðwÞ þ mj pji hD þ /i ifj1 ðwÞ ¼ 0:
ð7Þ
j¼1
Let gi1 ðwÞ ¼ hDifi1 ðwÞ, i 2 S E , wherever the derivative is defined, and K, U and W mark three n-dimensional diagonal matrices whose diagonal elements comprise of ðk1 ; . . . ; kn Þ, ð/1 ; . . . ; /n Þ, and ðw1 ; . . . ; wn Þ, respectively. Hence, introducing /i T 1 T b e¼B b U þ W1 K B ¼ Qi 8i ; B ¼ Qi 8i ; and B wi wi for i ¼ 1; . . . ; n as three n-dimensional square matrices, Eq. (7) can be expressed in the matrix form:
E. Mohebbi / European Journal of Operational Research 190 (2008) 562–570
hDiZ1 ðwÞ ¼ BZ1 ðwÞ;
567
ð8Þ T
where Z1 ðwÞ ¼ ½f11 ðwÞ; . . . ; fn1 ðwÞ; g11 ðwÞ; . . . ; gn1 ðwÞ , and 0nn Inn B¼ e ; B B with 0n·n and In·n representing the null and identity matrices, respectively. The general solution to the system of first-order linear differential equations depicted in Eq. (6) is of the form f0(w) = eAwf0(0) [20]. T Assuming that A is a diagonalizable matrix, let ci and Vi ¼ ½V 1i ; . . . ; V ni , i ¼ 1; . . . ; n, denote the ith eigenvalue and its corresponding eigenvector of A, respectively. As such, the general solution to Eq. (6) can be conveniently expressed as f 0 ðwÞ ¼ VeCw a w 2 ð0; rÞ;
ð9Þ
where V ¼ ½V1 ; . . . ; Vn , and C represents a diagonal matrix whose diagonal elements consist of (c1, . . . ,cn). Moreover, a ¼ ½a1 ; . . . ; an T is a column vector comprising of the constant coefficients involved in the solution. By the same token, assuming that B is diagonalizable, the general solution to Eq. (8) for w 2 ð0; rÞ and w 2 ½r; R can be characterized as Z1(w) = UeDwb and Z1(w) = U eDwc, respectively, where D is the diagonal matrix containing the eigenvalues of B; d1 ; . . . ; d2n , in its diagonal positions, and U = [U1, . . ., U2n] is a square matrix of size 2n whose columns denote the corresponding eigenvectors Ui, i ¼ 1; . . . ; 2n, of B. Accordingly, the constant coefficients associated with the functional forms of the general solution in these two state intervals T T are, respectively, denoted by b ¼ ½b1 ; . . . ; b2n and c ¼ ½c1 ; . . . ; c2n . Furthermore, it can be readily verified that f 1 ðwÞ ¼ !eDw b w 2 ð0; rÞ; ð10Þ and f 1 ðwÞ ¼ !eDw c w 2 ½r; R;
ð11Þ T
where f 1 ðwÞ ¼ ½f11 ðwÞ; . . . ; fn1 ðwÞ and ! ¼ ½Inn 0nn U. Clearly, a complete characterization of the limiting distribution of W(t) (and, therefore, of X(t)) as depicted above requires the determination of 6n unknowns; i.e., the elements of a, b, c, and f 0 ¼ ½f100 ; . . . ; fn00 . To accomplish this task we first note that substituting Eq. (10) in Eq. (2) yields W!b þ ðQT KÞf 0 ¼ 0n1 :
ð12Þ
Next, substituting Eqs. (9) and (10) in Eq. (1) followed by comparing the coefficients of common exponential terms, e/i w , i 2 S E , results in " # V ij a þ f 0 ¼ 0n1 ; ð13Þ /i þ cj |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} nn
where Vij denotes the ði; jÞth element of V. Similarly, comparing the coefficients of e/i w , i 2 S E , after inserting the general solution characterized by Eqs. (9)–(11) in Eqs. (3) and (4) leads to U ij b ¼ 0n1 ; ð14Þ / i þ dj |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflffl ffl} n2n
and
Z r Z r U ij ð/i þdj Þr ð/i þcj Þa ð/i þdj Þa e da a þ U ij e da b þ e c f 0 ¼ 0n1 ; V ij / þ d j a¼0 a¼0 i |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} nn
n2n
respectively, with Uij denoting the ði; jÞth element of U.
n2n
ð15Þ
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E. Mohebbi / European Journal of Operational Research 190 (2008) 562–570
The remaining two sets of relations can be obtained by considering the equilibrium of overall transition rates of the sample-paths of Wi0(t) and W i1 ðtÞ; i 2 S E , in steady state, which are characterized by Z r T þ Q F0 ðrÞ þ Wf 1 ð0 Þ K eUðraÞ dF0 ðaÞ ¼ 0n 1; ð16Þ a¼0
and QT ðF0 ðrÞ þ F1 ðRÞÞ ¼ 0n 1;
ð17Þ T
respectively, where Fj ðÞ ¼ ½F 1j ðÞ; . . . ; F nj , j ¼ 0; 1. Substituting Eqs. (9)–(11) in Eqs. (16) and (17) after some simplifications provides, respectively, 0 1 Z r B C V ij B T C ecj a da K ecj r Ca þ Kf 0 ¼ 0n1 ; ð18Þ BQ V ij @ A /i þ dj a¼0 |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} nn
n2n
and
Z r Z R dj a dj a e da a þ U ij e da b þ U ij e da c þ f 0 ¼ 0n1 : V ij a¼0 a¼0 a¼r |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} Z
r
cj a
nn
n2n
ð19Þ
n2n
Finally, the normalizing condition is n X ðF i0 ðrÞ þ F i1 ðRÞÞ ¼ 1;
ð20Þ
i¼1
which using the functional forms of the general solution, Eqs. (9)–(11), can be expressed in the matrix form 0 1 Z r Z r Z R B C ð21Þ ecj a da a þ U ij edj a da b þ U ij edj a da c þ f 0 C YB @ V ij A ¼ 1; a¼0 a¼r |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffla¼0 ffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} nn
n2n
n2n
0.16
1
0.14
0.9
0.12
0.8
0.1
0.7
f(w)
f(w)
where Y ¼ ½1; . . . ; 116n . Considering the above, Eq. (21) along with 5n relations chosen from those supplied by Eqs. (12)–(15), (18) and (19) provide a set of 6n linearly independent relations which is sufficient to determine the values of all the point mass probabilities and constant coefficients associated with the general solution.
0.08
0.6
0.06
0.5
0.04
0.4
0.02
0.3
0
0
2
4
w
6
8
0.2
0
2
4
6
8
w
Fig. 2. Sample plots of limiting density and cumulative distribution functions of W(t).
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Fig. 2 illustrates the plots of f(w) and F(w) for a sample parameters settings with n = 3 which include: r ¼ 2; R ¼ 8; k1 ¼ 10:7; k2 ¼ 3:6; k3 ¼ 0:7; w1 ¼ 75; w2 ¼ 25; w3 ¼ 5; /i ¼ 0:5; mi ¼ 1; pii ¼ 0; i ¼ 1; . . . ; 3, and pij ¼ 0:5; i 6¼ j; i ¼ 1; . . . ; 3; j ¼ 1; . . . ; 3. 5. Concluding remarks The operating environment of most manufacturing systems extends beyond the scope of an alternating twostate (up or down) stochastic process typically adopted in the literature. Nevertheless, due to a higher complexity, analytical treatment of production planning and control operations under the influence of a multi-state random environment remains largely unexplored. Following our earlier results [19], this note contributes to narrowing this gap by deriving the limiting distribution of the inventory level in a production–inventory system with limited storage capacity that is subject to stochastic multi-state Markov-modulated production and demand processes. Knowledge of such distribution serves as the critical enabler for formulating a variety of performance measures for evaluation purposes. The adopted two-parameter production control policy accommodates the possibility of incurring production set-up times/costs which are commonly observed in practice. Further generalization of the control policy to allow for state-dependent parameter(s) provides a direction for future research. Acknowledgement The author thanks the anonymous referees for their insightful comments which have greatly improved the presentation of this note. References [1] M. Berg, M.J.M. Posner, H. Zhao, Production–inventory systems with unreliable machines, Operations Research 42 (1994) 111–118. [2] P.H. Brill, System Point Theory in Exponential Queues, PhD Dissertation, University of Toronto, 1975. [3] P.H. Brill, Level Level crossing methods, in: S.I. Gauss, C.M. Harris (Eds.), Encyclopedia of Operations Research and Management Science, Kluwer Academic Publishers, 1996, pp. 338–340. [4] P.H. Brill, B.A. Chaouch, An EOQ model with random variations in demand, Management Science 41 (1995) 927–936. [5] 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. [6] F. Chen, J.S. Song, Optimal policies for multi-echelon problems with Markov-modulated demand, Operations Research 49 (2001) 226–234. [7] H. Chen, D.D. Yao, A fluid model for systems with random disruptions, Operations Research 40 (1992) S239–S247. [8] H. Groenevelt, L. Pintelon, A. Seidmann, Production batching with machine breakdowns and safety stocks, Operations Research 40 (1992) 959–971. [9] H. Groenevelt, L. Pintelon, A. Seidmann, Production lot sizing with machine breakdowns, Management Science 38 (1992) 104–123. [10] J.N.D. Gupta, R. Ruiz, J.W. Fowlers, S.J. Mason, Operational planning and control of semiconductor wafer production, Production Planning & Control 17 (2006) 639–647. [11] K.B. Hendricks, V.R. Singhal, Association between supply chain glitches and operating performance, Management Science 51 (2005) 695–711. [12] K.B. Hendricks, V.R. Singhal, The Effect of supply chain glitches on shareholder wealth, Journal of Operations Management 21 (2003) 501–522. [13] T. Itoh, H. Ishii, T. Nanseki, A model of crop planning under uncertainty in agricultural management, International Journal of Production Economics 81–82 (2003) 555–558. [14] B. Kamrad, R. Ernst, An economic model for evaluating mining and manufacturing ventures with output yield uncertainty, Operations Research 49 (2001) 690–699. [15] 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. [16] O. Kella, W. Whitt, A storage model with a two-state random environment, Operations Research 40 (1992) S257–S262. [17] R.R. Meyer, M.H. Rothkopf, S.A. Smith, Reliability and inventory in a production-storage system, Management Science 25 (1997) 799–807. [18] K. Moinzadeh, P. Aggarwal, Analysis of a production/inventory system subject to random disruptions, Management Science 43 (1997) 1577–1588. [19] E. Mohebbi, A production–inventory model with randomly changing environmental conditions, European Journal of Operational Research 174 (2006) 539–552.
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[20] B. Noble, Applied Linear Algebra, Prentice-Hall, Englewood Cliffs, NJ, 1969. [21] P.R. Parthasarathy, M. Sharafali, On the unutilized capacity of a production-storage system, Operations Research Letters 6 (1987) 233–238. [22] 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. [23] S. Sethi, F. Cheng, Optimality of (s, S) policies in inventory models with Markovian demand, Operations Research 45 (1997) 931–939. [24] V.R. Singhal, K.B. Hendricks, How supply chain glitches torpedo shareholder value, Supply Chain Management Review (January/ February) (2002) 18–24. [25] J.S. Song, P. Zipkin, Inventory control is a fluctuating demand environment, Operations Research 41 (1993) 351–370. [26] R.W. Wolff, Poisson arrivals see time averages, Operations Research 30 (1982) 223–231.
European Journal of Operational Research 190 (2008) 562–570 www.elsevier.com/locate/ejor
Short Communication
A note on a production control model for a facility with limited storage capacity in a random environment E. Mohebbi
*
Department of Management/MIS, College of Business, University of West Florida, Pensacola, FL 32514, USA Received 17 January 2006; accepted 17 June 2007 Available online 30 June 2007
Abstract This note presents an extension of our earlier paper on a production–inventory system with limited storage capacity in a multi-state random environment [i.e., E. Mohebbi, A production–inventory model with randomly changing environmental conditions, Eur. J Res. 174 (2006) 539–552]. More specifically, it assumes that the production rate as well as the occurrence rate and batch sizes of a compound Poisson demand stream are influenced by the state of a continuous-time homogeneous Markov chain representing the random environment. We present an adaptation of the system-point method of level crossings which yields the limiting distribution of inventory level (stock-on-hand) for a two-parameter production control policy when stockouts are lost. 2007 Elsevier B.V. All rights reserved. Keywords: Inventory; Production; Limited capacity; Random environments; Markov-modulated processes; Level crossings
1. Introduction Manufacturing systems are known for operating in random environments. A random environment is a common term used in the production control literature to highlight the random nature of operating conditions that play defining roles in characterizing the production and demand processes and as such, can significantly impact the performance of production–inventory systems. Such random changes in operating conditions may be attributed to a variety of factors ranging from equipment condition and breakdowns, production yield, raw material/subcomponent quality, production mix, workforce level and labor skills to economic and market conditions. The globalization of economy and liberalization of marketplace at an increasingly rapid pace has intensified the need for incorporating the resulting operational uncertainties and financial risks into the firms’ production and inventory control decisions. To that end, increased product variety and customization, frequent price
*
Fax: +1 850 474 2314. E-mail address: [email protected]
0377-2217/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.06.037
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discounts and promotions, shortened product life cycles, advancement in technology, geopolitical crises and environmental conditions, to name a few, are among the many factors that contribute to such uncertainties in today’s manufacturing world. (The reader is referred to [10,13,14], among others, for typical manifestations of environmental uncertainties in wafer manufacturing, agricultural and mining industries, respectively.) In addition, randomly occurring disruptive events such as equipment breakdowns, capacity limitations, material shortages, or strikes in most supply chains are known to have significant negative impacts on the financial bottom lines of the affected firms. Apple, Motorola and Sony reported problems with meeting their demands due to material shortages caused by unforeseen circumstances in 1995, 1999 and 2000, respectively [24]. A recent study indicates that while much of the emphasis on the increasingly popular supply chain management strategies seems to be on efficiency or reducing costs, overemphasis on efficiency can make the supply chain brittle and more susceptible to the risk of disruptions [11,12]. The study documents that based on a sample of 885 glitches reported by various companies, firms that experience disruptions face on average 6.92% lower sales growth, 10.66% growth in cost, and 13.88% growth in inventories. A large segment of the existing models in this area corresponds to inventory systems in randomly fluctuating demand environments where demand is characterized as a Markov-modulated process (e.g., [6,23,25]); i.e., the rate or distribution of demand is subject to change at random epochs in accordance with state transitions of a discrete-or continuous-time Markov chain which represents the environment. Another line of work within this context has focused on production control in production–inventory settings where a failure-prone production facility, which switches randomly between states available and unavailable, feeds into a storage system which faces the demand (e.g., [1,8,9,17,18,21]). Accordingly, the relationship between fluid queueing systems with deterministic or stochastic input and/or output flows modulated by two-state (up and down) environments, and production-storage systems in random environments has been explored in the queueing literature (e.g., [7,15,16,22]). The economic order quantity (EOQ) model in random environment has also received some attention in the literature [4]. This note presents an exact mathematical model for deriving the limiting (steady state) distribution of the inventory level (i.e., stock-on-hand) of a single item in a production–inventory system with limited storage capacity. We assume that the demand for this item arises according to a compound Poisson stream, and the production occurs continuously and uniformly over time until the storage facility reaches its maximum capacity upon which the production is halted. The production is restarted once the inventory level drops to or below a critical threshold. All shortages are lost. Furthermore, it is assumed that both demand and production processes are modulated by a multi-state random environment which is represented by a continuous-time homogeneous Markov chain. The model is an extension of our earlier work [19] in that it attempts to address a limitation embedded in the previous model by utilizing a two-parameter control policy. More specifically, the control policy introduced above is much broader than the single-parameter policy considered in [19] where the production of the item is halted whenever the inventory level reaches the maximum storage capacity and is immediately restarted at the next demand epoch. It should be clear that while the latter policy is only prevalent in circumstance where the production set-up cost/time is negligible, the former is a more realistic representation of most manufacturing environments where set-ups cannot be ignored. A special case of this problem includes a scenario involving several independent and failure-prone machines producing a single item to meet uncertain demand. Each machine must undergo repair independently upon failure to restore operation. The production on all machines is also halted due to storage capacity limitations, and is restarted once the inventory level crosses a threshold. We note that while the form of the optimal control policy for this type of production–inventory systems in general remains an open research question, the main contribution of this work in reference to our earlier paper [19] lies in its unique conflation of two main characteristics of such systems in an analytical framework which is largely unexplored: namely, a ‘‘multi-state’’ randomly changing environment which affects both production and demand, and a ‘‘two-parameter’’ control policy which accommodates the possibility of incurring a set-up time/cost for each production run. This note is organized as follows: Section 2 outlines our main assumptions; Section 3 contains the model formulation; Section 4 presents the solution process in detail, and is followed by concluding remarks in Section 5.
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2. System characteristics 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 > 0 units. (2) The storage facility is supplied by a production facility which can produce the item continuously and uniformly at the rate of wi ; i ¼ 1; . . . ; n, depending on the state of a continuous-time homogeneous Markov chain E fEðtÞ; t P 0g with discrete state space S E ¼ f1; . . . ; ng and transition probabilities pii0 for 0 i(5i ) and i0 2 S E , and pii = 0 for 8i 2 S E at any transition epoch. The sojourn time of E in each state i 2 S E is exponentially distributed with mean m1 i . (3) The random demand process is characterized by a Markov-modulated compound Poisson stream with an average intensity of ki and independent exponentially distributed batch sizes with mean /1 i , depending on the state of E at any point in time. (4) The production is halted whenever the inventory level in the storage facility reaches its maximum capacity R and is restarted as soon as the inventory level drops to or below a threshold level s(0 6 s < R) at a demand occurrence epoch. Letting r = R s, it should be clear that the two-parameter (s, R), or equivalently, (r, R), control policy considered here is a generalization of the single-parameter policy considered in [19] where production is stopped as soon as the inventory level reaches R and is resumed immediately after the next demand occurs. (5) The Markov process E is assumed to be ergodic. Moreover, transitions of E over time are independent of the inventory level of the storage facility; that is, transitions of E are independent 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. 3. Model formulation Let fX ðtÞ; t P 0g denote the inventory level at time t. Clearly, X ðtÞ 2 ½0; R. We define fW ðtÞ; t P 0g ¼ R X ðtÞ, as the slack storage capacity available at time t with W(t) 2 [0, R]. Note that X(t) under our set of assumptions can be readily characterized as a regenerative process possessing a limiting distribution. Accordingly, W(t) is a stationary process with W = limt!1W(t) in distribution. We utilize a level-crossing approach [2,5] to model the steady-state behavior of the production–inventory system and derive the limiting distribution of W(t), and hence, X(t). Knowledge of such distribution can then be conveniently used to formulate a variety of performance measures (e.g., average inventory level, average total cost/revenue per unit time, fill rate, etc.) for evaluation purposes [19]. Let F(Æ) and f(Æ), respectively, denote the limiting cumulative probability distribution and density functions of W(t). By level-crossing theory and ‘‘method of stages’’, observe that the evolution of the sample-path of W(t) over time can be viewed as the confluent of the sample-paths of Markovian processes Wi1(t) and Wi0(t), where fW i1 ðtÞ; t P 0g ffW ðtÞ; t P 0g \ fEðtÞ ¼ i; i 2 S E ; t P 0g \ fThe production facility is activegg, and fW i0 ðtÞ; t P 0g ffW ðtÞ; t P 0g \ fEðtÞ ¼ i; i 2 S E ; t P 0g \ fThe production facility is idlegg. Moreover, given the structure of the control policy, it follows that W i1 ðtÞ 2 ð0; R and Wi0(t) 2 [0, r), i 2 S E . Fig. 1 depicts typical sample-paths of these processes. Let fi j(Æ) and Fi j(Æ) denote the limiting density and cumulative distribution functions of W ij ðtÞ; i 2 S E and j ¼ 0; 1, respectively. Accordingly, 8 n P 1 P > > > fij ðwÞ w 2 ð0; rÞ; < i¼1 j¼0 f ðwÞ ¼ n > P > > w 2 ½r; R; : fi1 ðwÞ i¼1
Rw and F ðwÞ ¼ i¼1 fi00 þ a¼0 f ðaÞda, where fi00 ¼ dF i0 ð0Þ denotes the long-run point mass probability of having zero slack capacity (i.e., the storage facility is full and the production facility is idle) while E is in state i, i 2 S E . Pn
E. Mohebbi / European Journal of Operational Research 190 (2008) 562–570
565
Wι1(t) Lost sales R
φi-1
ψι r
0
t
Wι 0(t)
r
φi-1 0
t
Fig. 1. Sample-path tracings of W ij ðtÞ, i 2 S E , j ¼ 0; 1.
Our level-crossing approach toward formulating the sample-path tracings of the stochastic processes identified above is based on the conservation law of level crossings [3] which asserts the existence of an equilibrium condition between the long-run total rates at which these tracings enter and exit any chosen subset of states. Following this principle over a number of judiciously chosen state intervals and utilizing the PASTA [26] property lead to a series of equations which balance the long-run entrance and exit rates of sample-path tracings of W ij ðtÞ; i 2 S E , j ¼ 0; 1, processes in reference to the designated state intervals. We first consider state-space interval ½0; wÞ; w 2 ð0; rÞ, for W i0 ðtÞ 2 ½0; rÞ and obtain: Z w n X mj pji F j0 ðwÞ þ wi fi1 ð0þ Þ ¼ ki e/i ðwaÞ dF i0 ðaÞ þ mi F i0 ðwÞ: ð1Þ a¼0
j¼1
The first term on the left-hand side (LHS) represents the total rate of sample-path entering ½0; wÞ due to transitions of E, and the second term shows the entrance rate into ½0; wÞ due to downcrossing of level 0 upon system reaching its maximum storage capacity. On the right-hand side (RHS), the first term depicts the samplepath exit rate from ½0; wÞ due to a demand of sufficiently large size while the production facility is idle, and the second term captures its exit rate from the same interval due to transitions of E. Similarly, for W i0 ðtÞ ¼ 0; i 2 S E , in steady state, we have wi fi1 ð0þ Þ þ
n X
mj pji fj00 ¼ ðki þ mi Þfi00 ;
ð2Þ
j¼1
where the first LHS term illustrates the sample-path downcrossing rate of level 0 (while E is in state i) at points of continuity due to a continuous and linear production process (slope = wi) which depletes the slack storage capacity to the point where the storage facility reaches its full capacity and production is halted. The second LHS term expresses the total rate of sample-path transitions from level 0, with E being in state j 2 S E ðj 6¼ iÞ, to the designated level 0 where E is in state i due to environmental transitions. The RHS term renders the longrun rate of sample-path exit from this level caused by a demand occurrence (rate ki) or a change in the environmental state (rate mi). Considering W i1 ðtÞ 2 ð0; R, i 2 S E , the balance equations for state intervals ðw; rÞ, where w 2 ð0; rÞ, and ½r; wÞ with w 2 ½r; R, are established as Z w n X / ðwaÞ ki e i e/i ðraÞ fi1 ðaÞda þ wi fi1 ðrÞ þ mj pji ðF j1 ðrÞ F j1 ðwÞÞ a¼0
¼ mi ðF i1 ðrÞ F i1 ðwÞÞ þ ki
Z
j¼1 r
a¼w
e/i ðraÞ fi1 ðaÞda þ wi fi1 ðwÞ;
ð3Þ
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and wi fi1 ðwÞ þ þ ki
Z
n X
mj pji ðF j1 ðwÞ F j1 ðrÞÞ
j¼1 r
e
/i ðraÞ
e
/i ðwaÞ
a¼0
¼ mi ðF i1 ðwÞ F i1 ðrÞÞ þ ki
dF i0 ðaÞ þ ki
Z
r
e/i ðraÞ e/i ðwaÞ fi1 ðaÞda
a¼0
Z
w
e/i ðwaÞ fi1 ðaÞda þ wi fi1 ðrÞ;
ð4Þ
a¼r
respectively. The first and second LHS terms of Eq. (3), respectively, outline the rates at which the sample-path enters ðw; rÞ due to a demand of sufficiently large size and downcrossing of level r while the production facility is active and E is in state i. The third LHS term describes the total entrance rate into (w, r) due to transitions of E. By the same token, the first, second, and third RHS terms represent the sample-path exit rates from (w, r) due to a transition of E, a demand of sufficiently large size, and downcrossing of level w while the production facility is active, respectively. A similar line of reasoning can be applied to justify the structure of Eq. (4). The limiting distribution of W(t) is obtained by solving Eqs. (1)–(4). 4. Model solution Introducing hDi = hd/dwi and applying hDihD + /ii to Eq. (1), we obtain: ððki þ mi ÞhDi þ mi /i Þfi0 ðwÞ
n X
mj pji hD þ /i ifj0 ðwÞ ¼ 0;
ð5Þ
j¼1
or equivalently, hDif 0 ðwÞ ¼ Af 0 ðwÞ;
ð6Þ T
in the matrix form where f 0 ðwÞ ¼ ½f10 ðwÞ; . . . ; fn0 ðwÞ is an n-dimensional column vector, and 1
A ¼ ½K QT ½/i QTi 8i is a square matrix of size 2 m1 m1 p12 6 m2 p 6 21 m2 Q¼6 .. 6 .. 4 . . mn pn1
mn pn2
n with .. .
3 m1 p1n m2 p2n 7 7 7 .. 7 . 5
mn
representing the infinitesimal generator for E, and QTi denoting the ith row of QT, i ¼ 1; . . . ; n. Similarly, applying hDihD + /ii to Eqs. (3) and (4) results in the following differential equation over their corresponding state intervals:
n X wi hD2 i þ ðwi /i ki mi ÞhDi mi /i fi1 ðwÞ þ mj pji hD þ /i ifj1 ðwÞ ¼ 0:
ð7Þ
j¼1
Let gi1 ðwÞ ¼ hDifi1 ðwÞ, i 2 S E , wherever the derivative is defined, and K, U and W mark three n-dimensional diagonal matrices whose diagonal elements comprise of ðk1 ; . . . ; kn Þ, ð/1 ; . . . ; /n Þ, and ðw1 ; . . . ; wn Þ, respectively. Hence, introducing /i T 1 T b e¼B b U þ W1 K B ¼ Qi 8i ; B ¼ Qi 8i ; and B wi wi for i ¼ 1; . . . ; n as three n-dimensional square matrices, Eq. (7) can be expressed in the matrix form:
E. Mohebbi / European Journal of Operational Research 190 (2008) 562–570
hDiZ1 ðwÞ ¼ BZ1 ðwÞ;
567
ð8Þ T
where Z1 ðwÞ ¼ ½f11 ðwÞ; . . . ; fn1 ðwÞ; g11 ðwÞ; . . . ; gn1 ðwÞ , and 0nn Inn B¼ e ; B B with 0n·n and In·n representing the null and identity matrices, respectively. The general solution to the system of first-order linear differential equations depicted in Eq. (6) is of the form f0(w) = eAwf0(0) [20]. T Assuming that A is a diagonalizable matrix, let ci and Vi ¼ ½V 1i ; . . . ; V ni , i ¼ 1; . . . ; n, denote the ith eigenvalue and its corresponding eigenvector of A, respectively. As such, the general solution to Eq. (6) can be conveniently expressed as f 0 ðwÞ ¼ VeCw a w 2 ð0; rÞ;
ð9Þ
where V ¼ ½V1 ; . . . ; Vn , and C represents a diagonal matrix whose diagonal elements consist of (c1, . . . ,cn). Moreover, a ¼ ½a1 ; . . . ; an T is a column vector comprising of the constant coefficients involved in the solution. By the same token, assuming that B is diagonalizable, the general solution to Eq. (8) for w 2 ð0; rÞ and w 2 ½r; R can be characterized as Z1(w) = UeDwb and Z1(w) = U eDwc, respectively, where D is the diagonal matrix containing the eigenvalues of B; d1 ; . . . ; d2n , in its diagonal positions, and U = [U1, . . ., U2n] is a square matrix of size 2n whose columns denote the corresponding eigenvectors Ui, i ¼ 1; . . . ; 2n, of B. Accordingly, the constant coefficients associated with the functional forms of the general solution in these two state intervals T T are, respectively, denoted by b ¼ ½b1 ; . . . ; b2n and c ¼ ½c1 ; . . . ; c2n . Furthermore, it can be readily verified that f 1 ðwÞ ¼ !eDw b w 2 ð0; rÞ; ð10Þ and f 1 ðwÞ ¼ !eDw c w 2 ½r; R;
ð11Þ T
where f 1 ðwÞ ¼ ½f11 ðwÞ; . . . ; fn1 ðwÞ and ! ¼ ½Inn 0nn U. Clearly, a complete characterization of the limiting distribution of W(t) (and, therefore, of X(t)) as depicted above requires the determination of 6n unknowns; i.e., the elements of a, b, c, and f 0 ¼ ½f100 ; . . . ; fn00 . To accomplish this task we first note that substituting Eq. (10) in Eq. (2) yields W!b þ ðQT KÞf 0 ¼ 0n1 :
ð12Þ
Next, substituting Eqs. (9) and (10) in Eq. (1) followed by comparing the coefficients of common exponential terms, e/i w , i 2 S E , results in " # V ij a þ f 0 ¼ 0n1 ; ð13Þ /i þ cj |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} nn
where Vij denotes the ði; jÞth element of V. Similarly, comparing the coefficients of e/i w , i 2 S E , after inserting the general solution characterized by Eqs. (9)–(11) in Eqs. (3) and (4) leads to U ij b ¼ 0n1 ; ð14Þ / i þ dj |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflffl ffl} n2n
and
Z r Z r U ij ð/i þdj Þr ð/i þcj Þa ð/i þdj Þa e da a þ U ij e da b þ e c f 0 ¼ 0n1 ; V ij / þ d j a¼0 a¼0 i |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} nn
n2n
respectively, with Uij denoting the ði; jÞth element of U.
n2n
ð15Þ
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E. Mohebbi / European Journal of Operational Research 190 (2008) 562–570
The remaining two sets of relations can be obtained by considering the equilibrium of overall transition rates of the sample-paths of Wi0(t) and W i1 ðtÞ; i 2 S E , in steady state, which are characterized by Z r T þ Q F0 ðrÞ þ Wf 1 ð0 Þ K eUðraÞ dF0 ðaÞ ¼ 0n 1; ð16Þ a¼0
and QT ðF0 ðrÞ þ F1 ðRÞÞ ¼ 0n 1;
ð17Þ T
respectively, where Fj ðÞ ¼ ½F 1j ðÞ; . . . ; F nj , j ¼ 0; 1. Substituting Eqs. (9)–(11) in Eqs. (16) and (17) after some simplifications provides, respectively, 0 1 Z r B C V ij B T C ecj a da K ecj r Ca þ Kf 0 ¼ 0n1 ; ð18Þ BQ V ij @ A /i þ dj a¼0 |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} nn
n2n
and
Z r Z R dj a dj a e da a þ U ij e da b þ U ij e da c þ f 0 ¼ 0n1 : V ij a¼0 a¼0 a¼r |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} Z
r
cj a
nn
n2n
ð19Þ
n2n
Finally, the normalizing condition is n X ðF i0 ðrÞ þ F i1 ðRÞÞ ¼ 1;
ð20Þ
i¼1
which using the functional forms of the general solution, Eqs. (9)–(11), can be expressed in the matrix form 0 1 Z r Z r Z R B C ð21Þ ecj a da a þ U ij edj a da b þ U ij edj a da c þ f 0 C YB @ V ij A ¼ 1; a¼0 a¼r |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffla¼0 ffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} nn
n2n
n2n
0.16
1
0.14
0.9
0.12
0.8
0.1
0.7
f(w)
f(w)
where Y ¼ ½1; . . . ; 116n . Considering the above, Eq. (21) along with 5n relations chosen from those supplied by Eqs. (12)–(15), (18) and (19) provide a set of 6n linearly independent relations which is sufficient to determine the values of all the point mass probabilities and constant coefficients associated with the general solution.
0.08
0.6
0.06
0.5
0.04
0.4
0.02
0.3
0
0
2
4
w
6
8
0.2
0
2
4
6
8
w
Fig. 2. Sample plots of limiting density and cumulative distribution functions of W(t).
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Fig. 2 illustrates the plots of f(w) and F(w) for a sample parameters settings with n = 3 which include: r ¼ 2; R ¼ 8; k1 ¼ 10:7; k2 ¼ 3:6; k3 ¼ 0:7; w1 ¼ 75; w2 ¼ 25; w3 ¼ 5; /i ¼ 0:5; mi ¼ 1; pii ¼ 0; i ¼ 1; . . . ; 3, and pij ¼ 0:5; i 6¼ j; i ¼ 1; . . . ; 3; j ¼ 1; . . . ; 3. 5. Concluding remarks The operating environment of most manufacturing systems extends beyond the scope of an alternating twostate (up or down) stochastic process typically adopted in the literature. Nevertheless, due to a higher complexity, analytical treatment of production planning and control operations under the influence of a multi-state random environment remains largely unexplored. Following our earlier results [19], this note contributes to narrowing this gap by deriving the limiting distribution of the inventory level in a production–inventory system with limited storage capacity that is subject to stochastic multi-state Markov-modulated production and demand processes. Knowledge of such distribution serves as the critical enabler for formulating a variety of performance measures for evaluation purposes. The adopted two-parameter production control policy accommodates the possibility of incurring production set-up times/costs which are commonly observed in practice. Further generalization of the control policy to allow for state-dependent parameter(s) provides a direction for future research. Acknowledgement The author thanks the anonymous referees for their insightful comments which have greatly improved the presentation of this note. References [1] M. Berg, M.J.M. Posner, H. Zhao, Production–inventory systems with unreliable machines, Operations Research 42 (1994) 111–118. [2] P.H. Brill, System Point Theory in Exponential Queues, PhD Dissertation, University of Toronto, 1975. [3] P.H. Brill, Level Level crossing methods, in: S.I. Gauss, C.M. Harris (Eds.), Encyclopedia of Operations Research and Management Science, Kluwer Academic Publishers, 1996, pp. 338–340. [4] P.H. Brill, B.A. Chaouch, An EOQ model with random variations in demand, Management Science 41 (1995) 927–936. [5] 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. [6] F. Chen, J.S. Song, Optimal policies for multi-echelon problems with Markov-modulated demand, Operations Research 49 (2001) 226–234. [7] H. Chen, D.D. Yao, A fluid model for systems with random disruptions, Operations Research 40 (1992) S239–S247. [8] H. Groenevelt, L. Pintelon, A. Seidmann, Production batching with machine breakdowns and safety stocks, Operations Research 40 (1992) 959–971. [9] H. Groenevelt, L. Pintelon, A. Seidmann, Production lot sizing with machine breakdowns, Management Science 38 (1992) 104–123. [10] J.N.D. Gupta, R. Ruiz, J.W. Fowlers, S.J. Mason, Operational planning and control of semiconductor wafer production, Production Planning & Control 17 (2006) 639–647. [11] K.B. Hendricks, V.R. Singhal, Association between supply chain glitches and operating performance, Management Science 51 (2005) 695–711. [12] K.B. Hendricks, V.R. Singhal, The Effect of supply chain glitches on shareholder wealth, Journal of Operations Management 21 (2003) 501–522. [13] T. Itoh, H. Ishii, T. Nanseki, A model of crop planning under uncertainty in agricultural management, International Journal of Production Economics 81–82 (2003) 555–558. [14] B. Kamrad, R. Ernst, An economic model for evaluating mining and manufacturing ventures with output yield uncertainty, Operations Research 49 (2001) 690–699. [15] 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. [16] O. Kella, W. Whitt, A storage model with a two-state random environment, Operations Research 40 (1992) S257–S262. [17] R.R. Meyer, M.H. Rothkopf, S.A. Smith, Reliability and inventory in a production-storage system, Management Science 25 (1997) 799–807. [18] K. Moinzadeh, P. Aggarwal, Analysis of a production/inventory system subject to random disruptions, Management Science 43 (1997) 1577–1588. [19] E. Mohebbi, A production–inventory model with randomly changing environmental conditions, European Journal of Operational Research 174 (2006) 539–552.
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