A Model to Analyze Airbase Oil Inventory System with Endogenous Lead Time

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1 Quality Technology & Quantitative Management Vol. 1, No. 3, pp , 15 QTQM ICAQM 15 A Model to Analyze Airbase Oil Inventory System with Endogenous Lead Time Kyung-Hwan Choi 1, Mohan L. Chaudhry and Bong-Kyoo Yoon 3,* 1 Department of International Weapon Systems Contract, Defense Acquisition Program Administration, Seoul, Korea Department of Mathematics and Computer Science, Royal Military College of Canada, Ontario, Canada 3 Department of Operations Research, Korea National Defense University, Seoul, Korea (Received November 13, accepted May 14) Abstract: In this paper, we analyze airplane oil inventory problem of a base in the Republic of Korea Air Force (ROKAF) which should eep maintaining an appropriate inventory level of oil for the efficient operation of flights. In ROKAF, airplane oil supply is ordered (or delivered) at a constant time period, which means reorder point is not stochastic but deterministic. Since the airplane oil inspection time occupies large proportion of time between a reorder point and the time the oil is available for usage, it is imperative to analyze the inspection time which causes delay in fulfilling an order. In addition, if demands increase dramatically because of a war or an unexpected training, the reorder point could be shortened than in normal situations. In such abnormal circumstances, the shortened reorder point may require more delay for order fulfillment. In this setting, the inspection time endogenously determines the replenishment lead time (or supply lead time) and the pattern of the inventory. Meanwhile, since the procedure of airplane oil inspection is composed of several steps, it may be better that the distributions of inspection times are treated as phase-type distributions. In this context, we use D/PH/1/m queue and counting process of D-BMAP (Discrete-time Batch Marovian Arrival Process) with -class to analyze the delay (lead time) for order fulfillment. Using this model, we find a way to derive decision variables such as the number of inventory replenishments and average fill rate. Keywords: Airbase oil inventory, D/PH/1/m, D-BMAP, endogenous lead time, phase-type distribution, queueing theory. 1. Introduction I n many cases, product cannot be available for immediate use until it is inspected to decide whether it is appropriate for usage or not. Liewise, the airplane oil inspection is very important for the airbase to prevent an accident lie a drop or a breaout. Since the airplane oil inspection should be thorough, the airplane oil inspection in an airbase is composed of several phases. Moreover, the airplane oil inspection time determines the replenishment lead time (or supply lead time) and the pattern of the inventory. This inspection/replenishment lead time is affected by order quantity and inspection time, resulting in endogenous lead time which occurs by internal factors of system (e.g. production time delay). Most of the studies on the inventory systems have focused on exogenous lead time which occurs by exterior factors of system (e.g. delivery time delay, administration time delay). The model with exogenous lead time can be found in the literature [11, 14, 16-19] and references therein. However, very few results have been * Corresponding author. byoon1@gmail.com

2 314 Choi, Chaudhry and Yoon suggested for the model with endogenous lead time since it was first introduced by Boute et al. [4-6]. In this paper, we consider a discrete-time, single-item inspection and inventory system. The inventory is controlled by periodic review stoc policy and inspection time is stochastic variable with several steps. If we assume that the order fulfillment is the arrival of customer and the inspection is the service process, then the finite system capacity of inspection facility corresponds to D/PH/1/m queue, which is a special case of the PH/PH/1/m queue [1]. Because of the specific characteristics of the deterministic interarrival time by the scheduled plan, it is preferable to set up a discrete-time queueing model rather than a continuous-time model. Meanwhile, the Bernoulli arrival process and geometric distribution are not powerful enough to capture special features for the real world phenomena. In order to tacle this problem, the phase-type (PH) distribution and D-BMAP (Discrete-time Batch Marovian Arrival Process) are introduced. The D-BMAP generalizes the D-MAP (Discrete-time Marovian Arrival Process) by permitting batch arrivals. Phase-type distribution and D-BMAP have been used in order to capture the correlation of the arrival process. They, in fact, provide simple framewor to demonstrate how one can extend many simple results on geometric distribution to more complex models without losing computational tractability. Note that the concept of correlation was introduced by Chaudhry [7-8] in several queueing problems. The results have been generalized by the introduction of phase-type distributions by Neuts [15] and D-BMAP by Blondia and Casals [3], respectively. They have recently been further generalized by many authors. For details on the phase-type distribution, see Latouche and Ramaswami [13] and on the D-BMAP, see Herrmann [1], and Chaudhry et al. [9]. They have been used in a wide range of stochastic modeling applications in areas as diverse as telecommunications, biostatistics, queueing theory, survival analysis and so on. We present a new inventory model with endogenous lead time, order failure and inspection failure. As a result, this paper gives information about optimal reorder point and initial stoc to meet the service level in the inventory system. We also develop a method to analyze the delay (lead time) for order fulfillment using D/PH/1/m queue and the counting process of D-BMAP with -class. This paper is organized as follows. In Section, we describe the airplane oil inventory system and inspection procedure in ROKAF (Republic of Korea Air Force). In Section 3, we give a queueing model to determine the number of inventory replenishments. In Section 4, we present numerical results based on the actual data from one of the ROKAF s bases.. Model Description.1. Discrete Phase-type Distribution A discrete Phase-type random variable denoted by PH d (, C) is defined by the absorbing time into state in the discrete time Marov chain with initial probability vector and transition matrix P as follows. C c P 1. (1) We have that Cij and ci for 1 i, j n and that c + C1 = 1, where 1 is an n 1 column vector with all its entries equal one. We also have that α1 = 1. C is an n n

3 A Model to Analyze Airbase Oil Inventory System with Endogenous Lead Time 315 matrix, delineating the transition probabilities between the transient states and α is a stochastic 1 n vector, where i-th element [α] i is the probability that the process started in the transient state i. Let us discuss some important properties of discrete phase-type distributions according to Latouche and Ramaswami [13]. From the fact that P C 1 C 1 1 for, () we obtain the mass and cumulative probability function in Equation (3) given below. 1 Pr[ X ] C c for 1, Pr[ X ] 1 C1for. There exists a representation such that I C is nonsingular, which is equivalent to stating that absorption occurs from any phase, where I is the identity matrix. We then 1 have that [( I C) ] pq is the expected number of steps visiting phase q before absorption, given that the initial phase is p. Note that [ ] ij means ij-th element of a matrix. The probability generating function of PH (, C), (z) is also easy to obtain and is given by (z) = z PX d 1 (3) z( I zc) c for z 1. (4) By differentiating this successively with respect to z and setting z = 1, we obtain the factorial moments EXX [ ( 1) ( X1)]! ( IC) C 1 for 1. (5).. Discrete-time Batch Marovian Arrival Process (D-BMAP) with -class We develop the counting process of a D-BMAP with -class from the counting process of a D-BMAP given in Chaudhry et al. [9]. Consider that an m-state Marov chain governing the arrival process, D rs, r, s, is an m-dimensional probability matrix with an arrival of a batch of size r, s if r, s > and without an arrival if r, s =. [(D rs )] ij denotes the transition probability matrix whose ij-th element is the probability that the arrival of class-1 is r and the arrival of class- is s, respectively, while the phase of UMC (underlying Marov chain) varying from i to j during slots. Note that [ ] pq means entry in the p-th row and q-th column. We assume that arrivals of two customers are governed by a time-homogenous, stationary Marov chain with countable state space. The UMC is assumed to be irreducible and positive recurrent. Dˆ r s Drs is a stochastic matrix which has the stationary probability vector,, of the UMC such that 1 Dˆ 1 1. (6) The fundamental arrival rate of this process is given by ( r s) D 1. r s rs Let A1 (), A () and I (), respectively, denote the number of class-1, class- arrivals and the phase of the UMC during time interval (, ]. Then, the counting process of a D-BMAP with -class is a discrete-time three dimensional Marov chain { A 1 (), (), I ()} on the state space i, j : i,1 j m. A Let P ( ) n1, n be the m-dimensional transition probability matrix whose ij-th element is

4 316 Choi, Chaudhry and Yoon the probability that the number of class-1 and customers are n1 and n, respectively, ( ) P n, is defined by 1, n ij P ( ) () n, n Pr[A( ) n,a ( ) n,i( ) j A(),A,I( ) i], ij where Pr[ ] indicates probability of an event within parenthesis. In words, [ ] n ij is the 1, n probability that given an initial phase i of the UMC, n1, n customers arrive during time units with the phase of the UMC changing to j, we have I, n1 n and ( ) P 1 P D 1, = n n n n ( 1), (8) x, y n1x, ny, x y n1, n and 1 () () where I is the m-dimensional identity matrix with P, = I, Pn, n =, n 1 1 1, n, and ( ) ( ) P n 1, n =, <. Further, it is easy to see that the generating function of P n1, n is given by, ( ) n1 n n1 n P ( z1, z) Pn z1 z. 1, n z 1 z Dn 1, n (9) n1 n n1 n.3. Model Description The oil is ordered or supplied at a constant time period d. Due to the limitation of fuel storage tans, the frequency of ordering oil is higher than expected. Nevertheless, the construction of extra oil storage facility is difficult due to its expensive nature. Moreover, an increase in order quantity per period is also difficult due to transportation capacity and safety problem, which results in constant amount of order quantity. Therefore, the interval (or frequency) of order is the only control variable to cope with the uncertainty. The arriving oil at the base should be inspected before it is available for usage. Since the inspection is conducted by silled airmen whose maintaining cost is quite high, an airbase has just one inspection team. The arriving oil is processed one at a time on FIFO (First-In First-Out) basis and arriving orders while inspection team is busy must wait in a queue. The capacity of inspection facility is limited, which is denoted by m. The details of inspection procedure will be discussed in next section. Since the flight operation in ROKAF is carried out by the scheduled plan to balance the load of flights for each base, the demand per slot is assumed to be constant. The demand at slot, D is observed at the beginning of a slot, but it needs not fulfill until the end of the slot. Unfulfilled demand is bacordered. Inventory levels are reviewed at the end of every potential ordering slot. After demand is satisfied, an order at slot, O is placed to raise the inventory level. Since the demand is constant, it is natural that the amount of order is also constant. However, the order O cannot be placed when the inspection capacity is full. The inventory control system in our model is described in Figure 1. The arriving order at slot is inspected at the beginning of each slot. Inspection results are divided into inspection failure and success. In case that the inspection failure occurs, the oil returns to the supplier. Otherwise, the order is replenished and satisfies demand. After that, a manager decides whether to place an order to the supplier or not. The net stoc denoted by NS, is equal to the amount of initial inventory on hand plus all replenishment orders received so far minus total observed customer demands. Assuming that O and D are identical for, respectively, the net stoc after satisfying demand at slot, P ( ) (7)

5 A Model to Analyze Airbase Oil Inventory System with Endogenous Lead Time 317 NS NS O D NS o D, (1) x x x1 x1 where NS is the initial inventory level, is the number of inventory replenishments during slots. Note that the order cannot be placed if the capacity of inspection facility is full. Therefore, we can calculate as the function of several random variables as in Equation (11) given below. fo fi SI, (11) where f O is the number of order failure, f I is the number of inspection failure, S I is the number of orders under inspection or in the queue at slot. Once we derive stochastic distribution of the number of inventory replenishments,, then we can evaluate NS from Equation (1). To measure customer service, we use the average fill rate, which measures the proportion of demand that can immediately fulfill from the inventory on hand []. Hence, the average fill rate for slot can be calculated as follows: Average Fill Rate = NS y1 1 l y1 yd, NS y1 y1 y1 yd 1, Otherwise. (1).4. Oil Inspection Procedure Figure 1. Inventory control system. The quality of airplane oil plays a crucial role to guarantee the performance of a plane and prevent an accident. Considering the importance of oil, the procedure of airplane oil inspection in ROKAF is composed of four steps with feedbacs. The procedure of airplane oil inspection is described in Figure. Once oil arrives, there is an examination such as sealed, cover state chec and water examination with the naed eye. If first step is good, it proceeds to the second step called lower layer sampling test composed of precipitation reaction and freezing prevention. Third step is an all-layer test consisting of six inspection items such as conductivity and filtering test. Finally, they get a final receiving test. We assumed that the inspection went into the first step without any queueing if other steps fail. The oil must return to the supplier in case that inspection failure occurs at any step again. We assume that inspection failure does not lead to place an extra order. Since the oil inspection procedure is composed of several steps (or phases), it can be described properly by phase-type distribution than any other probability distributions.

6 318 Choi, Chaudhry and Yoon Step 1 : Naed eye test Failure Success Step : Lower layer sampling test Failure Success Step 3 : All layer sampling test Failure Success Step 4 : Receiving test Failure Supplier Figure. The procedure of airplane oil inspection. We assumed that the time of each inspection step has the geometric distribution with inspection completion probability r 1,..., r4. Let f1,..., f4 be the inspection failure probability of each step. We assume that all probabilities of retest are the same as those of initial test. We can model the above inspection procedure with PH(, S), 1 r1 r1(1 f1) 1 r r(1 f) r f 1 r3 r3(1 f3) r3f3 1 r4 r4 f4 S, 1 r1 r1(1 f1) 1 r r(1 f) 1 r3 r3(1 f3) 1 r4 where [1 ]. (13) 3. Determination of the Number of the Inventory Replenishments 3.1. Queueing Model To determine the number of inventory replenishments, we set up a queueing model, D/PH/1/m queue. The periodic review policy results in deterministic arrival time and multiple inspection steps with feedbac leading to the inspection time distributed with phase-type distribution. We also considered system capacity m reflecting on a real environment. Because of the specific characteristics of the arrival pattern, it is preferable to set up a discrete time queueing model rather than a continuous-time queueing model. To analyze the D/PH/1/m queue, we applied the PH/PH/1/m queue because D/PH/1/m queue is a special case of the PH/PH/1/m queue [1]. Therefore, the model in this paper has interarrival time with the phase-type distribution represented by (, C) of

7 A Model to Analyze Airbase Oil Inventory System with Endogenous Lead Time 319 dimension d whose mean is also d. 1 d 1 1 [1 ], C, c. d 1 1 d 1 The inspection completion and failure probability also have the phase-type distribution represented by (, S) with dimension 8. 1 r1 r1(1 f1) 1 r r(1 f) r f 1r3 r3(1 f r 3) 3 f 3 S 1 r4 r4 f 4, 1r1 r1(1 f1) 1 r r(1 f ) 1 r3 r3(1 f3) 1 r4 rf 1 1 r4(1 f4) s1, s, (15) rf 1 1 r f r3 f 3 r4 f4 r4 (1 f4 ) s 1 where is the absorbing probability vector in case of inspection failure, is the absorbing probability vector in case of inspection success. The state space describing the Marov chain representing the PH/PH/1/m queue is given by G = {(, j) (i, j, q), 1 i m, 1 j d, 1 q 8}, where (, j) represents the state when system is empty and the arrival is in phase j and (i, j, q) represents the state when there are i customers in the system (orders under inspection), the arrival is in phase j and service is in phase q. The transition probability matrix of the Marov chain defined by G that governs the system is given by B B B A A1 P A -1 A A1, (16) A1 A-1 E s (14)

8 3 Choi, Chaudhry and Yoon where B C, B1 ( c ), B C s, A1 ( c ) S, A ( c ) ( c ) CS, A ( Note that c = 1 C1 and s -1 C s), E= AA 1. = 1 S1. For more details on the transition probability matrix for a PH/PH/1/m, see Alfa and Zhao [1]. Let x = [ x, x1,, xm ] be the probability invariant vector associated with P. Then, we have xp x, xe=1. (17) The probability invariant vector x can be computed by the Matrix Geometric Algorithm whose details are found in Latouche and Ramaswami [13]. 3. Derivation of Using the Counting Process of D-BMAP To determine the number of inventory replenishments,, P should be divided into three transition probability matrices as in Figure 3. The first one is the transition probability matrix with order failure denoted by P f. The second is the transition probability matrix O containing inspection failure denoted by P f, and the third is the transition probability I matrix without any of order failure and inspection failure denoted by PN. Pf is the state O transition probability matrix where any order of oil is placed due to the system capacity being full. The nonzero entries of P f are the transition probabilities from the states (m-1 or O m, d-1, q) to the state (m, d, q ), 1 q, q 8 in P and other entries are all zero. Pf is the I state transition probability matrix that oil is returned to the supplier due to the inspection failure. The nonzero entries of P f are the transition probabilities from the states (1, j, q) to I the state (, j ), 1 j, j d, q = 1, 5, 6, 7, 8 for i = 1 and from the states (i, j, q) to the state (i-1, j, q ), 1 j, j d, 1 q, q = 1, 5, 6, 7, 8 for i > 1 in P (See, s 1 on p.319). The remainder of P except for and P is P. P fo f I N Figure 3. The decomposition of transition matrix P. ( ) Let Pn 1, nbe the transition probability matrix whose ij-th element is the probability that the number of class-1 and customers are n1 and n, respectively, while the phase of UMC (Underlying Marov Chain) varying from i to j during slots in D-BMAP. Conditioning on -th slot we have, ( ) ( 1) ( 1) ( 1) ( 1) n1, n n11, n1 11 n11, n 1 n1, n1 1 n1, n, P P D P D P D P D (18) where D rs is the transition probability matrix whose ij-th element is the probability that the arrival of class-1 is r and the arrival of class- is s, respectively, while the phase of UMC (underlying Marov chain) varying from i to j during slots.

9 A Model to Analyze Airbase Oil Inventory System with Endogenous Lead Time 31 1 Multiplying both sides of the Equation (19) by z1, z and summing them, we finally get, P ( z1, z ) ( D D1z1 D1z D 11zz 1 ). (19) Assume that order failure is class-1 customer and inspection failure is class- customer. Then, order failure and inspection failure cannot happen simultaneously in our model. Once inspection fails, one space of buffer becomes empty meaning that ordering is available again. Therefore, MGF (Matrix Generating Function) of the probability of order failure and inspection failure at slot is, ( ) 1, ) ( ) ( N f 1 O fi P ( z z D D z D z P P z P z ). Differentiating Equation () and setting z 1 z, we can obtain the probability that the number of order failure and inspection failure are n1 and n, respectively, while the phase of UMC (underlying Marov chain) varying from i to j at slot as in the Equation (1). ( ) n1, n ( ) n1 n 1 P ( z1, z)! ( n1n) n1 n, n1 n N Pf P o fi 1!! z1 z ( z1 z 1 )! n1! n! P P. (1) n n n n Since the first UMC phase in (, j) (i, j, q) is the number of customers (order under inspection), we can obtain the joint probability of the number of order failure, f O, the number of inspection failure, f I, and the number of orders under inspection, S I, at slot in Equation () given below. ( ) O 1 I I ] n1, n n n () Pr[ f n, f n, S P 1, () where is an initial probability with 1 (d + (8d) (m -1)) dimensional vector. is set of UMC phases of the number of customer being, that is, = {(, j) 1 j d} for = or {(, j, q) 1 j d, 1 q 8} for. Finally, we can get the number of inventory replenishments, γ as in Equation (11) and average fill rate as in Equation (1). 4. Numerical Results In this section, we present some numerical results. We have carried out numerical wor to analyze the number of inventory replenishments in one of the ROKAF s bases. We assumed that the unit of slot is a day because the usages of airplane oil or aviation operations tae place on daily basis. Table 1. Analysis of the results of the inspection completion and success probability. Distribution Phase-type Step 1 Step Step 3 Step 4 Geometric Inspection completion probability Inspection success probability Chi-square p-value.78.1

10 3 Choi, Chaudhry and Yoon An oil train arrives every two days, namely, customer arrives every other slot. We also collected and analyzed actual inspection time and inspection success probability from one of the ROKAF s bases in the second half of 11. Table 1 gives the statistical results on the assumption of the inspection completion and success probability whether it follows phase-type or geometric distribution. When inspection completion and success probability is assumed to have geometric distribution, the p-value for the assumption is below.5, which means the assumption seems wrong. Meanwhile, in case that inspection completion and success probability is assumed to have PH distribution, we can adopt the null hypothesis because p value is.78. Note that the parameter fitting for phase-type distribution is conducted by the EM (Expectation-Maximization) algorithm []. Histogram of real data and probability distributions fitted for the data are given in Figure 4. We can see that the phase-type distribution fits the data better than the geometric distribution as can be seen from Figure 4. Figure 4. The fitting comparison between phase-type and geometric distribution. Table gives several performance measures such as the number of order failure, inspection failure, the number of orders under inspection and the number of inventory replenishments, net stoc ( NS ) as well as average fill rate for D/PH/1/3 queue where, NS is 1, demand is 3, Order is 5. If you should maintain the positive net stoc for the slots, you need to pull up the net stoc after slot 18. We can decide that the optimal interarrival time (order interval) is 3 slots under the same conditions. As a result, we present a model which can give information to meet your service level about optimal order interval. 5. Conclusion In this paper, we considered a periodic review stoc controlled inspection-inventory system, where replenishment lead time is endogenously generated by an inspection system with finite capacities. We present a new inventory model with endogenous lead time, feedbac due to the inspection failure, and order failure. We also develop a method to analyze the delay (lead time) of order fulfillment using phase-type distribution and the counting process of D-BMAP with -class. In our analysis, we used discrete phase-type distributions that provide a simple framewor. With this analytical model, we analyze the optimal inventory policy for the ROKAF air base.

11 A Model to Analyze Airbase Oil Inventory System with Endogenous Lead Time 33 Slot The number of total order Table. Several performance measures at order slot. Order failure Inspection failure Under inspection The number of inventory replenishments Net stoc ( NS ) Average fill rate (%) Acnowledgements The third author acnowledges with thans the support provided by BISOM (Business Information Systems and Operations Management) department of University of North Carolina at Charlotte, where he held an Adjunct-Visiting scholarship during his sabbatical year and where part of this wor was done. This research is also supported (in part) by the C-ITRC Support Program (H , Development of KULAV Based Tactical Logistics Convergence System) funded by the Ministry of Science, ICT and Future Planning, Republic of Korea. The second author acnowledges with thans the partial support provided by NSERC. References 1. Alfa, A. S. and Zhao, Y. Q. (). Overload analysis of the PH/PH/1/K queue and the queue of the M/G/1/K type with very large K, ASIA Pacific Journal of Operational Research, 17(), Asmussen, S. (1996). Fitting phase-type distributions via the EM algorithm, Scandinavian Journal of Statistics, 3(4), Blondia, C. and Casals, O. (199). Statistical multiplexing of VBR sources: A matrix-analytic approach, Performance Evaluation, 16, Boute, R. N., Disney, S. M., Lambrecht, M. R., Velde, W. V. and Houdt, B. V. (8). A win-win solution for the bullwhip problem, Production Planning & Control: The Management of Operations, 19(7), Boute, R. N., Disney, S. M., Lambrecht, M. R., Velde, W. V. and Houdt, B. V. (9). Designing replenishment rules in a two-echelon supply chain with a flexible or an inflexible capacity strategy, International Journal of Production Economic, 119(1), Boute, R. N., Lambrecht, M. R. and Houdt, B. V. (7). Performance evaluation of a production/inventory system with periodic review and endogenous lead times, Naval Research Logistics, 54(4), Chaudhry, M. L. (1965). Correlated queueing, Journal of Canadian Operational Research Society, 3, Chaudhry, M. L. (1966). Some queueing with phase-type service, Operations Research, 14,

12 34 Choi, Chaudhry and Yoon 9. Chaudhry, M. L., Yoon, B. K. and Kim, N. K. (1). On the distribution of the number of customers in the D-BMAP/G a,b /1/M queue - A simple approach to a complex problem, INFOR, 48(), Herrmann, C. (1). The complete analysis of the discrete time finite D-BMAP/G/1/N queue, Performance Evaluation, 43, Kim, J. G., Chatfield, D. C., Harrison, T. P. and Hayaa, J. C. (6). Qualifying the bullwhip effect in a supply chain with stochastic lead time, European Journal of Operations Research, 173, Latouche, G. and Ramaswami, V. (1997). The PH/PH/1 queue at epochs of queue size change, Queueing System Theory Applications, 14, Latouche, G. and Ramaswami, V. (1999). Introduction to matrix analytic methods in stochastic modeling, SIAM, Philadelphia. 14. Lu, Y., Song, J. S. and Yao, D. D. (3). Order fill rate, lead time variability, and advance demand information in an assemble-to-order system, Operations Research, 51, Neuts, M. F. (1975). Probability Distributions of Phase Type, University of Louvain, Belgium, Song, J. S. (1994). The effect of lead-time uncertainty in a simple stochastic inventory model, Management Science, 4, Song, J. S. (1994). Understanding the lead-time effects in stochastic inventory systems with discounted costs, Operations Research Letter, 15, Song, J. S. and Yao, D. D. (). Performance analysis and optimization of assemble-to-order systems with stochastic lead times, Operations Research, 5, Van Nyen, P. L. M., Bertrand, J. W. M., Van Ooijen, H. P. Q. and Vandaele, N. J. (5). A heuristic to control integrated multi-product multi-machine production-inventory systems with job shop routings and stochastic arrival, set-up and processing times, Operations Research Spetrum, 7, Zipin, P. H. (). Foundations of Inventory Management, McGraw-Hill, New Yor. Authors Biographies: Kyung Hwan Choi is in charge of international contracts at DAPA (Defense Acquisition Program Administration) and has received Ph.D. degree from KNDU (Korea National Defense University) in 13. His main research areas are inventory management, stochastic modeling, performance evaluation and analysis. Mohan L. Chaudhry moved to Canada as a senior postdoctoral fellow at the University of Toronto before moving to the Royal Military College of Canada in 1967, where he is currently a senior professor. He has held an invited professor s position at George Mason University, USA in 1988, a distinguished professor s position at Korea Advanced Institute of Science and Technology in 1, Ghent University, Belgium in 4, and Indian Institute of Technology, Kharagpur, in 8. He has held associate editorship for several Operational Research Journals. In 3, Operational Research Society of Canada gave him an Award of Merit and in 5 a Service Award. Besides many scientific publications, he has co-authored a few boos among which A First Course in Bul Queues, has gained high reputation and is well quoted by many researchers. His current interests are in stochastic processes, applied probability and queueing theory. Bong-Kyoo Yoon, Associate Professor in the Dept. of Operations Research at Korea National Defense University (KNDU), has received Ph.D. degree in Industrial Engineering from Korea Advanced Institute of Science and Technology (KAIST), Republic of Korea. He provided consultancy on financial management and cost innovation to various global companies at IBM BCS. He has been woring for KNDU since 6. His major interest focuses on applying stochastic models to Logistics Optimization and Performance Innovation in public sector including defense industry.