Optimal basestock policy for the inventory system with periodic review, backorders and sequential lead times


 Ophelia Horn
 1 years ago
 Views:
Transcription
1 44 Int. J. Inventory Research, Vol. 1, No. 1, 2008 Optimal basestock policy for the inventory system with periodic review, backorders and sequential lead times Søren Glud Johansen Department of Operations Research, University of Aarhus, Ny Munkegade, Bldg. 530, DK8000 Aarhus C, Denmark Anders Thorstenson* CORAL Centre for OR Applications in Logistics, Department of Business Studies, Aarhus School of Business, University of Aarhus, Fuglesangs Allé 4, DK8210 Aarhus V, Denmark *Corresponding author Abstract: We extend wellknown formulae for the optimal base stock of the inventory system with continuous review and constant lead time to the case with periodic review and stochastic, sequential lead times. Our extension uses the notion of the extended lead time. The derived performance measures are exact for Poisson demands. Keywords: inventory control; optimisation; periodic review; sequential lead times; single echelon; single item; stochastic demand. Reference to this paper should be made as follows: Johansen, S.G. and Thorstenson, A. (2008) Optimal basestock policy for the inventory system with periodic review, backorders and sequential lead times, Int. J. Inventory Research, Vol. 1, No. 1, pp Biographical notes: Søren Glud Johansen is an Associate Professor at the Department of Operations Research, University of Aarhus, Denmark. His main research interests are in production economics, inventory control, Markov decision theory, and costing and pricing. He has published in Advances in Applied Probability, European Journal of Operational Research, International Journal of Production Economics, Journal of the Operational Research Society and Operations Research. Anders Thorstenson is a Professor and Director of the research centre CORAL at Aarhus School of Business, Denmark. His main research interests are in supply chain management, production and inventory control, operations management and performance management. He has published in European Copyright 2008 Inderscience Enterprises Ltd.
2 Optimal basestock policy 45 Journal of Operational Research, International Journal of Production Economics, International Journal of Production Research, Journal of the Operational Research Society and Managerial and Decision Economics. 1 Introduction We consider a singleitem inventory system with periodic review. We assume that the system is controlled by a basestock policy and that unsatisfied demands are backordered and satisfied as soon as possible. The inventory position (stock on hand + stock on order amount backordered) is monitored at each review instant, when the policy places a replenishment order to restore the base stock s. This type of system is characteristic of, e.g., frequently used spare parts in the automotive industry and some durable consumer goods in the retail industry. The periodic control feature might also be attractive for coordination purposes when the singleitem inventory system is embedded in a larger system consisting of multiple items and/or echelons. We assume that demands occur in continuous time and are generated by a Poisson process with rate λ. The holding and shortage costs are incurred accordingly as in Hadley and Whitin (1963, Section 53). It is common in models of periodic review systems to assume that demands occur only periodically. Such models can be analysed by direct analogy to continuous review systems; see, for example, Axsäter (2006, Section 5.12). However, as stated in Rosling (2002), demand arriving in continuous time is more realistic. The basestock policy with review interval R is commonly referred to as an (R, S) policy, where S = s. This policy is frequently used in practice, and it is often applied as a benchmark in theoretical studies. Furthermore, for the inventory system described and with a standard cost structure, the (R, S) policy is the optimal control policy (Zipkin, 2000, Section ). Note, however, that in our model costs are incurred in continuous time. Furthermore, we assume that the lead times L of replenishment orders are generated by an exogenous, sequential supply system (Zipkin, 2000, Section 7.4). This assumption implies that different replenishment orders do not cross in time. Similar ideas have been applied in, e.g., Hadley and Whitin (1963, Chapters 4 5), Kaplan (1970), and Ehrhardt (1984). The sequential supply system is formalised by Zipkin (1986). Our contribution is the provision of necessary and sufficient optimality conditions under stochastic, sequential lead times by using the notion of the extended lead time and the analogy to a continuousreview system. The rest of this paper is organised as follows. In Section 2 we introduce the notion of the extended lead time and characterise the probability mass function (pmf) of extended leadtime demand. Performance measures and optimality conditions for the periodic review system considered are provided in Section 3, which also includes a numerical example. Section 4 contains concluding remarks.
3 46 S.G. Johansen and A. Thorstenson 2 The extended lead time Like Zipkin (2000, Section 6.7.3) and Rao (2003), we model the times from the demand instants until the next review instant as realisations of a random variable U which, because of the Poisson demand process, is uniformly distributed over the interval from 0 to R. Similar approaches are used in Hadley and Whitin (1963, Section 53) and Axsäter (1993). The generic variable L' for the time from a demand instant until the delivery instant of the replenishment order triggered by this demand can be specified as U + L. Hadley Whitin, Zipkin and Rao assume that lead time L is constant, while Axsäter considers leadtime uncertainty due to possible shortages at the upstream level in a twoechelon system with no order crossing allowed. Hence, these approaches all involve special cases of the sequential supply system referred to in the Introduction. We call L' the extended lead time and observe that like the ordinary lead times L, which we allow to be stochastic, the extended lead times for customer demands can also be seen as generated by an exogenous, sequential supply system. This observation provides that the exact results derived for the corresponding inventory system with continuous review can be used to derive similar results for the system with periodic review (see also a similar remark in Yano, 1985). The results for the latter system are then obtained by focusing on customer demands and letting the lead times in the former system be distributed as L' rather than as L. This way the net inventory and hence the cost of the basestock policy in the periodic review system are replicated exactly by the continuous review system in which the inventory position is constant at the basestock level s. The generic variable D L for the demand during the extended lead time equals D U + D L, where D U is the generic variable for the demand during an interval of random length U and D L is the generic variable for the ordinary leadtime demand. The pmf of D U is R x y 1 λu λu λr 1 ( ) ( ) λr fu ( x) = e du e, x 0,1, R = = (1) x! y! 0 y= x+ 1 The pmf f U (x) is strongly unimodal. A proof is provided in the Appendix. Assuming that lead time L is gamma distributed, D L has the negative binomial distribution with parameters a = (E[L]) 2 /Var[L] and p = λ/(λ + a/e[l]) (Zipkin, 2000, Section ). Hence, if Var[L] > 0, we have the following useful recursion for the pmf of D L α 1+ y fl( y) = pfl( y 1), y = 1,2,, (2) y where f L (0) = (1 p) α. If Var[L] = 0, D L has the Poisson distribution with mean λe[l]. In either case, the pmf f L (y) is also strongly unimodal (Keilson and Gerber, 1971). Finally, because D L is the convolution of D U and D L, the pmf of D L is z f ( z) = f ( x) f ( z x), z = 0,1, (3) L U L x= 0 The following proposition summarises the conditions for strong unimodality of f L (z).
4 Optimal basestock policy 47 Proposition: If demand is Poisson and if (ordinary) lead times are gamma distributed, then the pmf specified in equation (3) of the extended leadtime demand is strongly unimodal. Proof: The result follows immediately from Theorem 2 in Keilson and Gerber (1971) which states that the pmf of a convolution of strongly unimodal pmfs is also strongly unimodal. Remark: Obviously, the proposition is true for any leadtime distribution that results in a strongly unimodal pmf of (ordinary) leadtime demand. 3 Performance measures Because of the equivalence to the continuousreview system with sequential extended lead times, important performance measures of the considered periodic inventory system can now be obtained using the standard arguments in Zipkin (2000, Corollary 7.4.3). Hence, the stockout frequency is A() s = Pr{ D s}, (4) L the average backorders are Bs ( ) = E[max{ D s,0}], (5) L and the average inventory is I() s = s E[ D ] + B() s = E[max{0, s D }]. (6) L L Let h and b denote the unit holding and backorder costs per unit time, respectively, and let π denote the cost incurred for each unit backordered. If there are no further costs, then the longrun average cost incurred per unit time is Cs () = hi() s+ bbs () + πλ As (). (7) The basestock model with this cost structure corresponds to Model 2, Case (ii) in Rosling (2002). Hence, based on his Proposition 22, we may conclude that C(s) is quasiconvex in s, because the pmf of D L is strongly unimodal, as stated in the Proposition above. (Quasiconvexity of C(s) is equivalent to the negative of C(s) being unimodal.) Then, for Cs ( ) = Cs ( + 1) Cs ( ) = ( h+ b) Pr{ D s} ( b+ πλ Pr{ D = s}), (8) the optimal basestock level is obtained as L L s* = min{ s C( s) > 0}. (9) An example is depicted in Figure 1 which shows that C(s) is quasiconvex in s. Figure 1 also shows the misrepresentation that occurs if lead times were assumed to be independent, implying by Palm s theorem (Palm, 1938) that the approximate cost rate
5 48 S.G. Johansen and A. Thorstenson C a (s) depends on the distribution of L' only through its mean E[L ] = R/2 + E[L]. Hence, the approximate cost rate is given as C () s = hi () s + bb () s + πλ A (), s (10) a a a a where the performance measures Aa(), s Ba() s and Ia() s are specified analogously to A(), s B() s and I() s but with (the stochastic) L replaced everywhere by its (constant) mean E[L ]. Figure 1 The basestock costrate function C(s) and its approximation C a (s) assuming independent lead times. Example with parameter values h = 2, b = 5, π = 15, λ = 10, E[L] = 2, Var[L] = 1 and R = 1 For the example depicted in Figure 1, the optimal basestock levels and their corresponding cost rates are shown in Table 1. This table also contains basestock levels and cost rates obtained in two special cases discussed below. The shortage costs have been chosen such that for sequential lead times the same optimal basestock level is obtained in all three cases. Note that in all cases the approximate approach assuming independent lead times yields a misrepresentation both in the optimal basestock level and in the cost rate. The relative magnitude of these misrepresentations is not negligible. In particular, the optimal basestock level and its associated cost rate are lower for independent lead times than for sequential lead times. This is a consequence of the fact that leadtime variability has no impact when lead times are independent, as also remarked in Svoronos and Zipkin (1991). A similar effect is noted in Robinson et al (2001) and in Bradley and Robinson (2005) for the periodic review and demand system with order crossing due to independent lead times. Table 1 Examples with fixed parameter values h = 2, λ = 10, E[L] = 2, Var[L] = 1 and R = 1 b = 5, π = 15 b = 21, π = 0 b = 0, π = 19 Example s* C(s*) C a (s*) s* C(s*) C a (s*) s* C(s*) C a (s*) Sequential lead times Independent lead times (approximation)
6 Optimal basestock policy 49 For π = 0, we obtain the extended standard solution b s* = min s Pr{ DL s} >. h+ b The exact and approximate cost rates in this special case are presented in Figure 2 for the example in Figure 1 with the adjusted shortage costs. In this case, it can easily be inferred from equation (8), and analogously for the approximate cost rate, that both cost rates are in fact convex discrete functions of the basestock level. This is also illustrated by the two cost functions shown in Figure 2. The optimal solutions and their associated costs are displayed in Table 1. In this particular example, the cost of using the approximate solution exceeds the optimal cost by approx. 29%. (11) Figure 2 The basestock costrate function C(s) and its approximation C a (s) assuming independent lead times. Example with parameter values h = 2, b = 21, π = 0, λ = 10, E[L] = 2, Var[L] = 1 and R = 1 As another special case, consider the case when b = 0 (and π > 0). Equation (9) may then be written as Pr( DL = s) h s* = min s <. Pr{ DL s} πλ The ratio Pr{ DL = s} Pr{ DL s}, is nonincreasing in s, because the pmf of D L is strongly unimodal (Rosling, 2002). Note that equation (12) is an extension to the case of periodic review and stochastic lead times of the expression found in the wellknown textbook by Silver et al. (1998, Section 8.3.1) in the case of continuous review and a constant lead time. The exact and approximate cost rates in this special case are presented in Figure 3 for the example in Figure 1 with the adjusted shortage costs. In this case, however, it cannot be concluded that the cost rates are convex discrete functions of the basestock level. This is also illustrated by the two cost functions shown in Figure 3. It is a consequence of having the unit backorder cost π > 0 in the expressions for the cost rates. The optimal solutions and their associated costs are displayed in Table 1. In this particular example, the cost of using the approximate solution exceeds the optimal cost by approx. 9%. (12)
7 50 S.G. Johansen and A. Thorstenson The optimality conditions in equations (9), (11), and (12) are intuitively appealing, since they are obtained from extending the standard continuousreview formulae by simply replacing D L, the demand during the ordinary lead time, by D L, the demand during the extended lead time. However, the numerical examples have illustrated that considering the extended lead time only through its mean, as when assuming that lead times are independent, may lead to considerable misrepresentation of both the optimal basestock level and the corresponding cost rate. Figure 3 The basestock costrate function C(s) and its approximation C a (s) assuming independent lead times. Example with parameter values h = 2, b = 0, π = 19, λ = 10, E[L] = 2, Var[L] = 1 and R = 1 4 Concluding remarks In this study the extended lead time has been defined as the time between a customer demand instant and the delivery instant of the replenishment order triggered by that demand. By applying this concept, exact performance measures for the basestock inventory system with continuous review have been extended to a system with periodic review and stochastic, sequential lead times. Simple conditions for optimisation have also been derived. The crucial assumption for these results is that lead times are sequential so that orders do not cross in time. The results obtained can be generalised further to accommodate compound Poisson demand (Johansen and Thorstenson, 2005). Acknowledgments This work was developed while the second author was visiting at the Department of Logistics, Hong Kong Polytechnic University. The authors wish to thank two anonymous referees whose comments significantly improved the presentation of the paper.
8 References Optimal basestock policy 51 Axsäter, S. (1993) Optimization of orderupto S policies in twoechelon inventory systems with periodic review, Naval Research Logistics, Vol. 40, pp Axsäter, S. (2006) Inventory Control, 2nd ed., Springer s International Series, New York. Bradley, J.R. and Robinson, L.W. (2005) Improved basestock approximations for independent stochastic lead times with crossover, Manufacturing and Service Operations Management, Vol. 7, No. 4, pp Ehrhardt, R. (1984) (s, S) policies for a dynamic inventory model with stochastic lead times, Operations Research, Vol. 32, pp Hadley, G. and Whitin, T.M. (1963) Analysis of Inventory Control Systems, PrenticeHall, Englewood Cliffs, NJ. Johansen, S.G. and Thorstenson, A. (2005) BaseStock Policy for the LostSales Inventory System with Periodic Review, Working Paper, available from the authors on request. Kaplan, R.S. (1970) A dynamic inventory model with stochastic lead times, Management Science, Vol. 16, No. 7, pp Keilson, J. and Gerber, H. (1971) Some results for discrete unimodality, Journal of the American Statistical Association, Vol. 66, pp Palm, C. (1938) Analysis of the Erlang traffic formula for busy signal assignment, Ericsson Technics, Vol. 5, pp Rao, U.S. (2003) Properties of the periodic review (R, T) inventory control policy for stationary, stochastic demand, Manufacturing and Service Operations Management, Vol. 5, No. 1, pp Robinson, L.W., Bradley, J.R. and Thomas, L.J. (2001) Consequences of order crossover under orderupto policies, Manufacturing and Service Operations Management, Vol. 3, No. 3, pp Rosling, K. (2002) Inventory cost rate functions with nonlinear shortage costs, Operations Research, Vol. 50, No. 6, pp Silver, E.A., Pyke, D.F. and Peterson, R. (1998) Inventory Management and Production Planning and Scheduling, 3rd ed., John Wiley & Sons, New York. Svoronos, A. and Zipkin, P.H. (1991) Evaluation of oneforone replenishment policies for multiechelon inventory systems, Management Science, Vol. 37, No. 1, pp Yano, C.A. (1985) New algorithms for (Q, r) systems with complete backordering using a fillrate criterion, Naval Research Logistics Quarterly, Vol. 32, pp Zipkin, P.H. (1986) Stochastic leadtimes in continuoustime inventory models, Naval Research Logistics Quarterly, Vol. 33, pp Zipkin, P.H. (2000) Foundations of Inventory Management, McGrawHill, Boston, MA. Appendix Lemma: The pmf f U (x) as defined in Section 2 is strongly unimodal. Proof: By Theorem 3 in Keilson and Gerber (1971), a necessary and sufficient condition for the pmf of the discrete variable D U to be strongly unimodal is that it is logconcave, i.e., that 2 [ fu( x)] fu( x 1) fu( x 1), x 1,2, + = (13)
9 52 S.G. Johansen and A. Thorstenson The proof is by contradiction: Assume that condition (13) is not true. From equation (1) we then have for some x that 2 y 1 y 1 y 1 ( λr) λr ( λr) λr ( λr) λr e < e e y= x+ 1 y! y= x+ 2 y! y= x y! ( λr) ( λr) ( λr) ( λr) ( x 1)! y! y! x! y y 1 ( λr) λr ( λr) λr e < e. x+ 1 y= x+ 1 y! y= x+ 1 ( y+ 1)! x y 1 y 1 x 1 λr λr e < e + y= x+ 1 y= x+ 2 As can be inferred from its last member, condition (14) is not satisfied. Hence, the assertion is false and condition (13) must be true. (14)
Note: Optimal basestock policy for the inventory system with periodic review, backorders and sequential lead times
WORKING PAPER L200602 Søren Glud Johansen & Anders Thorstenson Note: Optimal basestock polic for the inventor sstem with periodic review, backorders and sequential lead times Note: Optimal basestock
More informationSPARE PARTS INVENTORY SYSTEMS UNDER AN INCREASING FAILURE RATE DEMAND INTERVAL DISTRIBUTION
SPARE PARS INVENORY SYSEMS UNDER AN INCREASING FAILURE RAE DEMAND INERVAL DISRIBUION Safa Saidane 1, M. Zied Babai 2, M. Salah Aguir 3, Ouajdi Korbaa 4 1 National School of Computer Sciences (unisia),
More informationIssues in inventory control models with demand and supply uncertainty Thesis proposal
Issues in inventory control models with demand and supply uncertainty Thesis proposal Abhijit B. Bendre August 8, 2008 CORAL Centre for Operations Research Applications in Logistics Dept. of Business Studies,
More informationSingle item inventory control under periodic review and a minimum order quantity
Single item inventory control under periodic review and a minimum order quantity G. P. Kiesmüller, A.G. de Kok, S. Dabia Faculty of Technology Management, Technische Universiteit Eindhoven, P.O. Box 513,
More informationExact Fill Rates for the (R, S) Inventory Control with Discrete Distributed Demands for the Backordering Case
Informatica Economică vol. 6, no. 3/22 9 Exact Fill ates for the (, S) Inventory Control with Discrete Distributed Demands for the Backordering Case Eugenia BABILONI, Ester GUIJAO, Manuel CADÓS, Sofía
More informationEvaluating the Lead Time Demand Distribution for (r, Q) Policies Under Intermittent Demand
Proceedings of the 2009 Industrial Engineering Research Conference Evaluating the Lead Time Demand Distribution for (r, Q) Policies Under Intermittent Demand Yasin Unlu, Manuel D. Rossetti Department of
More informationRiskPooling Effects of Emergency Shipping in a TwoEchelon Distribution System
Seoul Journal of Business Volume 8, Number I (June 2002) RiskPooling Effects of Emergency Shipping in a TwoEchelon Distribution System Sangwook Park* College of Business Administration Seoul National
More informationA QUEUEINGINVENTORY SYSTEM WITH DEFECTIVE ITEMS AND POISSON DEMAND. bhaji@usc.edu
A QUEUEINGINVENTORY SYSTEM WITH DEFECTIVE ITEMS AND POISSON DEMAND Rasoul Hai 1, Babak Hai 1 Industrial Engineering Department, Sharif University of Technology, +98166165708, hai@sharif.edu Industrial
More informationOptimal replenishment for a periodic review inventory system with two supply modes
European Journal of Operational Research 149 (2003) 229 244 Production, Manufacturing and Logistics Optimal replenishment for a periodic review inventory system with two supply modes Chi Chiang * www.elsevier.com/locate/dsw
More informationA Programme Implementation of Several Inventory Control Algorithms
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume, No Sofia 20 A Programme Implementation of Several Inventory Control Algorithms Vladimir Monov, Tasho Tashev Institute of Information
More informationA SingleUnit Decomposition Approach to MultiEchelon Inventory Systems
A SingleUnit Decomposition Approach to MultiEchelon Inventory Systems Alp Muharremoğlu John N. sitsiklis July 200 Revised March 2003 Former version titled: Echelon Base Stock Policies in Uncapacitated
More informationA Decomposition Approach for a Capacitated, Single Stage, ProductionInventory System
A Decomposition Approach for a Capacitated, Single Stage, ProductionInventory System Ganesh Janakiraman 1 IOMSOM Group Stern School of Business New York University 44 W. 4th Street, Room 8160 New York,
More informationOptimizing Stochastic Supply Chains via Simulation: What is an Appropriate Simulation Run Length?
Optimizing Stochastic Supply Chains via Simulation: What is an Appropriate Simulation Run Length? ArreolaRisa A 1, FortunySantos J 2, VintróSánchez C 3 Abstract The most common solution strategy for
More informationThe Exact Fill Rate in a Periodic Review Base Stock System under Normally Distributed Demand
The Exact Fill ate in a Periodic eview Base Stock System under Normally Distributed Demand Edward A. Silver, a Diane P. Bischak a a Haskayne School of Business University of Calgary 500 University Drive
More informationNumerical Studies of Singlestage, Singleitem Inventory Systems with Lost Sales
Numerical Studies of Singlestage, Singleitem Inventory Systems with Lost Sales Ph.D. Thesis by Abhijit Bhagwan Bendre CORAL Centre for Operations Research Applications in Logistics Department of Business
More informationA SingleUnit Decomposition Approach to Multiechelon Inventory Systems
OPERAIONS RESEARCH Vol. 56, No. 5, September October 2008, pp. 089 03 issn 0030364X eissn 5265463 08 5605 089 informs doi 0.287/opre.080.0620 2008 INFORMS A SingleUnit Decomposition Approach to Multiechelon
More informationOptimizing Replenishment Intervals for TwoEchelon Distribution Systems with Fixed Order Costs
Optimizing Replenishment Intervals for TwoEchelon Distribution Systems with Fixed Order Costs Kevin H. Shang Sean X. Zhou Fuqua School of Business, Duke University, Durham, North Carolina 27708, USA Systems
More informationSTRATEGIC CAPACITY PLANNING USING STOCK CONTROL MODEL
Session 6. Applications of Mathematical Methods to Logistics and Business Proceedings of the 9th International Conference Reliability and Statistics in Transportation and Communication (RelStat 09), 21
More informationINTEGRATED OPTIMIZATION OF SAFETY STOCK
INTEGRATED OPTIMIZATION OF SAFETY STOCK AND TRANSPORTATION CAPACITY Horst Tempelmeier Department of Production Management University of Cologne AlbertusMagnusPlatz D50932 Koeln, Germany http://www.spw.unikoeln.de/
More informationAnalysis of a Production/Inventory System with Multiple Retailers
Analysis of a Production/Inventory System with Multiple Retailers Ann M. Noblesse 1, Robert N. Boute 1,2, Marc R. Lambrecht 1, Benny Van Houdt 3 1 Research Center for Operations Management, University
More informationA Comparison of the Optimal Costs of Two Canonical Inventory Systems
A Comparison of the Optimal Costs of Two Canonical Inventory Systems Ganesh Janakiraman 1, Sridhar Seshadri 2, J. George Shanthikumar 3 First Version: December 2005 First Revision: July 2006 Subject Classification:
More informationWe consider a singleproduct inventory system that serves multiple demand classes, which differ in their
MANAGEMENT SCIENCE Vol. 53, No. 9, September 2007, pp. 1486 1500 issn 00251909 eissn 15265501 07 5309 1486 informs doi 10.1287/mnsc.1070.0701 2007 INFORMS A SingleProduct Inventory Model for Multiple
More informationModeling Stochastic Inventory Policy with Simulation
Modeling Stochastic Inventory Policy with Simulation 1 Modeling Stochastic Inventory Policy with Simulation János BENKŐ Department of Material Handling and Logistics, Institute of Engineering Management
More informationInventory Control with Generalized Expediting
Inventory Control with Generalized Expediting Eric Logan Huggins huggins e@fortlewis.edu School of Business Administration Fort Lewis College Durango, CO 813013999 Tava Lennon Olsen olsen@wustl.edu Olin
More informationSTOCHASTIC PERISHABLE INVENTORY CONTROL SYSTEMS IN SUPPLY CHAIN WITH PARTIAL BACKORDERS
Int. J. of Mathematical Sciences and Applications, Vol. 2, No. 2, May 212 Copyright Mind Reader Publications www.journalshub.com STOCHASTIC PERISHABLE INVENTORY CONTROL SYSTEMS IN SUPPLY CHAIN WITH PARTIAL
More informationA ContinuousReview Inventory Model with Disruptions at Both Supplier and Retailer
A ContinuousReview Inventory Model with isruptions at Both Supplier and Retailer Lian Qi epartment of Business Administration, Missouri University of Science and Technology 107I Fulton Hall, Rolla, MO
More informationFIXED CHARGE UNBALANCED TRANSPORTATION PROBLEM IN INVENTORY POOLING WITH MULTIPLE RETAILERS
FIXED CHARGE UNBALANCED TRANSPORTATION PROBLEM IN INVENTORY POOLING WITH MULTIPLE RETAILERS Ramidayu Yousuk Faculty of Engineering, Kasetsart University, Bangkok, Thailand ramidayu.y@ku.ac.th Huynh Trung
More informationCompanies often face nonstationary demand due to product life cycles and seasonality, and nonstationary
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol. 14, No. 3, Summer 2012, pp. 414 422 ISSN 15234614 (print) ISSN 15265498 (online) http://dx.doi.org/10.1287/msom.1110.0373 2012 INFORMS SingleStage
More informationInformation Sharing to Reduce Fluctuations in Supply Chains: A Dynamic Feedback Approach
Information Sharing to Reduce Fluctuations in Supply Chains: A Dynamic Feedback Approach Baris Gunduz Yaman Barlas Ford Otosan Bogazici University Ankara Asf. 4.Km Department of Industrial Engineering
More informationResearch Article Stability Analysis for HigherOrder Adjacent Derivative in Parametrized Vector Optimization
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 510838, 15 pages doi:10.1155/2010/510838 Research Article Stability Analysis for HigherOrder Adjacent Derivative
More informationMathematical Modeling of Inventory Control Systems with Lateral Transshipments
Mathematical Modeling of Inventory Control Systems with Lateral Transshipments Lina Johansson Master Thesis Department of Industrial Management and Logistics Division of Production Management Lund university,
More informationA Simple Model of Price Dispersion *
Federal Reserve Bank of Dallas Globalization and Monetary Policy Institute Working Paper No. 112 http://www.dallasfed.org/assets/documents/institute/wpapers/2012/0112.pdf A Simple Model of Price Dispersion
More informationWe consider a twoechelon inventory system with a capacitated centralized production facility and several
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol. 16, No. 4, Fall 2014, pp. 561 577 ISSN 15234614 (print) ISSN 15265498 (online) http://dx.doi.org/10.1287/msom.2014.0494 2014 INFORMS Exact Analysis
More informationNonparametric adaptive age replacement with a onecycle criterion
Nonparametric adaptive age replacement with a onecycle criterion P. CoolenSchrijner, F.P.A. Coolen Department of Mathematical Sciences University of Durham, Durham, DH1 3LE, UK email: Pauline.Schrijner@durham.ac.uk
More informationOn spare parts optimization
On spare parts optimization Krister Svanberg Optimization and Systems Theory, KTH, Stockholm, Sweden. krille@math.kth.se This manuscript deals with some mathematical optimization models for multilevel
More informationOptimal and Heuristic Echelon ( r, nq, T ) Policies in Serial Inventory Systems with Fixed Costs
OPERATIONS RESEARCH Vol. 58, No. 2, March April 2010, pp. 414 427 issn 0030364X eissn 15265463 10 5802 0414 informs doi 10.1287/opre.1090.0734 2010 INFORMS Optimal and Heuristic Echelon ( r, nq, T )
More informationResearch Article TwoPeriod Inventory Control with Manufacturing and Remanufacturing under Return Compensation Policy
Discrete Dynamics in Nature and Society Volume 2013, Article ID 871286, 8 pages http://dx.doi.org/10.1155/2013/871286 Research Article TwoPeriod Inventory Control with Manufacturing and Remanufacturing
More informationWe introduce and analyze a model that explicitly considers the timing effect of intertemporal pricing
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol. 11, No. 4, Fall 29, pp. 613 629 issn 15234614 eissn 15265498 9 114 613 informs doi 1.1287/msom.18.241 29 INFORMS Inventory, Discounts, and the Timing
More informationBinomial lattice model for stock prices
Copyright c 2007 by Karl Sigman Binomial lattice model for stock prices Here we model the price of a stock in discrete time by a Markov chain of the recursive form S n+ S n Y n+, n 0, where the {Y i }
More informationAnalysis of Various Forecasting Approaches for Linear Supply Chains based on Different Demand Data Transformations
Institute of Information Systems University of Bern Working Paper No 196 source: https://doi.org/10.7892/boris.58047 downloaded: 16.11.2015 Analysis of Various Forecasting Approaches for Linear Supply
More informationA Synchronized Supply Chain for Reducing Decoupling Stock
A Synchronized Supply Chain for Reducing Decoupling Stock Jian Wang Shanghai University, China, jwang@t.shu.edu.cn Hiroaki Matsukawa Keio University, Japan, matsukawa@ae.keio.ac.jp Shane J. Schvaneveldt
More informationA SYSTEMS APPROACH TO OPTIMIZE A MULTI ECHELON REPAIRABLE ITEM INVENTORY SYSTEM WITH MULTIPLE CLASSES OF SERVICE
A SYSTEMS APPROACH TO OPTIMIZE A MULTI ECHELON REPAIRABLE ITEM INVENTORY SYSTEM WITH MULTIPLE CLASSES OF SERVICE by Paul Lee Ewing Jr. A thesis submitted to the Faculty and the Board of Trustees of the
More informationApplying Actual Usage Inventory Management Best Practice in a Health Care Supply Chain
Applying Actual Usage Inventory Management Best Practice in a Health Care Supply Chain Vijith Varghese #1, Manuel Rossetti #2, Edward Pohl #3, Server Apras *4, Douglas Marek #5 # Department of Industrial
More informationOn the (S, 1;S) lost sales inventory model with priority. October 1997. Abstract
On the (S, 1;S) lost sales inventory model with priority demand classes R. Dekker R.M. Hill y M.J. Kleijn October 1997 Abstract In this paper an inventory model with several demand classes, prioritised
More informationCoordinated Pricing and Inventory in A System with Minimum and Maximum Production Constraints
The 7th International Symposium on Operations Research and Its Applications (ISORA 08) Lijiang, China, October 31 Novemver 3, 2008 Copyright 2008 ORSC & APORC, pp. 160 165 Coordinated Pricing and Inventory
More informationTHE CONTROL OF AN INTEGRATED PRODUCTIONINVENTORY SYSTEM WITH JOB SHOP ROUTINGS AND STOCHASTIC ARRIVAL AND PROCESSING TIMES
THE ONTROL OF AN INTEGRATED RODUTIONINVENTORY SYSTEM WITH JOB SHO ROUTINGS AND STOHASTI ARRIVAL AND ROESSING TIMES LM Van Nyen 1 * JWM Bertrand 1 HG Van Ooijen 1 NJ Vandaele 2 1 2 Technische Universiteit
More informationOnline supplement On the Integrated Production and Distribution Problem with Bidirectional Flows
Online supplement On the Integrated Production and Distribution Problem with Bidirectional Flows Lei Lei Department of Supply Chain Management and Marketing Sciences, Rutgers University, 180 University
More informationStochastic Inventory Control
Chapter 3 Stochastic Inventory Control 1 In this chapter, we consider in much greater details certain dynamic inventory control problems of the type already encountered in section 1.3. In addition to the
More informationProductionInventory Systems with Imperfect Advance Demand Information and Updating
ProductionInventory Systems with Imperfect Advance Demand Information and Updating Saif Benjaafar William L. Cooper Setareh Mardan October 25, 2010 Abstract We consider a supplier with finite production
More informationInventory Management with Auctions and Other Sales Channels: Optimality of (s, S) Policies
Inventory Management with Auctions and Other Sales Channels: Optimality of (s, S) Policies Woonghee Tim Huh, Columbia University Ganesh Janakiraman, New York University Initial Version: December 17, 2004
More informationOPTIMIZATION AND OPERATIONS RESEARCH Vol. IV  Markov Decision Processes  Ulrich Rieder
MARKOV DECISIO PROCESSES Ulrich Rieder University of Ulm, Germany Keywords: Markov decision problem, stochastic dynamic program, total reward criteria, average reward, optimal policy, optimality equation,
More informationINCREASING FORECASTING ACCURACY OF TREND DEMAND BY NONLINEAR OPTIMIZATION OF THE SMOOTHING CONSTANT
58 INCREASING FORECASTING ACCURACY OF TREND DEMAND BY NONLINEAR OPTIMIZATION OF THE SMOOTHING CONSTANT Sudipa Sarker 1 * and Mahbub Hossain 2 1 Department of Industrial and Production Engineering Bangladesh
More informationAn improved online algorithm for scheduling on two unrestrictive parallel batch processing machines
This is the PrePublished Version. An improved online algorithm for scheduling on two unrestrictive parallel batch processing machines Q.Q. Nong, T.C.E. Cheng, C.T. Ng Department of Mathematics, Ocean
More informationOptimal Control of a ProductionInventory System with both Backorders and Lost Sales
Optimal Control of a ProductionInventory System with both Backorders and Lost Sales Saif Benjaafar, 1 Mohsen ElHafsi, 2 Tingliang Huang 3 1 Industrial and Systems Engineering, University of Minnesota,
More informationIn a multiitem inventory system, such as an assembletoorder manufacturing system or
OrderBased Backorders and Their Implications in MultiItem Inventory Systems JingSheng Song Graduate School of Management, University of California, Irvine, California 9697 jssong@uci.edu In a multiitem
More informationComponent commonality has been widely recognized as a key factor in achieving product variety at low
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol. 11, No. 3, Summer 2009, pp. 493 508 issn 15234614 eissn 15265498 09 1103 0493 informs doi 10.1287/msom.1080.0235 2009 INFORMS The Value of Component
More informationINVENTORY MODELS WITH STOCK AND PRICE DEPENDENT DEMAND FOR DETERIORATING ITEMS BASED ON LIMITED SHELF SPACE
Yugoslav Journal of Operations Research Volume 0 (00), Number, 5569 0.98/YJOR00055D INVENTORY MODELS WITH STOCK AND PRICE DEPENDENT DEMAND FOR DETERIORATING ITEMS BASED ON LIMITED SHELF SPACE ChunTao
More informationPerishable Items in MultiLevel Inventory Systems
Perishable Items in MultiLevel Inventory Systems Department of Industrial Management and Logistics, LTH Master Thesis presented by Yann Bouchery Double Degree Student Ecole Centrale de Lille (France)
More informationA Simulation to Illustrate PeriodicReview Inventory Control Policies
Spreadsheets in Education (ejsie) Volume 4 Issue 2 Article 4 12212010 A Simulation to Illustrate PeriodicReview Inventory Control Policies Matthew J. Drake Duquesne University, drake987@duq.edu Kathryn
More informationOptimum decisionmaking process for an Economic Order Quantity (EOQ) model with budget and floor space constraints
American Journal of Service Science and Management 2014; 1(5): 5357 Published online January 20, 2015 (http://www.openscienceonline.com/journal/ajssm) Optimum decisionmaking process for an Economic Order
More informationA discrete time Markov chain model for a periodic inventory system with oneway substitution
Faculty of Business and Economics A discrete time Markov chain model for a periodic inventory system with oneway substitution Yannick Deflem and Inneke Van Nieuwenhuyse DEPARTMENT OF DECISION SCIENCES
More informationThe Relation between Two Present Value Formulae
James Ciecka, Gary Skoog, and Gerald Martin. 009. The Relation between Two Present Value Formulae. Journal of Legal Economics 15(): pp. 6174. The Relation between Two Present Value Formulae James E. Ciecka,
More informationCase study of a batchproduction/inventory system E.M.M. Winands 1, A.G. de Kok 2 and C. Timpe 3
Case study of a batchproduction/inventory system E.M.M. Winands 1, A.G. de Kok 2 and C. Timpe 3 The plant of BASF under consideration consists of multiple parallel production lines, which produce multiple
More informationEffective Distribution Policies Utilizing Logistics Contracting
Effective Distribution Policies Utilizing Logistics Contracting HyunSoo Ahn Osman Engin Alper Philip Kaminsky Stephen M. Ross School of Business, University of Michigan, 701 Tappan Street, Ann Arbor,
More informationAn overview of inventory systems with several demand classes Marcel J. Kleijn Rommert Dekker September 1998 Econometric Institute Report 9838/A Abstract In this chapter we discuss inventory systems where
More informationThe life cycle of new products is becoming shorter and shorter in all markets. For electronic products, life
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol. 10, No. 2, Spring 2008, pp. 278 287 issn 15234614 eissn 15265498 08 1002 0278 informs doi 10.1287/msom.1070.0175 2008 INFORMS Strategic Inventory Placement
More informationInformation Sharing in Supply Chain Management: A Literature Review on Analytical Research
Information Sharing in Supply Chain Management: A Literature Review on Analytical Research Hyuncheol Paul Choi California State University, Fullerton, CA In this paper, we reviewed the area of upstream
More informationApproximation Algorithms for Stochastic Inventory Control Models
Approximation Algorithms for Stochastic Inventory Control Models (Abstract) Retsef Levi Martin Pál Robin O. Roundy David B. Shmoys School of ORIE, Cornell University, Ithaca, NY 14853, USA DIMACS Center,
More informationRetail Inventory Management with StockOut Based Dynamic Demand Substitution
Retail Inventory Management with StockOut Based Dynamic Demand Substitution Barış Tan a, Selçuk Karabatı a a College of Administrative Sciences and Economics, Koç University, Rumeli Feneri Yolu, Sariyer,
More informationAnalysis of Load Frequency Control Performance Assessment Criteria
520 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 3, AUGUST 2001 Analysis of Load Frequency Control Performance Assessment Criteria George Gross, Fellow, IEEE and Jeong Woo Lee Abstract This paper presents
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationA LOTSIZING PROBLEM WITH TIME VARIATION IMPACT IN CLOSEDLOOP SUPPLY CHAINS
A LOTSIZING PROBLEM WITH TIME VARIATION IMPACT IN CLOSEDLOOP SUPPLY CHAINS Aya Ishigaki*, ishigaki@rs.noda.tus.ac.jp; Tokyo University of Science, Japan Tetsuo Yamada, tyamada@uec.ac.jp; The University
More informationInventory Management  A Teaching Note
Inventory Management  A Teaching Note Sundaravalli Narayanaswami W.P. No.20140901 September 2014 INDIAN INSTITUTE OF MANAGEMENT AHMEDABAD380 015 INDIA Inventory Management  A Teaching Note Sundaravalli
More informationOptimal Expediting Policies for a Serial Inventory System with. Stochastic Lead Time and Visibility of Orders In Transit
Optimal Expediting Policies for a Serial Inventory System with Stochastic Lead Time and Visibility of Orders In Transit Chiwon Kim Diego Klabjan David SimchiLevi Abstract Recent supply chains face higher
More informationChapter 2: Binomial Methods and the BlackScholes Formula
Chapter 2: Binomial Methods and the BlackScholes Formula 2.1 Binomial Trees We consider a financial market consisting of a bond B t = B(t), a stock S t = S(t), and a calloption C t = C(t), where the
More informationInventory management in distribution systems case of an Indian FMCG company
Asia Pacific Management Review (2004) 9(1), 122 Inventory management in distribution systems case of an Indian FMCG company Subrata Mitra and A. K. Chatterjee (received April 2003; revision received October
More informationTHE DYING FIBONACCI TREE. 1. Introduction. Consider a tree with two types of nodes, say A and B, and the following properties:
THE DYING FIBONACCI TREE BERNHARD GITTENBERGER 1. Introduction Consider a tree with two types of nodes, say A and B, and the following properties: 1. Let the root be of type A.. Each node of type A produces
More informationAn Entropic Order Quantity (EnOQ) Model. with Post Deterioration Cash Discounts
Int. J. Contemp. Math. Sciences, Vol. 6,, no. 9, 93939 An Entropic Order Quantity (EnOQ Model with Post Deterioration Cash Discounts M. Pattnaik Dept. of Business Administration, Utkal University Bhubaneswar754,
More informationTraditional Inventory Models in an ERetailing Setting: A TwoStage Serial System with Space Constraints
Traditional Inventory Models in an ERetailing Setting: A TwoStage Serial System with Space Constraints Russell Allgor, Stephen Graves, and Ping Josephine Xu Amazon.com MIT Astract In an eretailing setting,
More informationFundamentals of Actuarial Mathematics
Fundamentals of Actuarial Mathematics S. David Promislow York University, Toronto, Canada John Wiley & Sons, Ltd Contents Preface Notation index xiii xvii PARTI THE DETERMINISTIC MODEL 1 1 Introduction
More information10 The Impact of the Number of Warehouses on Inventories in a Distribution System
10 The Impact of the Number of Warehouses on Inventories in a Distribution System Bernhard Fleischmann, University of Augsburg, bernhard.fleischmann@wiwi.uniaugsburg.de Abstract In the strategic design
More informationPROBABILITY AND STATISTICS. Ma 527. 1. To teach a knowledge of combinatorial reasoning.
PROBABILITY AND STATISTICS Ma 527 Course Description Prefaced by a study of the foundations of probability and statistics, this course is an extension of the elements of probability and statistics introduced
More informationA DISCRETE TIME MARKOV CHAIN MODEL IN SUPERMARKETS FOR A PERIODIC INVENTORY SYSTEM WITH ONE WAY SUBSTITUTION
A DISCRETE TIME MARKOV CHAIN MODEL IN SUPERMARKETS FOR A PERIODIC INVENTORY SYSTEM WITH ONE WAY SUBSTITUTION A Class Project for MATH 768: Applied Stochastic Processes Fall 2014 By Chudamani Poudyal &
More informationYuanjie He. 2011 Associate Professor, Technology and Operations Management Department, California State Polytechnic University, Pomona
Yuanjie He 3801 W. Temple Ave, Pomona, CA 91768 Office: (909) 8692458 Technology and Operations Management Department College of Business Administration California State Polytechnic University, Pomona
More informationInventory Control in MultiChannel Retail
Inventory Control in MultiChannel Retail Frank Schneider Supply Chain Management & Management Science University of Cologne Cologne, Germany frank.schneider@unikoeln.de Diego Klabjan Industrial Engineering
More informationarxiv:1112.0829v1 [math.pr] 5 Dec 2011
How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly
More informationJoint Audit and Replenishment Decisions for an Inventory System with Unrecorded Demands
Joint Audit and Replenishment Decisions for an Inventory System with Unrecorded Demands Woonghee Tim Huh Department of Industrial Engineering and Operations Research, Columbia University, New York, NY
More informationInventory Control with Overtime and Premium Freight
Inventory Control with Overtime and Premium Freight Eric Huggins Department of Industrial and Operations Engineering University of Michigan Ann Arbor, Michigan, USA hugginse@engin.umich.edu Tava Olsen
More informationSpare parts inventory control for an aircraft component repair shop
Spare parts inventory control for an aircraft component repair shop Willem van Jaarsveld and Twan Dollevoet Erasmus School of Economics, Erasmus University Rotterdam Working Paper, Econometric Institute
More informationChapter 21: The Discounted Utility Model
Chapter 21: The Discounted Utility Model 21.1: Introduction This is an important chapter in that it introduces, and explores the implications of, an empirically relevant utility function representing intertemporal
More informationSupplement to Call Centers with Delay Information: Models and Insights
Supplement to Call Centers with Delay Information: Models and Insights Oualid Jouini 1 Zeynep Akşin 2 Yves Dallery 1 1 Laboratoire Genie Industriel, Ecole Centrale Paris, Grande Voie des Vignes, 92290
More informationA note on optimal policies for a periodic inventory system with emergency orders
Computers & Operations Research 28 (2001) 93}103 A note on optimal policies for a periodic inventory system with emergency orders Chi Chiang* Department of Management Science, National Chiao Tung University,
More informationLecture 6. Inverse of Matrix
Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that
More informationOptimization of Stochastic Inventory Control with Correlated Demands. Roger Lederman
Optimization of Stochastic Inventory Control with Correlated Demands Roger Lederman Honors Thesis in Computer Science Advisors: Amy Greenwald Aaron Cohen 1 1 Introduction and Motivation An application
More informationA MANAGER S ROADMAP GUIDE FOR LATERAL TRANSSHIPMENT IN SUPPLY CHAIN INVENTORY MANAGEMENT
A MANAGER S ROADMAP GUIDE FOR LATERAL TRANSSHIPMENT IN SUPPLY CHAIN INVENTORY MANAGEMENT By implementing the proposed five decision rules for lateral transshipment decision support, professional inventory
More informatione.g. arrival of a customer to a service station or breakdown of a component in some system.
Poisson process Events occur at random instants of time at an average rate of λ events per second. e.g. arrival of a customer to a service station or breakdown of a component in some system. Let N(t) be
More informationReused Product Pricing and Stocking Decisions in ClosedLoop Supply Chain
Reused Product Pricing and Stocking Decisions in ClosedLoop Supply Chain Yuanjie He * California State Polytechnic University, Pomona, USA Abolhassan Halati California State Polytechnic University, Pomona,
More informationPeriodic Review Inventory Management with Contingent Use of Two Freight Modes with Fixed Costs
Submitted to Operations Research manuscript Periodic Review Inventory Management with Contingent Use of Two Freight Modes with Fixed Costs Aditya Jain Indian School of Business, aditya jain@isb.edu, Harry
More informationCost performance of traditional and vendor managed inventory approaches in hospital pharmaceutical supply chains
Cost performance of traditional and vendor managed inventory approaches in hospital pharmaceutical supply chains Sineenart Krichanchai* and Bart L. MacCarthy Operations Management and Information Systems
More informationAn Extension Model of Financiallybalanced BonusMalus System
An Extension Model of Financiallybalanced BonusMalus System Other : Ratemaking, Experience Rating XIAO, Yugu Center for Applied Statistics, Renmin University of China, Beijing, 00872, P.R. China Phone:
More information