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Dynamic control in multi-item production/inventory systems


We consider a production/inventory system consisting of one production line and multiple products. Finished goods are kept in stock to serve stochastic demand. Demand is fulfilled immediately if there is an item of the requested product in stock and otherwise it is backordered and fulfilled later. The production line is modeled as a non-preemptive single server and the objective is to minimize the sum of the average inventory holding costs and backordering costs. We investigate the structure of the optimal production policy, propose a new scheduling policy, and develop a method for calculating base stock levels under an arbitrary but given scheduling policy. The performance of the various production policies is evaluated in extensive numerical experiments.

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  • Adan I, Sleptchenko A, Van Houtum G (2009) Reducing costs of spare parts supply systems via static priorities. Asia-Pacific J Oper Res 26(04):559–585

    Article  Google Scholar 

  • Arreola-Risa A, Giménez-García V, Martínez-Parra J (2011) Optimizing stochastic production-inventory systems: a heuristic based on simulation and regression analysis. Eur J Oper Res 213(1):107–118

    Article  Google Scholar 

  • Bertsekas D (2007) Dynamic programming and optimal control, 3rd edn, vol. II. Athena Scientific, USA

  • De Vericourt F, Karaesmen F, Dallery Y (2000) Dynamic scheduling in a make-to-stock system: a partial characterization of optimal policies. Oper Res 48(5):811–819

    Article  Google Scholar 

  • DeCroix G, Arreola-Risa A (1998) Optimal production and inventory policy for multiple products under resource constraints. Manag Sci 44(7):950–961

    Article  Google Scholar 

  • Dusonchet F (2003) Dynamic scheduling for production systems operating in a random environment. PhD thesis, École Polytechnique Fédérale de Lausanne

  • Gavish B, Graves S (1980) A one-product production/inventory problem under continuous review policy. Oper Res 28(5):1228–1236

    Article  Google Scholar 

  • Ha A (1997) Optimal dynamic scheduling policy for a make-to-stock production system. Oper Res 45(1):42–53

    Article  Google Scholar 

  • Janakiraman G, Nagarajan M, Veeraraghavan S (2009) Simple policies for managing flexible capacity. Working paper

  • Janssen R, Blankers I, Moolenburgh E, Posthumus B (2014) The impact of 3-d printing on supply chain management. Tech. rep, TNO

  • Kat B, Avşar Z (2011) Using aggregate fill rate for dynamic scheduling of multi-class systems. Ann Oper Res 182(1):87–117

    Article  Google Scholar 

  • Khajavi S, Partanen J, Hölmstrom J (2014) Additive manufacturing in the spare parts supply chain. Comput Ind 65(1):50–63

    Article  Google Scholar 

  • Li L (1992) The role of inventory in delivery-time competition. Manag Sci 38(2):182–197

    Article  Google Scholar 

  • Liang W, Balcıoğlu B, Svaluto R (2013) Scheduling policies for a repair shop problem. Ann Oper Res 211(1):273–288

    Article  Google Scholar 

  • Lieckens KT, Colen PJ, Lambrecht MR (2013) Optimization of a stochastic remanufacturing network with an exchange option. Dec Supp Syst 54(4):1548–1557

    Article  Google Scholar 

  • Linebaugh K (2008) Honda’s flexible plants provide edge. Wall Street J 22:B1

  • Liu P, Huang SH, Mokasdar A, Zhou H, Hou L (2014) The impact of additive manufacturing in the aircraft spare parts supply chain: supply chain operation reference (scor) model based analysis. Prod Plann Control 25(13–14):1169–1181

    Article  Google Scholar 

  • Niño-Mora J (2007) Dynamic priority allocation via restless bandit marginal productivity indices. Top 15(2):161–198

    Article  Google Scholar 

  • Peña Perez A, Zipkin P (1997) Dynamic scheduling rules for a multiproduct make-to-stock queue. Oper Res 45(6):919–930

    Article  Google Scholar 

  • Porteus EL (2002) Foundations of stochastic inventory theory. Stanford University Press, Palo Alto

  • Sanajian N, Abouee-Mehrizi H, Balcıoğlu B (2010) Scheduling policies in the M/G/1 make-to-stock queue. J Oper Res Soc 61(1):115–123

    Article  Google Scholar 

  • Sobel M (1982) The optimality of full service policies. Oper Res 30(4):636–649

    Article  Google Scholar 

  • Steiger N, Wilson J (2001) Convergence properties of the batch means method for simulation output analysis. INFORMS J Comput 13(4):277–293

    Article  Google Scholar 

  • Van Houtum G, Adan I, Van der Wal J (1997) The symmetric longest queue system. Stoch Models 13(1):105–120

    Article  Google Scholar 

  • Veatch M, Wein L (1996) Scheduling a make-to-stock queue: index policies and hedging points. Oper Res 44(4):634–647

    Article  Google Scholar 

  • Wein L (1992) Dynamic scheduling of a multiclass make-to-stock queue. Oper Res 40(4):724–735

    Article  Google Scholar 

  • Whittle P (1988) Restless bandits: activity allocation in a changing world. J Appl Probab 25:287–298

    Article  Google Scholar 

  • Zheng Y, Zipkin P (1990) A queueing model to analyze the value of centralized inventory information. Oper Res 38(2):296–307

    Article  Google Scholar 

  • Zipkin P (1995) Performance analysis of a multi-item production-inventory system under alternative policies. Manag Sci 41(4):690–703

    Article  Google Scholar 

Download references


We kindly thank the associate editor and two anonymous referees for their helpful suggestions to improve the paper.

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Correspondence to H. G. H. Tiemessen.

Calculation of E(P|w)

Calculation of E(P|w)

Let P represent an exponentially distributed time variable with mean \( 1/\mu \). Consider a queuing system where customers arrive according to a Poisson distribution with constant rate \( \lambda \). In this appendix, we show that E(P|w) , the expectation of a time interval given that during this time interval w customer arrivals occur, is equal to \( (w+1)/(\lambda + \mu ) \). Using the definition of conditional expectation and the property that the number of Poisson arrivals during an exponentially distributed time interval with mean \( 1/\mu \) has a geometric distribution with success probability \( \frac{1}{1 + \frac{\lambda }{\mu }} \), we obtain the following expression for E(P|w) :

$$\begin{aligned} E(P|w)&= \frac{\int \limits _{0}^{\infty } t \quad \mu \mathrm {e}^{-\mu \, t} \quad \frac{(\lambda \, t)^w}{w!} \quad \mathrm {e}^{-\lambda \, t} \quad \mathrm {d}t}{(1 - \frac{1}{1 + \frac{\lambda }{\mu }})^w \quad (\frac{1}{1 + \frac{\lambda }{\mu }})} \end{aligned}$$

Rearranging terms, and using that \( \frac{(\lambda + \mu )^{w+2} \quad t^{w+1}}{(w+1)!} \quad \mathrm {e}^{-(\lambda + \mu ) \, t} \) is the probability density function of the Erlang distribution with parameters \( (w + 2) \) and \( (\lambda + \mu ) \), we can rewrite expression (13a) as follows:

$$\begin{aligned} E(P|w)= & {} \frac{\frac{\mu \, \lambda ^w \, (w+1)}{(\lambda + \mu )^{w+2}} \int \limits _{0}^{\infty } \frac{(\lambda + \mu )^{w+2} \quad t^{w+1}}{(w+1)!} \quad \mathrm {e}^{-(\lambda + \mu ) \, t}}{\left( 1 - \frac{1}{1 + \frac{\lambda }{\mu }}\right) ^w \quad \left( \frac{1}{1 + \frac{\lambda }{\mu }}\right) } \end{aligned}$$
$$\begin{aligned}= & {} \frac{\frac{\mu \, \lambda ^w \, (w+1)}{(\lambda + \mu )^{w+2}} }{\left( 1 - \frac{1}{1 + \frac{\lambda }{\mu }}\right) ^w \quad \left( \frac{1}{1 + \frac{\lambda }{\mu }}\right) } \end{aligned}$$

Finally, we obtain the desired result \( E(P|w) = (w+1)/(\lambda + \mu ) \) from (13c) via straightforward algebraic manipulations.

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Tiemessen, H.G.H., Fleischmann, M. & van Houtum, G.J. Dynamic control in multi-item production/inventory systems. OR Spectrum 39, 165–191 (2017).

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  • Inventory control
  • Dynamic scheduling
  • Simulation optimization