Inventory control with an order-time constraint: optimality, uniqueness and significance

Bensoussan, Alain; Moussawi-Haidar, Lama; Çakanyıldırım, M.

doi:10.1007/s10479-010-0791-1

Cited by 7 publications

(7 citation statements)

References 30 publications

(31 reference statements)

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“…3 Stochastic impulse control problems naturally arise in many areas of applications. Among these we refer to optimal control of exchange and interest rates (Cadenillas and Zapatero (1999), Mitchell et al (2014), Perera et al (2016), among others), portfolio optimization with fixed transaction costs (Korn (1999)), optimal inventory control (Bensoussan et al (2010), and Harrison et al (1983)), rational harvesting of renewable resources (Alvarez (2004)), and optimal dividend problems (Cadenillas et al (2006)).…”

Section: Introductionmentioning

confidence: 99%

On a strategic model of pollution control

Ferrari

Koch

2018

Ann Oper Res

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This paper proposes a strategic model of pollution control. A firm, representative of the productive sector of a country, aims at maximizing its profits by expanding its production. Assuming that the output of production is proportional to the level of pollutants' emissions, the firm increases the level of pollution. The government of the country aims at minimizing the social costs due to the pollution, and introduces regulatory constraints on the emissions' level, which then effectively cap the output of production. Supposing that the firm and the government face both proportional and fixed costs in order to adopt their policies, we model the previous problem as a stochastic impulse two-person nonzero-sum game. The state variable of the game is the level of the output of production which evolves as a general linearly controlled one-dimensional Itôdiffusion. Following an educated guess, we first construct a pair of candidate equilibrium policies and of corresponding equilibrium values, and we then provide a set of sufficient conditions under which they indeed realize an equilibrium. Our results are complemented by a numerical study when the (uncontrolled) output of production evolves as a geometric Brownian motion, and the firm's operating profit and the government's running cost functions are of power type. An analysis of the dependency of the equilibrium policies and values on the model parameters yields interesting new behaviors that we explain as a consequence of the strategic interaction between the firm and the government.

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Section: Introductionmentioning

confidence: 99%

On a strategic model of pollution control

Ferrari

Koch

2018

Ann Oper Res

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“…Some early work on continuous time models can be found in Bather (1966), Harrison et al (1983), Harrison (1985), Sulem (1986) and the references therein. More recent work include Fleischmann and Kuik (2003), Bensoussan et al (2005), Presman and Sethi (2006), Bensoussan et al (2010), Cadenillas et al (2010), Bensoussan (2011), Bensoussan et al (2013), Dai and Yao (2013a), and Dai and Yao (2013b), to name just a few. In Bensoussan et al (2013), a discrete inventory system is studied in which the unmet demand is lost and the excess inventory is subject to shrinkage.…”

Section: Introductionmentioning

confidence: 99%

Continuous inventory models of diffusion type: Long-term average cost criterion

Helmes¹,

Stockbridge²,

Zhu³

2017

Ann. Appl. Probab.

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This paper establishes conditions for optimality of an (s, S) ordering policy for the minimization of the long-term average cost of one-dimensional diffusion inventory models. The class of such models under consideration have general drift and diffusion coefficients and boundary points that are consistent with the notion that demand should tend to decrease the inventory level. Characterization of the cost of a general (s, S) policy as a function F of two variables naturally leads to a nonlinear optimization problem over the ordering levels s and S. Existence of an optimizing pair (s * , S * ) is established for these models. Using the minimal value F * of F , along with (s * , S * ), a function G is identified which is proven to be a solution of a quasi-variational inequality provided a simple condition holds. At this level of generality, optimality of the (s * , S * ) ordering policy is established within a large class of ordering policies such that local martingale and transversality conditions involving G hold. For specific models, optimality of an (s, S) policy in the general class of admissible policies can be established using comparison results. This most general optimality result is shown for the classical drifted Brownian motion inventory model with holding and fixed plus proportional ordering costs and for a geometric Brownian motion inventory model with fixed plus level-dependent ordering costs. However, for a drifted Brownian motion process with reflection at {0}, a new class of non-Markovian policies is introduced which have lower costs than the (s, S) policies. In addition, interpreting reflection at {0} as "justin-time" ordering, a necessary and sufficient condition is given that determines when just-in-time ordering is better than traditional (s, S) policies.MSC 2010 Classifications: 93E20, 90B05, 60H30

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“…Finally, as discussed in Section 2.2.3, the

( {s,S} )

optimality result in Bensoussan et al. (2010) with Poisson demand and order‐time constraints also holds for the average‐cost case. Moreover, Ormeci et al.…”

Section: Average Cost Criterionmentioning

confidence: 80%

“…The resulting inventory problem with these order‐time constraints is intricate, and Bensoussan et al. (2010) considered this problem with a Poisson demand process. Under linear holding and shortage costs and the standard fixed plus per‐unit ordering cost, they established the optimality of an

( {s,S} )

policy under the discounted and average cost objectives, using the QVI approach.…”

Section: Discounted Cost Criterionmentioning

confidence: 99%

A survey of stochastic inventory models with fixed costs: Optimality of (s, S) and (s, S)‐type policies—Continuous‐time case

Perera

Sethi

2023

Production and Operations Management

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Fixed costs of ordering items or setting up a production process arise in many real‐life scenarios. In their presence, the most widely used ordering policy in the stochastic inventory literature is the false(s,Sfalse)$(s,S)$ policy. Optimality of false(s,Sfalse)$(s,S)$ policies and false(s,Sfalse)$(s,S)$‐type policies have been examined for various inventory models, including those with the inventory level being reviewed in every period or continuously, finite and infinite horizons, discounted‐cost and average‐cost criteria, backlogging and lost‐sales practices, standard and generalized demands and/or costs, deterministic and stochastic lead times, single‐product and multi‐product settings, and coordinated pricing‐inventory decisions. We comprehensively survey the vast literature accumulated over seven decades in two papers. This paper is devoted to continuous‐time models, and the companion paper, also published in this journal issue, reviews discrete‐time models. We go over model specifications, proof techniques, specific results, and limitations of the articles published in the literature. We conclude each paper by providing corresponding suggestions for extensions and directions for future research.

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Inventory control with an order-time constraint: optimality, uniqueness and significance

Cited by 7 publications

References 30 publications

On a strategic model of pollution control

On a strategic model of pollution control

Continuous inventory models of diffusion type: Long-term average cost criterion

A survey of stochastic inventory models with fixed costs: Optimality of (s, S) and (s, S)‐type policies—Continuous‐time case

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