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Average Cost Optimality in Inventory Models with Markovian Demands: A Summary
 Journal of Optimization Theory and Applications
, 1997
"... This paper is concerned with longrun average cost minimization of a stochastic inventory problem with Markovian demand, fixed ordering cost, and convex surplus cost. The states of the Markov chain represent different possible states of the environment. Using a vanishing discount approach, a dyna ..."
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This paper is concerned with longrun average cost minimization of a stochastic inventory problem with Markovian demand, fixed ordering cost, and convex surplus cost. The states of the Markov chain represent different possible states of the environment. Using a vanishing discount approach, a dynamic programming equation and the corresponding verification theorem are established. Finally, the existence of an optimal statedependent (s; S) policy is proved. Key Words: Dynamic inventory model, Markov chain, dynamic programming, infinite horizon, average cost, ergodic cost, (s,S) policy 1 Introduction This paper studies a stochastic inventory problem with Markovian demand and fixed cost from the viewpoint of minimizing the longrun average cost of inventory/backlog and ordering. The purpose is to establish the dynamic programming equation or average cost optimality equation for the problem, prove the existence of an optimal feedback (or Markov) policy, and show that a feedback polic...
Multicriteria ranking of inventory replenishment policies in the presence of uncertainty in customer demand
 International Journal of Production Economics
, 1996
"... presence of uncertainty in customer demand ..."
Inventory control with generalized expediting. Operations Research, forthcoming
, 2009
"... Abstract We consider a singleitem, periodic review inventory control problem where discrete stochastic demand must be satisfied. When shortages occur, the unmet demand must be filled by some form of expediting. We allow a very general form for the cost structure of expediting. This expediting migh ..."
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Cited by 4 (1 self)
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Abstract We consider a singleitem, periodic review inventory control problem where discrete stochastic demand must be satisfied. When shortages occur, the unmet demand must be filled by some form of expediting. We allow a very general form for the cost structure of expediting. This expediting might include inhouse rush production, outsourcing, or even lost sales. However, we explicitly consider the case where expedited production is allowed to produce up to a positive inventory level. For the infinite horizon discounted problem, we characterize the structure of the optimal expediting policy and show that an (s, S) policy is optimal for regular production. In certain cases we demonstrate that it may indeed be optimal to use expedited production to build up inventory. A heuristic for policy calculation is given; a numerical study tests the heuristic and add insight into the results.
A simple and robust batchordering inventory policy for unobserveable demand. Working paper
, 2007
"... Generally, the derivation of an inventory policy requires the knowledge of the underlying demand distribution. Unfortunately, in many settings such as retail, demand is not completely observable in a direct way or inventory records may be inaccurate. A variety of factors, including the potential in ..."
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Cited by 2 (2 self)
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Generally, the derivation of an inventory policy requires the knowledge of the underlying demand distribution. Unfortunately, in many settings such as retail, demand is not completely observable in a direct way or inventory records may be inaccurate. A variety of factors, including the potential inaccuracy of inventory records, motivate retailers to seek replenishment policies with a fixed order quantity. We derive estimators of the first two moments of the (periodic) demand by means of renewal theoretical concepts. We then propose a regressionbased approximation to improve the quality of the estimators. These estimators are used in conjunction with the Power Approximation (PA) method of Ehrhardt and Mosier (1984) to obtain an (r,Q) replenishment policy. The proposed methodology is robust and easy to code into a spreadsheet application. A series of numerical studies are carried out to evaluate the accuracy and precision of the estimators, and to investigate the impact of the estimation on the optimality of the inventory policies. Our experiments indicate that the proposed (r,Q) policy is very close, with regard to the mean total cost per period, to the (s, S) policy obtained via the PA method when the demand process is fully observable.
ASAC 2008 Halifax, Nova Scotia
"... VMI (Vendor Managed Inventory) system is known as an effective mechanism to reduce the bullwhip effect between suppliers and buyers, by switching the responsibility of inventory management from buyers to suppliers. Although most attempts of VMI implementation have been successful, there were some fa ..."
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VMI (Vendor Managed Inventory) system is known as an effective mechanism to reduce the bullwhip effect between suppliers and buyers, by switching the responsibility of inventory management from buyers to suppliers. Although most attempts of VMI implementation have been successful, there were some failure cases such as in South Korea. VMI system was beneficial only to the buyer and hurt the longrun relationship in the supply chain. This paper discusses the possible factors to result in the unsatisfactory VMI implementation.
This research is partially supported by NSERC grant OGP0005527
"... This paper reports results of a study of inventory policies for Canadian Tire Pacific Associates which operates 21 retail stores and a warehouse in the vicinty of Vancouver, British Columbia and maintains stock of over 30,000 products. We describe a periodic review, fixed lead time, singleproduct, s ..."
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This paper reports results of a study of inventory policies for Canadian Tire Pacific Associates which operates 21 retail stores and a warehouse in the vicinty of Vancouver, British Columbia and maintains stock of over 30,000 products. We describe a periodic review, fixed lead time, singleproduct, singlefacility model with random demand, lost sales and service constraints which we developed for potential application. The model utilizes empirical demand data to calculate the long run average cost of inventory and service level for a given (s,S) policy. We provide a search algorithm to quickly locate an optimal policy based on an updating scheme for the transition probability matrix of the underlying Markov chain, bounds on S and monotonicity assumptions on the cost and service level functions . We compare the computed policies to those currently in use on a test bed of 420 products and find that stores currently hold inventories which are 40% to 50% higher than those determined by our model. We estimate that implementing the proposed policies for the entire system would result in annual savings of between $5.5 and $7 million.
The Classical Papers of Iglehart (1963) and Veinott and Wagner (1965) Revisited
 Journal of Optimization and Applications
, 1999
"... This paper revisits the classical papers of Iglehart (1963) and Veinott and Wagner (1965) devoted to stochastic inventory problems with the criterion of longrun average cost minimization. We indicate some of the assumptions that are implicitly used without verification in their stationary distribut ..."
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This paper revisits the classical papers of Iglehart (1963) and Veinott and Wagner (1965) devoted to stochastic inventory problems with the criterion of longrun average cost minimization. We indicate some of the assumptions that are implicitly used without verification in their stationary distribution approach to the problems, and provide the missing (nontrivial) verification. In addition to completing their analysis, we examine the relationship between the stationary distribution approach and the dynamic programming approach to the averagecost stochastic inventory problems. This research was supported in part by a Feodor Lynengrant provided by the A. von Humboldt Foundation and NSERC Grant A4619. Thanks are due to Youhua (Frank) Chen, HansJoachim Girlich, Arie Harel and Dmitry Krass for helpful comments. y Technical Staff Member, HewlettPackard Laboratories, Palo Alto, California. z Professor, School of Management,The University of Texas at Dallas, Richardson, Texas. 1 1 ...
Simulation for the Optimization of (s, S) Inventory System and KrushKuhnTucker Testing
, 2007
"... We consider the simulation of constrained optimization problem, the (s, S) inventory system with stochastic lead time and a service level constraint. We allow the orders to cross in time which makes the problem more complicated. We establish the (s, S) inventory model by using Arean and find the es ..."
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We consider the simulation of constrained optimization problem, the (s, S) inventory system with stochastic lead time and a service level constraint. We allow the orders to cross in time which makes the problem more complicated. We establish the (s, S) inventory model by using Arean and find the estimators by OptQuest. We try to solve several issues: 1) what the true optimal values of (s, S) are in this specified conditions; 2) whether the OptQuest can find the optimal values; 3) how we can prove the outcomes are the estimators of true optimal values. In our conclusion, we give the true optimal estimator of (s*, S*) pairs estimated by Brute Force. Further, we prove that OptQuest can be used in solving the stochastic constrained optimization problem effectively. By testing estimators’ KKT conditions under the method of Bettonvil [16], we prove that we can find the good estimator of (s*, S*).
“Evaluation and optimization of innovative production systems of goods and services” ANALYSIS OF ORDERUPTOLEVEL INVENTORY SYSTEMS WITH COMPOUND POISSON DEMAND
"... ABSTRACT: We analyse a single echelon single item inventory system where the demand and the lead time are stochastic. Demand is modelled as a compound Poisson process and the stock is controlled according to a continuous time orderupto level policy. We propose a new method for determining the opti ..."
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ABSTRACT: We analyse a single echelon single item inventory system where the demand and the lead time are stochastic. Demand is modelled as a compound Poisson process and the stock is controlled according to a continuous time orderupto level policy. We propose a new method for determining the optimal orderupto level for a cost oriented inventory systems where unfilled demands are backordered. The conditions under which the system behaves like a MakeToOrder setting are also discussed. By means of a numerical investigation, we show that the proposed method provides very good results. It is also shown to outperform another approximate solution provided in the literature. Our work allows insights to be gained on stock control related issues for both fast and slow moving Stock Keeping Units.