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This paper has a double purpose. First, it provides a historical analysis of the developments leading to the first “Stanford Studies”, a collection of papers which probably can be considered even today as the single most fundamental reference in the mathematical handling of inventories. From this point of view, the paper is about people and their works. Secondly, we give an insight into the interrelation of mathematics and inventory modelling in a historical context. We will sketch, on the one hand, the development of the mathematical tools for inventory modelling first of all in statistics, probability theory and stochastic processes but also in game theory and dynamic programming up to the 1950s. Furthermore, we report how inventory problems have motivated the improvement of mathematical disciplines such as Markovian decision theory and optimal control of stochastic systems to provide a new basis of inventory theory in the second half of our century.

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... The parallel development of these fields since the beginning of the second half of the 20th century is broadly recognized. For example, the abstract of the historical essay by Girlich and Chikan [37] on the history of inventory control studies states: "... we report how inventory problems have motivated the improvement of mathematical disciplines such as Markovian decision theory and optimal control of stochastic systems to provide a new basis of inventory theory in the second half of our century." However, over a long period of time there was a gap between the modeling needs for inventory control, that require mathematical methods for the analysis of infinite-state controlled stochastic systems with unbounded action sets and weakly continuous transition probabilities, and available results for the corresponding models for MDPs. ...

... In words, u(x) is the largest number such that u(x) ≤ lim inf n→∞ u αn (y n ) for all sequences {y n → x} and {α n → 1−}. (11) with u defined in (37). Thus, equalities (12) hold for this policy φ. ...

... If the cost function c is inf-compact, then the functions v α , u, andũ are inf-compact as well; see Theorem 5.1 for the proof of this fact for v α and Feinberg et al. [27,Theorem 4(e) and Corollary 2] for u andũ. We denote by A * u (x) the sets defined in (37), when the function u is replaced withũ. In addition, if the one-step cost function c is inf-compact, the minima of the functions v α possess additional properties. ...

This tutorial describes recently developed general optimality conditions for Markov Decision Processes that have significant applications to inventory control. In particular, these conditions imply the validity of optimality equations and inequalities. They also imply the convergence of value iteration algorithms. For total discounted-cost problems only two mild conditions on the continuity of transition probabilities and lower semi-continuity of one-step costs are needed. For average-cost problems, a single additional assumption on the finiteness of relative values is required. The general results are applied to periodic-review inventory control problems with discounted and average-cost criteria without any assumptions on demand distributions. The case of partially observable states is also discussed.

... De acuerdo con Girlich y Chikán [6], el desarrollo conjunto de las teorías de inventarios y la aplicación de las matemáticas y la estadística se inició desde los años 50 cuando la Oficina de Investigación Naval de California destinaron recursos para la investigación en el área. Desde ese entonces, la diversidad de trabajos de investigación y extensión que se ha desarrollado es amplia, mediante la utilización de herramientas técnicas, clásicas y modernas. ...

... Otra de las formas de tratamiento de la aleatoriedad de la demanda es la modelación estocástica, la cual, a pesar de no ser muy conocida en el medio industrial nacional, ha sido utilizada como técnica desde la década de los veinte [6,15]. Entre los trabajos más recientes, Chen [16] analiza un modelo de revisión periódico y horizonte infinito para un producto, en el cual se toman decisiones de fijación de precios y de producción e inventarios simultáneamente. ...

In this paper, we review inventory management models for designing inventory policies for final products and raw materials in supply chains, considering random demand and lead times. (1) Random Demand Models, (2) Random Lead Times Models, (3) Inventory Policy Models, and (4) Integrated Inventory Management Model. For each section we present summary tables describing the main characteristcs of models found in the literature. Special emphasis is placed on the lack of methodologies for modeling random issues of the system. Research and development opportunities in the context of the Colombian industry are also identified.

... Desde la práctica, el problema se ha abordado mediante la implementación de software y de sistemas de información [1]. Desde la investigación, la toma de decisiones se ha apoyado en técnicas de las matemáticas y la estadística [2] y de la investigación de operaciones [3], [4] Sin embargo, existe una creciente brecha entre la realidad de las empresas y los avances logrados en la investigación. Mientras que el ambiente en el cual los empresarios deben tomar decisiones es complejo [5], los desarrollos investigativos parten de supuestos reduccionistas en búsqueda de soluciones plausibles al problema. ...

... Dado que se buscó indagar por los métodos para gestionar inventarios de productos terminados, producto en proceso y materias primas, fue necesario seleccionar de la población las empresas cuya actividad económica involucrara la producción y distribución de bienes. Del mismo modo, se seleccionaron las empresas de los siguientes sectores: (1) alimentos para el ser humano, (2) fármacos, (3) plásticos, (4) textiles. ...

The growing gap between inventory-control research and practice evidences the need to create approaching mechanisms, so alternatives for improvement could be offered from research, in order to manage the complexity of inventories in supply chains. This paper presents a diagnostic of the inventory management of final products, work in process and raw materials, developed with the medium companies of the Valle of Aburrá, Colombia, in the foods, pharmaceutical, plastic and apparel sectors. Results evidence opportunities of improvement and confirm the need of taking the engineering methodologies to a more realistic implementation, that allows to involve the dynamics of the regional supply chains.

... De acuerdo con Girlich y Chikán [6], el desarrollo conjunto de las teorías de inventarios y la aplicación de las matemáticas y la estadística se inició desde los años 50 cuando la Oficina de Investigación Naval de California destinaron recursos para la investigación en el área. Desde ese entonces, la diversidad de trabajos de investigación y extensión que se ha desarrollado es amplia, mediante la utilización de herramientas técnicas, clásicas y modernas. ...

... Otra de las formas de tratamiento de la aleatoriedad de la demanda es la modelación estocástica, la cual, a pesar de no ser muy conocida en el medio industrial nacional, ha sido utilizada como técnica desde la década de los veinte [6,15]. Entre los trabajos más recientes, Chen [16] analiza un modelo de revisión periódico y horizonte infinito para un producto, en el cual se toman decisiones de fijación de precios y de producción e inventarios simultáneamente. ...

... MSC 2010 Classifications: 93E20, 90B05, 60H30 Stochastic control motivated by inventory models has a long history and is a common theme in the literature. Early work on discrete time models may be found in the Stanford Studies (see, e.g., Arrow et al. (1958)); for an excellent survey, see Girlich and Chikán (2001). 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. ...

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 " just-in-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

... There is a significant amount of work in the literature on characterizing the optimal inventory control policy under numerous settings. We refer the reader to Girlich and Chikán (2001), which provides a historical analysis of development of the mathematical tools for inventory modeling such as the ones in probability theory, stochastic processes, game theory and dynamic programming. ...

In this study, we investigate a single-item, periodic-review inventory problem where the production capacity is limited and unmet demand is backordered. We assume that customer demand in each period is a stationary, discrete random variable. Linear holding and backorder cost are charged per unit at the end of a period. In addition to the variable cost charged per unit ordered, a positive fixed ordering cost is incurred with each order given. The optimization criterion is the minimization of the expected cost per period over a planning horizon. We investigate the infinite horizon problem by modeling the problem as a discrete-time Markov chain. We propose a heuristic for the problem based on a particular solution of this stationary model, and conduct a computational study on a set of instances, providing insight on the performance of the heuristic.

... In other words, a supply chain is a network of retailers, distributors, transporters, storage facilities and suppliers that take part in the sale, delivery and production of a particular product that satisfies a specific need. The stock-level modelling could be considered based on the 'Stanford studies' (Chikan and Girlich, 1999). Complementing this position, Guardiola et al. (2007) study the coordination of actions and the allocation of profit in supply chains under decentralised control, in which a single supplier supplies several retailers with goods for replenishment of stocks. ...

Fictitious Play (FP) concept, based on Game Theory, can be used for developing models to characterise conditions in supply chain. The focus of this paper is on using an FP algorithm as a learning methodology to analyse performance of a two-echelon supply chain, highlighting one-echelon retailers' processes for making decisions about replenishments. FP methodology has not been applied well to supply chain conditions. This paper provides an approach using FP to analyse retailers' two-echelon supply chain problems to create equilibrium policies to improve systems' performance.

... The mathematical modeling of inventory has a long history. For a wide-ranging description of its early formation centered around the Stanford Studies see Girlich and Chikán (2001); in particular, this survey points out the early work of Dvoretzky et al. (1953) and Scarf (1959) and others which address the optimality of (s, S) policies for their discrete models. Harrison et al. (1983) considers a drifted Brownian motion inventory model but restricted the inventory process to non-negative states and proves optimality of an (0, q, Q, S) 1 policy; unlike ours, their model allows for an instantaneous reduction of inventory. ...

This paper develops a new approach to the solution of impulse control problems for continuous inventory models under a discounted cost criterion. The analysis imbeds this stochastic problem in two different infinite-dimensional linear programs, parametrized by the initial inventory level $x_0$, by concentrating on particular functions and capturing the expected (discounted) behavior of the inventory level process and ordering decisions as measures. The first imbedding then naturally leads to the minimization of a nonlinear function representing the cost associated with an $(s,S)$ ordering policy and an optimizing pair determines optimal levels $(s^*,S^*)$. The lower bound arising from this imbedding is tight when $x_0 \geq s^*$ but is a strict lower bound when $x_0 < s^*$. Solving the first linear program determines the value function in the ``no order'' region and is critical to the formulation of the second linear program. The dual of the second linear program is then solved to provide a tight lower bound for all $x_0$, in particular for $x_0 < s^*$, and thereby completely determines the value function. Existence of an optimal $(s,S)$ policy in the admissible class of ordering policies (and its characterization) is a consequence of the method, not an {\em a priori}\/ assumption. Also of note in this approach is that it solves piecewise in two regions the family of linear programs parametrized by $x_0$. No smoothness of the value function is required; instead, the level of smoothness results from its construction using the particular functions from which the linear programs are derived. This paper places minimal assumptions on a general stochastic differential equation model for the inventory level and illustrates the approach on two examples.

... The main references are Whitin (1957), Arrow et al. ( 1958) and Scarf et al. (1963). (See Girlich and Chikan 2001, for an analysis of developments in this era.) ...

The significance of inventories in business operations have never been denied. The actual role of inventories, however, is changing over time, as required by the business environment. This paper provides empirical background to the thesis, which says that the role of inventories in the "Golden Era" of inventory research, which was in the 1950s, was significantly different from that of today because of fundamental changes in business. This development requires new approaches in research as well. After a summary of the antecedents, the results of a survey are analysed, and they support the above thesis. The lack of difference between the inventory performance measured by the turnover rate of those companies, whose managers accept and those who deny the birth of the new paradigm calls attention to the need for the elaboration of a more complex inventory performance measurement.

... The effect of intra-supply chain competition on supply chain performance in general and inventory policies in particular is well studied for a static framework. Extensive reviews include a historical analysis of Girlich and Chiká n (2001) in the development of mathematical approaches to inventories management using classical optimization and game theory; discussions on integrated inventory models (Goyal and Gupta, 1989); game theory in supply chains (Cachon and Netessine, 2004); competition and coordination (Leg, Parlar, 2005;Banerjee et al., 2007;Lee and Byong-Duk, 2010). Numerous papers are devoted to the effect of competition on supply chains. ...

... There is a significant amount of work in the literature on characterizing the optimal inventory control policy under numerous settings. We refer the reader to Girlich and Chikán (2001) , which provides a historical analysis of development of the mathematical tools for inventory modeling such as the ones in probability theory, stochastic processes, game theory and dynamic programming. ...

Keywords: Capacity constraint, stochastic demand, fixed ordering cost, All-or-Nothing policy. Thesis (M.S.)--Middle East Technical University, 2003.

This part is an introduction of the concept and role of inventories in the economy. We explain why holding inventories is a necessary component of economic activity and why item-level, firm-level and national-level inventories present themselves as natural focuses of analysis. Reasons of holding inventories on these three levels are discussed. Item-level inventories are hold to meet specific demand components, firm-level inventories are subject to company management as an important contribution to smooth and cost-effective operation, while national inventories are aggregates of lower-level inventories influenced by the structure of the economy and economic policy factors. Our attention is focused in this book on national inventories, three characteristics of which can be analysed: level, change and fluctuation. We provide reasons why we find analysing long-term trends of national inventories as a useful contribution to our general knowledge of operation of the economies.

This paper considers the evolution of the mathematical study of management problems in the military, in industrial production systems and in bureaucracy, through the analysis of the cultural sources of operations research. OR is sometimes considered a branch of applied mathematics, or a synonymous of management science or of industrial engineering. In a few decades of impetous growth, OR has tried to maintain a balance between the expectations (a rational, ready answer to very complex problems in practice) and the systematic development of a theoretical corpus; and OR practitioners have often criticized the OR "ecosystem" and the current research trends. Historical studies of OR have considered several partial aspects, such as national contexts (English, American, French, Russian), mathematical techniques, application contexts (main characters and institutions in the military, in industry). History of mathematics offers a general cultural framework in which OR can be better understood as a cultural project that shares with biomathematics and mathematical economics a common root in the Enlightenment views of a mathematical rationality that can be found in natural phenomena or can be used to control human and social behaviour; as a cultural project typical of the 20 th century as the age of a mathematical "systems approach"; as a cultural project that was stimulated by the development of mathematical modelling and by the new ideas of the early 20 th century mathematics.

The study aims to improve the current level of service to the customer (75 %) which breached the target set of 95 %. Using the scientific method, the investigation was based on a diagnosis and it was detected the lack of an inventory policy, among the main causes of failure. The demand for the product was studied and various forecasting methods were evaluated according to its performance. It was proposed a system of periodic review RS, which was considered the most appropriate, since it provides greater flexibility in its initial implementation process and monitoring, in addition to be favorable in terms of times and costs. The pilot implementation of the model (6 weeks), covered more effectively the demand of the product, which increased the level of service up to 87 %, improved earnings at $ 675,458.08 and allowed affirm the relevance of the proposal.

This paper analyses some of the connections between macroeconomic inventory data and other GDP characteristics. Until now this issue has got relatively low interest, though we believe it can be important in understanding macroeconomic phenomena.After a brief summary of previous research, six hypotheses are formulated regarding tendencies of the ratio of inventory investment to GDP and its relationship with the level of development, growth and fluctuations of GDP. Elementary statistical methods are applied on an OECD database, containing 14 of the most developed economies of the world. The analysis mostly supports the hypotheses, however—due to the limitations by the quality of the database and the methodology used—this support is not very reliable. This calls attention to the need for further research on the subject.

This paper is an attempt to formulate a new paradigm for inventory research, based on recent developments in business and the global economy. I begin with an analysis of the traditional paradigm as developed in the 1950s and its relationship to the economic and business environment of the time. Then, changes in this environment which have led to the characteristics of today's economy and business are described. Considering these characteristics, I derive a new inventory research paradigm. Several research areas—some leading to new directions—are identified on the basis of the new paradigm.

Optimal inventory policy is first derived for a simple model in which the future (and constant) demand flow and other relevant quantities are known in advance. This is followed by the study of uncertainty models — a static and a dynamic one — in which the demand flow is a random variable with a known probability distribution. The best maximum stock and the best reordering point are determined as functions of the demand distribution, the cost of making an order, and the penalty of stock depletion.

We analyse the optimal ordering policy for impulse control of a one-product inventory system subject to a demand modelised by a diffusion process. The purpose is to minimize the expected discounted cost that includes a fixed set-up cost and linear costs of purchase, storage and shortage. The optimal cost is explicitly obtained as the smoothest solution of a Quasi-Variational Inequality derived from the optimal principle of Dynamic Programming. The optimal s, S policy is determined as the unique solution of a system of algebraic equations.

We consider a discrete review, single product, dynamic inventory model. New conditions are found for the optimality of an $(s,S)$-policy which generalize those of Scarf (1960) and Veinott (1966). Moreover, we obtain as a special case a result very similar to that of Porteus (1971) without any assumption on the probability distribution of demand. The analysis is based on a general concept of convexity which includes convex, functions in the usual sense, and monotone functions as special cases. The paper builds on ideas of Veinott.

A standard periodic review, stochastic, dynamic multiproduct inventory model is considered in which the ordering cost consists of linear portions for each product and a setup cost. This setup cost is incurred if an order is placed for any number of products. There is no separate setup cost incurred for each product ordered. The author shows that there exists an optimal ( sigma ,S) policy. Such a policy does not order when the initial stock level is in sigma and orders up to the vector level S otherwise (provided that such an order is feasible). Furthermore, the set sigma is an upper layer (equivalently, an increasing set) with respect to a certain partial ordering. Such a policy reduces to an (s,S) policy in the single product case, in which case the conditions are very similar to those of Veinott and amount to a special case of the model of Schael.

"This is the classic work upon which modern-day game theory is based. What began more than sixty years ago as a modest proposal that a mathematician and an economist write a short paper together blossomed, in 1944, when Princeton University Press published Theory of Games and Economic Behavior. In it, John von Neumann and Oskar Morgenstern conceived a groundbreaking mathematical theory of economic and social organization, based on a theory of games of strategy. Not only would this revolutionize economics, but the entirely new field of scientific inquiry it yielded--game theory--has since been widely used to analyze a host of real-world phenomena from arms races to optimal policy choices of presidential candidates, from vaccination policy to major league baseball salary negotiations. And it is today established throughout both the social sciences and a wide range of other sciences. This sixtieth anniversary edition includes not only the original text but also an introduction by Harold Kuhn, an afterword by Ariel Rubinstein, and reviews and articles on the book that appeared at the time of its original publication in the New York Times, tthe American Economic Review, and a variety of other publications. Together, these writings provide readers a matchless opportunity to more fully appreciate a work whose influence will yet resound for generations to come.

This paper discusses the structure and formulation of dynamic programing models. R. Bellman's “Principle of Optimality” is reexamined for deterministic models. A formulation of the general stochastic dynamic programing model ie given and applied to an inventory model.

The inventory problem for continuous time is studied under the following assumptions about the demand process (1) an arbitrary distribution of the length of intervals between successive demands; (2) a distribution of the quantity demanded which is independent of the last quantity demanded and any previous events but may depend on the time elapsed since the last demand; (3) unfilled orders are backlogged. The delivery time is fixed. Costs considered are fixed ordering costs and proportional costs of purchase, storage and shortage. The loss function and the equations for reordering point and minimal ordering quantity are derived. Formulae are calculated for the Poisson, stuttering Poisson, geometric, negative binomial, Gamma and compounded distributions.

The purpose of this paper is to discuss a number of functional equations which arise in the "optimal inventory" problem. This is a particular case of the general problem of ordering in the face of an uncertain future demand. Actually, an important aspect of the problem is that of determining a suitable criterion of cost, one which is both realistic and analytically malleable.

In past decades there have been occasional upsurges of intensive interest in inventory control problems, sometimes in the aftermath of forced inventory liquidation. For the most part, the literature consisted of a few articles in business journals that had but little impact on the current business behavior of the time. In the last few years we have witnessed another upsurge of interest which far surpasses any of its predecessors, with respect to the quantity and quality of the work accomplished as well as its overall impact on the business community. This vigorous research and the interest it has aroused have been made possible by parallel development in research and in business. Statisticians and economists have become interested in industrial problems concomitantly with the increased attention in business to techniques of advanced management. The appearance of this journal is an example of this harmony of interests that has come into being. The fact that modern statistical methods have been applied successfully in several instances has had much influence on the interest level of both research workers and industrialists.