Studies in the Mathematical Theory of Inventory and Production

Edited by Kenneth J. Arrow, Samuel Karlin, and Herbert E. Scarf
Stanford University Press, 1958
340pp

This book is the first to explore and formulate the general mathematical concepts behind efficient inventory and production control. In a series of interrelated papers, inventory policies are studied under a wide variety of assumptions likely to be met in practice.

The extensive introduction considers the historical background of modern inventory theory, and nature and structure of inventory problems, and certain features common to all inventory models. In addition, the authors provide summaries of their results and describe their approach to the analysis of inventory problems.

Part II is devoted to deterministic inventory processes. It presents methods of devising optimal solutions to four characteristic industrial problems: production over time with increasing marginal costs, production "smoothing," production planning without storage, and expansion of the capacity of a firm.

Part III, on stochastic inventory processes, offers detailed, integrated discussions of the static inventory model, the Arrow-Harris-Marschak dynamic model, and the dynamic model with the addition of a time lag factor. The last three chapters of this Part consider applications of the dynamic model.

Part IV introduces several new concepts, including the concept of the operating characteristics of an inventory policy and that of the stationary solution of an inventory problem. These concepts are particularly suited to the solution of inventory problems when the determination of optimal policies is prohibitively difficulty. Explicit techniques are given for obtaining numerical solutions.

The mathematics used represent new developments in the calculus of variations and in the theory of stochastic processes. The developments in stochastic theory relate to both queueing and renewal problems, and considerably improve existing results.

The bibliography on inventory theory forms an up-to-date supplement to the earlier Whitin and Gourary-Lewis-Neeland bibliographies.

PREFACE

The research papers in this collection treat a number of mathematical and conceptual problems in the analysis of business decisions about inventories and production. The results presented in these papers have been obtained during the last three years and appear in print for the first time in this volume.

Each chapter is signed by its author or authors and may be read independently of other chapters. We suggest however, that Chapters 1 and 2 be read first, since they serve to place the following chapters in their proper setting with respect to previous work, and to furnish the reader with a survey of the many common features of all inventory models. In the following chapters, we have frequently omitted detailed explanation of the models whose significance seems to us clearly presented in Chapter 2.

In Chapter 3, the reader will find summaries of the results of the subsequent chapters.

Most of the research reported in this book was done at Stanford University, with the support of the Office of Naval Research. Chapter 12 was prepared at The RAND Corporation. Chapter 13 was prepared at the Cowles Foundation for Research in Economics, Yale University, with the support of the Office of Naval Research. To all these organizations, we wish to express our gratitude.

We also with to thank Professor Albert H. Bowker, Director of the Applied Mathematics and Statistics Laboratory at Stanford University, both for doing so much to encourage the development of mathematical research in the social sciences at Stanford and for first suggesting that we assemble a book on inventory and production theory. John Gessford and Donald Roberts spend considerable time reading several of the chapters in their first drafts, and made a number of valuable suggestions. Discussions with Ronald Pyke were very helpful in developing some of the work on stochastic processes in this volume. Finally we must thank Caroleanne Roberts, Laura Staggers, and Sharon Steck for converting our appalling handwriting into typescript.

Kenneth J. Arrow
Samuel Karlin
Herbert Scarf

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