# Inventory theory

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Material theory (or more formally the mathematical theory of inventory and production) is the sub-specialty within operations research and operations management that is concerned with the design of production/inventory systems to minimize costs: it studies the decisions faced by firms and the military in connection with manufacturing, warehousing, supply chains, spare part allocation and so on and provides the mathematical foundation for logistics. The inventory control problem is the problem faced by a firm that must decide how much to order in each time period to meet demand for its products. The problem can be modeled using mathematical techniques of optimal control, dynamic programming and network optimization. The study of such models is part of inventory theory.

Operations research, or operational research (OR) in British usage, is a discipline that deals with the application of advanced analytical methods to help make better decisions. Further, the term operational analysis is used in the British military as an intrinsic part of capability development, management and assurance. In particular, operational analysis forms part of the Combined Operational Effectiveness and Investment Appraisals, which support British defense capability acquisition decision-making.

Operations management is an area of management concerned with designing and controlling the process of production and redesigning business operations in the production of goods or services. It involves the responsibility of ensuring that business operations are efficient in terms of using as few resources as needed and effective in terms of meeting customer requirements. Operations management is primarily concerned with planning, organizing and supervising in the contexts of production, manufacturing or the provision of services.

Inventory or stock is the goods and materials that a business holds for the ultimate goal of resale.

## Issues

One issue is infrequent large orders vs. frequent small orders. Large orders will increase the amount of inventory on hand, which is costly, but may benefit from volume discounts. Frequent orders are costly to process, and the resulting small inventory levels may increase the probability of stockouts, leading to loss of customers. In principle all these factors can be calculated mathematically and the optimum found.

A stockout, or out-of-stock (OOS) event is an event that causes inventory to be exhausted. While out-of-stocks can occur along the entire supply chain, the most visible kind are retail out-of-stocks in the fast-moving consumer goods industry. Stockouts are the opposite of overstocks, where too much inventory is retained.

In sales, commerce and economics, a customer is the recipient of a good, service, product or an idea - obtained from a seller, vendor, or supplier via a financial transaction or exchange for money or some other valuable consideration.

A second issue is related to changes in demand (predictable or random) for the product. For example, having the needed merchandise on hand in order to make sales during the appropriate buying season(s). A classic example is a toy store before Christmas: if the items are not on the shelves, they cannot be sold. And the wholesale market is not perfect' there can be considerable delays, particularly with the most popular toys. So, the entrepreneur or business manager will buy speculatively. Another example is a furniture store. If there is a six-week, or more, delay for customers to receive merchandise, some sales will be lost. A further example is a restaurant, where a considerable percentage of the sales are the value-added aspects of food preparation and presentation, and so it is rational to buy and store somewhat more to reduce the chances of running out of key ingredients. The situation often comes down to two key questions: confidence in the merchandise selling, and the benefits accruing if it does?

A toy store or toy shop, is a retail business specializing in selling toys.

Christmas is an annual festival, commemorating the birth of Jesus Christ, observed primarily on December 25 as a religious and cultural celebration among billions of people around the world. A feast central to the Christian liturgical year, it is preceded by the season of Advent or the Nativity Fast and initiates the season of Christmastide, which historically in the West lasts twelve days and culminates on Twelfth Night; in some traditions, Christmastide includes an octave. Christmas Day is a public holiday in many of the world's nations, is celebrated religiously by a majority of Christians, as well as culturally by many non-Christians, and forms an integral part of the holiday season centered around it.

Furniture refers to movable objects intended to support various human activities such as seating, eating (tables), and sleeping. Furniture is also used to hold objects at a convenient height for work, or to store things. Furniture can be a product of design and is considered a form of decorative art. In addition to furniture's functional role, it can serve a symbolic or religious purpose. It can be made from many materials, including metal, plastic, and wood. Furniture can be made using a variety of woodworking joints which often reflect the local culture.

A third issue comes from the view that inventory also serves the function of decoupling two separate operations. For example, work in process inventory often accumulates between two departments because the consuming and the producing department do not coordinate their work. With improved coordination this buffer inventory could be eliminated. This leads to the whole philosophy of Just In Time, which argues that the costs of carrying inventory have typically been underestimated, both the direct, obvious costs of storage space and insurance, but also the harder-to-measure costs of increased variables and complexity, and thus decreased flexibility, for the business enterprise.

Work in process (WIP), work in progress (WIP), goods in process, or in-process inventory are a company's partially finished goods waiting for completion and eventual sale or the value of these items. These items are either just being fabricated or waiting for further processing in a queue or a buffer storage. The term is used in production and supply chain management.

## Inventory models

The mathematical approach is typically formulated as follows: a store has, at time ${\displaystyle k}$, ${\displaystyle x_{k}}$ items in stock. It then orders (and receives) ${\displaystyle u_{k}}$ items, and sells ${\displaystyle w_{k}}$ items, where ${\displaystyle w}$ follows a given probability distribution. Thus:

${\displaystyle x_{k+1}=x_{k}+u_{k}-w_{k}}$
${\displaystyle u_{k}\geq 0}$

Whether ${\displaystyle x_{k}}$ is allowed to go negative, corresponding to back-ordered items, will depend on the specific situation; if allowed there will usually be a penalty for back orders. The store has costs that are related to the number of items in store and the number of items ordered:

${\displaystyle c_{k}=c(x_{k},u_{k})}$. Often this will be in additive form: ${\displaystyle c_{k}=p(x_{k})+h(u_{k})}$

The store wants to select ${\displaystyle u_{k}}$ in an optimal way, i.e. to minimize

${\displaystyle \sum _{k=0}^{T}c_{k}.}$

Many other features can be added to the model, including multiple products (denoted ${\displaystyle x_{ik}}$), upper bounds on inventory and so on. Inventory models can be based on different assumptions: [1] [2]

Demand is the quantity of a good that consumers are willing and able to purchase at various prices during a given period of time.

In mathematics, computer science and physics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given starting condition or initial state.

The word stochastic is an adjective in English that describes something that was randomly determined. The word first appeared in English to describe a mathematical object called a stochastic process, but now in mathematics the terms stochastic process and random process are considered interchangeable. The word, with its current definition meaning random, came from German, but it originally came from Greek στόχος (stókhos), meaning 'aim, guess'.

### Classic models

Although the number of models described in the literature is immense, the following is a list of classics:

## Related Research Articles

A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data. A statistical model represents, often in considerably idealized form, the data-generating process.

Material requirements planning (MRP) is a production planning, scheduling, and inventory control system used to manage manufacturing processes. Most MRP systems are software-based, but it is possible to conduct MRP by hand as well.

In the field of mathematical optimization, stochastic programming is a framework for modeling optimization problems that involve uncertainty. Whereas deterministic optimization problems are formulated with known parameters, real world problems almost invariably include some unknown parameters. When the parameters are known only within certain bounds, one approach to tackling such problems is called robust optimization. Here the goal is to find a solution which is feasible for all such data and optimal in some sense. Stochastic programming models are similar in style but take advantage of the fact that probability distributions governing the data are known or can be estimated. The goal here is to find some policy that is feasible for all the possible data instances and maximizes the expectation of some function of the decisions and the random variables. More generally, such models are formulated, solved analytically or numerically, and analyzed in order to provide useful information to a decision-maker.

In inventory management, economic order quantity (EOQ) is the order quantity that minimizes the total holding costs and ordering costs. It is one of the oldest classical production scheduling models. The model was developed by Ford W. Harris in 1913, but R. H. Wilson, a consultant who applied it extensively, and K. Andler are given credit for their in-depth analysis.

In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random and independent, and statistically identical over different time intervals of the same length. A Lévy process may thus be viewed as the continuous-time analog of a random walk.

A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as unstable stock prices or physical systems subject to thermal fluctuations. Typically, SDEs contain a variable which represents random white noise calculated as the derivative of Brownian motion or the Wiener process. However, other types of random behaviour are possible, such as jump processes.

The economic lot scheduling problem (ELSP) is a problem in operations management and inventory theory that has been studied by a large number of researchers for more than 50 years. The term was first used in 1958 by professor Jack D. Rogers of Berkeley, who extended the economic order quantity model to the case where there are several products to be produced on the same machine, so that one must decide both the lot size for each product and when each lot should be produced. The method illustrated by Jack D. Rogers draws on a 1956 paper from Welch, W. Evert. The ELSP is a mathematical model of a common issue for almost any company or industry: planning what to manufacture, when to manufacture and how much to manufacture.

The newsvendormodel is a mathematical model in operations management and applied economics used to determine optimal inventory levels. It is (typically) characterized by fixed prices and uncertain demand for a perishable product. If the inventory level is , each unit of demand above is lost in potential sales. This model is also known as the newsvendor problem or newsboy problem by analogy with the situation faced by a newspaper vendor who must decide how many copies of the day's paper to stock in the face of uncertain demand and knowing that unsold copies will be worthless at the end of the day.

The economic production quantity model determines the quantity a company or retailer should order to minimize the total inventory costs by balancing the inventory holding cost and average fixed ordering cost. The EPQ model was developed by E.W. Taft in 1918. This method is an extension of the economic order quantity model. The difference between these two methods is that the EPQ model assumes the company will produce its own quantity or the parts are going to be shipped to the company while they are being produced, therefore the orders are available or received in an incremental manner while the products are being produced. While the EOQ model assumes the order quantity arrives complete and immediately after ordering, meaning that the parts are produced by another company and are ready to be shipped when the order is placed.

Robust optimization is a field of optimization theory that deals with optimization problems in which a certain measure of robustness is sought against uncertainty that can be represented as deterministic variability in the value of the parameters of the problem itself and/or its solution.

In the theory of stochastic processes, the filtering problem is a mathematical model for a number of state estimation problems in signal processing and related fields. The general idea is to establish a "best estimate" for the true value of some system from an incomplete, potentially noisy set of observations on that system. The problem of optimal non-linear filtering was solved by Ruslan L. Stratonovich, see also Harold J. Kushner's work and Moshe Zakai's, who introduced a simplified dynamics for the unnormalized conditional law of the filter known as Zakai equation. The solution, however, is infinite-dimensional in the general case. Certain approximations and special cases are well understood: for example, the linear filters are optimal for Gaussian random variables, and are known as the Wiener filter and the Kalman-Bucy filter. More generally, as the solution is infinite dimensional, it requires finite dimensional approximations to be implemented in a computer with finite memory. A finite dimensional approximated nonlinear filter may be more based on heuristics, such as the Extended Kalman Filter or the Assumed Density Filters, or more methodologically oriented such as for example the Projection Filters, some sub-families of which are shown to coincide with the Assumed Density Filters.

In actuarial science and applied probability ruin theory uses mathematical models to describe an insurer's vulnerability to insolvency/ruin. In such models key quantities of interest are the probability of ruin, distribution of surplus immediately prior to ruin and deficit at time of ruin.

The dynamic lot-size model in inventory theory, is a generalization of the economic order quantity model that takes into account that demand for the product varies over time. The model was introduced by Harvey M. Wagner and Thomson M. Whitin in 1958.

In mathematics, the slow manifold of an equilibrium point of a dynamical system occurs as the most common example of a center manifold. One of the main methods of simplifying dynamical systems, is to reduce the dimension of the system to that of the slow manifold—center manifold theory rigorously justifies the modelling. For example, some global and regional models of the atmosphere or oceans resolve the so-called quasi-geostrophic flow dynamics on the slow manifold of the atmosphere/oceanic dynamics, and is thus crucial to forecasting with a climate model.

In queueing theory, a discipline within the mathematical theory of probability, a D/M/1 queue represents the queue length in a system having a single server, where arrivals occur at fixed regular intervals and job service requirements are random with an exponential distribution. The model name is written in Kendall's notation. Agner Krarup Erlang first published a solution to the stationary distribution of a D/M/1 and D/M/k queue, the model with k servers, in 1917 and 1920.

In queueing theory, a discipline within the mathematical theory of probability, an M/D/c queue represents the queue length in a system having c servers, where arrivals are determined by a Poisson process and job service times are fixed (deterministic). The model name is written in Kendall's notation. Agner Krarup Erlang first published on this model in 1909, starting the subject of queueing theory. The model is an extension of the M/D/1 queue which has only a single server.

In applied probability, an assemble-to-order system is a model of a warehouse operating a build to order policy where products are assembled from components only once an order has been made. The time to assemble a product from components is negligible, but the time to create components is significant.

The base stock model is a statistical model in inventory theory. In this model inventory is refilled one unit at a time and demand is random. If there is only one replenishment, then the problem can be solved with the newsvendor model.

Supersymmetric theory of stochastic dynamics or stochastics (STS) is an exact theory of stochastic (partial) differential equations (SDEs), the class of mathematical models with the widest applicability covering, in particular, all continuous time dynamical systems, with and without noise. The main utility of the theory from the physical point of view is a rigorous theoretical explanation of the ubiquitous spontaneous long-range dynamical behavior that manifests itself across disciplines via such phenomena as 1/f, flicker, and crackling noises and the power-law statistics, or Zipf's law, of instantonic processes like earthquakes and neuroavalanches. From the mathematical point of view, STS is interesting because it bridges the two major parts of mathematical physics – the dynamical systems theory and topological field theories. Besides these and related disciplines such as algebraic topology and supersymmetric field theories, STS is also connected with the traditional theory of stochastic differential equations and the theory of pseudo-Hermitian operators.

## References

1. Zipkin Paul H., Foundations of Inventory Management, Boston: McGraw Hill, 2000, ISBN   0-256-11379-3
2. W. Hopp, M. Spearman, Factory Physics, 3rd ed. Waveland Press, 2011
• International Journal of Inventory Research is an academic journal on inventory theory publishing current research.

Classic books that established the field are:

• Kenneth J. Arrow, Samuel Karlin, and Herbert E. Scarf: Studies in the Mathematical Theory of Inventory and Production, Stanford University Press, 1958
• Thomson M. Whitin, G. Hadley, Analysis of Inventory Systems, Englewood Cliffs: Prentice-Hall 1963

Many university courses in inventory theory use one or more of the following current textbooks:

• Axsaeter, Sven. Inventory Control. Norwell, MA: Kluwer, 2000. ISBN   0-387-33250-2
• Porteus, Evan L. Foundations of Stochastic Inventory Theory. Stanford, CA: Stanford University Press, 2002. ISBN   0-8047-4399-1
• Silver, Edward A., David F. Pyke, and Rein Peterson. Inventory Management and Production Planning and Scheduling, 3rd ed. Hoboken, NJ: Wiley, 1998. ISBN   0-471-11947-4
• Simchi-Levi, David, Xin Chen, and Julien Bramel. The Logic of Logistics: Theory, Algorithms, and Applications for Logistics Management, 2nd ed. New York: Springer Verlag, 2004. ISBN   0-387-22199-9
• Tempelmeier, Horst. Inventory Management in Supply Networks, 3rd. Edition, Norderstedt (Books on Demand) 2011, ISBN   3-8423-4677-8
• Zipkin, Paul H. Foundations of Inventory Management. Boston: McGraw Hill, 2000. ISBN   0-256-11379-3