Inventory theory

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 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 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.

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]

• Nature of demand: constant, deterministically time-varying or stochastic
• Costs: variable versus fixed
• Flow of time: discrete versus continuous
• Lead time: deterministic or stochastic
• Time horizon: finite versus infinite (T=+∞)
• Presence or absence of back-ordering
• Production rate: infinite, deterministic or random
• Presence or absence of quantity discounts
• Imperfect quality
• Capacity: infinite or limited
• Products: one or many
• Location: one or many
• Echelons: one or many

• Safety stock
• Inventory optimization
• Inventory management software
• Supply chain management
• Warehouse management system

• 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
• Sethi, S.P., Yan, H., and Zhang, H., Inventory and Supply Chain Management with Forecast Updates, in series International Series in Operations Research & Management Science, Springer, NY, NY, 2005.(310 pages - ISBN 1-4020-8123-5)
• Beyer, D., Cheng, F., Sethi, S.P., and Taksar, M.I., Markovian Demand Inventory Models, in series: International Series in Operations Research and Management Science, Springer, New York, NY, 2010. (253 pages - ISBN 978-0-387-71603-9)