Inventory models

Summary: Mathematical inventory control models help businesses make decisions, and they are widely studied in the discipline of operations research.

In an ideal world, retail stores would stock all of the products that customers are interested in buying and stock these in sufficient quantity to cater to all customers. In reality, store area is limited and a company would not benefit by stocking an excess of each product. The problem, then, is to calculate the optimal amount of supply.

These decisions take into consideration how many units should be kept so that most, if not all, customers can be served on a particular day, because if customers do not find what they want, they will shop elsewhere. At the same time, a store does not want too many units on hand, as there are costs attached to storing excess units, and they may remain unsold, which also reduces profit.

The problem can be considered in manufacturing, where a product consists of many small components, and a business has to decide how many components it must order and store so that fabrication runs smoothly. Similar examples exist in service industries and military ships. Mathematical inventory control models help businesses make decisions, and they are widely studied in the mathematical discipline of operations research.

Mathematics of Inventory

Computational logistics is a mathematical and businesses field concerned with planning the flow and storage of goods, services, or information from the point of origin to the point of use. One key planning consideration is the trade-off between transport and inventory costs, a factor recognized at least as early as the mid-1880s. Mathematicians, computer scientists, and others continue to develop new inventory management and optimization models as well as the algorithms and software necessary to implement them. Mathematician Samuel Karlin was awarded the John von Neumann Theory Prize in 1987, as well as the National Medal of Science in 1989, for diverse mathematical contributions, including inventory theory.

Inventory models used to calculate optimal order quantities and reorder points, often broadly called economic order quantity (EOQ) models, existed long before the arrival of the computer. Advances in mathematical methods and computer technology have facilitated more realistic models that account for more variables. Optimizing inventory depends on factors such as storage space, storage cost, demand rate, time between demands, cost of ordering, time for retrieving stored item or receiving an ordered item, discounts for bulk orders, and many other real-world costs.

Just-in-time models are based on the idealized principle that items are available exactly when they are needed, with zero storage time or delay. Just-in-time inventory management and lean manufacturing ideas existed as far back as Henry Ford’s Model T factories but became widely feasible in the late twentieth century with advances in technology that affected variables, like the lead time required to place an order for more stock. Reduction of process variability, using better monitoring, waste reduction, or inventory buffers, are typically seen as key to achieving optimal models under this system. A just-in-time model can save money by reducing inventory, but tighter constraints make them consequently more vulnerable to disruptions that violate the constraints.

Many basic EOQ models are simplified by assuming that variables such as demand are fixed or uniform across some period of time. These deterministic models are easy to solve analytically but may produce unrealistic results. They are often useful for theoretical study or businesses with greater variability tolerances. Many variables that influence inventories, such as demand and delay times for orders of new goods, are more realistically modeled as random variables. As a result, inventory models are often probabilistic or stochastic. Constraints tend to be operationalized as costs. For example, the physical area available for storage, such as square footage of shelf space or warehouse volume, can be reformulated as a cost constraint by calculating a cost per unit area or volume. Cost may also be parameterized into components like procurement and maintenance costs. Markov chains and linear programming techniques are useful for formulating and solving various types of inventory models. Statistical methods are used to obtain valid data for modeling and simulations.

Bibliography

Cachon, Gerard, and Christian Terwiesch. Matching Supply With Demand. New York: McGraw-Hill, 2008.

Luenberger, David G., and Yinyu Ye. Linear and Nonlinear Programming. 3rd ed. New York: Springer, 2010.

Porteus, Evan. Foundations of Stochastic Inventory Theory. Stanford, CA: Stanford Business Books, 2002.

Sethi, Suresh P., et al. Inventory and Supply Chain Management With Forecast Updates. New York: Springer, 2010.

Sherbrooke, Craig. Optimal Inventory Modeling of Systems. 2nd ed. New York: Springer, 2004.