Material theory (or more formally the
mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
theory of inventory and production) is the sub-specialty within
operations research
Operations research () (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a branch of applied mathematics that deals with the development and application of analytical methods to improve management and ...
and
operations management
Operations management is concerned with designing and controlling the production (economics), production of good (economics), goods and service (economics), services, ensuring that businesses are efficiency, efficient in using resources to meet ...
that is concerned with the design of production/
inventory
Inventory (British English) or stock (American English) is a quantity of the goods and materials that a business holds for the ultimate goal of resale, production or utilisation.
Inventory management is a discipline primarily about specifying ...
systems to minimize
cost
Cost is the value of money that has been used up to produce something or deliver a service, and hence is not available for use anymore. In business, the cost may be one of acquisition, in which case the amount of money expended to acquire it i ...
s: it studies the decisions faced by firms and the military in connection with
manufacturing
Manufacturing is the creation or production of goods with the help of equipment, labor, machines, tools, and chemical or biological processing or formulation. It is the essence of the
secondary sector of the economy. The term may refer ...
,
warehousing
A warehouse is a building for storing goods. Warehouses are used by manufacturers, importers, exporters, wholesalers, transport businesses, customs, etc. They are usually large plain buildings in industrial parks on the rural–urban fringe, out ...
,
supply chain
A supply chain is a complex logistics system that consists of facilities that convert raw materials into finished products and distribute them to end consumers or end customers, while supply chain management deals with the flow of goods in distri ...
s,
spare part
A spare part, spare, service part, repair part, or replacement part, is an interchangeable part that is kept in an inventory and used for the repair or Refurbishment (electronics), refurbishment of defective equipment/units. Spare parts are an i ...
allocation and so on and provides the mathematical foundation for
logistics
Logistics is the part of supply chain management that deals with the efficient forward and reverse flow of goods, services, and related information from the point of origin to the Consumption (economics), point of consumption according to the ...
. 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
Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations ...
,
dynamic programming and
network optimization. The study of such models is part of inventory theory.
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
stockout
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 (e.g. ...
s, leading to loss of
customer
In sales, commerce, and economics, a customer (sometimes known as a Client (business), client, buyer, or purchaser) is the recipient of a Good (economics), good, service (economics), service, product (business), product, or an Intellectual prop ...
s. In principle all these factors can be calculated mathematically and the
optimum
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfiel ...
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
A toy store or toy shop is a type of retail business specializing in selling toys; usually ones marketed towards children.
History
The first toy store was founded in 1760 by William Hamley in London, under the name of "Noah's Ark", later re ...
before
Christmas
Christmas is an annual festival commemorating Nativity of Jesus, the birth of Jesus Christ, observed primarily on December 25 as a Religion, religious and Culture, cultural celebration among billions of people Observance of Christmas by coun ...
: 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
Furniture refers to objects intended to support various human activities such as seating (e.g., Stool (seat), stools, chairs, and sofas), eating (table (furniture), tables), storing items, working, and sleeping (e.g., beds and hammocks). Furnitur ...
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
Value added is a term in economics for calculating the difference between market value of a product or service, and the sum value of its constituents. It is relatively expressed by the demand curve, supply-demand curve for specific units of sale. ...
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
Work in process or work-in-process, (WIP), work in progress (WIP), goods in process, or in-process inventory refers to a company's partially finished goods waiting for completion and eventual sale, or the value of these items. The term is used in ...
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, including 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.
Inventory models
The mathematical approach is typically formulated as follows:
a store has, at time
,
items in stock. It then orders (and receives)
items, and sells
items, where
follows a given probability distribution. Thus:
:
:
Whether
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:
:
. Often this will be in additive form:
The store wants to select
in an optimal way, i.e. to minimize
:
Many other features can be added to the model, including multiple products (denoted
), upper bounds on inventory and so on. Inventory models can be based on different assumptions:
[W. Hopp, M. Spearman, ''Factory Physics'', 3rd ed. Waveland Press, 2011]
*Nature of
demand
In economics, demand is the quantity of a goods, good that consumers are willing and able to purchase at various prices during a given time. In economics "demand" for a commodity is not the same thing as "desire" for it. It refers to both the desi ...
: constant,
deterministically time-varying or
stochastic Stochastic (; ) is the property of being well-described by a random probability distribution. ''Stochasticity'' and ''randomness'' are technically distinct concepts: the former refers to a modeling approach, while the latter describes phenomena; i ...
*
Cost
Cost is the value of money that has been used up to produce something or deliver a service, and hence is not available for use anymore. In business, the cost may be one of acquisition, in which case the amount of money expended to acquire it i ...
s:
variable versus
fixed
Fixed may refer to:
* ''Fixed'' (EP), EP by Nine Inch Nails
* ''Fixed'' (film), an upcoming animated film directed by Genndy Tartakovsky
* Fixed (typeface), a collection of monospace bitmap fonts that is distributed with the X Window System
* Fi ...
*Flow of
time
Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...
:
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
* Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
* Discrete group, ...
versus
continuous
*
Lead time
A lead time is the latency between the initiation and completion of a process. For example, the lead time between the placement of an order and delivery of new cars by a given manufacturer might be between 2 weeks and 6 months, depending on vari ...
: deterministic or stochastic
*
Time horizon
A time horizon, also known as a planning horizon, is a fixed point of time in the future at which point certain processes will be evaluated or assumed to end. It is necessary in an accounting, finance or risk management regime to assign such a f ...
: finite versus infinite (T=+∞)
*Presence or absence of
back-ordering
*
Production rate: infinite, deterministic or random
*Presence or absence of quantity
discounts
*Imperfect
quality
Quality may refer to:
Concepts
*Quality (business), the ''non-inferiority'' or ''superiority'' of something
*Quality (philosophy), an attribute or a property
*Quality (physics), in response theory
*Energy quality, used in various science discipli ...
*
Capacity: infinite or limited
*
Products: one or many
*
Location
In geography, location or place is used to denote a region (point, line, or area) on Earth's surface. The term ''location'' generally implies a higher degree of certainty than ''place'', the latter often indicating an entity with an ambiguous bou ...
: one or many
*Echelons: one or many
Classic models
Although the number of models described in the literature is immense, the following is a list of classics:
* Infinite fill rate for the part being produced:
Economic order quantity
Economic order quantity (EOQ), also known as financial purchase quantity or economic buying quantity, is the order quantity that minimizes the total holding costs and ordering costs in inventory management. It is one of the oldest classical pro ...
model, a.k.a. Wilson EOQ Model
* Constant fill rate for the part being produced:
Economic production quantity model
* Orders placed at regular intervals:
fixed time period model
* Demand is random, only one replenishment: classical
Newsvendor model
The newsvendor (or newsboy or single-periodWilliam J. Stevenson, Operations Management. 10th edition, 2009; page 581 or salvageable) model is a mathematical model in operations management and applied economics used to determine Inventory optimizati ...
* Demand is random, continuous replenishment:
Base stock model
* Continuous replenishment with backorders:
(Q,r) model
* Demand varies deterministically over time:
Dynamic lot size model or Wagner-Whitin model
* Demand varies deterministically over time:
Silver–Meal heuristic
* Several products produced on the same machine:
Economic lot scheduling problem
See also
*
Inventory control
Inventory control or stock control is the process of managing stock held within a warehouse, store or other storage location, including auditing actions concerned with "checking a shop's stock". These processes ensure that the right amount of suppl ...
*
Strategic inventory
*
Safety stock
Safety stock is a term used by logistics, logisticians to describe a level of extra inventory, stock which is maintained to mitigate the risk of stockouts, which can be caused, for example, by shortfalls in raw material availability or uncertainty ...
*
Inventory optimization
*
Inventory management software
An inventory management software is a software system for tracking inventory levels, orders, sales and deliveries. It can also be used in the manufacturing industry to create a work order, bill of materials and other production-related documents. ...
*
Supply chain management
In commerce, supply chain management (SCM) deals with a system of procurement (purchasing raw materials/components), operations management, logistics and marketing channels, through which raw materials can be developed into finished produc ...
*
Warehouse management system
References
Further reading
* International Journal of Inventory Research is an
academic journal
An academic journal (or scholarly journal or scientific journal) is a periodical publication in which Scholarly method, scholarship relating to a particular academic discipline is published. They serve as permanent and transparent forums for the ...
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:
* Silver, Edward A., David F. Pyke, and Rein Peterson. Inventory Management and Production Planning and Scheduling, 3rd ed. Hoboken, NJ: Wiley, 1998.
* Zipkin, Paul H. Foundations of Inventory Management. Boston: McGraw Hill, 2000.
* Axsaeter, Sven. Inventory Control. Norwell, MA: Kluwer, 2000.
* Porteus, Evan L. Foundations of Stochastic Inventory Theory. Stanford, CA: Stanford University Press, 2002.
* 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.
* 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 - )
* 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 - )
Inventory optimization
* Tempelmeier, Horst. Inventory Management in Supply Networks, 3rd. Edition, Norderstedt (Books on Demand) 2011,
* Snyder, Lawrence V. Fundamentals of Supply Chain Theory, 2nd ed. Hoboken, NJ: John Wiley & Sons, Inc, 2019.
* Rossi, Roberto. Inventory Analytics. Cambridge, UK: Open Book Publishers, 2021. {{ISBN, 978-1-800-64176-1