Inventory control problem
Encyclopedia
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
.
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 pre-Christmas. If one does not have the items on the shelves, one will not make the sales. 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 on spec. Another example is a furniture store. If there is a six week, or more, delay for customers to get merchandise, some sales will be lost. And yet another 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. With all these examples, the situation often comes down to these two key questions: How confident are you that the merchandise will sell, and how much upside is there if it does?
And 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 under-estimated, 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.
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.
Optimal control
Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and his collaborators in the Soviet Union and Richard Bellman in the United States.-General method:Optimal...
, dynamic programming
Dynamic programming
In mathematics and computer science, dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. It is applicable to problems exhibiting the properties of overlapping subproblems which are only slightly smaller and optimal substructure...
and network optimization
Flow network
In graph theory, a flow network is a directed graph where each edge has a capacity and each edge receives a flow. The amount of flow on an edge cannot exceed the capacity of the edge. Often in Operations Research, a directed graph is called a network, the vertices are called nodes and the edges are...
. The study of such models is part of inventory theory
Inventory theory
Inventory theory is the sub-specialty within operations research that is concerned with the design of production/inventory systems to minimize costs...
.
Concepts
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 stock-outs, 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 pre-Christmas. If one does not have the items on the shelves, one will not make the sales. 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 on spec. Another example is a furniture store. If there is a six week, or more, delay for customers to get merchandise, some sales will be lost. And yet another 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. With all these examples, the situation often comes down to these two key questions: How confident are you that the merchandise will sell, and how much upside is there if it does?
And 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 under-estimated, 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.
Equations
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.See also
- InventoryInventoryInventory means a list compiled for some formal purpose, such as the details of an estate going to probate, or the contents of a house let furnished. This remains the prime meaning in British English...
- Inventory control systemInventory control systemAn inventory control system is a process for managing and locating objects or materials. In common usage, the term may also refer to just the software components......
- Inventory management softwareInventory management softwareInventory management software is a computer-based system for tracking product 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. Companies use inventory management software to...
- Operations managementOperations managementOperations management is an area of management concerned with overseeing, designing, and redesigning business operations in the production of goods and/or services. It involves the responsibility of ensuring that business operations are efficient in terms of using as little resources as needed, and...
- Supply chain managementSupply chain managementSupply chain management is the management of a network of interconnected businesses involved in the ultimate provision of product and service packages required by end customers...
- Warehouse management systemWarehouse management systemA warehouse management system, or WMS, is a key part of the supply chain and primarily aims to control the movement and storage of materials within a warehouse and process the associated transactions, including shipping, receiving, putaway and picking...