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| 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 [[flow network|network optimization]]. The study of such models is part of [[inventory theory]].
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| ==Concepts==
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| 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.
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| 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?
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| 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 (business)|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.
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| ==Equations==
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| The mathematical approach is typically formulated as follows:
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| A store has, at time <math>k</math>, <math>x_k</math> items in stock. It then orders (and receives) <math>u_k</math> items, and sells <math>w_k</math> items, where <math>w</math> follows a given probability distribution. Thus
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| : <math> x_{k+1} = x_k + u_k - w_k</math>.
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| : <math> u_k \ge 0 </math>
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| Whether <math>x_k</math> 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.
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| The store has costs that are related to the number of items in store and the number of items ordered:
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| :<math>c_k = c(x_k, u_k)</math>. Often this will be in additive form: <math>c_k = p(x_k) + h(u_k)</math>
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| The store wants to select <math>u_k</math> in an optimal way, i.e. to minimize
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| :<math> \sum_{k=0}^{\infty} c_k </math>.
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| Many other features can be added to the model, including multiple products, denoted <math>x_{ik}</math>, upper bounds on inventory and so on.
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| ==See also==
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| * [[Inventory]]
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| * [[Inventory control system]]
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| * [[Inventory management software]]
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| * [[Operations management]]
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| * [[Supply chain management]]
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| * [[Warehouse management system]]
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| [[Category:Mathematical optimization]]
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| [[Category:Inventory]]
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