Inventory control problem: Difference between revisions

Revision as of 00:49, 2 March 2009

Revision as of 01:00, 2 March 2009

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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

: <math> x_{k+1} = x_k + u_k - w_k</math>.

: <math> x_ku_k \ge 0 </math>

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.

The store has costs that are related to the number of items in store and the number of items ordered:

:<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>

The store wants to select <math>u_k</math> in an optimal way, i.e. to minimize

:<math> \sum_{k=0}^{\infty} c_k </math>.

Many other features can be added to the model, including multiple products, denoted <math>x_{ik}</math>, upper bounds on inventory and so on.

==Mathematics and Judgement: The Best of Both Worlds==