OPTIMISATION OF THE STOCK STRUCTURE OF A SINGLE STOCK ITEM TAKING INTO ACCOUNT STOCK QUANTITY CONSTRAINTS, USING A LAGRANGE MULTIPLIER

Background: Optimisation in the area of stock management is most often performed in relation to cycle stock. The classic example here is the Harris-Wilson formula for calculating the economic order quantity. Often these models are not subject to any constraints imposed on the optimised quantities. However, in practice, taking such constraints into account is important. The application of the so-called Lagrange multiplier is helpful here, but the examples of its application usually refer to the multi-position sets of stock items (e.g. the search for the optimum structure of stock of material groups in the case of capital constraints). This paper attempts to optimise the structure of the stock (cycle stock vs. safety stock) for a single stock item.

Methods: To achieve the objective of determining the optimum stock structure for the various conditions under which stock replenishment is implemented, a general model has been built, a component of which is a Lagrange function containing the constraint conditions for the solution. Next, this model has been implemented in the form of an EXCEL spreadsheet application.

Results: The result of solving the optimization task based on the proposed model is a system of equations, the solution of which (with the help of the EXCEL application) allows to determine the optimum value of the Lagrange multiplier, on the basis of which the components of the inventory structure and other related quantities (service level indicators and costs, such as stock replenishment, stock maintenance and stock deficit costs) are calculated. This has been illustrated using a fictitious example, which at the same time made it possible to observe certain general relationships between the adopted constraints and the recorded quantities.

Conclusions: Two types of conclusions can be presented. The first type concerns the approach itself. The possibility of determining the optimum structure of the stock (cycle stock vs. safety stock) depending on various values characterising the adopted stock replenishment system as well as the adopted limitations has been demonstrated. The second type of conclusions results from the presented example of application of the method for the assumed ranges of changes of selected quantities.