## Stochastic Inventory Management in a Shortage

Read this article. The study indicates that a stochastic inventory management system should be preceded by determining the economic feasibility of the shortage. How does using real-time statistics help determine purchasing?

### Literature review and problem statement

The theory of optimal inventory management was born in the 19th century, when in scientific writings the analytical models of calculating the optimal order quantity under the conditions of deterministic demand were elaborated. A distinctive feature was a certain idealization of supply and stable replenishment conditions. The new stage of the inventory management theory relates to the rapid development of logistics, which some scientists reference as a science of control of the stock movement. Some authors, separately investigated the processes of formation and optimization of safety stocks in the conditions of the uncertainty of demand. The authors of "Methods of solving nonlinear extremal problems," developed the so-called stochastic quasi-gradient method of solving the problems of convex stochastic programming. Modifications of this method were considered in the works. At the same time, the use of the developed mathematical methods in business is limited, on the one hand, by the existence of assumptions and simplification of real conditions, and on the other – the difficulties of obtaining the source data for calculations. It should be noted that the researchers have paid significant attention to the economic essence of inventories and their influence on the financial results of the enterprises. The author of "Efficient inventory management;" claims that effective inventory management allows companies to better meet the needs of consumers and thus achieve high ROI in warehousing. On the other hand, globalization of supply chains and attempts to satisfy the needs of consumers perfectly led to the creation of considerable excess stocks in the channels of distribution of produced goods, which was named the "bullwhip effect". That is why scientific researches emphasizes the need to assess the risk of stock accumulation and the erroneous development of some methods of calculating the size of safety stock aimed to avoid the deficit. A planned shortage or delay of supply sometimes helps to lower the overall cost of the system. In "Computers & Industrial Engineering;", the author presented a model of economic production quantity with planned backorders and proved its economic efficiency. The author used the analytic geometry and algebra for inventory management models with the shortage. Two types of the costs of breach of obligations were used: one – linear, others – fixed.

The work which investigates the enlarged range of inventory control system costs, including costs of damage to goods, costs of extraordinary supply in case of shortage, loss from lost sales is interesting. The authors of "Analysis of an EOQ inventory model with partial backordering and non-linear unit holding cost;" developed a model of managing single-product inventory, considering a partial shortage. It is assumed that in case of shortage, a part of clients will not wait for the goods to come but will go to the competitor, which would affect the general expenditures. In "Annals of Operations Research;", the shortage is viewed as a valid, set value. The authors developed a model with time-dependent demand and storage costs for building optimal stock replenishment policies. Some authors, for example, determined the optimum shortage for specific objects (poultry farms), which determined the use of specific costs groups and additional partial criteria.

Thus, the analysis of modern literature on planning the inventory shortage testifies that the inclusion of appropriate costs in the general model allows considering not only the direct expenses of the system but also lost sales and costs of the unplanned supply. The expansion of the range of expenses can significantly affect the value of inventory management system parameters.

The research in the area of stochastic inventory management systems is represented by a wide range of works. In particular, the authors of "Applied Mathematical Modelling;" developed an optimizing model for supply chain, considering the uncertain demand, production cost, distribution of transportation costs, shortage losses and other parameters.

Due to global research, it is possible to estimate the share of the authors, who paid attention to the problems of stock shortage, as well as stochastic models of inventory management in 2008–2018. The authors examined 56 publications that were included in 32 journals of ScienceDirect and Scopus. Approximately 15 % of the works related to stochastic models of inventory management, about 12 % touched the issues of economically viable shortage. None of the works contained, on the one hand, the determination of economically viable shortage, and on the other – the determination of stochastic inventory management system parameters considering the optimum level of service.

Thus, a review of scientific literature and problems related to stock optimization at different enterprises showed a high degree of development of appropriate mathematical methods and models. On the other hand, there is a low level of practical use due to the need to account for many factors influencing the formation and holding of inventory in exact conditions. The problem of determination of inventory management system parameters under the conditions of uncertainty and economically justified deficit by the criterion of minimization of the general expenses of the enterprise requires a scientific basis and methodological support.