On an Inventory Control Model with Random Lead Time and Random Demand

A single-product inventory control model with different distributions of demand for goods and random lead time is considered. The proposed model uses the regenerative approach. It gives possibility to find optimal values of reorder point and order quantity. Criterion of optimization is minimum of average expenses for goods holding, ordering and losses from deficit per a time unit. Suggested model will be examined in solving practical tasks for Latvian Railway company. INTRODUCTION Transport system in the Latvian Republic is an important economic field of the country, which together with the communication industry provides about 21% of the state national gross output and 45% of currency incomes. The Latvian railway as a part of the transport industry of Latvia is a large and complex system and has a number of specific features, which add sufficiently to the efficiency of its performance. Among them we should note the following: a complex, farflung railways network; large distances between related objects; high dynamics of processes; dependence on random factors; high requirements for reliability and safety of transportation systems; and large financial, labor and material resources. Therefore, even small errors in management and planning the process of transportation result in a considerable decrease of the efficiency of the railway company performance and of the quality of the clients’ service and in big financial losses, which, no doubt, negatively affects the development of Latvian economics as a whole. One of the most important problems for railways’ enterprises is to maintain the rolling stock in working conditions. We have to keep up some stock of rather wide range of spare parts and equipment for rolling stock maintenance. If we increase the stock, it makes more expensive the railway transport services; on the other hand it eliminates the losses from the idle rolling stock, expecting reparations and technical services. It is a quite complicated mathematical task to find the optimal solution for the necessary stock, if you are working in the railways industry. The inventory control management model should take into account the random demand for the spare parts and consider the whole supply chain of the companies, which have influence on spare parts deliveries “manufacturer – supplier – intermediate company – transport company”. For the construction the effective inventory control system authors propose to use the modern mathematical tools: inventory control theory, time series analysis, and forecasting methods (Kopytov et al., 2004). In the given paper the inventory control model for one type of product is considered. Different types of stochastic inventory models are considered in literature (Prabhu 1967; Chopra and Meindl 2001; Ross 1992; Ryzhikov 2001). In practice it is common for inventory manager to answer two basic questions: how much to order and when to order, and there are many different types of inventory control models which provide the decision-maker with a satisfactory solution. In previous authors’ works the main attention was paid to economical aspects of inventory problem. We have to take into account that the sum of costs for goods ordering, holding and losses from deficit should be minimal. In the given paper we suppose inventory model using the same economical criteria. This paper is based on the previous paper (Kopytov and Greenglaz 2004), but here we consider random demand of goods and lead time with distributions for which some constraints are fulfilled. DESCRIPTION OF THE MODEL We consider a single-product inventory control model under the following conditions. The demand for goods Dτ for period of time τ after goods delivering is random variable with known distribution ) (x fDτ . We suppose that if m and σ are the mean and standard deviation for X=D1, then m ⋅ τ and σ τ ⋅ are mean and standard deviation for demand Dτ. We suppose in particular that the distribution of the demand for goods Dτ may be: