This paper presents a mathematical model for an inventory control system in which customers' demands and suppliers' service time are considered as stochastic parameters. The proposed problem is solved through queuing theory for a single item. In this case, transitional probabilities are calculated in steady state. Afterward, the model is extended to the case of multi-item inventory systems. Then, to deal with the complexity of this problem, a new heuristic algorithm is developed. Finally, the presented bi-level inventory-queuing model is implemented as a case study in Electroestil Company.

Considering the competition in today's business environment, tactical planning of a supply chain becomes more complex than before. In many multi-product inventory systems, substitution flexibility can improve profits. This paper aims to prepare a comprehensive substitution inventory model, where an inventory system with two substitute products with ignorable lead time has been considered, and effects of simultaneous ordering have been examined. In this paper, demands of customers for both of the products have been regarded as stochastic parameters, and queuing theory has been used to construct a mathematical model. The model has been coded by C??, and it has been analyzed due to a real example, where the results indicate efficiency of proposed model.

Cross-docking is a logistic concept, which plays an important role in supply chain management by decreasing inventory holding, order packing, transportation costs and delivery time. Paying attention to these concerns, and importance of the congestion in cross docks, we present a mixed-integer model to optimize the location and design of cross docks at the same time to minimize the total transportation and operating costs. The model combines queuing theory for design aspects, for that matter, we consider a network of cross docks and customers where two M/M/c queues have been represented to describe operations of indoor trucks and outdoor trucks in each cross dock. To prepare a perfect illustration for performance of the model, a real case also has been examined that indicated effectiveness of the proposed model. Rate of indoor demand for cross dock t k O1t Rate of outdoor flows that need to be transported from cross dock t to another cross dock k O2t Outdoor flows go through cross dock t and customers

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