Mathematical Model for Supply Planning of Multi-level Assembly Systems with Stochastic Lead Times O. Ben Ammar*, H. Marian*, D. Wu** , ***, A. Dolgui* *École Nationale Supérieure des Mines, EMSE-FAYOL, CNRS UMR6158, LIMOS, F-42023 Saint-Étienne, France (e-mail:{obenammar, marian, dolgui}@emse.fr) **RiskLab, University of Toronto, Toronto Canada (e-mail: DWu@Rotman.Utoronto.Ca) *** School of Management, University of Science and Technology of China, Anhui Province, P.R. China Abstract: The deal for industrial companies is to satisfy their customers with the best quality, the most reliable lead times, and the lowest costs. One of the keys to achieve it is in the inventory control in the Supply Chain. Moreover, it is mandatory to possess necessary components in order to produce the requested products by the due date. But, as the components lead time is an unpredictable parameter, it is difficult to take it into account correctly. In this paper we deal with this question considering that each component has a fixed unit inventory cost; the finished product has an inventory cost and a backlogging cost per unit of time. Then, a general mathematical model for supply planning of multi- level assembly system is presented to calculate the expected value of the total cost which equals to the sum of the inventory holding costs for the components, the backlogging and the inventory holding costs for the finished product. Keywords: Stochastic lead time, multi-level assembly systems. 1. INTRODUCTION AND RELATED PUBLICATIONS The MRP system has been designed for production planning and supply management in certain environments. But today, the supply chain is becoming very vulnerable and far to being deterministic (Dolgui and Prodhon, 2007). Planners need to adapt themselves to changes in economic conditions (changes in costs increase in prices of raw materials, etc.) and to technical problems (machines breakdowns, limited capacity, delay of transport, etc.). Nevertheless the MRP method is still very used and useful. Wazed et al. (2009) identified the major factors of uncertainty in a real manufacturing environment as demand, supplier lead time, quality and capacity. Van Kampen Tim et al. (2010) specified that various techniques, such as safety stocks and safety lead times, can be needed to control the supply variability and to lead to better anticipation of uncertainties. The literature review identified several states of the art in the field of MRP parameterization under uncertainties (Damand et al. (2011), Dolgui and Prodhon (2007) and Koh et al. (2002). Different types of supply systems are identified: serial supply and assembly systems. For assembly systems, several problems with a one or multi-period model have been studied in several papers before. In the literature few researchers have modeled lead times as discrete random variables. Dolgui et al. (1995) and Dolgui (2001) proposed a multi-period model for one-level assembly systems under fixed demand and stochastic lead times. Several types of finished products were considered and the lot for lot policy was suggested. Holding and backlogging costs for each item were considered. Each finished product is assembled using several components. The authors proposed an approach based on the coupling of simulation models and an integer linear programming. It calculates the number of components of each type to be ordered at the beginning of each period as well as the number of products to be assembled during each period. A multi-period model was proposed in Dolgui and Ould- Louly. (2002). Markov chains and Newsboy model were used to study a one-level assembly systems under components lead times uncertainties. The demand was considered as known, the production capacity was assumed unlimited and the case of Lot-for-Lot policy was treated. The criterion considered is the total average cost that equals to the sum of the average holding cost for the components and the average backlogging cost for the finished product. This model gives the optimal values of the safety stocks when the component lead times are i.i.d. random variables and the unit holding costs are the same for all types of components. The same problem was examined by Ould-Louly and Dolgui (2004) for the case of the Periodic Order Quantity (POQ) policy. In the article (Ould Louly et al., 2008) a Branch and Bound approach was applied to solve the same problem but for Lot-For-Lot policy with service level constraint. More than one paper has been published by the same authors (Ould Louly and Dolgui 2011, Ould Louly et al., 2013) dealing with supply lead time variability in one-level assembly systems.