# IE Seminar: “On the End of Life-Inventory Problem,” Prof. Hans Frenk (Sabancı University), EA-409, 1:40PM March 8 (EN)

Title: On the End of Life-Inventory Problem

Prof. Hans Frenk, Department of Engineering and Natural Sciences, Sabanci University

March 8th 2019, Friday 13:40 p.m.
EA-409

In this presentation we discuss the end-of-life inventory problem (cf.[3]) for a supplier of a product in its final phase of the service life cycle. The final phase starts when the production of the product terminates and lasts until the warranty or service obligation for the last sold item expires. In general the final phase of the product is much longer than the time the product is in production. Due to uncertainty of the availability of spare parts for the out-of production product the supplier places at the beginning of this final phase a final order for these spare parts. At any time during the final phase the supplier may also decide to switch to an alternative and more cost effective service policy. This alternative policy may be in the form of replacing defective items with a substitutable product or offering discounts/rebates on a new generation product. In this setup, the objective is to find a final order quantity and a time to switch to an alternative policy which will minimize the total expected discounted costs of the supplier. The switching time can be a deterministic time to be determined at the start of the final phase (cf.[1]) (static model) or in the most general case a stopping time based on the realized demand for spare parts (dynamic model). In this presentation, we discuss both the static and dynamic version of this problem under a general cost structure in a continuous-time framework where the demand process for spare parts is given by a non-homogeneous Poisson process. We show for the static version that for any given deterministic stopping time the function is discrete convex in the order quantity and so it is easy to determine for each deterministic stopping time the optimal order quantity. Next to this we then apply a one dimensional Lipschitz optimization procedure over the set of deterministic stopping times to find an approximate optimal solution. For the continuous time dynamic version we also show by means of a discretization procedure (cf.[2]) how to compute the value function and determine the optimal policy among the set of all stopping times and the corresponding optimal stopping sets. This research is a joint work with former Ph.D student Sonya Javadi, Semih Onur Sezer and Morteza Pourakbar (RSM Erasmus University Rotterdam).

Bio: Hans Frenk joined the Faculty of Engineering and Natural Sciences at Sabanci University in 2009. He obtained his master degree (1979) from the Mathematics Department of Utrecht Univer-sity, the Netherlands and his Ph.D degree (1984) from Erasmus University Rotterdam. His Ph.D thesis was on the rate of convergence of the renewal measure to the Lebesgue measure (renewal theory and regenerative processes) for interarrival times having a subexponential distribution using the theory of Banach algebras. Next to this he also obtained a Bachelor degree in Econometrics (1982) from Erasmus University Rotterdam. Through the years his main research interest is in optimization, convex and quasiconvex analysis, stochastic processess and stochastic control prob-lems and applying those techniques to problems in Management Science and Engineering (main-tenance, inventory control, revenue management and pricing, health care). Recently his reserch interest is in the statistical estimation of probability laws of stochastic processess. Some of his papers jointly written with coauthors appeared in Production and Operations Management, Man-agement Science, Operations Research, Transportation Science, Advances of Applied Probability, Journal of Applied Probability, Journal of Optimization Theory and Applications. Mathematics of Operations Research, Mathematical Programming and Discrete Applied Mathematics.