On the generalized last time buy problem: an application of pseudo-deterministic stopping times to end-of life management

Document Type : Original Article

Authors

JBG Frenk ¹

Mustafa Hekimoğlu ²

Ayda Amniattalab ³

¹ ‎‎Industrial Engineering Department‎, ‎Faculty of Engineering‎, ‎Gebze Technical University,‎ ‎Kocaeli‎, Turkey

² ‎Department of Industrial Engineering‎, ‎Kadir Has University, ‎Istanbul‎, ‎ Turkey

³ ‎‎Department of Engineering and Natural Sciences‎, ‎Sabancı University, ‎Gebze, Turkey‎

10.22067/smps.2025.92293.1042

Abstract

‎The End-of-life phase is the longest phase of the different life cycles of a product‎. ‎This phase‎, ‎characterized by a last time buy order‎, ‎is known for increasing supply risk and decreasing demand rate‎.  In this paper‎, ‎starting initially with a repair replacement policy the manufacturer optimizes the order size of the last time buy and selects a time switching to another policy substituting failed components.The defective items stochastic arrival process is given by a non-homogenous Poisson process and the switch time is modeled as a stopping time of this demand process‎. ‎Since the optimal switching time within the class of stopping times can be difficult to implement‎, ‎we introduce in this paper the set of so-called pseudo-deterministic stopping times being the minimum of a deterministic stopping time and the time to depletion of the spare part inventory‎. ‎We show that the optimal pseudo-deterministic stopping time satisfies some nice properties under realistic assumptions on the arrival rate function of the Poissonian defective items arrival process and the substitution cost function of the alternative policy‎. ‎Using these properties an efficient algorithm is proposed to determine the optimal pseudo-deterministic stopping time‎. ‎Although in general an optimal stopping time (optimal among the class of all stopping times) does not belong to the class of pseudo-deterministic stopping times‎, ‎we numerically show and explain in the presence of high penalty costs that the objective value of the optimal pseudo-deterministic stopping time is close to the objective value of the optimal stopping time‎.

Keywords

End-of-life inventory problem

‎Martingales‎

‎Non-homogeneous Poisson process‎

‎Spare parts inventory management

Stopping time

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