On the end-of-life inventory problem

We consider the so-called End-of-Life inventory problem for a manufacturer of spare parts in the final phase of the service life cycle. The final phase starts when the part production is terminated and continues until the last service contract expires. One of the most popular tactics to cope with this problem is to place a sufficient volume of spare parts at the beginning of the final phase which is called the nal order quantity. Then the repair-replacement policy serves the costumers by repairing or replacing the defective items. On the other hand, nowadays, a considerable price erosion happens for the products while repair and service costs stay steady over time. If so, it is more cost effective to consider an alternative policy to meet the service demands after some time. This policy may offer the costumers a new product of similar type or a discount on a next generation product. In this setup, the purpose is to find an optimal pair of final order quantity and switching time to an alternative policy which minimizes the total expected discounted costs. We study this problem under the static and dynamic approaches which require different mathematical techniques

On the principle of Lagrange in optimization theory and its application in transportation and location problems

In mathematical optimzation, the Lagrangian approach is a general method to find an optimal solution of a finite( infinite) dimensional constrained continuous optimization problem. This method has beeen introduced by the Italian mathematician Joseph-Louıs Lagrange in 1775 in a series of letters to Euler.This approach became known under the name the principle of Lagrange and was also applied much later to integer programming problems The basic idea behind this method is to replace a constraianed optimization problem by a sequence of easier solvable optimzation probelems having fewer constraints and penalizing the deletıon of the original constraints by replacing the original objective function. To select the best penalization,the so-called Lagrangian dual function needs to be optimized and a possible algorithm to do so is called the subgradient algorithm. This method is discussed in detail at the end of this chapter.The Lagrangian approach led to the introduction of dual optimization problems in nonlinear programming and recently to the development of interior point methods and the identification of polynomially solvable casses of continuous optimiAlso it had its impact on how to construct algorithms to generate approximate solutions of integer programming problems.In this chapter, we discuss in the first part the main ideas behind this approach for any type of finite dimensional optimization problem. In the remaining parts of this chapter we focus in more detail how this approach is used in continuous optimization problems and show its full impact on the so-called K-convex continuous optimzation problems. Also we consider its application within linear integer programming problems and show how it is used to solve these type of problems. To illustrate its application to the well-known integer programming problems we consider in the final section its application to some vehicle routing and location models. As such this chapter shouldf be regarded as an introduction to duality theory for less mathematically oriented readers proving at the same time most of the results using the simplest possible proofs

On computing the multivariate poisson probability distribution

Within the theory of non-negative integer valued multivariate infinitely divisible distributions, the multivariate Poisson distribution plays a key role. As in the univariate case, any non-negative integer valued infinitely divisible multivariate distribution can be approximated by a multivariate distribution belonging to the compound Poisson family. The multivariate Poisson distribution is an important member of this family. In recent years, the multivariate Poisson distributions also has gained practical importance, since they serve as models to describe counting data having a positive covariance structure. However, due to the computational complexity of computing the multivariate Poisson probability mass function (pmf) and its corresponding cumulative distribution function (cdf), their use within these counting models is limited. Since most of the theoretical properties of the multivariate Poisson probability distribution seem already to be known, the main focus of this paper is on proposing more efficient algorithms to compute this pmf. Using a well known property of a Poisson multivariate distributed random vector, we propose in this paper a direct approach to calculate this pmf based on finding all solutions of a system of linear Diophantine equations. This new approach complements an already existing procedure depending on the use of recurrence relations existing for the pmf. We compare our new approach with this already existing approach applied to a slightly different set of recurrence relations which are easier to evaluate. A proof of this new set of recurrence relations is also given. As a result, several algorithms are proposed where some of them are based on the new approach and some use the recurrence relations. To test these algorithms, we provide an extensive analysis in the computational section. Based on the experiments in this section, we conclude that the approach finding all solutions of a set of linear Diophantine equations is computationally more efficient than the approach using the recurrence relations to evaluate the pmf of a multivariate Poisson distributed random vector

An optimal stopping approach for the end-of-life inventory problem

We consider the end-of-life inventory problem for the supplier of a product in its final phase of the service life cycle. This phase starts when the production of the items stops and continues until the warranty of the last sold item expires. At the beginning of this phase the supplier places a final order for spare parts to serve customers coming with defective items. 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 substitutable products or offering discounts/rebates on the new generation ones. 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 is a stopping time and is based on the realization of the arrival process of defective items. In this paper, we study this problem under a general cost structure in a continuous-time framework where the arrival of customers is given by a non-homogeneous Poisson process. We show in detail how to compute the value function, and illustrate our approach on numerical examples

An optimal stopping approach for the end-of-life inventory problem

Sezer, Semih Onur/0000-0003-4215-7703We consider the end-of-life inventory problem for the supplier of a product in its final phase of the service life cycle. This phase starts when the production of the items stops and continues until the warranty of the last sold item expires. At the beginning of this phase the supplier places a final order for spare parts to serve customers coming with defective items. 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 substitutable products or offering discounts/rebates on the new generation ones. 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 is a stopping time and is based on the realization of the arrival process of defective items. In this paper, we study this problem under a general cost structure in a continuous-time framework where the arrival of customers is given by a non-homogeneous Poisson process. We show in detail how to compute the value function, and illustrate our approach on numerical examples

An exact static solution approach for the service parts end of life inventory problem

This paper studies the spare parts end-of-life inventory problem that happens after the discontinuation of part production. A final ordering quantity is set such that the service process is sustained until all service obligations expire. Also, the price erosion of substitutable or new generation products over time makes it economically justifiable to consider switching to an alternative service policy for repair such as swapping the old product with a new one. This requires the joint optimization of the final order quantity and the time to switch from repair to an alternative service policy. To the best of our knowledge, the problem has not been optimally solved yet either in its static or dynamic formulation. In the current paper, we solve its static version as a bi-level optimization problem. We investigate the convexity of the objective function and give a computationally efficient algorithm to find an exact optimal solution up to any given numerical error level ϵ > 0. We illustrate our approach on some numerical examples and compare our results with earlier works on this problem