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Following our previous paper (Brahimi, N., Dauzère-Pérès, S., Najid, N. M., and Nordli, A. Single item lot sizing problems. European Journal of Operational Research, 168(1):1-16, 2006), we present an updated and extended survey of Single-Item Lot-Sizing Problems with focus on publications from 2004 to 2016. Exact and heuristic solution procedures are surveyed. A concise and comprehensive summary of different extensions of the problem is given. The classification of the extensions is based on different characteristics such as resource limitations, assumptions on demand and cost structure. The large number of surveyed papers shows the increased interest of researchers in lot-sizing problems in general and in single-item problems in particular. The survey and the proposed classification should help researchers to identify new research topics, to propose relevant problems and/or novel solution approaches.Since the publication of the seminal work of Wagner and Whitin (1958) in 1958, a lot of research has been conducted on the Single-Item Lot-Sizing Problem (SILSP) and its extensions. The SILSP is interesting in itself to model some tactical production and distribution planning problems, e.g. when planning the replenishment of one raw material with fixed and variable ordering and transportation costs. It is also important to efficiently solve the SILSP because it appears as a subproblem in the solution procedures of many complex lot-sizing problems such as capacitated multi-item problems.Extensions of the SILSP include production capacity, remanufacturing, backlogging, lost sales, demand and production time windows, bounded inventory, perishability, etc. The large number of publications on the SILSP and the large variety of its applications and extensions call for a literature review to put together all the relevant references and classify them. This will allow researchers in the field to more easily identify research trends and gaps to be filled in lot sizing in particular and in production ACCEPTED MANUSCRIPT and distribution planning in general. This survey is an extension and an update of a previous survey published by three of the authors in European Journal of Operational Research in 2006(Brahimi et al. (2006). Since this survey was published, there has been a growing interest in the study of the SILSP.There were at least 100 publications on the SILSP and its extensions in the past 8 years. The objective of this paper is to update the previous literature review and enrich it with the new research in the area. To avoid repetition and for space reasons, most of the references and details in (Brahimi et al. (2006)) are omitted.The SILSP can be defined as a planning problem in which there is time-varying demand for a single product over a planning horizon of T periods. The objective is to determine periods where production will take place and the quantities that have to be produced in these periods. The total production should satisfy the demands while minimizing the total costs. The basic costs are the unit producti...

This research concerns a new family of capacitated multi-item lot-sizing problems, namely, lot-sizing problems with time windows. Two classes of the problem are analyzed and solved using different Lagrangian heuristics. Capacity constraints and a subset of time window constraints are relaxed resulting in particular single-item time window problems that are solved in polynomial time. Other relaxations leading to the classical Wagner-Whitin problem are also tested. Several smoothing heuristics are implemented and tested, and their results are compared. The gaps between lower and upper bounds for most problems are very small (less than 1%). Moreover, the proposed algorithms are robust and do not seem to be too affected when different parameters of the problem are varied.

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