Innovative Applications of O.R. Pricing, relaxing and fixing under lot sizing and scheduling Luis Guimarães a,b,⇑ , Diego Klabjan b , Bernardo Almada-Lobo a a INESC TEC, Faculdade de Engenharia, Universidade do Porto, Porto, Portugal b Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, USA article info Article history: Received 14 March 2012 Accepted 17 April 2013 Available online 28 April 2013 Keywords: Lot sizing and scheduling Sequence-dependent setups Non-triangular setups Column generation MIP-based heuristics abstract We present a novel mathematical model and a mathematical programming based approach to deliver superior quality solutions for the single machine capacitated lot sizing and scheduling problem with sequence-dependent setup times and costs. The formulation explores the idea of scheduling products based on the selection of known production sequences. The model is the basis of a matheuristic, which embeds pricing principles within construction and improvement MIP-based heuristics. A partial explora- tion of distinct neighborhood structures avoids local entrapment and is conducted on a rule-based neigh- bor selection principle. We compare the performance of this approach to other heuristics proposed in the literature. The computational study carried out on different sets of benchmark instances shows the ability of the matheuristic to cope with several model extensions while maintaining a very effective search. Although the techniques described were developed in the context of the problem studied, the method is applicable to other lot sizing problems or even to problems outside this domain. Ó 2013 Elsevier B.V. All rights reserved. 1. Introduction In many production environments, production planning prob- lems involve the determination of production lot sizes and se- quence of different products on a single capacitated machine. Production lot sizes are driven by deterministic demand over the planning horizon. Switching between production runs of two dif- ferent products triggers operations, such as machine adjustments and cleaning procedures, which consume scarce production time and can cause costs due, for example, to losses in materials. Under these conditions, production sequencing must explicitly take into account for these sequence-dependent setup times and costs. In this context, the need for simultaneous lot sizing and scheduling decisions arises. Production plans are created with the objective of minimizing the overall costs consisting mainly of holding and setup costs, while satisfying the available capacity in each time period from which the expenditure in setup times is deducted. Examples of industries where these decisions must be taken concurrently are chemicals, drugs and pharmaceuticals, pulp and paper, textiles, foundries, glass container, and food and beverage, among many others (see Clark et al. (2011)). Tackling real world problems requires to address special cases that may occur by introducing additional features into mathemati- cal models. Among these realistic features are changeovers that do not respect the triangle inequality. When setups obey the triangle inequality with respect to both the setup time and costs, i.e. it is more efficient to change directly between two products than via a third product, at most one setup for each product per time period oc- curs. In some industries, contamination occurs when changing from one product to another implying additional cleaning operations. If a ‘cleaning’ or shortcut product can absorb contamination while being produced, replacing the cleaning operations, non-triangular setups appear. In their presence, models have to allow for more than one production run of each product per time period as it potentially re- duces setup times and costs. Many examples of this type are known in the chemical, pharmaceutical, food and dyeing industries. Mixed integer programming (MIP) models are unable to solve relevant size instances of the problem, suffering from its computa- tional intractability (they are NP-hard by Bitran and Yanasse (1992)). State-of-the-art optimization engines either fail to gener- ate feasible solutions to this problem or take a prohibitively large amount of computational time, as the computational experiments presented herein attest. Therefore, solving this class of problems requires the use of efficient solution approaches. Mathematical pro- gramming-based heuristics (Ball, 2011), also known as matheuris- tics (Maniezzo et al., 2010), are algorithms which integrate exact and heuristic search techniques. Exact algorithms probably achieve optimal or quasi-optimal solutions, yet the size of tractable problems is limited. On the other hand, metaheuristics (heuristic search) are tailored to solve large-scale combinatorial optimization problems exploring large size neighborhoods efficiently. The under- laying idea of matheuritics is to seek the best trade-off between the efficacy of exact approaches and the efficiency of metaheuristics. 0377-2217/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2013.04.030 ⇑ Corresponding author at: INESC TEC, Faculdade de Engenharia, Universidade do Porto, Porto, Portugal. Tel.: +351 22 508 21 82. E-mail addresses: guimaraes.luis@fe.up.pt (L. Guimarães), d-klabjan@northwes- tern.edu (D. Klabjan), almada.lobo@fe.up.pt (B. Almada-Lobo). European Journal of Operational Research 230 (2013) 399–411 Contents lists available at SciVerse ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor