On the equivalence of strong formulations for capacitated multi-level lot sizing problems with setup times

Several mixed integer programming formulations have been proposed for modeling capacitated multi-level lot sizing problems with setup times. These formulations include the so-called facility location formulation, the shortest route formulation, and the inventory and lot sizing formulation with (l,S) inequalities. In this paper, we demonstrate the equivalence of these formulations when the integrality requirement is relaxed for any subset of binary setup decision variables. This equivalence has significant implications for decomposition-based methods since same optimal solution values are obtained no matter which formulation is used. In particular, we discuss the relax-and-fix method, a decomposition-based heuristic used for the efficient solution of hard lot sizing problems. Computational tests allow us to compare the effectiveness of different formulations using benchmark problems. The choice of formulation directly affects the required computational effort, and our results therefore provide guidelines on choosing an effective formulation during the development of heuristic-based solution procedures

An optimization framework for solving capacitated multi-level lot-sizing problems with backlogging

This paper proposes two new mixed integer programming models for capacitated multi-level lot-sizing problems with backlogging, whose linear programming relaxations provide good lower bounds on the optimal solution value. We show that both of these strong formulations yield the same lower bounds. In addition to these theoretical results, we propose a new, effective optimization framework that achieves high quality solutions in reasonable computational time. Computational results show that the proposed optimization framework is superior to other well-known approaches on several important performance dimensions

A priori reformulations for joint rolling-horizon scheduling of materials processing and lot-sizing problem

In many production processes, a key material is prepared and then transformed into different final products. The lot sizing decisions concern not only the production of final products, but also that of material preparation in order to take account of their sequence-dependent setup costs and times. The amount of research in recent years indicates the relevance of this problem in various industrial settings. In this paper, facility location reformulation and strengthening constraints are newly applied to a previous lot-sizing model in order to improve solution quality and computing time. Three alternative metaheuristics are used to fix the setup variables, resulting in much improved performance over previous research, especially regarding the use of the metaheuristics for larger instances

Modeling Industrial Lot Sizing Problems: A Review

In this paper we give an overview of recent developments in the field of modeling single-level dynamic lot sizing problems. The focus of this paper is on the modeling various industrial extensions and not on the solution approaches. The timeliness of such a review stems from the growing industry need to solve more realistic and comprehensive production planning problems. First, several different basic lot sizing problems are defined. Many extensions of these problems have been proposed and the research basically expands in two opposite directions. The first line of research focuses on modeling the operational aspects in more detail. The discussion is organized around five aspects: the set ups, the characteristics of the production process, the inventory, demand side and rolling horizon. The second direction is towards more tactical and strategic models in which the lot sizing problem is a core substructure, such as integrated production-distribution planning or supplier selection. Recent advances in both directions are discussed. Finally, we give some concluding remarks and point out interesting areas for future research

A Conceptual Model for Integrating Transport Planning: MRP IV

In this article, a conceptual model, called MRP IV, is proposed in order to serve as a reference to develop a new production technology that integrates material planning decisions, production resource capacities and supply chain transport for the purpose of avoiding the suboptimization of these plans which, today, are usually generated sequentially and independently. This article aim is twofold: (1) it identifies the advances and deficiencies in the MRP calculations, mainly based on the dynamic multi-level capacitated lot-sizing problem (MLCLSP); and (2) it proposes a conceptual model, defining the inputs, outputs, modeling and solution approaches, to overcome the deficiencies identified in current MRP systems and act as a baseline to propose resolution models and algorithms required to develop MRP IV as a decision-making system. © 2012 IFIP International Federation for Information Processing. Mula, J.; Díaz-Madroñero Boluda, FM.; Peidro Payá, D. (2012). A conceptual model for integrating transport planning: MRP IV. En IFIP Advances in Information and Communication Technology. Springer. (384):54-65. doi:10.1007/978-3-642-33980-6_7 Senia 54 65 384 Document type: Part of book or chapter of boo

A theoretical and computational analysis of lot-sizing in remanufacturing with separate setups

Due to the stricter government regulations on end-of-life product treatment and the increasing public awareness towards environmental issues, remanufacturing has been a significantly growing industry over the last decades, offering many potential business opportunities. In this paper, we investigate a crucial problem apparent in this industry, the remanufacturing lot-sizing problem with separate setups. We first discuss two reformulations of this problem, and remark an important property with regards to their equivalence. Then, we present a theoretical investigation of a related subproblem, where our analysis indicates that a number of flow cover inequalities are strong for this subproblem under some general conditions. We then investigate the computational effectiveness of the alternative methods discussed for the original problem. Detailed numerical results are insightful for the practitioner, indicating that in particular when the return variability increases or when the remanufacturing setup costs decrease relevant to manufacturing setup costs, the flow covers can be very effective

Integrated Production-Inventory Models in Steel Mills Operating in a Fuzzy Environment

Despite the paramount importance of the steel rolling industry and its vital contributions to a nation’s economic growth and pace of development, production planning in this industry has not received as much attention as opposed to other industries. The work presented in this thesis tackles the master production scheduling (MPS) problem encountered frequently in steel rolling mills producing reinforced steel bars of different grades and dimensions. At first, the production planning problem is dealt with under static demand conditions and is formulated as a mixed integer bilinear program (MIBLP) where the objective of this deterministic model is to provide insights into the combined effect of several interrelated factors such as batch production, scrap rate, complex setup time structure, overtime, backlogging and product substitution, on the planning decisions. Typically, MIBLPs are not readily solvable using off-the-shelf optimization packages necessitating the development of specifically tailored solution algorithms that can efficiently handle this class of models. The classical linearization approaches are first discussed and employed to the model at hand, and then a hybrid linearization-Benders decomposition technique is developed in order to separate the complicating variables from the non-complicating ones. As a third alternative, a modified Branch-and-Bound (B&B) algorithm is proposed where the branching, bounding and fathoming criteria differ from those of classical B&B algorithms previously established in the literature. Numerical experiments have shown that the proposed B&B algorithm outperforms the other two approaches for larger problem instances with savings in computational time amounting to 48%. The second part of this thesis extends the previous analysis to allow for the incorporation of internal as well as external sources of uncertainty associated with end customers’ demand and production capacity in the planning decisions. In such situations, the implementation of the model on a rolling horizon basis is a common business practice but it requires the repetitive solution of the model at the beginning of each time period. As such, viable approximations that result in a tractable number of binary and/or integer variables and generate only exact schedules are developed. Computational experiments suggest that a fair compromise between the quality of the solutions and substantial computational time savings is achieved via the employment of these approximate models. The dynamic nature of the operating environment can also be captured using the concept of fuzzy set theory (FST). The use of FST allows for the incorporation of the decision maker’s subjective judgment in the context of mathematical models through flexible mathematical programming (FMP) approach and possibilistic programming (PP) approach. In this work, both of these approaches are combined where the volatility in demand is reflected by a flexible constraint expressed by a fuzzy set having a triangular membership function, and the production capacity is expressed as a triangular fuzzy number. Numerical analysis illustrates the economical benefits obtained from using the fuzzy approach as compared to its deterministic counterpart

Lot-Sizing of Several Multi-Product Families

Production planning problems and its variants are widely studied in operations management and optimization literature. One variation that has not garnered much attention is the presence of multiple production families in a coordinated and capacitated lot-sizing setting. While its single-family counterpart has been the subject of many advances in formulations and solution techniques, the latest published research on multiple family problems was over 25 years ago (Erenguc and Mercan, 1990; Mercan and Erenguc, 1993). Chapter 2 begins with a new formulation for this coordinated capacitated lot-sizing problem for multiple product families where demand is deterministic and time-varying. The problem considers setup and holding costs, where capacity constraints limit the number of individual item and family setup times and the amount of production in each period. We use a facility location reformulation to strengthen the lower bound of our demand-relaxed model. In addition, we combine Benders decomposition with an evolutionary algorithm to improve upper bounds on optimal solutions. To assess the performance of our approach, single-family problems are solved and results are compared to those produced by state-of-the-art heuristics by de Araujo et al. (2015) and Süral et al. (2009). For the multi-family setting, we first create a standard test bed of problems, then measure the performance of our heuristic against the SDW heuristic of Süral et al. (2009), as well as a Lagrangian approach. We show that our Benders approach combined with an evolutionary algorithm consistently achieves better bounds, reducing the duality gap compared to other single-family methods studied in the literature. Lot-sizing problems also exist within a vendor-managed-inventory setting, with production-planning, distribution and vehicle routing problems all solved simultaneously. By considering these decisions together, companies achieve reduced inventory and transportation costs compared to when these decisions are made sequentially. We present in Chapter 3 a branch-and-cut algorithm to tackle a production-routing problem (PRP) consisting of multiple products and customers served by a heterogeneous fleet of vehicles. To accelerate the performance of this algorithm, we also construct an upper bounding heuristic that quickly solves production-distribution and routing subproblems, providing a warm-start for the branch-and-cut procedure. In four scenarios, we vary the degree of flexibility in demand and transportation by considering split deliveries and backorders, two settings that are not commonly studied in the literature. We confirm that our upper bounding procedure generates high quality solutions at the root node for reasonably-sized problem instances; as time horizons grow longer, solution quality degrades slightly. Overall costs are roughly the same in these scenarios, though cost proportions vary. When backorders are not allowed (Scenarios 1 and 3), inventory holding costs account for over 90% of total costs and transportation costs contribute less than 0.01%. When backorders are allowed (Scenarios 2 and 4), most of the cost burden is shouldered by production, with transportation inching closer to 0.1% of total costs. In our fifth scenario for the PRP with multiple product families, we employ a decomposition heuristic for determining dedicated routes for distribution. Customers are clustered through k-means++ and a location-alloction subproblem based on their contribution to overall demand, and these clusters remain fixed over the entire planning horizon. A routing subproblem dictates the order in which to visit customers in each period, and we allow backorders in the production-distribution routine. While the branch-and-cut algorithm for Scenarios 1 through 4 quickly finds high quality solutions at the root node, Scenario 5's dedicated routes heuristic boasts high vehicle utilization and comparable overall costs with minimal computational effort

Combining Column Generation and Lagrangian Relaxation

Although the possibility to combine column generation and Lagrangian relaxation has been known for quite some time, it has only recently been exploited in algorithms. In this paper, we discuss ways of combining these techniques. We focus on solving the LP relaxation of the Dantzig-Wolfe master problem. In a first approach we apply Lagrangian relaxation directly to this extended formulation, i.e. no simplex method is used. In a second one, we use Lagrangian relaxation to generate new columns, that is Lagrangian relaxation is applied to the compact for-mulation. We will illustrate the ideas behind these algorithms with an application in Lot-sizing. To show the wide applicability of these techniques, we also discuss applications in integrated vehicle and crew scheduling, plant location and cutting stock problems.column generation;Lagrangean relaxation;cutting stock problem;lotsizing;vehicle and crew scheduling