Mixed integer programming in production planning with backlogging and setup carryover : modeling and algorithms

This paper proposes a mixed integer programming formulation for modeling the capacitated multi-level lot sizing problem with both backlogging and setup carryover. Based on the model formulation, a progressive time-oriented decomposition heuristic framework is then proposed, where improvement and construction heuristics are effectively combined, therefore efficiently avoiding the weaknesses associated with the one-time decisions made by other classical time-oriented decomposition algorithms. Computational results show that the proposed optimization framework provides competitive solutions within a reasonable time

A heuristic approach for big bucket multi-level production planning problems

Multi-level production planning problems in which multiple items compete for the same resources frequently occur in practice, yet remain daunting in their difficulty to solve. In this paper, we propose a heuristic framework that can generate high quality feasible solutions quickly for various kinds of lot-sizing problems. In addition, unlike many other heuristics, it generates high quality lower bounds using strong formulations, and its simple scheme allows it to be easily implemented in the Xpress-Mosel modeling language. Extensive computational results from widely used test sets that include a variety of problems demonstrate the efficiency of the heuristic, particularly for challenging problems

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 review of discrete-time optimization models for tactical production planning

This is an Accepted Manuscript of an article published in International Journal of Production Research on 27 Mar 2014, available online: http://doi.org/10.1080/00207543.2014.899721[EN] This study presents a review of optimization models for tactical production planning. The objective of this research is to identify streams and future research directions in this field based on the different classification criteria proposed. The major findings indicate that: (1) the most popular production-planning area is master production scheduling with a big-bucket time-type period; (2) most of the considered limited resources correspond to productive resources and, to a lesser extent, to inventory capacities; (3) the consideration of backlogs, set-up times, parallel machines, overtime capacities and network-type multisite configuration stand out in terms of extensions; (4) the most widely used modelling approach is linear/integer/mixed integer linear programming solved with exact algorithms, such as branch-and-bound, in commercial MIP solvers; (5) CPLEX, C and its variants and Lindo/Lingo are the most popular development tools among solvers, programming languages and modelling languages, respectively; (6) most works perform numerical experiments with random created instances, while a small number of works were validated by real-world data from industrial firms, of which the most popular are sawmills, wood and furniture, automobile and semiconductors and electronic devices.This study has been funded by the Universitat Politècnica de València projects: ‘Material Requirement Planning Fourth Generation (MRPIV)’ (Ref. PAID-05-12) and ‘Quantitative Models for the Design of Socially Responsible Supply Chains under Uncertainty Conditions. Application of Solution Strategies based on Hybrid Metaheuristics’ (PAID-06-12).Díaz-Madroñero Boluda, FM.; Mula, J.; Peidro Payá, D. (2014). A review of discrete-time optimization models for tactical production planning. International Journal of Production Research. 52(17):5171-5205. doi:10.1080/00207543.2014.899721S51715205521

A relax-and-fix with fix-and-optimize heuristic applied to multi-level lot-sizing problems

In this paper, we propose a simple but efficient heuristic that combines construction and improvement heuristic ideas to solve multi-level lot-sizing problems. A relax-and-fix heuristic is firstly used to build an initial solution, and this is further improved by applying a fix-and-optimize heuristic. We also introduce a novel way to define the mixed-integer subproblems solved by both heuristics. The efficiency of the approach is evaluated solving two different classes of multi-level lot-sizing problems: the multi-level capacitated lot-sizing problem with backlogging and the two-stage glass container production scheduling problem (TGCPSP). We present extensive computational results including four test sets of the Multi-item Lot-Sizing with Backlogging library, and real-world test problems defined for the TGCPSP, where we benchmark against state-of-the-art methods from the recent literature. The computational results show that our combined heuristic approach is very efficient and competitive, outperforming benchmark methods for most of the test problems

Lot-Sizing Problem for a Multi-Item Multi-level Capacitated Batch Production System with Setup Carryover, Emission Control and Backlogging using a Dynamic Program and Decomposition Heuristic

Wagner and Whitin (1958) develop an algorithm to solve the dynamic Economic Lot-Sizing Problem (ELSP), which is widely applied in inventory control, production planning, and capacity planning. The original algorithm runs in O(T^2) time, where T is the number of periods of the problem instance. Afterward few linear-time algorithms have been developed to solve the Wagner-Whitin (WW) lot-sizing problem; examples include the ELSP and equivalent Single Machine Batch-Sizing Problem (SMBSP). This dissertation revisits the algorithms for ELSPs and SMBSPs under WW cost structure, presents a new efficient linear-time algorithm, and compares the developed algorithm against comparable ones in the literature. The developed algorithm employs both lists and stacks data structure, which is completely a different approach than the rest of the algorithms for ELSPs and SMBSPs. Analysis of the developed algorithm shows that it executes fewer number of basic actions throughout the algorithm and hence it improves the CPU time by a maximum of 51.40% for ELSPs and 29.03% for SMBSPs. It can be concluded that the new algorithm is faster than existing algorithms for both ELSPs and SMBSPs. Lot-sizing decisions are crucial because these decisions help the manufacturer determine the quantity and time to produce an item with a minimum cost. The efficiency and productivity of a system is completely dependent upon the right choice of lot-sizes. Therefore, developing and improving solution procedures for lot-sizing problems is key. This dissertation addresses the classical Multi-Level Capacitated Lot-Sizing Problem (MLCLSP) and an extension of the MLCLSP with a Setup Carryover, Backlogging and Emission control. An item Dantzig Wolfe (DW) decomposition technique with an embedded Column Generation (CG) procedure is used to solve the problem. The original problem is decomposed into a master problem and a number of subproblems, which are solved using dynamic programming approach. Since the subproblems are solved independently, the solution of the subproblems often becomes infeasible for the master problem. A multi-step iterative Capacity Allocation (CA) heuristic is used to tackle this infeasibility. A Linear Programming (LP) based improvement procedure is used to refine the solutions obtained from the heuristic method. A comparative study of the proposed heuristic for the first problem (MLCLSP) is conducted and the results demonstrate that the proposed heuristic provide less optimality gap in comparison with that obtained in the literature. The Setup Carryover Assignment Problem (SCAP), which consists of determining the setup carryover plan of multiple items for a given lot-size over a finite planning horizon is modelled as a problem of finding Maximum Weighted Independent Set (MWIS) in a chain of cliques. The SCAP is formulated using a clique constraint and it is proved that the incidence matrix of the SCAP has totally unimodular structure and the LP relaxation of the proposed SCAP formulation always provides integer optimum solution. Moreover, an alternative proof that the relaxed ILP guarantees integer solution is presented in this dissertation. Thus, the SCAP and the special case of the MWIS in a chain of cliques are solvable in polynomial time

Design of a network of reusable logistic containers

In this paper, we consider the management of the return flows of empty logistic containers that accumulate at the customer’s sites. These containers must be brought back to the factories in order to sustain future expeditions. We consider a network composed of several factories and several customers in which the return flows are independent of the delivery flows. The models and their solutions aim at finding to which factory the contain- ers have to be brought back and at which frequency. These frequencies directly define the volume of logistic containers to hold in the network. We consider fixed transportation costs depending on the locations of the customers and of the factories and linear holding costs for the inventory of logistic containers. The analysis also provides insight on the benefit of pooling the containers among different customers and/or factories.supply chain management, returnable items, reverse logistic, economic order quantity, network design

Variable neighborhood search for the multi-level capacitated lotsizing problem

Das dynamische mehrstufige kapazitierte Losgrößenproblem (MLCLSP) behandelt im Rahmen der Produktionsplanung die wichtige Entscheidung über die optimalen Losgrößen, angefangen bei Endprodukten über Komponenten bis hin zu Rohstoffen, bei gleichzeitiger Berücksichtigung beschränkter Kapazitäten der zur Produktion benötigten Ressourcen. Da es sich um ein NP-schweres Problem handelt, stoßen exakte Lösungsverfahren an ihre Grenzen, sobald die Problemdimensionen ein größeres – man könnte durchaus sagen realistisches – Ausmaß erreichen. In der Praxis dominieren deshalb Methoden, die die Losgrößen der einzelnen Produkte sequenziell festlegen und überdies etwaige Kapazitätsbeschränkungen im Nachhinein, falls überhaupt, berücksichtigen. In der Literatur finden sich zahlreiche approximative Ansätze zur Lösung dieses komplexen betriebswirtschaftlichen Problems. Lokale Suche und auf ihr basierende Metaheuristiken stellen vielversprechende Werkzeuge dar, um die Defizite der aktuell eingesetzten Trial-and-Error Ansätze zu beheben und letzten Endes zulässige sowie kostenoptimale Produktionspläne zu erstellen. Die in dieser Diplomarbeit vorgestellte Studie beschäftigt sich mit lokalen Suchverfahren für das MLCLSP. Acht Nachbarschaftsstrukturen, die sich aus einer Veränderung der Rüstvariablen ergeben, werden präsentiert und evaluiert. Grundlegende Optionen bei der Gestaltung eines iterativen Verbesserungsverfahrens, wie beispielsweise unterschiedliche Schrittfunktionen oder die temporäre Berücksichtigung unzulässiger Lösungen, werden getestet und verglichen. Obwohl nur die Switch Nachbarschaft, die durch das Ändern einer einzigen Rüstvariable definiert wird, wirklich überzeugende Resultate liefert, können die übrigen Nachbarschaftsstrukturen durchaus als Perturbationsmechanismen im Rahmen einer Variablen Nachbarschaftssuche (VNS) zum Einsatz kommen. Die Implementierung dieser Metaheuristik, geprägt von den Ergebnissen der einfachen lokalen Suchverfahren, kann allerdings nicht vollkommen überzeugen. Die entwickelte VNS Variante kann die Lösungsgüte anderer zum Vergleich herangezogener Lösungsverfahren nicht erreichen und benötigt relativ lange Laufzeiten. Andererseits sind die Ergebnisse mit einer durchschnittlichen Abweichung zur besten bekannten Lösung von etwa vier Prozent über sämtliche untersuchte Problemklassen weit entfernt von einem Totalversagen. Es überwiegt der Eindruck, dass es sich um eine robuste Methode handelt, die in der Lage ist, Lösungen von hoher, teils sehr hoher Qualität nicht nur in Ausnahmefällen zu liefern. Etwaige Nachjustierungen könnten das Verfahren durchaus zu einem ernstzunehmenden Konkurrenten für bereits existierende Lösungsmethoden für das MLCLSP machen.The Multi-Level Capacitated Lotsizing Problem (MLCLSP) depicts the important decision in production planning of determining adequate lot sizes from final products onward, to subassemblies, parts and raw materials, all the while assuming limited capacities of the resources employed for manufacture. It is an NP-hard problem where exact methods fail in solving larger – one could say realistic – problem instances. Sequential approaches that tackle the problem item by item and postpone capacity considerations dominate current practice; approximate solution methods abound throughout the literature. Local search and metaheuristics based on it constitute a class of approximate methods well-equipped to take on the challenge of eventually replacing the trial-and-error process that impedes manufacturing companies in establishing feasible and cost-minimal production plans. This thesis presents a study of local search based procedures for solving the MLCLSP. Eight different neighborhood structures, resulting from manipulations of the setup variables, are devised and evaluated. Fundamental options when designing an iterative improvement algorithm, such as best-improvement versus first-improvement step functions or the inclusion of infeasible solutions during the search are explored and compared. Although only the Switch move, which alters the value of a single setup value, is convincing as a stand-alone neighborhood structure, the other neighborhoods can in any case be employed for the perturbation of solutions during the shaking step of a Variable Neighborhood Search (VNS). The implementation of this metaheuristic, shaped by the findings from testing the basic local search variants, led to mixed results. The procedure designed to tackle the MLCLSP cannot outperform the compared heuristics. Neither does it produce results that are terribly off – the average gap to the best known solutions settles around four percent over all problem classes tested. Nonetheless, the impression is supported that the VNS procedure is a robust method leading to good, sometimes even very good solutions at a regular basis that is amenable to further adjustments and thus eventually becoming a serious competitor for existing methods dealing with multi-level capacitated lotsizing decisions

Improved formulations, heuristics and metaheuristics for the dynamic demand coordinated lot-sizing problem

Coordinated lot sizing problems, which assume a joint setup is shared by a product family, are commonly encountered in supply chain contexts. Total system costs include a joint set-up charge each time period any item in the product family is replenished, an item set-up cost for each item replenished in each time period, and inventory holding costs. Silver (1979) and subsequent researchers note the occurrence of coordinated replenishment problems within manufacturing, procurement, and transportation contexts. Due to their mathematical complexity and importance in industry, coordinated lot-size problems are frequently studied in the operations management literature. In this research, we address both uncapacitated and capacitated variants of the problem. For each variant we propose new problem formulations, one or more construction heuristics, and a simulated annealing metaheuristic (SAM). We first propose new tight mathematical formulations for the uncapacitated problem and document their improved computational efficiency over earlier models. We then develop two forward-pass heuristics, a two-phase heuristic, and SAM to solve the uncapacitated version of the problem. The two-phase and SAM find solutions with an average optimality gap of 0.56% and 0.2% respectively. The corresponding average computational requirements are less than 0.05 and 0.18 CPU seconds. Next, we propose tight mathematical formulations for the capacitated problem and evaluate their performance against existing approaches. We then extend the two-phase heuristic to solve this more general capacitated version. We further embed the six-phase heuristic in a SAM framework, which improves heuristic performance at minimal additional computational expense. The metaheuristic finds solutions with an average optimality gap of 0.43% and within an average time of 0.25 CPU seconds. This represents an improvement over those reported in the literature. Overall the heuristics provide a general approach to the dynamic demand lot-size problem that is capable of being applied as a stand-alone solver, an algorithm embedded with supply chain planning software, or as an upper-bounding procedure within an optimization based algorithm. Finally, this research investigates the performance of alternative coordinated lotsizing procedures when implemented in a rolling schedule environment. We find the perturbation metaheuristic to be the most suitable heuristic for implementation in rolling schedules

Adaptive genetic algorithm based on fuzzy reasoning for the multilevel capacitated lot-sizing problem with energy consumption in synchronizer production

The multilevel capacitated lot-sizing problem (MLCLSP) is a vital theoretical problem of production planning in discrete manufacturing. An improved algorithm based on the genetic algorithm (GA) is proposed to solve the MLCLSP. Based on the solution results, the distribution of energy consumption in a synchronous production case is analyzed. In the related literature, the GA has become a much-discussed topic in solving these kinds of problems. Although the standard GA can make up for the defects of the traditional algorithm, it will lead to the problems of unstable solution results and easy local convergence. For these reasons, this research presents an adaptive genetic algorithm based on fuzzy theory (fuzzy-GA) to solve the MLCLSP. Firstly, the solving process of the MLCLSP with the fuzzy-GA is described in detail, where algorithms for key technologies such as the capacity constraint algorithm and the algorithm of solving fitness value are developed. Secondly, the auto-encoding of decision variables for MLCLSPs is studied; within this, the decision variables of whether to produce or not are encoded into a hierarchical structure based on the bill of material; combined with external demand, the decision variables of lot-sizing are constructed. Thirdly, the adaptive optimization process of parameters of the GA for the MLCLSP based on fuzzy theory is expounded, in which membership function, fuzzy rule, and defuzzification of the MLCLSP is mainly presented. Experimental studies using the processed dataset collected from a synchronizer manufacturer have demonstrated the merits of the proposed approach, in which the energy consumption distribution of the optimized production plan is given. The optimal lot-sizing is closer to the average value of the optimal value compared with the standard GA, which indicates that the proposed fuzzy-GA approach has better convergence and stability