An O(Tˆ3) algorithm for the capacitated lot sizing problem with minimum order quantities

This paper explores a single-item capacitated lot sizing problem with minimum order quantity, which plays the role of minor set-up cost. We work out the necessary and suffcient solvability conditions and apply the general dynamic programming technique to develop an O(T³) exact algorithm that is based on the concept of minimal sub-problems. An investigation of the properties of the optimal solution structure allows us to construct explicit solutions to the obtained sub-problems and prove their optimality. In this way, we reduce the complexity of the algorithm considerably and confirm its efficiency in an extensive computational study. --production planning,capacitated lot sizing problem,single item,minimum order quantities,capacity constraints,dynamic programming

A New Dantzig-Wolfe Reformulation And Branch-And-Price Algorithm For The Capacitated Lot Sizing Problem With Set Up Times

The textbook Dantzig-Wolfe decomposition for the Capacitated LotSizing Problem (CLSP),as already proposed by Manne in 1958, has animportant structural deficiency. Imposingintegrality constraints onthe variables in the full blown master will not necessarily givetheoptimal IP solution as only production plans which satisfy theWagner-Whitin condition canbe selected. It is well known that theoptimal solution to a capacitated lot sizing problem willnotnecessarily have this Wagner-Whitin property. The columns of thetraditionaldecomposition model include both the integer set up andcontinuous production quantitydecisions. Choosing a specific set upschedule implies also taking the associated Wagner-Whitin productionquantities. We propose the correct Dantzig-Wolfedecompositionreformulation separating the set up and productiondecisions. This formulation gives the samelower bound as Manne'sreformulation and allows for branch-and-price. We use theCapacitatedLot Sizing Problem with Set Up Times to illustrate our approach.Computationalexperiments are presented on data sets available from theliterature. Column generation isspeeded up by a combination of simplexand subgradient optimization for finding the dualprices. The resultsshow that branch-and-price is computationally tractable andcompetitivewith other approaches. Finally, we briefly discuss how thisnew Dantzig-Wolfe reformulationcan be generalized to other mixedinteger programming problems, whereas in theliterature,branch-and-price algorithms are almost exclusivelydeveloped for pure integer programmingproblems.branch-and-price;Lagrange relaxation;Dantzig-Wolfe decomposition;lot sizing;mixed-integer programming

A hybrid heuristic for the multi-plant capacitated lot sizing problem with setup carry-over

This paper addresses the capacitated lot sizing problem (CLSP) with a single stage composed of multiple plants, items and periods with setup carry-over among the periods. The CLSP is well studied and many heuristics have been proposed to solve it. Nevertheless, few researches explored the multi-plant capacitated lot sizing problem (MPCLSP), which means that few solution methods were proposed to solve it. Furthermore, to our knowledge, no study of the MPCLSP with setup carry-over was found in the literature. This paper presents a mathematical model and a GRASP (Greedy Randomized Adaptive Search Procedure) with path relinking to the MPCLSP with setup carry-over. This solution method is an extension and adaptation of a previously adopted methodology without the setup carry-over. Computational tests showed that the improvement of the setup carry-over is significant in terms of the solution value with a low increase in computational time.FAPES

Solution methods for lot-sizing problems

In dieser Diplomarbeit lege ich meinen Fokus auf das Losgrößenproblem, das ein Teil der Materialbedarfsplanung ist. Das Losgrößenproblem versucht die Lagerhaltungs-, Rüst-, und Produktionskosten zu minimieren und dabei die notwendige Nachfrage zu bedienen. Da jede produzierende Firma in einer globalisierten Welt immer mehr auf Produktionsentscheidungen und Kosten achtet, ist das Losgrößenproblem von größtmöglicher Bedeutung. Nach einem theoretischen Überblick über das Losgrößenproblem werden zwei verschiedene Varianten des Losgrößenproblems abgedeckt: Erstens das sogenannte Multi-Level Capacitated Lot-Sizing (MLCLS) Problem und zweitens das sogennante Capacitated Lot-Sizing Problem with Linked Lot Sizes (CLSPL). Das CLSPL ist ein Big-Bucket-Modell, das es erlaubt Rüstzustände in die nächste Periode mitzunehmen. Desweiteren werden zwei mathematische Formulierungen auf ihre Effektivität getestet. Der von mir benutzte Lösungsansatz ist ein hybrider Algorithmus der das gegebene Problem in mehrere kleinere Subprobleme zerlegt. Dies Subprobleme werden dann mit CPLEX gelöst. Eine Ant Colony Optimization(ACO)-Metaheuristik wird dann angewendet um die Reihung der Losgrößen zu bestimmen und die Zerlegung in Subprobleme zu verbessern. Mein Lösungsansatz funktioniert sehr gut mit mittelgroßen Instanzen, hat aber Schwierigkeiten bezüglich der Lösungsgüte bei großen Instanzen. Gute Resultate werden für das CLSPL erreicht.In this thesis I focus my attention on the lot-sizing problem, which is part of the material requirements planning (MRP). A lot-sizing problem intends to minimize the inventory, setup, and production costs while meeting the required demand. Since producing firms in the globalized economy more and more pay attention to production decisions and costs the lot sizing problem is of prime importance. After a theoretical overview of the lot-sizing problems two different types of the lot-sizing problem are covered in this thesis: Firstly, the multi-level capacitated lot-sizing problem (MLCLS), and secondly the capacitated lot-sizing problem with linked lot sizes (CLSPL). The CLSPL is a big-bucket model that allows to carry over setup states from one period to the next. Furthermore, I test two different mathematical formulations for effectiveness. The solution approach I use is a hybrid algorithm which decomposes the given problem into multiple smaller subproblems. These subproblems are then solved by CPLEX. An Ant Colony Optimization (ACO) algorithm is then applied to determine the lot-sizing sequence and to improve the decomposition. My approach for the MLCLS problem works very well with medium-sized instances, but has difficulties with respect to solution quality when solving large-sized test instances. Good results are obtained for the CLSPL problem

Improvement to an existing multi-level capacitated lot sizing problem considering setup carryover, backlogging, and emission control

This paper presents a multi-level, multi-item, multi-period capacitated lot-sizing problem. The lot-sizing problem studies can obtain production quantities, setup decisions and inventory levels in each period fulfilling the demand requirements with limited capacity resources, considering the Bill of Material (BOM) structure while simultaneously minimizing the production, inventory, and machine setup costs. The paper proposes an exact solution to Chowdhury et al. (2018)\u27s[1] developed model, which considers the backlogging cost, setup carryover & greenhouse gas emission control to its model complexity. The problem contemplates the Dantzig-Wolfe (D.W.) decomposition to decompose the multi-level capacitated problem into a single-item uncapacitated lot-sizing sub-problem. To avoid the infeasibilities of the weighted problem (WP), an artificial variable is introduced, and the Big-M method is employed in the D.W. decomposition to produce an always feasible master problem. In addition, Wagner & Whitin\u27s[2] forward recursion algorithm is also incorporated in the solution approach for both end and component items to provide the minimum cost production plan. Introducing artificial variables in the D.W. decomposition method is a novel approach to solving the MLCLSP model. A better performance was achieved regarding reduced computational time (reduced by 50%) and optimality gap (reduced by 97.3%) in comparison to Chowdhury et al. (2018)\u27s[1] developed model

A New Dantzig-Wolfe Reformulation And Branch-And-Price Algorithm For The Capacitated Lot Sizing Problem With Set Up Times

The textbook Dantzig-Wolfe decomposition for the Capacitated Lot Sizing Problem (CLSP),as already proposed by Manne in 1958, has an important structural deficiency. Imposingintegrality constraints on the variables in the full blown master will not necessarily give theoptimal IP solution as only production plans which satisfy the Wagner-Whitin condition canbe selected. It is well known that the optimal solution to a capacitated lot sizing problem willnot necessarily have this Wagner-Whitin property. The columns of the traditionaldecomposition model include both the integer set up and continuous production quantitydecisions. Choosing a specific set up schedule implies also taking the associated Wagner-Whitin production quantities. We propose the correct Dantzig-Wolfe decompositionreformulation separating the set up and production decisions. This formulation gives the samelower bound as Manne's reformulation and allows for branch-and-price. We use theCapacitated Lot Sizing Problem with Set Up Times to illustrate our approach. Computationalexperiments are presented on data sets available from the literature. Column generation isspeeded up by a combination of simplex and subgradient optimization for finding the dualprices. The results show that branch-and-price is computationally tractable and competitivewith other approaches. Finally, we briefly discuss how this new Dantzig-Wolfe reformulationcan be generalized to other mixed integer programming problems, whereas in the literature,branch-and-price algorithms are almost exclusively developed for pure integer programmingproblems

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

A heuristic procedure for solving multi-plant, multi-item, multi-period capacitated lot-sizing problems

This paper presents a heuristic procedure for solving multi-plant, multi-item, capacitated lot sizing problems with inter-plant transfers. The solution procedure uses the solution for the uncapacitated problem as a starting point. A smoothing routine has been employed to remove capacity violations. The smoothing routine consists of two modules. Extensive experimentation has been conducted comparing the heuristic solution procedure and LINDO. The heuristic has been implemented on IBM 3090 mainframe using FORTRAN

On alternative mixed integer programming formulations and LP-based heuristics for lot-sizing with setup times

We address the multi-item, capacitated lot-sizing problem (CLSP) encountered in environments where demand is dynamic and to be met on time. Items compete for a limited capacity resource, which requires a setup for each lot of items to be produced causing unproductive time but no direct costs. The problem belongs to a class of problems that are difcult to solve. Even the feasibility problem becomes combinatorial when setup times are considered. This difculty in reaching optimality and the practical relevance of CLSP make it important to design and analyse heuristics to nd good solutions that can be implemented in practice. We consider certain mixed integer programming formulations of the problem and develop heuristics including a curtailed branch and bound, for rounding the setup variables in the LP solution of the tighter formulations. We report our computational results for a class of instances taken from literature

A Tabu List-Based Algorithm for Capacitated Multilevel Lot-Sizing with Alternate Bills of Materials and Co-Production Environments

[EN] The definition of lot sizes represents one of the most important decisions in production planning. Lot-sizing turns into an increasingly complex set of decisions that requires efficient solution approaches, in response to the time-consuming exact methods (LP, MIP). This paper aims to propose a Tabu list-based algorithm (TLBA) as an alternative to the Generic Materials and Operations Planning (GMOP) model. The algorithm considers a multi-level, multi-item planning structure. It is initialized using a lot-for-lot (LxL) method and candidate solutions are evaluated through an iterative Material Requirements Planning (MRP) procedure. Three different sizes of test instances are defined and better results are obtained in the large and medium-size problems, with minimum average gaps close to 10.5%.This paper shows the results of the project entitled "Algoritmo heuristico basado en listas tabu para la planificacion de la produccion en sistemas multinivel con listas de materiales alternativas y entornos de coproduccion" supported by Universidad de la Costa and Universitat Politecnica de Valencia.Romero-Conrado, AR.; Coronado-Hernandez, J.; Rius-Sorolla, G.; García Sabater, JP. (2019). A Tabu List-Based Algorithm for Capacitated Multilevel Lot-Sizing with Alternate Bills of Materials and Co-Production Environments. Applied Sciences. 9(7):1-17. https://doi.org/10.3390/app9071464S11797Karimi, B., Fatemi Ghomi, S. M. T., & Wilson, J. M. (2003). The capacitated lot sizing problem: a review of models and algorithms. Omega, 31(5), 365-378. doi:10.1016/s0305-0483(03)00059-8Martí, R., & Reinelt, G. (2010). Heuristic Methods. Applied Mathematical Sciences, 17-40. doi:10.1007/978-3-642-16729-4_2Barany, I., Van Roy, T. J., & Wolsey, L. A. (1984). Strong Formulations for Multi-Item Capacitated Lot Sizing. Management Science, 30(10), 1255-1261. doi:10.1287/mnsc.30.10.1255Eppen, G. D., & Martin, R. K. (1987). Solving Multi-Item Capacitated Lot-Sizing Problems Using Variable Redefinition. Operations Research, 35(6), 832-848. doi:10.1287/opre.35.6.832Maes, J., McClain, J. O., & Van Wassenhove, L. N. (1991). Multilevel capacitated lotsizing complexity and LP-based heuristics. European Journal of Operational Research, 53(2), 131-148. doi:10.1016/0377-2217(91)90130-nBuschkühl, L., Sahling, F., Helber, S., & Tempelmeier, H. (2008). Dynamic capacitated lot-sizing problems: a classification and review of solution approaches. OR Spectrum, 32(2), 231-261. doi:10.1007/s00291-008-0150-7Drexl, A., & Kimms, A. (1997). Lot sizing and scheduling — Survey and extensions. European Journal of Operational Research, 99(2), 221-235. doi:10.1016/s0377-2217(97)00030-1Glock, C. H., Grosse, E. H., & Ries, J. M. (2014). The lot sizing problem: A tertiary study. International Journal of Production Economics, 155, 39-51. doi:10.1016/j.ijpe.2013.12.009KUIK, R., SALOMON, M., VAN WASSENHOVE, L. N., & MAES, J. (1993). LINEAR PROGRAMMING, SIMULATED ANNEALING AND TABU SEARCH HEURISTICS FOR LOTSIZING IN BOTTLENECK ASSEMBLY SYSTEMS. IIE Transactions, 25(1), 62-72. doi:10.1080/07408179308964266Standard Price List—AMPLhttps://ampl.com/products/standard-price-list/Seeanner, F., Almada-Lobo, B., & Meyr, H. (2013). Combining the principles of variable neighborhood decomposition search and the fix&optimize heuristic to solve multi-level lot-sizing and scheduling problems. Computers & Operations Research, 40(1), 303-317. doi:10.1016/j.cor.2012.07.002Hung, Y.-F., & Chien, K.-L. (2000). A multi-class multi-level capacitated lot sizing model. Journal of the Operational Research Society, 51(11), 1309-1318. doi:10.1057/palgrave.jors.2601026Kang, Y., Albey, E., & Uzsoy, R. (2018). Rounding heuristics for multiple product dynamic lot-sizing in the presence of queueing behavior. Computers & Operations Research, 100, 54-65. doi:10.1016/j.cor.2018.07.019BERRETTA, R., FRANÇA, P. M., & ARMENTANO, V. A. (2005). METAHEURISTIC APPROACHES FOR THE MULTILEVEL RESOURCE-CONSTRAINED LOT-SIZING PROBLEM WITH SETUP AND LEAD TIMES. Asia-Pacific Journal of Operational Research, 22(02), 261-286. doi:10.1142/s0217595905000510KIMMS, A. (1996). Competitive methods for multi-level lot sizing and scheduling: tabu search and randomized regrets. International Journal of Production Research, 34(8), 2279-2298. doi:10.1080/00207549608905025Sabater, J. P. G., Maheut, J., & Garcia, J. A. M. (2013). A new formulation technique to model materials and operations planning: the generic materials and operations planning (GMOP) problem. European J. of Industrial Engineering, 7(2), 119. doi:10.1504/ejie.2013.052572Maheut, J., & Sabater, J. P. G. (2013). Algorithm for complete enumeration based on a stroke graph to solve the supply network configuration and operations scheduling problem. Journal of Industrial Engineering and Management, 6(3). doi:10.3926/jiem.550Rius-Sorolla, G., Maheut, J., Coronado-Hernandez, J. R., & Garcia-Sabater, J. P. (2018). Lagrangian relaxation of the generic materials and operations planning model. Central European Journal of Operations Research, 28(1), 105-123. doi:10.1007/s10100-018-0593-0Maheut, J., Garcia-Sabater, J. P., & Mula, J. (2012). The Generic Materials and Operations Planning (GMOP) Problem Solved Iteratively: A Case Study in Multi-site Context. IFIP Advances in Information and Communication Technology, 66-73. doi:10.1007/978-3-642-33980-6_8Maheut, J. P. D. (s. f.). Modelos y Algoritmos Basados en el Concepto Stroke para la Planificación y Programación de Operaciones con Alternativas en Redes de Suministro. doi:10.4995/thesis/10251/29290Glover, F. (1989). Tabu Search—Part I. ORSA Journal on Computing, 1(3), 190-206. doi:10.1287/ijoc.1.3.190Glover, F., Taillard, E., & Taillard, E. (1993). A user’s guide to tabu search. Annals of Operations Research, 41(1), 1-28. doi:10.1007/bf02078647Chelouah, R., & Siarry, P. (2000). Tabu Search applied to global optimization. European Journal of Operational Research, 123(2), 256-270. doi:10.1016/s0377-2217(99)00255-6Raza, S. A., Akgunduz, A., & Chen, M. Y. (2006). A tabu search algorithm for solving economic lot scheduling problem. Journal of Heuristics, 12(6), 413-426. doi:10.1007/s10732-006-6017-7Cesaret, B., Oğuz, C., & Sibel Salman, F. (2012). A tabu search algorithm for order acceptance and scheduling. Computers & Operations Research, 39(6), 1197-1205. doi:10.1016/j.cor.2010.09.018Li, X., Baki, F., Tian, P., & Chaouch, B. A. (2014). A robust block-chain based tabu search algorithm for the dynamic lot sizing problem with product returns and remanufacturing. Omega, 42(1), 75-87. doi:10.1016/j.omega.2013.03.003Li, J., & Pan, Q. (2015). Solving the large-scale hybrid flow shop scheduling problem with limited buffers by a hybrid artificial bee colony algorithm. Information Sciences, 316, 487-502. doi:10.1016/j.ins.2014.10.009Hindi, K. S. (1995). Solving the single-item, capacitated dynamic lot-sizing problem with startup and reservation costs by tabu search. Computers & Industrial Engineering, 28(4), 701-707. doi:10.1016/0360-8352(95)00027-xHindi, K. S. (1996). Solving the CLSP by a Tabu Search Heuristic. Journal of the Operational Research Society, 47(1), 151-161. doi:10.1057/jors.1996.13Gopalakrishnan, M., Ding, K., Bourjolly, J.-M., & Mohan, S. (2001). A Tabu-Search Heuristic for the Capacitated Lot-Sizing Problem with Set-up Carryover. Management Science, 47(6), 851-863. doi:10.1287/mnsc.47.6.851.9813Glover, F. (1990). Tabu Search—Part II. ORSA Journal on Computing, 2(1), 4-32. doi:10.1287/ijoc.2.1.4Overview for Create General Full Factorial Designhttps://support.minitab.com/en-us/minitab/18/help-and-how-to/modeling-statistics/doe/how-to/factorial/create-factorial-design/create-general-full-factorial/before-you-start/overview/Perttunen, J. (1994). On the Significance of the Initial Solution in Travelling Salesman Heuristics. Journal of the Operational Research Society, 45(10), 1131-1140. doi:10.1057/jors.1994.183Elaziz, M. A., & Mirjalili, S. (2019). A hyper-heuristic for improving the initial population of whale optimization algorithm. Knowledge-Based Systems, 172, 42-63. doi:10.1016/j.knosys.2019.02.010Chen, C.-F., Wu, M.-C., & Lin, K.-H. (2013). Effect of solution representations on Tabu search in scheduling applications. Computers & Operations Research, 40(12), 2817-2825. doi:10.1016/j.cor.2013.06.003Tabu List Based Algorithm Datasetshttps://github.com/alfonsoromeroc/tlba-gmo