Solving Lotsizing Problems on Parallel Identical Machines Using Symmetry Breaking Constraints

Production planning on multiple parallel machines is an interesting problem, both from a theoretical and practical point of view. The parallel machine lotsizing problem consists of finding the optimal timing and level of production and the best allocation of products to machines. In this paper we look at how to incorporate parallel machines in a Mixed Integer Programming model when using commercial optimization software. More specifically, we look at the issue of symmetry. When multiple identical machines are available, many alternative optimal solutions can be created by renumbering the machines. These alternative solutions lead to difficulties in the branch-and-bound algorithm. We propose new constraints to break this symmetry. We tested our approach on the parallel machine lotsizing problem with setup costs and times, using a network reformulation for this problem. Computational tests indicate that several of the proposed symmetry breaking constraints substantially improve the solution time, except when used for solving the very easy problems. The results highlight the importance of creative modeling in solving Mixed Integer Programming problems.Mixed Integer Programming;Formulations;Symmetry;Lotsizing

Solving Lotsizing Problems on Parallel Identical Machines Using Symmetry Breaking Constraints

Production planning on multiple parallel machines is an interesting problem, both from a theoretical and practical point of view. The parallel machine lotsizing problem consists of finding the optimal timing and level of production and the best allocation of products to machines. In this paper we look at how to incorporate parallel machines in a Mixed Integer Programming model when using commercial optimization software. More specifically, we look at the issue of symmetry. When multiple identical machines are available, many alternative optimal solutions can be created by renumbering the machines. These alternative solutions lead to difficulties in the branch-and-bound algorithm. We propose new constraints to break this symmetry. We tested our approach on the parallel machine lotsizing problem with setup costs and times, using a network reformulation for this problem. Computational tests indicate that several of the proposed symmetry breaking constraints substantially improve the solution time, except when used for solving the very easy problems. The results highlight the importance of creative modeling in solving Mixed Integer Programming problems

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

Reformulation and decomposition of integer programs

In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.Integer program, Lagrangean relaxation, column generation, branch-and-price, extended formulation, Benders' algorithm

A Local Search Algorithm for Clustering in Software as a Service Networks

In this paper we present and analyze a model for clustering in networks that offer Software as a Service (SaaS). In this problem, organizations requesting a set of applications have to be assigned to clusters such that the costs of opening clusters and installing the necessary applications in clusters are minimized. We prove that this problem is NP-hard, and model it as an Integer Program with symmetry breaking constraints. We then propose a Tabu search heuristic for situations where good solutions are desired in a short computation time. Extensive computational experiments are conducted for evaluating the quality of the solutions obtained by the IP model and the Tabu Search heuristic. Experimental results indicate that the proposed Tabu Search is promising.integer programming;complexity theory;Tabu Search;software as a service

Breaking symmetries to rescue Sum of Squares in the case of makespan scheduling

The Sum of Squares (\sos{}) hierarchy gives an automatized technique to create a family of increasingly tight convex relaxations for binary programs. There are several problems for which a constant number of rounds of this hierarchy give integrality gaps matching the best known approximation algorithms. For many other problems, however, ad-hoc techniques give better approximation ratios than \sos{} in the worst case, as shown by corresponding lower bound instances. Notably, in many cases these instances are invariant under the action of a large permutation group. This yields the question how symmetries in a formulation degrade the performance of the relaxation obtained by the \sos{} hierarchy. In this paper, we study this for the case of the minimum makespan problem on identical machines. Our first result is to show that $\Omega(n)$ rounds of \sos{} applied over the \emph{configuration linear program} yields an integrality gap of at least $1.0009$, where $n$ is the number of jobs. Our result is based on tools from representation theory of symmetric groups. Then, we consider the weaker \emph{assignment linear program} and add a well chosen set of symmetry breaking inequalities that removes a subset of the machine permutation symmetries. We show that applying $2^{\tilde{O}(1/\varepsilon^2)}$ rounds of the SA hierarchy to this stronger linear program reduces the integrality gap to $1+\varepsilon$, which yields a linear programming based polynomial time approximation scheme. Our results suggest that for this classical problem, symmetries were the main barrier preventing the \sos{}/ SA hierarchies to give relaxations of polynomial complexity with an integrality gap of~$1+\varepsilon$. We leave as an open question whether this phenomenon occurs for other symmetric problems

A Mathematical Approach to Paint Production Process Optimization

As the global paint market steadily grows, finding the most effective processing model to increase production capacity will be the best way to enhance competitiveness. Therefore, this study proposes two production environments commonly used in the paint industry: build-to-order (BTO) and the variation of a configuration-to-order (CTO), called group production, to schedule paint production. Mixed-Integer Linear Program (MILP) was solved using genetic algorithms (GA) to analyze two production environments with various products, different set-up times, and different average demand for each product. The models determine the number of batches, the size and product of each batch, and the batch sequence such that the makespan is minimized. Several numerical instances are presented to analyze the proposed models. The experimental results show that BTO production completes products faster than group production when products are simple (low variety). However, group production is more applicable to manufacturing diverse products (high variety) and mass production (high volume). Finally, the number of colors has the most significant impact on the two models, followed by the number of product types, and finally the average demand

Operations planning test bed under rolling horizons, multiproduct,multiechelon, multiprocess for capacitated production planning modelling with strokes

[EN] One of the problems when conducting research in mathematical programming models for operations planning is having an adequate database of experiments that can be used to verify advances and developments with enough factors to understand different consequences. This paper presents a test bed generator and instances database for a rolling horizons analysis for multiechelon planning, multiproduct with alternatives processes, multistroke, multicapacity with different stochastic demand patterns to be used with a stroke-like bill of materials considering production costs, setup, storage and delays for operations management. From the analysis of the operations planning obtained from this test bed, it is concluded that a product structure with an alternative process obtains the lowest total cost and the highest service level. In addition, decreasing seasonal demand could present a lower total cost than constant demand, but would generate a worse service level. This test bed will allow researchers further investigation so as to verify improvements in forecast methods, rolling horizons parameters, employed software, etc.Rius-Sorolla, G.; Maheut, J.; Estelles Miguel, S.; García Sabater, JP. (2021). Operations planning test bed under rolling horizons, multiproduct,multiechelon, multiprocess for capacitated production planning modelling with strokes. Central European Journal of Operations Research. 29:1289-1315. https://doi.org/10.1007/s10100-020-00687-5S1289131529Araujo SA, Arenales MN, Clark A (2007) Joint rolling-horizon scheduling of materials processing and lot-sizing with sequence-dependent setups. J Heuristics 13(4):337–358. https://doi.org/10.1007/s10732-007-9011-9ASIC (2018) Clúster de cálculo: Rigel. http://www.upv.es/entidades/ASIC/catalogo/857893normalc.html. Accessed date 22 July 2018Baker KR (1977) An experimental study of the effectiveness of rolling schedules in production planning. Decis Sci 8(1):19–27. https://doi.org/10.1111/j.1540-5915.1977.tb01065.xBarrett RT, LaForge RL (1991) A study of replanning frequencies in a material requirements planning system. Comput Oper Res 18(6):569–578. https://doi.org/10.1016/0305-0548(91)90062-VBehnamian J, Fatemi Ghomi SMT (2014) A survey of multi-factory scheduling. J Intell Manuf 27:1–19. https://doi.org/10.1007/s10845-014-0890-yBillington PJ, McClain JO, Thomas LJ (1983) Mathematical programming approaches to capacity-constrained MRP systems: review, formulation and problem reduction. Manag Sci 29(10):1126–1141. https://doi.org/10.1287/mnsc.29.10.1126Blackburn JD, Millen RA (1980) Heuristic lot-sizing performance in a rolling-schedule environment. Decis Sci 11(4):691–701. https://doi.org/10.1111/j.1540-5915.1980.tb01170.xCao Y (2015) Long-distance procurement planning in global sourcing. Ecole Centrale Paris. https://tel.archives-ouvertes.fr/tel-01154871/. Accessed 15 Dec 2017Carlson RC, Beckman SL, Kropp DH (1982) The effectiveness o extending the horizon in rolling production scheduling. Decis Sci 13(1):129–146. https://doi.org/10.1111/j.1540-5915.1982.tb00136.xChand S, Hsu VN, Sethi S (2002) Forecast, solution, and rolling horizons in operations management problems: a classified bibliography. Manuf Serv Oper Manag 4(1):25–43. https://doi.org/10.1287/msom.4.1.25.287Coronado-Hernández JR (2016) Análisis del efecto de algunos factores de complejidad e incertidumbre en el rendimiento de las Cadenas de Suministro. Propuesta de una herramienta de valoración basada en simulación. Universitat Politècnica de València, Valencia (Spain). https://doi.org/10.4995/Thesis/10251/61467de Sampaio RJB, Wollmann RRG, Vieira PFG (2017) A flexible production planning for rolling-horizons. Int J Prod Econ 190:31–36. https://doi.org/10.1016/j.ijpe.2017.01.003DeYong GD, Cattani KD (2016) Fenced in? Stochastic and deterministic planning models in a time-fenced, rolling-horizon scheduling system. Eur J Oper Res 251(1):85–95. https://doi.org/10.1016/j.ejor.2015.11.006Federgruen A, Tzur M (1994) Minimal forecast horizons and a new planning procedure for the general dynamic lot sizing model: nervousness revisited. Oper Res 42(3):456–468. https://doi.org/10.1287/opre.42.3.456Fisher ML, Ramdas K, Zheng YS (2001) Ending inventory valuation in multiperiod production scheduling. Manag Sci 47(5):679–692. https://doi.org/10.1287/mnsc.47.5.679.10485Garcia-Sabater JP, Maheut J, Garcia-Sabater JJ (2009. A capacitated material requirements planning model considering delivery constraints: a case study from the automotive industry. In: 2009 international conference on computers and industrial engineering, IEEE, pp 378–383. https://doi.org/10.1109/ICCIE.2009.5223806Garcia-Sabater JP, Maheut J, Garcia-Sabater JJ (2012) A two-stage sequential planning scheme for integrated operations planning and scheduling system using MILP: the case of an engine assembler. Flex Serv Manuf J 24(2):171–209. https://doi.org/10.1007/s10696-011-9126-zGarcia-Sabater JP, Maheut J, Marin-Garcia JA (2013) A new formulation technique to model materials and operations planning: the generic materials and operations planning (GMOP) problem. Eur J Ind Eng 7(2):119. https://doi.org/10.1504/EJIE.2013.052572Hair JF, Prentice E, Cano D (1999) Análisis multivariante. Prentice-Hall, Upper Saddle RiverHozak K, Hill JA (2009) Issues and opportunities regarding replanning and rescheduling frequencies. Int J Prod Res 47(18):4955–4970. https://doi.org/10.1080/00207540802047106Hsu CH, Yang HC (2017) Real-time near-optimal scheduling with rolling horizon for automatic manufacturing cell. IEEE Access 5:3369–3375. https://doi.org/10.1109/ACCESS.2016.2616366Jans R (2009) Solving lot-sizing problems on parallel identical machines using symmetry-breaking constraints. Inf J Comput 21(1):123–136. https://doi.org/10.1287/ijoc.1080.0283Karimi B, Fatemi Ghomi SMT, Wilson JM (2003) The capacitated lot sizing problem: a review of models and algorithms. Omega 31(5):365–378. https://doi.org/10.1016/S0305-0483(03)00059-8Kimms A (1997) Multi-level lot sizing and scheduling, vol 53. Physica-Verlag, Heidelberg. https://doi.org/10.1007/978-3-642-50162-3Kleindorfer P, Kunreuther H (1978) Stochastic horizons for the aggregate planning problem. Manag Sci 24(5):485–497. https://doi.org/10.1287/mnsc.24.5.485Kumar BK, Nagaraju D, Narayanan S (2016) Supply chain coordination models: a literature review. Indian J Sci Technol. https://doi.org/10.17485/ijst/2016/v9i38/86938Lalami I, Frein Y, Gayon JP (2017) Production planning in automotive powertrain plants: a case study. Int J Prod Res 55(18):5378–5393. https://doi.org/10.1080/00207543.2017.1315192Lee HL, Padmanabhan V, Whang S (1997) The bullwhip effect in supply chains 1. Sloan Manag Rev Assoc 38(3):93–102. https://doi.org/10.1287/mnsc.43.4.546Lee DU, Villasenor JD, Luk W, Leong PHW (2006) A hardware Gaussian noise generator using the box-muller method and its error analysis. IEEE Trans Comput 55(6):659–671. https://doi.org/10.1109/TC.2006.81Lv Y, Zhang J, Qin W (2017) A genetic regulatory network-based method for dynamic hybrid flow shop scheduling with uncertain processing times. Appl Sci 7(1):23. https://doi.org/10.3390/app7010023Maheut J (2013) Modelos y Algoritmos Basados en el Concepto Stroke para la Planificación y Programación de Operaciones con Alternativas en Redes de Suministro. Universitat Politècnica de València, Valencia (Spain). https://doi.org/10.4995/Thesis/10251/29290Maheut J, Garcia-Sabater JP (2011) La matriz de operaciones y materiales y la matriz de operaciones y recursos, un nuevo enfoque para resolver el problema GMOP basado en el concepto del stroke. Direccion y Organizacion 45:46–57Maheut J, Garcia-sabater JP, Mula J (2012) A supply chain operations lot-sizing and scheduling model with alternative operations. In: Sethi SP, Bogataj M, Ros-McDonnell L (eds) Industrial engineering: innovative networks. Springer, London, pp 309–316. https://doi.org/10.1007/978-1-4471-2321-7Meindl B, Templ M (2012) Analysis of commercial and free and open source solvers for linear optimization problems. Common Tools and Harmonized Methodologies for SDC in the ESS, pp 1–13. http://neon.vb.cbs.nl/cascprivate/..%5Ccasc%5CESSNet2%5Cdeliverable_solverstudy.pdf. Accessed 15 Dec 2016Meyr H (2002) Simultaneous lotsizing and scheduling on parallel machines. Eur J Oper Res 139(2):277–292. https://doi.org/10.1016/S0377-2217(01)00373-3Narayanan A, Robinson P (2010) Evaluation of joint replenishment lot-sizing procedures in rolling horizon planning systems. Int J Prod Econ 127(1):85–94. https://doi.org/10.1016/j.ijpe.2010.04.038Nedaei H, Mahlooji H (2014) Joint multi-objective master production scheduling and rolling horizon policy analysis in make-to-order supply chains. Int J Prod Res 52(9):2767–2787. https://doi.org/10.1080/00207543.2014.884732Newman M (2005) Power laws, Pareto distributions and Zipf’s law. Contemp Phys 46(5):323–351. https://doi.org/10.1080/00107510500052444Omar MK, Bennell JA (2009) Revising the master production schedule in a HPP framework context. Int J Prod Res 47(20):5857–5878. https://doi.org/10.1080/00207540802130803Pérez C (2002) Estadística práctica con Statgraphics®. PEARSON EDUCACIÓN, S. A, MadridPoler R, Mula J (2011) Forecasting model selection through out-of-sample rolling horizon weighted errors. Expert Syst Appl 38(12):14778–14785. https://doi.org/10.1016/j.eswa.2011.05.072Prasad PSS, Krishnaiah Chetty OV (2001) Multilevel lot sizing with a genetic algorithm under fixed and rolling horizons. Int J Adv Manuf Technol 18(7):520–527. https://doi.org/10.1007/s0017010180520Rafiei R, Gaudreault J, Bouchard M, Santa-Eulalia L (2012) A reactive planning approach for demand-driven wood remanufacturing industry: a real-scale application, vol 71. CIRRELT, MontrealRafiei R, Nourelfath M, Gaudreault J, Santa-Eulalia LA, Bouchard M (2014) A periodic re-planning approach for demand-driven wood remanufacturing industry: a real-scale application. Int J Prod Res 52(14):4198–4215. https://doi.org/10.1080/00207543.2013.869631Ramezanian R, Fallah Sanami S, Shafiei Nikabadi M (2017) A simultaneous planning of production and scheduling operations in flexible flow shops: case study of tile industry. Int J Adv Manuf Technol 88(9–12):2389–2403. https://doi.org/10.1007/s00170-016-8955-zRodriguez MA, Montagna JM, Vecchietti A, Corsano G (2017) Generalized disjunctive programming model for the multi-period production planning optimization: an application in a polyurethane foam manufacturing plant. Comput Chem Eng 103:69–80. https://doi.org/10.1016/j.compchemeng.2017.03.006Sahin F, Narayanan A, Robinson EP (2013) Rolling horizon planning in supply chains: review, implications and directions for future research. Int J Prod Res 51(18):5413–5436. https://doi.org/10.1080/00207543.2013.775523Sethi S, Sorger G (1991) A theory of rolling horizon decision making. Ann Oper Res 29(1):387–415. https://doi.org/10.1007/BF02283607Simpson NC (2001) Questioning the relative virtues of dynamic lot sizing rules. Comput Oper Res 28(9):899–914. https://doi.org/10.1016/S0305-0548(00)00015-0Stadtler H (2000) Improved rolling schedules for the dynamic single-level lot-sizing problem. Manag Sci 46(2):318–326. https://doi.org/10.1287/mnsc.46.2.318.11924Stadtler H (2003) Multilevel lot sizing with setup times and multiple constrained resources: internally rolling schedules with lot-sizing windows. Oper Res 51(3):487–502. https://doi.org/10.1287/opre.51.3.487.14949Tiacci L, Saetta S (2012) Demand forecasting, lot sizing and scheduling on a rolling horizon basis. Int J Prod Econ 140:803–814. https://doi.org/10.1016/j.ijpe.2012.02.007Trigeiro WW (1987) A dual-cost heuristic for the capacitated lot sizing problem. IIE Trans 19(1):67–72. https://doi.org/10.1080/07408178708975371Wolsey LA (2002) Solving multi-item lot-sizing problems with an MIP solver using classification and reformulation. Manage Sci 48(12):1587–1602. https://doi.org/10.1287/mnsc.48.12.1587.442Xie J, Zhao X, Lee TS (2003) Freezing the master production schedule under single resource constraint and demand uncertainty. Int J Prod Econ 83(1):65–84. https://doi.org/10.1016/S0925-5273(02)00262-1Yıldırım I, Tan B, Karaesmen F (2005) A multiperiod stochastic production planning and sourcing problem with service level constraints. OR Spectrum 27(2–3):471–489. https://doi.org/10.1007/s00291-005-0203-0Zhao X, Xie J (1998) Multilevel lot-sizing heuristics and freezing the master production schedule in material requirements planning systems. Prod Plan Control 9(4):371–384. https://doi.org/10.1080/095372898234109Zoller K, Robrade A (1988) Efficient heuristics for dynamic lot sizing. Int J Prod Res 26(2):249–265. https://doi.org/10.1080/00207548808947857Zulkafli NI, Kopanos GM (2017) Integrated condition-based planning of production and utility systems under uncertainty. J Clean Prod 167:776–805. https://doi.org/10.1016/j.jclepro.2017.08.15

A Local Search Algorithm for Clustering in Software as a Service Networks

In this paper we present and analyze a model for clustering in networks that offer Software as a Service (SaaS). In this problem, organizations requesting a set of applications have to be assigned to clusters such that the costs of opening clusters and installing the necessary applications in clusters are minimized. We prove that this problem is NP-hard, and model it as an Integer Program with symmetry breaking constraints. We then propose a Tabu search heuristic for situations where good solutions are desired in a short computation time. Extensive computational experiments are conducted for evaluating the quality of the solutions obtained by the IP model and the Tabu Search heuristic. Experimental results indicate that the proposed Tabu Search is promising