ABSTRACT
In some industrial contexts, such as in the molded pulp, paper, furniture and electro fused grain industries, items or final products are obtained by processes that can produce several types of products simultaneously. These processes are considered here as any specific mode of operation or configuration of a production system that can produce several different items simultaneously and in varied quantities. The production planning and scheduling in these industrial contexts involve decisions of: i) configuration selection of these processes; ii) production lot sizing of selected configurations; and iii) decisions of scheduling these selected configurations. In this work, a mixed integer programming model (MIP) is presented to adequately represent this integrated problem, called the general lot sizing and scheduling problem with process configuration selection. In order to achieve effective production plans at acceptable computational times in practice, RelaxandFix and FixandOptimize heuristics based on this model are also presented. Computational experiments were performed solving the model with an optimization solver and applying the proposed heuristics in examples inspired in molded pulp, furniture and electro fused grain packaging companies. The results show that MIPheuristics can solve the problem more effectively (good quality solution in acceptable time) than simply using the solver for the mathematical model.
Keywords:
Lot sizing and scheduling; process configuration selection; mixed integer programming; MIPheuristics
1 INTRODUCTION
The shortterm production planning problem for intermittent processes, in general, is a lot sizing and scheduling problem. Lot sizing involves decisions of how much to produce to meet the demand of the final products, respecting the production capacities and considering the costs involved. When the change of production from one product to another consumes resources (time, material, labor) and generates costs dependent on the production sequence of the lots, the scheduling decisions of the lots become more important. As resource consumption influences lot sizing, it is increasingly common to integrate these decisions into decision making (Clark et al., 201110 CLARK A, ALMADALOBO B & ALMEDER C. 2011. Editorial: Lot Sizing and Scheduling, Industrial Extensions and Research Opportunities, Special Issue on Lot Sizing and Scheduling. International Journal of Production Research, 49: 24572461.). In some literature reviews, such as in Drexl & Kimms (199716 DREXL A & KIMMS A. 1997. Lot sizing and schedulingsurvey and extensions. European Journal of Operational Research , 99: 221235.), Jans & Degraeve (200830 JANS R & DEGRAEVE Z. 2008. Modeling industrial lot sizing problems: a review. International Journal of Production Research , 46: 16191643.) and Copil et al. (201712 COPIL K, WÖRBELAUER M, MEYR H & TEMPELMEIER H. 2017. Simultaneous lotsizing and scheduling problems: a classification and review of models. OR spectrum , 39: 164.), the integration of lot sizing problems and scheduling can be observed.
In several industrial contexts, such as the chemical, oil, paper and cardboard, furniture, molded pulp, electro fused grains, wood and foundry industries, among others, products demanded are, in general, manufactured using production processes that produce several products simultaneously (Johnson & Montgomery, 197431 JOHNSON LA & MONTGOMERY DC. 1974. Operations research in production planning, scheduling, and inventory control. vol. 6. Wiley New York.). These processes are considered here as any specific mode of operation or configuration of a production system that can produce several different items simultaneously and in varied quantities. For example, in a furniture company, a sheet of wood can be cut in several ways, i.e., there are several possible cutting patterns to cut the sheets and to obtain a set of items from each pattern. A cutting pattern can be configured in cutting equipment to obtain a single type of item in several quantities, or several types of items in different quantities of each type. In the latter case, each cutting pattern is configured as a simultaneous production process of several types of items. To produce all the items demanded in a planning horizon, a combination of process configurations (cutting patterns) must be selected to cut the sheets (Melega et al., 201842 MELEGA GM, DE ARAUJO SA & JANS R. 2018. Classification and literature review of integrated lotsizing and cutting stock problems. European Journal of Operational Research , 271: 119.).
In another example, in an electro fused grain company, the raw material (quartz and petroleum coke, for example) is transformed into grains through a series of industrial processes. These grain are classified according to their size by a set of different vibrating sieves, which together separate the grains into different granulometric ranges. That is, each set of sieves allows the simultaneous production of several grain sizes and in different quantities of each size, similar to the example of the cutting pattern in the furniture company. To produce the grains in the required sizes and quantities, various process configurations (i.e., various combinations of different vibrating sieves) must be selected for use (Luche et al., 200937 LUCHE JRD, MORABITO R & PUREZA V. 2009. Combining process selection and lot sizing models for production scheduling of electro fused grains. AsiaPacific Journal of Operational Research, 26: 421443.).
Selecting the configuration and scheduling of these configurations can influence planning and production control decisions, that is, the quantity and variety of products produced, the consumption of resources and the use of capacities. The problem of this production planning can be called a general lot sizing and scheduling problem with process configuration selection. Note that this problem involves configuration selection decisions of these processes, production lot sizing decisions of the selected configurations and scheduling decisions of these configurations. Each production lot contains several types of products being produced simultaneously. It should be noted that this problem is different from the general problem of lot sizing and scheduling, where each lot contains only one type of product.
Several examples of industries that use processes with simultaneous production of different products have already been studied in the literature. However, these papers are applied to specific cases of these industries, considering the particularities of their production processes and other specificities of their planning environments. In general, the solution methods are also customized and specific to the cases studied and difficult to adapt to the cases of other industries, whether specialist or mathematical programming heuristics. Therefore, the objective of this study is to present a mathematical modelling and solution methods for the general lot sizing and scheduling problem with process configuration selection. To the best of our knowledge, there are few studies addressing this problem in the literature in contrast to the traditional lot sizing and scheduling problem, which has been extensively studied.
The main contributions of this paper are: (i) the presentation of an integer mixed programming model (MIP) that includes the objectives and constraints of the problem integrated with process configuration selection. The model must consider inventory balancing constraints to meet demands, capacity constraints and identification of changes in production process configurations. These process configuration changes result in times and costs dependent on the production sequence of the configurations. The proposed model represents the decision making for this type of problem and presents the traditional constraints adapted from the lot sizing and scheduling problem, besides the process configuration selection decisions. The objective function aims to minimize inventory costs, delay and change of production process settings. (ii) The proposal of a set of heuristics based on mathematical programming as the solution method for the model. These heuristics can be easily adapted when additional specificities need to be considered in the problem and incorporated into the mathematical model. Several solving strategies are investigated that comprise constructive and improvement heuristics, specifically the MIPheuristics RelaxandFix and FixandOptimize.
In Section 2, we discuss how a process and its configurations compose the industrial environment of a process industry. The lot sizing and scheduling problem with process configuration selection is defined in Section 3 and a general mathematical model is proposed. MIPheuristic solution methods are proposed for the problem in Section 4. The results of the computational tests, with examples inspired by molded pulp packaging, furniture and electro fused grain companies, are presented in Section 5. Finally, the conclusions and future directions of the research are presented in Section 6.
2 LOTSIZING AND SCHEDULING PROBLEM WITH PROCESS CONFIGURATION SELECTION
Process industries are those that add value to materials by mixing, separating, forming or chemical reactions (Kopanos & Puigjaner, 201933 KOPANOS GM & PUIGJANER L. 2019. Solving LargeScale Production Scheduling and Planning in the Process Industries. Springer.). Processes can be continuous or batchbased and generally require strict process control and high investments. Fransoo & Rutten (199423 FRANSOO JC & RUTTEN WG. 1994. A typology of production control situations in process industries. International Journal of Operations & Production Management, 14: 4757.) classify process industries as presented in Table 1. The production flow differentiates between batch production and continuous production. Process industries that produce in batches have a large set of types of items produced in small quantities. On the other hand, continuous production flow presents a low variety of items produced in large volumes. The difference between the items is small since the processing of the items follows very similar production routings and, in general, the value added in the item is low.
Typology for Process Industries. Fransoo & Rutten (199423 FRANSOO JC & RUTTEN WG. 1994. A typology of production control situations in process industries. International Journal of Operations & Production Management, 14: 4757.).
In these industries, the same process can produce more than one type of product and one type of product can be produced by several alternative process configurations. Figures 1, 2 and 3 show examples of this feature in process industries. Figure 1 presents an example of a process used in the molded pulp packaging industry. This type of industry uses a set of molds to produce packaging for eggs and fruits, made from pulp produced from recyclable paper. In this example, the selected process configuration can produce 3 different types of products at each mold stamping: two packs for 12 eggs, one pack for 24 eggs and two packs for 4 fruits. Other mold configurations can be used to produce different types of products and/or in other quantities. An alternative configuration could produce only 2 different types of products at each stamping: three packs for 12 eggs and four packs for 4 fruits. Having different possible configurations, the sequence of the configurations and the time of use of each one should be defined.
Figures 2 and 3 show the case where the process configuration is a cutting pattern that cuts larger units (raw material) into smaller units ordered by customers. Figure 2 depicts the case of a furniture industry, where a sheet of wood (or a pack of sheets) is cut according to a twodimensional cutting pattern for the production of items used in furniture assembly. Note that in this process three types of items are produced in different quantities. Figure 3 shows a onedimensional cutting pattern (configuration) in a paper industry, where a large reel of paper is cut into smallersized reels to obtain different products.
Note that, in each case, there may be many possible process configurations to be used, and the process configuration selection decision must be integrated with the remaining production planning decisions. Unlike the flexibility problem of Fiorotto et al. (201821 FIOROTTO DJ, JANS R & DE ARAUJO SA. 2018. Process flexibility and the chaining principle in lot sizing problems. International Journal of Production Economics , 204: 244263.), which studies different known paths in the production process among the available machines, our proposal addresses the different configurations of a process (mixing, separation, conformation or chemical reactions) that are known a priori.
The flexibility of the selection of these configurations and the integration with lot sizing and scheduling decisions can allow important reductions in inventory and waste of materials.
2.1 Lot sizing and scheduling
The lot sizing and scheduling problem has been presented in the literature in different types of industries, such as beverages (Toscano et al., 201955 TOSCANO A, FERREIRA D & MORABITO R. 2019. A decomposition heuristic to solve the twostage lot sizing and scheduling problem with temporal cleaning. Flexible Services and Manufacturing Journal , 31: 142173., ^{2020}56 TOSCANO A, FERREIRA D & MORABITO R. 2020. Formulation and MIPheuristics for the lot sizing and scheduling problem with temporal cleanings. Computers & Chemical Engineering, 142: 107038.), foundries (Furtado et al., 201925 FURTADO MGS, CAMARGO VC & TOLEDO FM. 2019. The production planning problem of orders in small foundries. RAIROOperations Research, 53: 15511561.; de Araujo et al., 200813 DE ARAUJO SA, ARENALES MN & CLARK AR. 2008. Lot sizing and furnace scheduling in small foundries. Computers & Operations Research , 35: 916932.), textile (Camargo et al., 20147 CAMARGO VC, TOLEDO FM & ALMADALOBO B. 2014. HOPSHammingOriented Partition Search for production planning in the spinning industry. European Journal of Operational Research , 234: 266277.), glass packaging (Fachini et al., 201817 FACHINI RF, ESPOSTO KF & CAMARGO VCB. 2018. A framework for development of advanced planning and scheduling (APS) systems in glass container industry. Journal of Manufacturing Technology Management, .), animal nutrition (Clark et al., 201011 CLARK AR, MORABITO R & TOSO EAV. 2010. Production setupsequencing and lot sizing at an animal nutrition plant through atsp subtour elimination and patching. Journal of Scheduling, 13: 111121.), poultry (Boonmee & Sethanan, 20166 BOONMEE A & SETHANAN K. 2016. A GLNPSO for multilevel capacitated lot sizing and scheduling problem in the poultry industry. European Journal of Operational Research , 250: 652665.), paper (Leao et al., 201734 LEAO AA, FURLAN MM & TOLEDO FM. 2017. Decomposition methods for the lot sizing and cuttingstock problems in paper industries. Applied Mathematical Modelling, 48: 250268.), pulp (Furlan et al., 201524 FURLAN M, ALMADALOBO B, SANTOS M & MORABITO R. 2015. Unequal individual genetic algorithm with intelligent diversification for the lotscheduling problem in integrated mills using multiplepaper machines. Computers & Operations Research , 59: 3350.), food (Claassen et al., 20169 CLAASSEN G, GERDESSEN J, HENDRIX EM & VAN DER VORST JG. 2016. On production planning and scheduling in food processing industry: Modelling nontriangular setups andproduct decay. Computers & Operations Research, 76: 147154.), among others. The complexity of these problems is influenced by the characteristics of the production system, such as time horizon, number of levels in product production, number of products, capacity and resource constraints, damage to items in inventory, demand, presence of exchanges, lack of products, among others (Karimi et al., 200332 KARIMI B, GHOMI SF & WILSON J. 2003. The capacitated lot sizing problem: a review of models and algorithms. Omega, 31: 365378.).
In the manufacturing of some products such as feed, soft drinks, beer, foundry among others, the production sequence influences their costs and production times (Ferreira et al., 201218 FERREIRA D, CLARK AR, ALMADALOBO B & MORABITO R. 2012. Singlestage formulations for synchronised twostage lot sizing and scheduling in soft drink production. International Journal of Production Economics, 136: 255265.; Toledo et al., 200954 TOLEDO CFM, FRANÇA PM, MORABITO R & KIMMS A. 2009. Multipopulation genetic algorithm to solve the synchronized and integrated twolevel lot sizing and scheduling problem. International Journal of Production Research , 47: 30973119.). This problem can be solved in two phases: in the first, the quantity to be produced is defined, considering demand, production capacity and inventories (lot sizing problem). In the second phase, the sequence of this production is defined, considering the times and costs of production changeovers between products. In this case, the changeover times and costs depend on the defined sequence of production of the lots. For a better solution approach, several authors (see Drexl & Kimms (199716 DREXL A & KIMMS A. 1997. Lot sizing and schedulingsurvey and extensions. European Journal of Operational Research , 99: 221235.) and Copil et al. (201712 COPIL K, WÖRBELAUER M, MEYR H & TEMPELMEIER H. 2017. Simultaneous lotsizing and scheduling problems: a classification and review of models. OR spectrum , 39: 164.)) present models considering the integrated problem of lot sizing and scheduling, which define what to produce, lot quantities and the sequence of the lots. In general, each lot refers to the quantity produced of a single type of product at a time.
The lot sizing and scheduling problem can be represented by several different mathematical models, depending on the characteristics of the problem, as can be seen in Drexl & Kimms (199716 DREXL A & KIMMS A. 1997. Lot sizing and schedulingsurvey and extensions. European Journal of Operational Research , 99: 221235.). Among the models, the General Lot sizing and Scheduling Problem (GLSP) is a mathematical model that integrates the lot sizing and scheduling of several products in a single machine, subject to capacity constraints (Karimi et al., 200332 KARIMI B, GHOMI SF & WILSON J. 2003. The capacitated lot sizing problem: a review of models and algorithms. Omega, 31: 365378.). According to AlmadaLobo et al. (20154 ALMADALOBO B, CLARK A, GUIMARÃES L, FIGUEIRA G & AMORIM P. 2015. Industrial insights into lot sizing and scheduling modeling. Pesquisa Operacional, 35: 439464.), the GLSP, together with CLSD (Capacitated Lot sizing Problem with sequencedependent setups), are the most studied models in the literature. Fleischmann & Meyr (199722 FLEISCHMANN B & MEYR H. 1997. The general lot sizing and scheduling problem.OperationsResearchSpektrum, 19: 1121.) present GLSP considering a planning horizon of T periods, which are divided into a set of S _{t}  subperiods. A single item is produced by subperiods, therefore it is easy to differentiate the production sequence. This model has been used in several papers for this facility (for example: Ferreira et al. (200819 FERREIRA D, MORABITO R & RANGEL S. 2008. Um modelo de otimização inteira mista e heurísticas relax and fix para a programação da produção de fábricas de refrigerantes de pequeno porte. Produção, 18: 7688.), Toso et al. (200957 TOSO EA, MORABITO R & CLARK AR. 2009. Lot sizing and sequencing optimisation at an animalfeed plant. Computers & Industrial Engineering , 57: 813  821.), Martínez et al. (201641 MARTÍNEZ KY, TOSO EA & MORABITO R. 2016. Production planning in the molded pulp packaging industry. Computers & Industrial Engineering , 98: 554566.) and others).
2.2 Process configuration selection
Lot sizing and scheduling models used for industries with discrete production can be adapted to process industries to address the production specificity following alternative process configurations. Some authors have already studied some formulations and methods of resolution for the lot sizing and scheduling problem with process configuration selection for different industries. Sahinidis & Grossmann (199247 SAHINIDIS N & GROSSMANN IE. 1992. Reformulation of the multiperiod MILP model for capacity expansion of chemical processes. Operations Research, 40: S127S144.) present a model for lot sizing and scheduling and process configuration selection for the chemical industry. Lu & Qi (201135 LU L & QI X. 2011. Dynamic lot sizing for multiple products with a new joint replenishment model. European Journal of Operational Research , 212: 7480.) presented models for the chemical and poultry industries and proposed heuristics. Gaudreault et al. (201126 GAUDREAULT J, FRAYRET JM, ROUSSEAU A & D’AMOURS S. 2011. Combined planning and scheduling in a divergent production system with coproduction: A case study in the lumber industry. Computers & Operations Research , 38: 12381250.) proposed two mathematical models for the problem of process configuration selection and its sequencing in the wood industry, the first based on mixed integer programming and the second on constraint programming.
In oil refineries (Shi et al., 201450 SHI L, JIANG Y, WANG L & HUANG D. 2014. Refinery production scheduling involving operational transitions of mode switching under predictive control system. Industrial & Engineering Chemistry Research, 53: 81558170.; Persson et al., 200444 PERSSON JA, GÖTHELUNDGREN M, LUNDGREN JT & GENDRON B. 2004. A tabu search heuristic for scheduling the production processes at an oil refinery. International Journal of Production Research , 42: 445471.; GötheLundgren et al., 200227 GÖTHELUNDGREN M, LUNDGREN JT & PERSSON JA. 2002. An optimization model for refinery production scheduling. International Journal of Production Economics , 78: 255270.), production decisions involve selecting process configurations for each production unit over the planning horizon to determine the quantities of diesel, gasoline and other types of products that depend on the process configurations. Chunpeng & Gang (20098 CHUNPENG L & GANG R. 2009. A strategy for the integration of production planning and scheduling in refineries under uncertainty. Chinese Journal of Chemical Engineering, 17: 113127.) proposed two strategies to integrate production planning and process configuration scheduling in refineries, the first uses a rolling horizon and the second uses a twostage decomposition. In the electro fused grain industry, the size of the manufactured grain depends on the set of sieves used. Luche & Morabito (200536 LUCHE J & MORABITO R. 2005. Otimização na programação da produção de grãos eletrofundidos: Um estudo de caso. Gestão & Produção, 12: 135149.) proposed a mathematical model that consists of a combination of the problem of process configuration selection and the lot sizing problem. A constructive heuristic was also proposed as a solution method for this problem in Luche et al. (200937 LUCHE JRD, MORABITO R & PUREZA V. 2009. Combining process selection and lot sizing models for production scheduling of electro fused grains. AsiaPacific Journal of Operational Research, 26: 421443.).
Martínez et al. (201641 MARTÍNEZ KY, TOSO EA & MORABITO R. 2016. Production planning in the molded pulp packaging industry. Computers & Industrial Engineering , 98: 554566.) and Martínez et al. (201840 MARTÍNEZ KP, MORABITO R & TOSO EAV. 2018. A coupled process configuration, lotsizing and scheduling model for production planning in the molded pulp industry. International Journal of Production Economics , 204: 227243.) studied a molded pulp packaging industry and proposed models aimed at reducing setup and inventory costs. Due to the large number of existing molding patterns, which are difficult to enumerate in advance, the selection of the process configuration is also a model decision. Martínez et al. (201939 MARTÍNEZ KP, ADULYASAK Y, JANS R, MORABITO R & TOSO EAV. 2019. An exact optimization approach for an integrated process configuration, lotsizing, and scheduling problem. Computers & Operations Research , 103: 310323.) proposed solution methods to solve the models based on BranchandCheck. In the furniture, paper and steel industries this integration also occurs regarding lot sizing, scheduling and process configuration selection with the generation of configurations. In these cases, items are obtained by cutting larger objects following a cutting pattern. Each cutting pattern is a process configuration and is usually done by solving several cutting and packaging problems. These studies can be found in Gramani et al. (200928 GRAMANI M, FRANÇA P & ARENALES M. 2009. A Lagrangian relaxation approach to a coupled lotsizing and cutting stock problem. International Journal of Production Economics , 119: 219227.) and Alem & Morabito (20132 ALEM D & MORABITO R. 2013. Riskaverse twostage stochastic programs in furniture plants. OR spectrum, 35: 773806.) for the furniture industry, in Poltroniere et al. (200846 POLTRONIERE SC, POLDI KC, TOLEDO FMB & ARENALES MN. 2008. A coupling cutting stocklot sizing problem in the paper industry. Annals of Operations Research , 157: 91104.) for the paper industry and in Nonås & Thorstenson (200843 NONÅS SL & THORSTENSON A. 2008. Solving a combined cuttingstock and lotsizing problem with a column generating procedure. Computers & Operations Research , 35: 33713392.) for the steel industry. The proposed solution methods are Lagrangean heuristics, column generation methods and specialist heuristics for subproblems.
2.3 Research proposal
In this paper, the mathematical model proposed for the general lot sizing and scheduling problem with process configuration selection is in line with the basis of the models presented in the literature. The GLSP is adapted to include process configuration selection decisions and, thus, several products can be produced per subperiod. Each lot is related to the production time of a process according to the selected configuration (instead of the produced quantity of an item, as in GLSP). That is, the quantity of items is determined by the usage time of each of the configurations of the production process and the scheduling of configurations must be defined in the planning horizon. In this case, it can be stated that the set of configurations is known a priori.
The proposed solution methods for the problem are mathematical programming heuristics which are generally easy to adapt to possible changes in the mathematical model. In the literature, although some heuristics of the RelaxandFix and FixandOptimize types can be found, most of the solution methods are specific to the problem of each industry addressed. Therefore, we propose a set of constructive and improvement heuristics combinations based on mathematical programming.
3 PROBLEM DEFINITION
The lot sizing and scheduling problem with process configuration selection aims to determine the quantity of items to be produced during a planning horizon to meet a known demand. The production of the items occurs by processes, therefore it is necessary to select the configuration of production processes that will be used. It is worth mentioning that all possible configurations are known a priori. The case of generating new configurations is not dealt with in this paper and depends largely on specific characteristics of the production process involved. For example, generating process configurations in molded pulp packaging industries involves several specificities of the equipment and is not a simple task, as discussed in Martínez et al. (201641 MARTÍNEZ KY, TOSO EA & MORABITO R. 2016. Production planning in the molded pulp packaging industry. Computers & Industrial Engineering , 98: 554566., ^{2018}40 MARTÍNEZ KP, MORABITO R & TOSO EAV. 2018. A coupled process configuration, lotsizing and scheduling model for production planning in the molded pulp industry. International Journal of Production Economics , 204: 227243.). The planning horizon comprises T  time periods, which are divided into St  subperiods. In each subperiod, only one configuration of the production process can be used, but several types of items can be produced. In the case of changeover between two different production process configurations during the planning horizon, there is production time consumption and the changeover also incurs costs dependent on the production sequence. The total time consumed in a period, which includes the setup times of the machines and the production of the items, is limited to the time available in the period.
The production plan must consider that inventories and backlogs are allowed but they incur costs. Consequently, a production plan with minimum costs of inventory and backlogging of items and setup of production processes is desired. By considering the characteristics of the problem, the GLSP (General Lot sizing and Sequencing Problem) with some adaptations can represent the problem described. After presenting the model, some considerations are made about the differences from the classic GLSP model.
Consider the following indexes, parameters and decision variables:
Subject to
The objective function (1) aims to minimize the costs of inventory, backlogging and change of production process configurations. Constraints (2) define the inventory balance for each product i in each period t. Constraints (3) ensure that process usage time and machine setup time are limited by capacity in each period t. In case of production using configuration j in subperiod s, the constraints (4) impose that the machine must be prepared for this process configuration in this subperiod. Constraints (5) impose that the machine is prepared for one configuration in each subperiod of the planning horizon. Constraints (6)(8) are replacements for constraints
Constraints (6) relate the configuration j prepared in the last subperiod of the period
By simplicity and without loss of generality, the costs are considered period independent. If the characteristic of the problem requires that the costs be different from period to period, cost parameters such as h _{it} , b _{it} and c _{jkt} and changes according to the objective function can be considered.
As can be seen, variables I and B are interpreted identically to the classic GLSP model. However, variables X , Y and Z in the classic GLSP represent respectively: the lot sizes of the products, the setup to produce the products and, the changeover of setups for the products in the machines. In the extension proposed in this paper, variables X , Y and Z represent respectively: the usage time of a specific process configuration, the setup for a specific configuration, the changeover of setups between two different configurations. The demand balance constraints are also modified, where the lot sizing variables X are multiplied by a factor which represents the number of units of each product obtained by each process per time unit.
4 MIPHEURISTICS
A heuristic is a set of steps that aims to achieve a good quality solution in a short computational time. Karimi et al. (200332 KARIMI B, GHOMI SF & WILSON J. 2003. The capacitated lot sizing problem: a review of models and algorithms. Omega, 31: 365378.) divide heuristics for lot sizing problems into two categories: specialized heuristics and heuristics based on mathematical programming.
Despite the similarity of the lot sizing and scheduling problem and the problem addressed in this paper, the specialized heuristics are not easily adapted to the case that includes the process configuration selection. When adapting, the configuration that produces the demanded products must be selected. In order to avoid an accumulation of stocks, the configuration should not produce large quantities of nondemanded items. In addition, the configuration considering the changeover times and costs between these two configurations used must be selected.
4.1 Heuristics based on mathematical programming
Heuristics based on mathematical programming, or hybrid heuristics, combine heuristics with exact mathematical programming methods. These heuristics usually provide good solutions to the lot sizing and scheduling problem, they are more general and can be easily adapted to different problems. However, in general they are more difficult to implement than specialized heuristics due to the necessary technical concepts and have a higher computational complexity for real problems (Maes & Wassenhove, 198838 MAES J & WASSENHOVE LV. 1988. MultiItem SingleLevel Capacitated Dynamic LotSizing Heuristics: A General Review. Journal of the Operational Research Society, 39: 9911004.; Karimi et al., 200332 KARIMI B, GHOMI SF & WILSON J. 2003. The capacitated lot sizing problem: a review of models and algorithms. Omega, 31: 365378.). According to Pochet & Wolsey (200645 POCHET Y & WOLSEY LA. 2006. Production planning by mixed integer programming. Springer Science & Business Media.), this method consists of modifying the mixed integer programming (MIP) through relaxation of constraints, relaxation of integrality or fixing some variables. By understanding the use of the MIP, these heuristics are also called MIPheuristics. Within this category of heuristics, the most common for the lot sizing and scheduling problems are the RelaxandFix and the FixandOptimize heuristics. They are explored as solution methods for the present general lot sizing and scheduling problem with process configuration selection.
4.1.1 RelaxandFix
This is a constructive heuristic based on exact mathematical programming methods that was proposed by Dillenberger et al. (199414 DILLENBERGER C, ESCUDERO LF, WOLLENSAK A & ZHANG W. 1994. On practical resource allocation for production planning and scheduling with period overlapping setups. European Journal of Operational Research , 75: 275286.). The integer variables are partitioned into P sets, where Q _{p} is the set of variables of the partition p, and
Subject to
After solving all the P iterations, if all the p subproblems are feasible, then
The partitioning of decision variables could vary, but the most common in the literature for the lot sizing problem is based on time periods. The lot sizing and scheduling problem with process configuration selection presented in the previous section has only the setup state variables (Y _{js} ) as integers. In this case, we consider
RelaxandFix Forward: Algorithm 1 represents the heuristic. In a periodbased variable partitioning, the forward strategy starts fixing the variables from the first period with the (p = 1) partition and ends in the last period with the partition (
RelaxandFix Backward: In the backward strategy, the fixing of variable values starts from the partition of the last period
RelaxandFix Overlapping: In this strategy, the difference is in the number of variables of the partition that have the values fixed in each iteration. The proposal is to fix values for half of the variables, for example, in iteration p the second half of the variables of period p − 1 and the first half of the subperiods of period p. The second half of the variables of period p remains integer and free to be optimized again together with the next period. This strategy is similar to RelaxandFix Forward, however, constraints (17) in model (12)(19) are changed to fix values in the appropriate variables.
RelaxandFix Minimizes Backlogs: the strategy is similar to Forward. However, at the end of each iteration, it is checked if there is a backlog in meeting demand. If there is, the iteration is solved again with the freedom to optimize the variables of the last period previously fixed. This strategy is done until there are no more backlogs, or the variables of all periods are free for optimization (dos Santos Diz et al., 201915 DOS SANTOS DIZ GS, HAMACHER S & OLIVEIRA F. 2019. A robust optimization model for the maritime inventory routing problem. Flexible Services and Manufacturing Journal, 31: 675701.). In Algorithm 1, after line 5, the following loop is inserted:
In this paper, these four strategies of RelaxandFix heuristics and other applications in combination with improvement heuristics are investigated and compared with each other, as described in Table 3 in the section on computational experiments.
4.1.2 FixandOptimize
FixandOptimize is an improvement heuristic, that is, it performs improvement movements in a previously given solution. The procedure consists of using the MIP model by fixing the values of the decision variables and releasing a variable partition to be optimized along with all the continuous variables. The integer variables of the model are partitioned into P sets, where Qp is the set of variables of partition p, where p = 1, 2, ..., P. In any iteration of the method, the integer variables are fixed to the best incumbent solution
Subject to
FixandOptimize is well used in lot sizing and scheduling problem, mainly in combination with other constructive heuristics, such as RelaxandFix which provides an initial solution. The papers by Sahling et al. (200948 SAHLING F, BUSCHKÜHL L, TEMPELMEIER H & HELBER S. 2009. Solving a multilevel capacitated lot sizing problem with multiperiod setup carryover via a fixandoptimize heuristic. Computers & Operations Research, 36: 25462553.), Helber & Sahling (201029 HELBER S & SAHLING F. 2010. A fixandoptimize approach for the multilevel capacitated lot sizing problem. International Journal of Production Economics , 123: 247256.), Baldo et al. (20145 BALDO TA, SANTOS MO, ALMADALOBO B & MORABITO R. 2014. An optimization approach for the lot sizing and scheduling problem in the brewery industry. Computers & Industrial Engineering, 72: 5871.), Toledo et al. (201553 TOLEDO CFM, DA SILVA ARANTES M, HOSSOMI MYB, FRANÇA PM & AKARTUNALI K. 2015. A relaxandfix with fixandoptimize heuristic applied to multilevel lotsizing problems. Journal of heuristics, 21: 687717.), Tempelmeier & Copil (201652 TEMPELMEIER H & COPIL K. 2016. Capacitated lot sizing with parallel machines, sequencedependent setups, and a common setup operator. OR spectrum , 38: 819847.) and Soler et al. (201951 SOLER WA, POLDI KC & SANTOS MO. 2019. Capacitated lot sizing and scheduling with order acceptance and delivery time windows: mathematical model and a MIPbased heuristic. Pesquisa Operacional , 39: 471496.) are examples of the combined application of RelaxandFix and FixandOptimize. The partitioning of decision variables can vary, but the most common in the literature for the lot sizing problem is based on time periods. The lot sizing and scheduling problem with process configuration selection, presented in the previous section, has only the setup state variables (Y _{js} ) defined as integers. In this case, the partitioning can be based on configurations (K partitions) or, in a classic way, based on periods (T partitions).
In this paper, FixandOptimize is experimented with the two options of variable partitioning, by configurations and by periods. As an initial solution is required, FixandOptimize is tested and compared in combination with RelaxandFix heuristics in several combinations, as described in Table 3 in the next section.
5 COMPUTATIONAL EXPERIMENTS
The computational experiments performed with the model and heuristics aim to produce results to compare the quality of the solutions obtained and the computational time to find them.
Next, tests for the resolution of the MIP model and various combinations of MIPheuristics are reported. The RelaxandFix heuristics and their combinations with the FixandOptimize heuristics with partitions of the configurations and periods are tested. The experiments of five combinations of RelaxandFix and FixandOptimize heuristics are reported. According to preliminary tests, in which 8 combinations were tested, the combinations presented below seemed more promising and only they were investigated in the study.
The instances used in computational tests represent three types of process industries and are divided into three groups each. The instances of the molded pulp packaging company were based on the real data presented in Martínez et al. (201939 MARTÍNEZ KP, ADULYASAK Y, JANS R, MORABITO R & TOSO EAV. 2019. An exact optimization approach for an integrated process configuration, lotsizing, and scheduling problem. Computers & Operations Research , 103: 310323.). The G1 and G2 groups are data cutouts, while in G3 group the data are randomly generated based on practice. The instances of the furniture company are cutouts of the instances presented in Alem et al. (20103 ALEM DJ, MUNARI PA, ARENALES MN & FERREIRA PAV. 2010. On the cutting stock problem under stochastic demand. Annals of Operations Research, 179: 169186.).
The instances of the electrofused grain company are cutouts of the instances presented in Luche et al. (200937 LUCHE JRD, MORABITO R & PUREZA V. 2009. Combining process selection and lot sizing models for production scheduling of electro fused grains. AsiaPacific Journal of Operational Research, 26: 421443.). The cutout aims to consider a single process with a single machine.
Table 2 shows the number of variables and parameters present in the instances of each group, where T is the number of periods, S is the number of subperiods, “Continuous” is the number of continuous variables and “Binary” is the number of binary variables.
5.1 Computational results
The computational tests were performed on a machine with an Intel i7 processor containing 16 GB of RAM. The execution time limit of CPLEX v.12.5 for each method and for each of the instances was 3, 600 seconds. For heuristics, the total time was divided equally into the total number of iterations (partitions) required by the method. Thus, instances that contain more partitions provide less time per iteration, since the number of iterations is related to the number of partitions. With less time available for one iteration, the solver may not find the optimal solution for that partition. This may imply worse quality solutions at the end of the iterations when compared to other solutions that had longer solution time available. Similarly, when we compare the use of one method with P partitions with another method that additionally uses P partitions in an improvement procedure, there is no guarantee that the method with the improvement procedure has better solutions. This is because we cannot guarantee that each iteration finds the optimal solution.
To better understand the results, the instances were separated by industry and the objective function values, Gap and execution time are reported in Tables 4, 5 and 6. These indicators are provided for each of the MIPheuristics tested and listed in Table 3.
Besides the results for each one of the instances, Table 7 presents a general comparison considering the averages of all the instances for each indicator. In addition, the averages of the instances by industry are also reported.
Molded pulp packaging industry
In Table 4, for each of the methods, the following is presented: “OF” indicating the value found for the objective function; “Dsv” indicating the relative difference of the value of the objective function of the method from the MIP (
Table 4 shows that CPLEX finds the optimal solution of the MIP solely for several instances of group G1with a small number of periods and configurations. On the other hand, the solver stopped due to the time limit for most of the instances of the G3 group, presented high optimality gaps. Among the MIPheuristics, the (B) RelaxandFix Backward is the method that presented the worst performance. The FixandOptimize methods with configuration partitions using RelaxandFix Overlapping heuristics (G) or RelaxandFix that minimizes backlogs (H) presented the best performances for the groups of instances of the molded pulp packaging industry.
Regarding the computational time, the (B) RelaxandFix Backward method consumes, in general, the shortest time  on average 407 seconds. On the other hand, the (G) FixandOptimize with configuration partitions combined with RelaxandFix Overlapping has the highest average consumption, with 977 seconds.
Furniture Industry
Similarly to the previous table, the results shown in Table 5 present the same indicators used for comparing solution methods for the furniture industry.
Table 5 shows that CPLEX stops by time limit for all instances solved by MIP solely. The G1 group solutions have a 8% gap of the optimal solution, the G2 group has more than 13% and the G3 results have more than 47%. Once more, the (B) RelaxandFix Backward is the method that presented the worst performance. The FixandOptimize methods with configuration partitions using RelaxandFix that minimize backlogs (H) or RelaxandFix Overlapping (G) presented the best performances for the groups of instances of the furniture industry.
Regarding the computational time, the (B) RelaxandFix Backward method consumes, on average, a shorter time  139 seconds. On the other hand, the (H) FixandOptimize with configuration partitions using RelaxandFix that minimizes backlogs has the highest average runtime consumption, with 1793 seconds.
Electrofused grain industry
Table 6 presents the results for the tests with the instances of the electro fused grain industry. The same indicators as the previous tables are used for comparison.
Table 6 shows that CPLEX stops by time limit for all instances solved by mathematical modeling (MIP). The G1 group solutions have a 10% gap of the optimal solution, the G2 group has more than 20% and the G3 results have more than 60% of the optimality. Once more, (B) RelaxandFix Backward is the method with the worst performance. The FixandOptimize methods with configuration partitions using RelaxandFix, which minimizes backlogs (H) or RelaxandFix Overlapping (G), presented the best performances for the groups of instances of the molded pulp packaging industry.
Regarding computational time, the (C) RelaxandFix Forward and (H) FixandOptimize methods with configuration partitions combined with RelaxandFix that minimize backlogs consume, on average, less time  549 and 553, seconds respectively. On the other hand, (G) FixandOptimize with configuration partitions combined with RelaxandFix Overlapping has the highest average computational time consumption, at 1417 seconds.
Overall comparison
By analyzing the results of the previous tables, it can be understood that the mathematical modeling resolution (MIP) using CPLEX found optimal solutions only for some small instances (considering the number of binary variables). For larger instances, the solver stops by the time limit and presents high optimality gaps. This result justifies the use of heuristic methods of resolution for this model.
Using Table 7, we can analyze the averages by groups of instances and overall average by method. To support the solution quality analysis, we use a color scale over the “Dsv” column that must be observed per group of instances. The greener the cell, the smaller the “Dsv” value. On the other hand, the redder the cell, the higher the value of “Dsv”. For example, for the G1 group of instances of the molded pulp packaging industry, the highest result for “Dsv” is (B) RelaxandFix Backward method and the lowest results are methods (F), (G) and (H). Note that the table is divided into two parts so that the results can be visualized better.
Among the methods that most present “Dsv” values in green, and as we can see through the “Average” lines of the table, the (H) FixandOptimize method with configuration partitions that use RelaxandFix that minimizes backlogs is what presents the better results. The negative value indicates that the method found solutions that were, on average, 0.7% better than the solutions reported when MIP solely was solved with CPLEX. In addition, we can note that CPLEX consumed, on average, 3217 seconds to solve the MIP and FixandOptimize with configuration partitions that use RelaxandFix that minimizes backlogs consumed, on average, 1085 seconds to present slightly better results.
If we only compare the RelaxandFix methods, the method that minimizes delay (E) and with Overlapping (D) presents solutions that are, on average, 3% higher than the values found with the MIP model. If we compare only the FixandOptimize type methods, the best results are found using configuration partitions, particularly when the RelaxandFix methods that minimize backlogs (H) or the RelaxandFix Overlapping methods (G) are combined. It is a fact that by adding a solution improvement heuristic, such as FixandOptimize, run times increase. However, runtimes still represent 1/3 of the time consumed to solve the MIP model with CPLEX.
The overlapping strategies have more binary variables to be optimized at once in relation to the other strategies. Therefore, its processing time is considerably longer. Strategies that minimize backlogs are similar in principle to overlapping strategies in order to increase the number of integer variables not fixed to correct decisions fixed in previous iterations. Thus, by correcting badly fixed variables, both the strategies of overlapping and with minimization of the backlogs presented more effective results (relationship of computational time and quality of solution).
6 CONCLUSIONS
In this paper, we study the lot sizing and scheduling problem with process configuration selection. We propose a general mathematical model and solution methods based on mathematical programming: the MIPheuristics. The literature presents several applied studies of this problem considering the various specificities of the real problems. However, general mathematical modeling is still rare and solution methods are specific to the problem or adapted directly from the problem without configuration selection. The proposed mathematical model for the lot sizing and sequencing problem comprises the selection of which configurations are used for the item production. This problem is common in process industries, for example in furniture manufacturing, molded pulp packaging and electro fused grain industries, which were used as references in this study.
Several MIPheuristic strategies and combinations are proposed and compared using computational test results. Three sets of instances based on data from the furniture, molded pulp packaging and electro melted grain industries literature were used. The best strategy is to use the combination of MIPheuristics FixandOptimize with configuration partitions that use RelaxandFix, which minimizes backlogs. They present better results than the resolution of the mathematical model using CPLEX and in one third of the execution time. In general, the proposed model, as it stands or with some adaptations, has the potential to adequately represent the production planning environment of these and other process industries in practice. The FixandOptimize heuristics with configuration partitions that use RelaxandFix that minimize backlogs is a good candidate to be used in practice to solve the problem.
An interesting perspective for future research would be to investigate other formulations such as CLSD (Capacitated Lot sizing Problem with sequencedependent setups), valid inequalities for formulations and other more refined methods to solve larger instances of the problem, e.g., hybrid methods combining exact methods with metaheuristics. An exact method with the potential to solve this problem more effectively would be to adapt the algorithm BranchandCheck using cuts based on the Benders logic explored in Martínez et al. (201939 MARTÍNEZ KP, ADULYASAK Y, JANS R, MORABITO R & TOSO EAV. 2019. An exact optimization approach for an integrated process configuration, lotsizing, and scheduling problem. Computers & Operations Research , 103: 310323.). The model and solution methods presented presume that all process configurations are known a priori. In cases where this is not reasonable, another interesting perspective for future research would be to reformulate the model. The possible configurations of these processes can be implicitly described in the model. Or, alternatively, developing solution methods with column generation procedures to implicitly determine the best process configurations in each problem. This generation will depend largely on the specific characteristics of the process and equipment involved and could be done through more sophisticated methods based on DantzigWolfe decomposition with column generation and BranchandPrice methods.
Other interesting future research would be to extend the model and solution methods to consider more general problem situations, for example, with multiple production lines in parallel (multimachines) and with multiple production stages (multilevel). Moreover, developing approaches to consider uncertainties in problem parameters, based on stochastic programming methods and robust optimization, for example. In some cases, companies have difficulties in accurately estimating the economic loss due to lack or backlogs in meeting demand. In these cases, the proposal of biobjective approaches, considering the tradeoff between the backlogs in meeting demand and the costs of product inventories and configuration changes, could be useful to support production planning. Finally, it would be interesting to better evaluate the impact of the practical implementation of the proposed solution approaches to the lot sizing and scheduling problem with process configuration selection in real situations of these and other process industries.
ACKNOWLEDGEMENTS
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior  Brasil (CAPES)  Finance Code 001. We would also like to thank CNPq and FAPESP (2016/018601).
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Publication Dates

Publication in this collection
11 June 2021 
Date of issue
2021
History

Received
16 June 2020 
Accepted
10 Mar 2021