Conceptual Framework for the Optimization of Capacitated Lot-Sizing and Scheduling Problem

Conceptual Framework for the Optimization of Capacitated Lot-Sizing and Scheduling Problem Juan Pablo Fiesco, Ana Esteso(B), M. M. E. Alemany, and Raúl Poler Research Centre on Production Management and Engineering (CIGIP), Universitat Politècnica de València (UPV), Camino de Vera s/n, 46022 Valencia, Spain {jfiesco,aesteso,mareva,rpoler}@cigip.upv.es Abstract. The complex nature of the capacitated lot-sizing problem, particularly within the scheduling, requires a holistic approach that considers multiple factors and their intricate interactions. In this context, a novel conceptual framework (CF) to support the combined Capacitated Lot-Sizing and Scheduling Problem (CLSSP) through mathematical modelling is proposed. The CF is developed through a rigorous methodology that combines data collection, data analysis, literature review and conceptual framework definition. It is composed by seven dimensions, with different categories, in turn made up of elements related to different aspects of the problem and its modelling. The CF serves a dual purpose: as a comprehensive tool for the structured analysis of existing models facilitating the gaps identification and as a guide for proposing novel mathematical programming models to address the combined complexities of the CLSSP including those not yet covered. Keywords: Conceptual Framework · Capacitated-Lot Sizing · Scheduling · Mathematical programming · Optimization 1 Introduction Optimization of simultaneously lot-sizing and scheduling becomes crucial as production quantities and capacity utilization hinge on the process configurations, their durations, and the sequencing employed (Martínez et al., 2019). This dual focus is attracting attention from both academy and industry (Copil et al. 2017) under the name of Capacitated Lot-Sizing and Scheduling Problem (CLSSP) that can be considered as an extension of the Capacitated Lot-Sizing Problem (CLSP). The CLSSP complexity makes necessary a framework to characterize the problem under study, to systematically revise and classify existing optimization models and to identify gaps with the aim of developing novel models to cover them. The CF originality lies in the fact that it is structured to facilitate the formulation of mathematical programming models, drawing attention to important aspects. Most of them can be identified as rarely addressed supporting, therefore, the creation of innovative solutions. The rest of the paper is structured as follows. In Sect. 2 the rigorous research methodology applied to derive the CF is presented. Section 3 describes in detail the novel CF. Meanwhile, in Sect. 4 the main conclusions and future research lines are outlined. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2025 R. Carrasco-Gallego et al. (Eds.): CIO 2024, LNDECT 239, pp. 276–281, 2025. https://doi.org/10.1007/978-3-031-82334-3_47 Conceptual Framework for the Optimization 277 2 Research Methodology The research methodology proposed by Lorente-Leyva et al. (2024), that combines the methods of Seuring & Müller (2008) and Esteso et al. (2018), is followed to develop the CF. This methodology encompasses the following steps: 1. Material collection. It involves defining and delimiting the material to be collected. In this paper, keywords such as “Capacitated Lot-Sizing Problem”, “CLSP”, “Capacitated Lot-sizing and Scheduling Problem”, “CLSSP”, “Mathematical Programming”, “MILP”, “Optimization”, “Heuristic”, “Metaheuristic”, “Matheuristic” and “Hybrid” were used to conduct searches in the Web of Science and Scopus scientific databases. Only papers including models for the CLSSP written in English were considered. 2. Bibliographic data processing and visualisation. It involves analysing the formal aspects of the material collected, such as publication dates and sources. 3. Tentative definition of the CF. A draft of the CF is established by identifying dimensions, categories, and elements from previous literature reviews such as Comelli et al. (2008), Copiletal. (2017), Karimi et al. (2003), Quadt & Kuhn (2008), and Robinson et al. (2009). Novel aspects have been included based on the author’s knowledge of the problem and the research objective oriented to formulate optimization models. 4. Structured literature review. It involves analysing and evaluating materials according to the initial draft of the CF. In this paper, 34 papers were analysed to enrich the initial CF. 5. Refined CF and Final proposal. The initial draft is refined by integrating new dimensions and relevant categories identified during t he review process until its final proposal. 3 Conceptual Framework for CLSSP The application of the previous methodology results in the following CF used for the optimization of the CLSSP through mathematical programming. This CF encompasses seven dimensions, divided into categories and elements (Fig. 1). The “SC physical characteristics” dimension comprises three categories describing the physical attributes of the Supply Chain (SC). “Sector” defines the scope in which the model is applied (e.g. chemical, paper, textile, plastic). “SC stages” details key SC operations covered by the model: procurement, production, inventory, distribution, and sale. “Shop floor configuration” specifies the production’s area setup: single-machine, parallel machines, flowshop, or jobshop. The “Planning and scheduling characteristics” dimension is composed by five categories. “Number of products” distinguishes between single-product and multi-product scenarios. “Product characteristics” identifies dependencies or incompatibilities of products. “Sequence types” can be linear sequences, lacking a specific order of products during production, or cyclic sequences, which facilitate simple and safe changeovers for operators (Tousain & Bosgra, 2006). Within cyclic sequences, they can be non-adaptive, which adhere to a fixed pattern (Liberopoulos et al., 2013), and adaptive, which allow for adjustments in the pattern. “Setup type” comprises sequence-dependent setups, where the preparation time or cost varies with the production sequence, sequence-independent 278 J. P. Fiesco et al. Fig. 1. Conceptual framework for CLSSP setups unaffected by the production order, quantity-dependent setups, where the preparation time or cost rises with batch size, and quantity-independent setups, not tied to the quantity of production, and carry-over, where it is possible to continue production from the previous period into the current period without the need for a new setup, thus reducing costs and setup times (Karimi et al., 2003). “Temporal aspects” identifies the planning horizon and its division into a single or multiple periods, and the granularity of time intervals categorized as “Small bucket” when producing one product per period and resource, or “Big bucket” when producing multiple products per period and resource (Comelli et al., 2008; Karimi et al., 2003). In the “Decisional aspects” dimension, decisions influencing the CLSSP are considered, representing the decision variables of the model. Key decisions include the lot sizing, the allocation of jobs to resources, their sequencing and timing, the inventory management, sale decisions, and workforce-related decisions such as overtime and shifts. Additional decisional aspects such as outsourcing, unmet demand, and deferred demand are included in “Other decisions”. “Constraints” dimension includes categories representing potential limitations and minimum requirements across production, inventory, transportation, and sales planning. These include constraints related to product attributes, such as minimum batch sizes, product dependencies, incompatibilities, and sequencing rules. “Workforce” constraints have limits on overtime hours, the maximum number of shifts, and regulations governing shift scheduling (Sahling et al., 2009). “Inventory” constraints involve limitations on capacity, allocation, and mini-mum stock levels over the planning horizon (Popović et al., 2023). “Transport” constraints affect capacity and minimum quantity requirements (Martínez et al., 2019). “Market” constraints may involve meeting total demand or a minimum percentage thereof and the ability to defer a maximum demand rate (Larroche et al., 2022). “Machines” constraints relate to production capacity, allocation limitations, and considerations regarding maintenance scheduling. Conceptual Framework for the Optimization 279 The “Objective” dimension is divided into three categories aligned with sustainability: economic, environmental and social. Common CLSSP objectives include minimising inventory and setup costs, which fall into the economic category. Environmental objectives typically focus on reducing pollution and resource consumption (Retel Helmrich et al., 2015). In addition, a novel objective that has not been widely addressed in CLSSP models is customer satisfaction or employment, which represents the social dimension. The “Modelling Approach” dimension provides a comprehensive framework for understanding the mathematical programming models. The “Mathematical programming model type” category i ncludes Continuous linear programming, Integer linear programming, Binary linear programming, Mixed integer linear programming, and Nonlinear programming. The “Solution method” categorises the approaches employed in solving the models such as exact solutions, heuristics, metaheuristics, and hybrid methods. The “Model validation” identifies the validation process used: Real case, Case study, and Numerical example. Finally, in the “Uncertainty Modelling” dimension, uncertainty aspects i n modelling are examined, considering whether the modelling context is deterministic or uncertain. In the uncertain context case, define which parameters are subject to uncertainty, such as demand or processing time, and specify uncertainty modelling approaches, such as deterministic-based, probabilistic, possibilistic, or scenario-based (Supithak et al., 2010; Curcio et al., 2018). 4 Conclusions and Future Research Lines A novel CF for the CLSSP has been proposed following a research methodology comprised of five s teps. The proposed CF serves a dual purpose. It can be employed as a guide for proposing new mathematical programming models and derivatives to address the combined problem of Capacitated Lot-Sizing and Scheduling. Alternatively, it serves as a comprehensive framework for reviewing existing models in order to identify existing gaps and future research lines. As a prospective research direction, a comprehensive state of the art of models addressing the CLSSP through the proposed CF should be conducted. In the literature review undertaken during the development of the CF methodology, an initial exploration of this state of the art has been initiated, providing a basis for potential expansion and subsequent publication. In this initial analysis, promising future research lines have been identified concerning the proposition of models to support the CLSSP, such as the necessity of modeling cyclic production sequences, particularly emphasizing adaptive cyclic sequences, incorporating dependencies and incompatibilities among products, and addressing uncertainty in the manufacturing sector. Additionally, aligning the models with sustainability is suggested, with a heightened focus on environmental and social aspects, which have received limited attention. Acknowledgements. This research has been partially funded by Horizon Europe Ref. 101057294 “AI-Driven I ndustrial Equipment Product Life Cycle Boosting Agility, Sustainability, and Resilience (AIDEAS)” and Ref. 101070076 “Optimizing Production And Logistic Resources In 280 J. P. Fiesco et al. 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