Mixed integer linear programming models for minimizing ergonomic risk dispersion in an assembly line at the Nissan Barcelona factory

Mixed integer linear programming models for minimizing ergonomic risk dispersion in an assembly line at the Nissan Barcelona factory Joaquín Bautista-Valhondo · Rocío Alfaro-Pozo 1 Mixed integer linear programming models for minimizing ergonomic risk dispersion in an assembly line at the Nissan Barcelona factory Joaquín Bautista-Valhondo 1[0000-0002-2214-4991] , Rocío Alfaro-Pozo 2[0000-0001-8214-1875] Abstract We present a variant of the approach to the assembly line balancing problems, with the aim of reducing the ergonomic risk for operators of mixed-model assembly lines (MILP-3). Specifically, the MILP-3 model is focused on minimizing the average range between ergonomic risk values of workstations. Using a case study from Nissan’s plant in Barcelona, not only are the differences between levels of ergonomic risk of stations reduced, but we attempt to reduce the average maximum ergonomic risk of the assembly line. The new model is compared with two others, MILP-1 and MILP-2, which minimize the average maximum ergonomic risk and the average absolute deviation of the risks, respectively. Keywords: Assembly line balancing; Mixed-model assembly line; Ergonomic risk; Mixed integer linear programming. 1Joaquín Bautista-Valhondo (e-mail: [email protected]) IOC-ETSEIB, Universitat Politècnica de Catalunya, 08028 Barcelona, Spain 2Rocío Alfaro-Pozo (e-mail: ralfar[email protected]) EAE Business School, 08015 Barcelona, Spain Mixed integer linear programming models for minimizing ergonomic risk dispersion in an assembly line at the Nissan Barcelona factory Joaquín Bautista-Valhondo · Rocío Alfaro-Pozo 2 Modelos de Programación Lineal Entera Mixta para minimizar la dispersión del riesgo ergonómico en una línea de montaje de la fábrica Nissan Barcelona Joaquín Bautista-Valhondo 1[0000-0002-2214-4991] , Rocío Alfaro-Pozo 2[0000-0001-8214-1875] Resumen Se presenta una variante del problema de equilibrado de líneas de montaje de modelos mixtos cuyo objetivo es reducir la dispersión del riesgo ergonómico sujeto a restricciones de ciclo productivo y de área lineal disponible en las estaciones de trabajo (MILP-3). En concreto, el modelo MILP-3 tiene el objetivo de minimizar el rango medio del riesgo ergonómico que se determina a partir de los riesgos asociados por factores a las estaciones de trabajo de la línea de producción. Se aplica el nuevo modelo a un caso de estudio de la planta de motores de Nissan localizada en Barcelona para el que se emplea el conjunto de instancias Nissan-9Eng. Se realiza una experiencia computacional con el solver CPLEX para comparar el comportamiento de MILP-3 frente a otros dos modelos presentados en trabajos anteriores: (i) MILP-1 cuyo objetivo es minimizar el riesgo máximo promediado a partir de un conjunto de factores de riesgo, y (ii) MILP-2 centrado en minimizar la media de las desviaciones absolutas de los riesgos ergonómicos presentes en las estaciones de trabajo para cada factor. En nuestro caso de estudio y limitando a CPLEX con tiempo de CPU de 1000 segundos por instancia y modelo, la explotación del modelo MILP-3 ofrece excelentes resultados reduciendo las diferencias de los niveles de riesgo entre las estaciones de la línea, tanto en rango del riesgo ergonómico como en su desviación tipo. Además, si atendemos al criterio de minimización del máximo riesgo ergonómico promediado por factores, MILP-3 proporciona los mejores resultados en ganancia unitaria media frente a los modelos MILP-1 y MILP-2. Palabras clave: Equilibrado de líneas de montaje; Línea de montaje de modelos mixtos; Riesgo ergonómico; Programación lineal entera mixta. 1 Joaquín Bautista-Valhondo (e-mail: [email protected]) IOC-ETSEIB, Universitat Politècnica de Catalunya, 08028 Barcelona, Spain 2 Rocío Alfaro-Pozo (e-mail: [email protected]) EAE Business School, 08015 Barcelona, Spain Mixed integer linear programming models for minimizing ergonomic risk dispersion in an assembly line at the Nissan Barcelona factory Joaquín Bautista-Valhondo · Rocío Alfaro-Pozo 3 1 Preliminaries Currently, there are many manufacturing systems that operate mixed-model assembly lines. This type of production system facilitates flexible mass production, where different type of products, and with different features, levels of component consumption, and resource usage levels, must be assembled or disassembled without incurring excessive costs. In the automotive industry, major auto-assemblers have begun to overhaul some of their previously specialized car-assembly plants into flexible factories in order to produce several vehicle models on the same production line (Eynan and Dong 2012, Moreno and Terwiesch 2015). Competition and customer demands, clearly evident in this industry, drive the proliferation of product varieties (AlGeddawy and ElMaraghy 2010). However, such flexibility supposes two main problems with respect to establishing the configuration of the line and product sequence. Indeed, these issues have been discussed at length in literature under the names assembly line balancing problems or ALBPs (Salveson 1955; Baybars 1986; Scholl and Becker 2006; Boysen et al. 2007, 2008; Battaïa and Dolgui 2013) and mixed-model sequencing problems or MMSP (Miltenburg 1989; Yano and Rachamadugu 1991; Bautista et al. 1996; Boysen et al. 2009; Bautista and Cano 2011; Dörmer et al. 2015; Bautista-Valhondo 2016; Bautista and Alfaro-Pozo 2018; Bautista-Valhondo and Alfaro-Pozo 2018a). The first problems are focused on assigning the set of tasks or operations needed to manufacture the products to the set of workstations that make up the line, in accordance with an optimization criterion. The second ones consist of determining the manufacturing order of product types that make up the production plan, in order to maximize line productivity. With reference to ALB problems, one realistic variant is the time and space assembly line balancing problem (TSALBP) (Bautista and Pereira 2007; Chica et al. 2010). It considers the availability of space in the stations on the line in order to make operations more productive. Further, it makes use of a multi-objective problem definition (Greco 2005) to search for a set of optimal solutions to three optimization criteria: (i) number of stations 𝑚𝑚, (ii) cycle time 𝑐𝑐, and (iii) linear area of the workstations 𝐴𝐴. However, the latest research does not only include the productive and physical aspects of the assembly line, but also aspects related to: − Uncertainty in the input attributes of the tasks, such as operation time, caused by defining interval values or by setting different plausible scenarios with a set of possible values for the input attributes depending on historical data (Simaria et al. 2009; Xu and Xiao 2011; Dolgui and Kovalev 2012; Gurevsky et al. 2012, 2013) − The robustness of the assembly line configuration to mitigate the uncertainty defined by a set of possible demand scenarios or different demand plans (Chica et al. 2013, 2016; Li and Gao 2014; Papakostas et al. 2014; Chica et al. 2018) − Human resources, such as the ergonomic risks or the comfort of the production line (Otto and Scholl 2011; Bautista et al. 2013, Bautista, Batalla-García and Alfaro-Pozo 2016, Bautista, Alfaro-Pozo and Batalla-García 2016; Bortolini 2017; Otto and Battaïa 2017; Bautista-Valhondo et al. 2018; Bautista-Valhondo and Alfaro-Pozo 2018b) Mixed integer linear programming models for minimizing ergonomic risk dispersion in an assembly line at the Nissan Barcelona factory Joaquín Bautista-Valhondo · Rocío Alfaro-Pozo 4 In our framework, an ergonomically comfortable assembly line involves setting the maximum risk to a minimum level for any operator from the assembly line, as well as achieving a balanced sharing of ergonomic risks between the set of workstations. Somatic comfort refers to the set of physical demands to which workers are exposed during the workday. They can potentially cause muscle contractions that then induce chronic pain. There are several methods that analyze different risk factors to evaluate ergonomic risks that include postural loads, repetitive movements, and manual handling. In response to postural loads, workers may adopt inappropriate, asymmetrical, or uncomfortable postures during the workday. These postures can cause stress to the worker’s anatomy. The frequently used methods to analyze these types of ergonomic risk factors include the rapid upper limb assessment or RULA (Manghisi et al. 2017) and the Ovako working posture analysis system or OWAS (Brandl et al. 2017). Additionally, workers can perform activities that involve effort and rapid or repetitive movements of a muscle group. Repeated movements of the upper limbs can cause long-term musculoskeletal injuries. To assess the ergonomic risk involved in this type of movement, the occupational repetitive action or OCRA checklist (Rosecrance et al. 2017) is frequently used. In manual handling, some tasks performed by workers involve lifting, moving, pushing, holding, and transporting objects that can cause physical damage. The Revised NIOSH Lifting Equation (Arjmand et al. 2015), from the National Institute for Occupational Safety and Health, is a frequently used method to analyze this risk factor. When assessing the ergonomic risk in a workstation of an assembly line, one of the main drawbacks is the lack of unification of the disparate methods mentioned above. The specialization of each method to a single muscular disorder complicates the evaluation and designation of an ergonomic level of risk given to a task or set of tasks assigned to an assembly line workstation. For this reason, similar to the work of Bautista, Alfaro-Pozo and Batalla- García (2016), we propose a unified classification of risk levels in four categories: − Category 1: Acceptable level of risk. No action is required because there is no risk to the worker. − Category 2: Low/moderate level of risk. An analysis of the workstation is necessary. Corrective actions are recommended for its improvement in the immediate future. − Category 3: High level of risk. An analysis and improvement of the tasks assigned to the workstation are required immediately, as is medical supervision. Regular medical checks on workers are also recommended. − Category 4: Unacceptable level of risk. This requires an immediate modification of the workstation, its tasks, and the methods used. The continuity of workers in a job with this category of risk level can lead to serious bodily harm. Obviously, the evaluation and subsequent assignment of the level of risk (according to these categories) of a specific task with respect to a workstation must be established by an expert with knowledge in ergonomics, as well as the methods and processing times appropriate to the assembly line. With these considerations in mind, the remainder of this paper is structured as follows: In section 2, we briefly formalize the assembly line balancing problems with temporal, spatial, and ergonomic risk attributes. In section 3, we propose using mixed integer linear programming to model the problems under examination in this study. In section 4, we perform a computational experiment to analyze the behavior of the generated models with the help of a Mixed integer linear programming models for minimizing ergonomic risk dispersion in an assembly line at the Nissan Barcelona factory Joaquín Bautista-Valhondo · Rocío Alfaro-Pozo 5 case study on the Nissan plant’s engine. Finally, section 5 outlines the conclusions, and proposals for future research. 2 Problems TSALB-erg: TSALB with ergonomic risks Formalization: TSALB-erg is a family of ALB problems that establishes a surjective application between the elements of a set 𝐽𝐽 of indivisible tasks (𝑛𝑛 elements) and the elements of a set 𝐾𝐾 of workstations (𝑚𝑚 elements, with ≤𝑛𝑛 ). The tasks in group 𝐽𝐽 are classified into exclusive classes called workstations 𝑆𝑆𝑘𝑘 (𝑆𝑆𝑘𝑘⊆ 𝐽𝐽), which satisfy 𝐽𝐽=⋃𝑆𝑆𝑘𝑘𝑘𝑘∈𝐾𝐾 and 𝑆𝑆𝑘𝑘∩𝑆𝑆𝑘𝑘′=∅,∀{𝑘𝑘,𝑘𝑘′}∈𝐾𝐾. Each task 𝑗𝑗∈𝐽𝐽 is assigned to a single workstation 𝑘𝑘∈𝐾𝐾, and has a set 𝑃𝑃𝑗𝑗 of direct preceding tasks that must be completed before the task 𝑗𝑗 is started. Each task 𝑗𝑗∈𝐽𝐽 requires a processing time for its execution 𝑡𝑡𝑗𝑗> 0 that is determined as a function of the manufacturing technologies and employed resources. Each station 𝑘𝑘∈𝐾𝐾 has a workload time 𝑡𝑡(𝑆𝑆𝑘𝑘) that is equal to the sum of the processing times of its assigned tasks, and cannot exceed the cycle time of the assembly line 𝑐𝑐. Each task 𝑗𝑗∈𝐽𝐽 requires a linear area calculation that must be performed, that is, 𝑎𝑎𝑗𝑗≥0, which is determined as a function of the spatial needs of the workers, robots, and the parts of the product. Each station 𝑘𝑘∈𝐾𝐾 has a workload linear area 𝑎𝑎(𝑆𝑆𝑘𝑘) that is equal to the sum of the linear areas of its assigned tasks, and cannot exceed the available space or linear area assigned to each workstation 𝐴𝐴. In addition, each task 𝑗𝑗∈𝐽𝐽 has an associated ergonomic risk 𝑅𝑅𝜙𝜙,𝑗𝑗≥0 that depends on the risk factor 𝜙𝜙∈Φ and the processing time 𝑡𝑡𝑗𝑗. Each station 𝑘𝑘∈𝐾𝐾 has a workload ergonomic risk 𝑅𝑅𝜙𝜙(𝑆𝑆𝑘𝑘) for the factor 𝜙𝜙∈Φ that is equal to the sum of the ergonomic risks of its assigned tasks, and cannot exceed the maximum ergonomic risk for the risk factor 𝜙𝜙∈Φ, 𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚. The purpose of the problems in the TSALB-erg family is to address assigning all tasks to workstations in order to achieve maximum efficiency regarding some of the considered attributes, while all constraints imposed are fulfilled. In this work, we will focus on minimizing the ergonomic risk of the line and its dispersion between workstations. To formalize this purpose, three mathematical models adapted to mixed integer linear programming (MILP) are presented here. 3 MILP models for minimizing the ergonomic risk and its dispersion in lines with fixed number of workstations There are different ways to address the balancing problem in order to obtain comfortable line configurations in terms of ergonomics. − Simultaneously minimizing the maximum ergonomic risk and the risk differences between workstations using a multi-objective model − Prioritizing one objective over the other one − Solving the problem mono-objectively, and assessing the other objective afterwards Mixed integer linear programming models for minimizing ergonomic risk dispersion in an assembly line at the Nissan Barcelona factory Joaquín Bautista-Valhondo · Rocío Alfaro-Pozo 6 Accordingly, this work addresses three mono-objective mathematical models that aim at minimizing: (i) the average maximum ergonomic risk, (ii) The average absolute deviation of the ergonomic risk, and (iii) the average range of the ergonomic risks of workstations. Thus, the size of the smallest interval that contains all the ergonomic risks of the workstations is measured. BASIC NOMENCLATURE Parameters: 𝐽𝐽 Set of elemental tasks (𝑗𝑗= 1, … , |𝐽𝐽|) ; 𝑛𝑛=|𝐽𝐽| 𝐾𝐾 Set of workstations (𝑘𝑘= 1, … , |𝐾𝐾|) Φ Set of ergonomic risk factors (𝜙𝜙= 1, … , |Φ|) 𝑡𝑡𝑗𝑗 Processing time of the elemental task 𝑗𝑗∈𝐽𝐽 at normal activity levels 𝑎𝑎𝑗𝑗 Linear area required by the elemental task 𝑗𝑗∈𝐽𝐽 𝜒𝜒𝜙𝜙,𝑗𝑗 Category of task 𝑗𝑗∈𝐽𝐽 associated with the risk factor 𝜙𝜙∈Φ 𝑅𝑅𝜙𝜙,𝑗𝑗 Ergonomic risk of task 𝑗𝑗∈𝐽𝐽 associated with the risk factor 𝜙𝜙∈Φ , 𝑅𝑅𝜙𝜙,𝑗𝑗=𝑡𝑡𝑗𝑗∙ 𝜒𝜒𝜙𝜙,𝑗𝑗 𝑃𝑃𝑗𝑗 Set of direct precedent tasks of task 𝑗𝑗∈𝐽𝐽 𝑐𝑐 Cycle time: standard time assigned to each station to process its workload (𝑆𝑆𝑘𝑘) 𝑚𝑚 Number of workstations 𝑚𝑚=|𝐾𝐾| , which is known and fixed 𝐴𝐴 Available space or linear area assigned to each workstation 𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚 Average ergonomic risk for the risk factor 𝜙𝜙∈Φ , 𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚≡1 𝑚𝑚∙∑𝑅𝑅𝜙𝜙,𝑗𝑗 |𝐽𝐽| 𝑗𝑗=1 𝑅𝑅𝑚𝑚𝑚𝑚𝑚𝑚 Average ergonomic risk of the line or ideal ergonomic risk of each workstation 𝑅𝑅𝑚𝑚𝑚𝑚𝑚𝑚≡1 |Φ|∙∑𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚 |Φ| 𝜙𝜙=1 Variables: 𝑥𝑥𝑗𝑗,𝑘𝑘 Binary variable equal to 1 if the elemental task 𝑗𝑗∈𝐽𝐽 is assigned to the workstation 𝑘𝑘∈𝐾𝐾 , and to 0 otherwise 𝑆𝑆𝑘𝑘 Workload of station 𝑘𝑘∈𝐾𝐾 : set of tasks assigned to 𝑘𝑘∈𝐾𝐾:𝑆𝑆𝑘𝑘=�𝑗𝑗∈𝐽𝐽:𝑥𝑥𝑗𝑗,𝑘𝑘= 1� 𝑅𝑅𝜙𝜙(𝑆𝑆𝑘𝑘) Ergonomic risk for the factor 𝜙𝜙∈Φ associated with the workload 𝑆𝑆𝑘𝑘 (𝑘𝑘∈ 𝐾𝐾),𝑅𝑅𝜙𝜙(𝑆𝑆𝑘𝑘)=∑𝑅𝑅𝜙𝜙,𝑗𝑗𝑗𝑗∈𝑆𝑆𝑘𝑘 𝑅𝑅(𝑆𝑆𝑘𝑘) Average ergonomic risk associated with the workload 𝑆𝑆𝑘𝑘 (𝑘𝑘∈𝐾𝐾) with respect to the full set of ergonomic risk factors Φ , 𝑅𝑅(𝑆𝑆𝑘𝑘)≡1 |Φ|∙∑𝑅𝑅𝜙𝜙(𝑆𝑆𝑘𝑘) |Φ| 𝜙𝜙=1 𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚 Maximum ergonomic risk for the risk factor 𝜙𝜙∈Φ , 𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚 = max 𝑘𝑘∈𝐾𝐾 𝑅𝑅𝜙𝜙(𝑆𝑆𝑘𝑘) 𝑅𝑅𝑚𝑚𝑚𝑚𝑚𝑚 Average maximum ergonomic risk with respect to the full set of ergonomic risk factors Φ,𝑅𝑅𝑚𝑚𝑚𝑚𝑚𝑚≡1 |Φ|∙∑𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚 |Φ| 𝜙𝜙=1 =1 |Φ|∑max 𝑘𝑘∈𝐾𝐾𝑅𝑅𝜙𝜙(𝑆𝑆𝑘𝑘) |Φ| 𝜙𝜙=1 𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚 Minimum ergonomic risk for the risk factor 𝜙𝜙∈Φ , 𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚 = min 𝑘𝑘∈𝐾𝐾𝑅𝑅𝜙𝜙(𝑆𝑆𝑘𝑘) . Mixed integer linear programming models for minimizing ergonomic risk dispersion in an assembly line at the Nissan Barcelona factory Joaquín Bautista-Valhondo · Rocío Alfaro-Pozo 7 𝑅𝑅𝑚𝑚𝑚𝑚𝑚𝑚 Average minimum ergonomic risk with respect to all sets of ergonomic risk factors Φ , 𝑅𝑅𝑚𝑚𝑚𝑚𝑚𝑚≡ 1 |Φ|∙∑𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚 |Φ| 𝜙𝜙=1 =1 |Φ|∑min 𝑘𝑘∈𝐾𝐾𝑅𝑅𝜙𝜙(𝑆𝑆𝑘𝑘) |Φ| 𝜙𝜙=1 𝛿𝛿𝜙𝜙,𝑘𝑘 + Ergonomic risk excess associated with the risk factor 𝜙𝜙∈Φ at workstation 𝑘𝑘∈𝐾𝐾 with respect to the average (ideal) value 𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚 , 𝛿𝛿𝜙𝜙,𝑘𝑘 +=𝑚𝑚𝑎𝑎𝑥𝑥�0, 𝑅𝑅𝜙𝜙(𝑆𝑆𝑘𝑘)−𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚� . 𝛿𝛿𝜙𝜙,𝑘𝑘 − Ergonomic risk defect associated with the risk factor 𝜙𝜙∈Φ at workstation 𝑘𝑘∈𝐾𝐾 with respect to the average (ideal) value 𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚 , 𝛿𝛿𝜙𝜙,𝑘𝑘 −=𝑚𝑚𝑎𝑎𝑥𝑥�0, 𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚−𝑅𝑅𝜙𝜙(𝑆𝑆𝑘𝑘)� . 3.1 MODEL FOR MINIMIZING THE AVERAGE MAXIMUM ERGONOMIC RISK MILP-1 · min 𝑅𝑅𝑚𝑚𝑚𝑚𝑚𝑚 min 𝑍𝑍=𝑅𝑅𝑚𝑚𝑚𝑚𝑚𝑚≡1 |Φ|∙� 𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚 |Φ| 𝜙𝜙=1 (1) Subject to: � 𝑥𝑥𝑗𝑗,𝑘𝑘= 1 𝑚𝑚 𝑘𝑘=1 ∀𝑗𝑗= 1, . , |𝐽𝐽| (2) � 𝑡𝑡𝑗𝑗𝑥𝑥𝑗𝑗,𝑘𝑘≤𝑐𝑐 |𝐽𝐽| 𝑗𝑗=1 ∀𝑘𝑘= 1, . , 𝑚𝑚 (3) � 𝑎𝑎𝑗𝑗𝑥𝑥𝑗𝑗,𝑘𝑘≤𝐴𝐴 |𝐽𝐽| 𝑗𝑗=1 ∀𝑘𝑘= 1, . , 𝑚𝑚 (4) � 𝑘𝑘�𝑥𝑥𝑚𝑚,𝑘𝑘−𝑥𝑥𝑗𝑗,𝑘𝑘� 𝑚𝑚 𝑘𝑘=1 ≤0 ∀{𝑖𝑖,𝑗𝑗}⊆𝐽𝐽:𝑖𝑖∈𝑃𝑃𝑗𝑗 (5) � 𝑥𝑥𝑗𝑗,𝑘𝑘 |𝐽𝐽| 𝑗𝑗=1 ≥1 ∀𝑘𝑘= 1, . , 𝑚𝑚 (6) 𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚−� 𝑅𝑅𝜙𝜙,𝑗𝑗∙𝑥𝑥𝑗𝑗,𝑘𝑘 |𝐽𝐽| 𝑗𝑗=1 ≥0 ∀𝑘𝑘= 1, . , 𝑚𝑚 ∀𝜙𝜙= 1, . , |Φ| (7) 𝑥𝑥𝑗𝑗,𝑘𝑘∈{0,1} ∀𝑗𝑗= 1, . , |𝐽𝐽| ∀𝑘𝑘= 1, . , 𝑚𝑚 (8) 𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚≥0 ∀𝜙𝜙= 1, . , |Φ| (9) The objective function (1) expresses the minimization of the average maximum ergonomic risk. Constraint (2) forces the assignment of all tasks. Constraints (3) and (4) impose the maximum limitation of the workload time and the maximum linear area allowed by each station. Constraint (5) corresponds to the precedence task bindings, while constraint (6) ensures that there are no empty workstations. Constraint (7) determines the maximum ergonomic risk associated with the workload at each workstation and with each ergonomic factor analyzed. Finally, constraints (8) and (9) necessitate that the assigned variables be binary and the maximum ergonomic risk variables for the risk factors be non-negative. Mixed integer linear programming models for minimizing ergonomic risk dispersion in an assembly line at the Nissan Barcelona factory Joaquín Bautista-Valhondo · Rocío Alfaro-Pozo 8 3.2 MODEL FOR MINIMIZING THE AVERAGE ABSOLUTE DEVIATION OF ERGONOMIC RISK MILP-2 · min 𝐴𝐴𝐴𝐴𝐴𝐴(𝑅𝑅): min 𝑍𝑍=1 𝑚𝑚|Φ|� � �𝛿𝛿𝜙𝜙,𝑘𝑘 ++𝛿𝛿𝜙𝜙,𝑘𝑘 −� 𝑚𝑚 𝑘𝑘=1 |Φ| 𝜙𝜙=1 (10) Subject to: � 𝑥𝑥𝑗𝑗,𝑘𝑘= 1 𝑚𝑚 𝑘𝑘=1 ∀𝑗𝑗= 1, . , |𝐽𝐽| (11) � 𝑡𝑡𝑗𝑗𝑥𝑥𝑗𝑗,𝑘𝑘≤𝑐𝑐 |𝐽𝐽| 𝑗𝑗=1 ∀𝑘𝑘= 1, . , 𝑚𝑚 (12) � 𝑎𝑎𝑗𝑗𝑥𝑥𝑗𝑗,𝑘𝑘≤𝐴𝐴 |𝐽𝐽| 𝑗𝑗=1 ∀𝑘𝑘= 1, . , 𝑚𝑚 (13) � 𝑘𝑘�𝑥𝑥𝑚𝑚,𝑘𝑘−𝑥𝑥𝑗𝑗,𝑘𝑘� 𝑚𝑚 𝑘𝑘=1 ≤0 ∀{𝑖𝑖,𝑗𝑗}⊆𝐽𝐽:𝑖𝑖∈𝑃𝑃𝑗𝑗 (14) � 𝑥𝑥𝑗𝑗,𝑘𝑘 |𝐽𝐽| 𝑗𝑗=1 ≥1 ∀𝑘𝑘= 1, . , 𝑚𝑚 (15) � 𝑅𝑅𝜙𝜙,𝑗𝑗∙𝑥𝑥𝑗𝑗,𝑘𝑘 |𝐽𝐽| 𝑗𝑗=1 =𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚+𝛿𝛿𝜙𝜙,𝑘𝑘 +−𝛿𝛿𝜙𝜙,𝑘𝑘 − ∀𝑘𝑘= 1, . , 𝑚𝑚 ∀𝜙𝜙= 1, . , |Φ| (16) 𝑥𝑥𝑗𝑗,𝑘𝑘∈{0,1} ∀𝑗𝑗= 1, . , |𝐽𝐽| ∀𝑘𝑘= 1, . , 𝑚𝑚 (17) 𝛿𝛿𝜙𝜙,𝑘𝑘 +,𝛿𝛿𝜙𝜙,𝑘𝑘 −≥0 ∀𝑘𝑘= 1, . , 𝑚𝑚 ∀𝜙𝜙= 1, . , |Φ| (18) In the MILP-2 (min 𝐴𝐴𝐴𝐴𝐴𝐴(𝑅𝑅)) model, it is obvious that the constraint blocks (11)–(15) and (17) consecutively match formulas (2)–(6) and (8) of the MILP-1 (min 𝑅𝑅max ) model. The changes that are added by considering the absolute deviations are: − The objective function (10) expresses the minimization of the average absolute deviation of the ergonomic risk with respect to the average ergonomic risk of the line − Restriction (16) determines the ergonomic risk excess and defect associated with the risk factor 𝜙𝜙∈Φ at workstation 𝑘𝑘∈𝐾𝐾 with respect to the ideal value 𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚 − Condition (18) forces the deviation variables (𝛿𝛿𝜙𝜙,𝑘𝑘 +,𝛿𝛿𝜙𝜙,𝑘𝑘 −) to be non-negative 3.3 MODEL FOR MINIMIZING THE AVERAGE RANGE OF THE ERGONOMIC RISK MILP-3 · min 𝐴𝐴𝑅𝑅(𝑅𝑅): min 𝑍𝑍=1 |Φ|∙� �𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚−𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚� |Φ| 𝜙𝜙=1 (19) Subject to: Mixed integer linear programming models for minimizing ergonomic risk dispersion in an assembly line at the Nissan Barcelona factory Joaquín Bautista-Valhondo · Rocío Alfaro-Pozo 9 � 𝑥𝑥𝑗𝑗,𝑘𝑘= 1 𝑚𝑚 𝑘𝑘=1 ∀𝑗𝑗= 1, . , |𝐽𝐽| (20) � 𝑡𝑡𝑗𝑗𝑥𝑥𝑗𝑗,𝑘𝑘≤𝑐𝑐 |𝐽𝐽| 𝑗𝑗=1 ∀𝑘𝑘= 1, . , 𝑚𝑚 (21) � 𝑎𝑎𝑗𝑗𝑥𝑥𝑗𝑗,𝑘𝑘≤𝐴𝐴 |𝐽𝐽| 𝑗𝑗=1 ∀𝑘𝑘= 1, . , 𝑚𝑚 (22) � 𝑘𝑘�𝑥𝑥𝑚𝑚,𝑘𝑘−𝑥𝑥𝑗𝑗,𝑘𝑘� 𝑚𝑚 𝑘𝑘=1 ≤0 ∀{𝑖𝑖,𝑗𝑗}⊆𝐽𝐽:𝑖𝑖∈𝑃𝑃𝑗𝑗 (23) � 𝑥𝑥𝑗𝑗,𝑘𝑘 |𝐽𝐽| 𝑗𝑗=1 ≥1 ∀𝑘𝑘= 1, . , 𝑚𝑚 (24) 𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚−� 𝑅𝑅𝜙𝜙,𝑗𝑗∙𝑥𝑥𝑗𝑗,𝑘𝑘 |𝐽𝐽| 𝑗𝑗=1 ≥0 ∀𝑘𝑘= 1, . , 𝑚𝑚 ∀𝜙𝜙= 1, . , |Φ| (25) 𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚−� 𝑅𝑅𝜙𝜙,𝑗𝑗∙𝑥𝑥𝑗𝑗,𝑘𝑘≤ |𝐽𝐽| 𝑗𝑗=1 0 ∀𝑘𝑘= 1, . , 𝑚𝑚 ∀𝜙𝜙= 1, . , |Φ| (26) 𝑥𝑥𝑗𝑗,𝑘𝑘∈{0,1} ∀𝑗𝑗= 1, . , |𝐽𝐽| ∀𝑘𝑘= 1, . , 𝑚𝑚 (27) 𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚,𝑅𝑅𝜙𝜙 𝑚𝑚𝑚𝑚𝑚𝑚≥0 ∀𝜙𝜙= 1, . , |Φ| (28) In the MILP-3 (min 𝐴𝐴𝑅𝑅(𝑅𝑅)) model, the constraint blocks (20)–(25) and (27) consecutively match formulas (2)–(8) of the MILP-1 (min 𝑅𝑅max ) model. The changes that are added by considering the range of the ergonomic risks are: − The objective function (19) expresses the minimization of the average range of the ergonomic risks of workstations, that is, 𝑅𝑅𝑚𝑚𝑚𝑚𝑚𝑚−𝑅𝑅𝑚𝑚𝑚𝑚𝑚𝑚 − Restriction (26) determines the minimum ergonomic risk associated with the workload at each workstation and with each ergonomic factor analyzed − Condition (28) forces the maximum and minimum ergonomic risk variables for the risk factors to be non-negative 4 Computational experiment 4.1 DATA The computational experience is focused on analyzing the performance of the mathematical model proposed in this work, MILP-3, against the mathematical models MILP-1 and MILP-2 proposed in Bautista et al. (2016b). Like Bautista et al. 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Management Science 37(5), 572-586. https://doi.org/10.1287/mnsc.37.5.572 Mixed integer linear programming models for minimizing ergonomic risk dispersion in an assembly line at the Nissan Barcelona factory Joaquín Bautista-Valhondo · Rocío Alfaro-Pozo 19 Appendix A 𝑗𝑗∈𝐽𝐽 Precedent tasks: 𝑃𝑃𝑗𝑗 𝑗𝑗∈𝐽𝐽 Precedent tasks: 𝑃𝑃𝑗𝑗 𝑗𝑗∈𝐽𝐽 Precedent tasks: 𝑃𝑃𝑗𝑗 1 - 48 46 95 94 2 3, 31 49 42, 43 96 93, 95, 99 3 1 50 47, 48, 49 97 93, 95, 99 4 3, 5 51 47, 48, 49 98 92 5 1 52 47, 48, 49 99 89, 90, 91 6 4, 5 53 47, 48, 49 100 98, 99 7 1 54 47, 48, 49 101 98, 99 8 1 55 47, 48, 49 102 100, 101 9 1 56 47, 48, 49 103 100, 101 10 1 57 50, 51, 52, 53, 54, 55, 56 104 102, 103 11 1 58 57, 59, 60 105 106 12 11 59 41 106 100, 101 13 1 60 42, 43 107 100, 101, 104 14 1, 13 61 57, 58 108 100, 101, 104 15 9, 10, 11, 13, 14 62 61 109 108 16 9, 10, 11, 13, 14 63 57 110 108 17 9, 10, 11, 13, 14 64 57 111 11, 109 18 9, 10, 11, 13, 14 65 61, 62, 63, 64 112 11, 109 19 9, 10, 11, 13, 14 66 61, 62, 63, 64 113 108 20 9, 10, 11, 13, 14 67 66 114 113 21 9, 10, 11, 13, 14 68 65, 67 115 113 22 26, 27 69 68 116 111, 112, 114, 115 23 26, 27 70 67 117 118 24 26, 27 71 68 118 116 25 26, 27 72 68 119 116 26 15, 16, 17, 18, 19, 20, 21 73 71, 72 120 119 27 15, 16, 17, 18, 19, 20, 21 74 68, 69, 70, 73 121 105, 107, 117, 120 28 22, 23, 24, 25 75 74 122 121 29 28 76 74 123 122 30 29 77 75 124 123 31 6, 7, 8, 30 78 79 125 124 32 31 79 74 126 125 33 32 80 76, 77, 78 127 126 34 32 81 76, 77, 78 128 12, 117 35 36 82 80, 81 129 126 36 32 83 82 130 127, 128, 129 37 32, 35 84 83 131 12, 117 38 33, 34, 36, 37 85 75, 84 132 131 39 33, 34, 36, 37 86 82 133 130 40 33, 34, 36, 37 87 82 134 132 41 38, 39, 40 88 84 135 134 42 38, 39, 40 89 88 136 135 43 38, 39, 40 90 88 137 136 44 41, 42, 43 91 85, 86, 87, 88 138 136 45 41, 42, 43 92 89, 90, 91 139 137, 138 46 44, 45 93 92 140 133, 139 47 46 94 89, 90, 91 Table 4: Instance 𝜀𝜀= 1 from the Nissan-9Eng’s Set of Demand Plans: Set of elemental tasks (𝑗𝑗= 1, … ,140), and subsets of immediate precedent tasks of task 𝑗𝑗: 𝑃𝑃𝑗𝑗 (𝑗𝑗= 1, … , |𝐽𝐽|). Mixed integer linear programming models for minimizing ergonomic risk dispersion in an assembly line at the Nissan Barcelona factory Joaquín Bautista-Valhondo · Rocío Alfaro-Pozo 20 𝑗𝑗∈𝐽𝐽 𝑡𝑡𝑗𝑗 𝑎𝑎𝑗𝑗 𝜒𝜒𝜙𝜙,𝑗𝑗 𝑗𝑗∈𝐽𝐽 𝑡𝑡𝑗𝑗 𝑎𝑎𝑗𝑗 𝜒𝜒𝜙𝜙,𝑗𝑗 𝑗𝑗∈𝐽𝐽 𝑡𝑡𝑗𝑗 𝑎𝑎𝑗𝑗 𝜒𝜒𝜙𝜙,𝑗𝑗 1 60.00 300 1 48 35.00 50 3 95 20.00 50 3 2 75.00 200 2 49 5.00 50 3 96 10.00 50 3 3 20.00 50 1 50 15.00 50 3 97 5.00 50 3 4 60.00 100 1 51 25.00 0 3 98 80.00 0 2 5 20.00 50 1 52 30.00 0 3 99 30.00 0 3 6 60.00 150 1 53 15.00 0 3 100 10.00 50 2 7 45.00 100 2 54 15.00 0 3 101 10.00 50 2 8 10.00 50 2 55 20.00 0 3 102 20.00 50 2 9 20.00 50 2 56 10.00 0 3 103 30.00 50 2 10 30.00 50 2 57 10.00 50 3 104 5.00 0 3 11 15.00 50 2 58 20.00 50 2 105 30.00 50 2 12 15.00 50 2 59 5.00 0 3 106 25.00 50 2 13 15.00 100 1 60 20.00 50 3 107 5.00 0 3 14 10.00 50 2 61 45.00 100 2 108 5.00 0 2 15 8.00 100 2 62 30.00 50 2 109 5.00 50 2 16 8.00 50 2 63 30.00 50 2 110 5.00 0 2 17 80.00 100 2 64 10.00 50 2 111 10.00 0 2 18 40.00 50 2 65 5.00 0 2 112 10.00 0 2 19 5.00 50 2 66 10.00 50 2 113 15.00 50 2 20 5.00 50 2 67 15.00 50 2 114 20.00 0 2 21 5.00 50 2 68 60.00 150 2 115 20.00 0 2 22 7.00 50 2 69 10.00 50 2 116 45.00 100 2 23 7.00 50 2 70 30.00 100 2 117 20.00 50 2 24 30.00 50 2 71 10.00 50 2 118 25.00 0 2 25 30.00 50 2 72 10.00 50 2 119 25.00 0 2 26 5.00 50 2 73 40.00 150 2 120 20.00 50 2 27 5.00 50 2 74 25.00 50 2 121 45.00 150 2 28 30.00 100 2 75 10.00 50 2 122 15.00 50 1 29 10.00 50 2 76 10.00 100 2 123 10.00 50 1 30 15.00 100 2 77 15.00 50 2 124 10.00 0 1 31 10.00 0 2 78 15.00 50 2 125 20.00 100 1 32 15.00 50 2 79 15.00 50 2 126 30.00 50 2 33 30.00 100 3 80 10.00 50 2 127 10.00 50 2 34 10.00 50 3 81 10.00 100 2 128 25.00 50 2 35 5.00 50 3 82 10.00 0 2 129 30.00 50 2 36 25.00 100 2 83 20.00 50 2 130 30.00 75 2 37 15.00 0 3 84 10.00 0 2 131 40.00 50 2 38 5.00 50 3 85 20.00 50 3 132 25.00 100 1 39 5.00 50 3 86 25.00 50 2 133 25.00 50 1 40 5.00 50 3 87 20.00 50 2 134 20.00 50 1 41 60.00 50 3 88 15.00 25 3 135 15.00 50 1 42 15.00 150 3 89 20.00 50 3 136 20.00 50 1 43 15.00 150 3 90 30.00 50 3 137 30.00 50 2 44 25.00 50 3 91 20.00 50 3 138 30.00 50 2 45 25.00 50 3 92 25.00 50 3 139 15.00 100 2 46 5.00 50 3 93 10.00 50 3 140 120.00 0 1 47 35.00 50 3 94 5.00 50 3 Table 5: Instance 𝜀𝜀= 1 from the Nissan-9Eng’s Set of Demand Plans: Elemental tasks (𝑗𝑗= 1, … ,140), processing time of tasks (𝑡𝑡𝑗𝑗), linear area required by the tasks (𝑎𝑎𝑗𝑗), and category of tasks (𝜒𝜒𝜙𝜙,𝑗𝑗) associated with the risk factor 𝜙𝜙.