New criteria for configuration of cellular manufacturing considering product mix variation
Introduction
Cellular Manufacturing (CM) is an implementation scheme of the Group Technology (GT) concept for designing manufacturing systems. The main idea of GT, which was first introduced by Mitrofanov (1966), is to identify similar parts in terms of manufacturing processes and characteristics, and then grouping machines based on their applications to these production plans. Emerging global competition and shorter product life cycles, there has been a transfer in product demands to mid-volume and mid-variety mixes. Job shop and flow shop configurations cannot handle such needs efficiently and flexibly. Thus, CMSs have emerged to deal with such production requirements with favorable results (Wemmerlov & Hyer, 1989). The most important benefits of CMSs are reduction in: setup time, through put time, work-in-process inventories, and material handling costs, also increment in flexibility, production control, and product quality. (Heragu, 1994, Wemmerlov and Hyer, 1989). In design of a CMS, similar parts are grouped together into part families and machine types into manufacturing cells so that part families can be processed within a manufacturing cell as much as possible. Cell Formation Problem (CFP) is the major issue in designing CMSs. Most researches have considered the CFP in a zero-one part-machine incidence matrix. Rearrangement of rows and columns in the matrix can form part families and machine cells such that the optimal assignment is met.
In the last three decades, many solution methods such as mathematical programming, heuristics, meta-heuristics, neural networks, and clustering methods have been developed to address the CFP, mainly using zero–one part-machine incidence matrix as the input data (Paydar et al., 2011, Paydar and Saidi-Mehrabad, 2013, Singh, 1993, Yin and Yasuda, 2006). Giri, Srinivas, and Mouli (2007) introduced the machine sequence assumption to the CFP, then developed a heuristic algorithm which finds, at first step, an optimal machine sequence maximizing the overall flow of parts between the machines, secondly, an optimal number of cells by regarding the total inter-cell movements using the obtained machine sequence. They showed that the grouping of parts and machines based on the overall machine sequence leads to assignments with a minimum number of movements between the cells. Mahdavi, Paydar, Solimanpur, and Heidarzade (2009) minimized the number of voids and exceptional elements using a genetic algorithm and benchmarked the method against some of the most cited approaches in the literature. Paydar, Mahdavi, Sharafuddin, and Solimanpur (2010) reformulated the CFP as a multiple departures single destination multiple travelling salesman problem (MDmTSP), then developed a solution methodology based on simulated annealing to solve the formulated model. Paydar and Saidi-Mehrabad (2013) developed a hybrid metaheuristic algorithm based on genetic algorithm and variable neighborhood search to solve the CFP.
According to the nature of CMS in producing in mid-volume and mid-variety, it seems very likely that not only product demands may change but also products themselves may alter with time. In this paper, part mix variation is defined as periodically variation in the part-machine incidence matrix which is mostly assumed to be as a constant; even for the case of dynamic or stochastic CFP (Egilmez, Süer, & Huang, 2012). Product mix variation changes the machine-part incidence matrix periodically leading to alterations in design of a CMS. However, it brings to mind to modify the CMS periodically to meet the new processing requirements, but this approach may not be feasible for all cases (machine weights, machine attaching to the workshop floor, machine initial settings, etc.) (Seifoddini & Djassemi, 1996). Therefore, one approach is to initially configure the manufacturing cells considering just workers and machines (working teams). Then, after the required product mix in each period is known, part families can be assigned to these working teams optimally. Fig. 1 shows the flowchart of the proposed approach in this paper toward cell configuration in the case of product mix variation.
In this model the manpower plays a key role in configuring the manufacturing cells. Here, two different sides of the manpower are studied: (1) a worker’s skills to work with different machines and (2) a worker’s preferences to select their co-workers. The first assumption minimizes the prospective movements of workers among the manufacturing cells. The second assumption is a new concept that may enhance the performance of CMSs in the long run through, for example, creating a friendly environment, workers cooperation and coordination, exchange of experiences, system synergy and contribution, etc. Due to the important impact of worker element in this paper, what follows are some recent works on manpower association with design of CMSs.
Süer and Dagli (2005) developed a model for a type of CM called “labor-intensive” CMS, in where some of the operations can be performed on multiple identical machines by using multiple manning. The labor-intensive CMS was first introduced by Süer and Bera (1998) for describing manufacturing environments in where the system output is strongly dependent on the operator performance over machine performance. To model this concept, they considered two objective functions: (1) determining an optimal product-sequencing with the aim of minimizing the total intra-cell manpower movements and, (2) cell loading process to minimize makespan and also machine and space requirements. Mahdavi, Aalaei, Paydar, and Solimanpur (2012) developed a mathematical model for arrangement of a three-dimensional part- machine-worker incidence matrix that minimizes the total number of exceptional elements and voids. Süer, Kamat, Mese, and Huang (2013) developed a model toward manpower allocation and cell loading with minimizing the part total tardiness criterion in a labor-intensive CMS. The main decisions of the proposed model contain determining: the number of cells, the number of workers assigned to each cell, how workers will be assigned to different operations of parts, how to assign parts to cells, and the part sequence in each cell. Bootaki, Mahdavi, and Paydar (2015) addressed a dynamic CFP with a robust design approach and objective functions of minimizing the total inter-cell movements and maximizing the machine and worker utilization.
The previous studies mostly consider the operating aspects of manpower in CMSs like: minimizing inter- and intra-cell movements, work load balancing, minimizing/maximizing worker idleness/utilization, etc. There are some other papers addressing other aspects of manpower in designing manufacturing cells, like: training, team working, skills, motivations, and communications. Askin and Huang (2001) introduced an integer programming model to deal with worker assignment and training problem simultaneously. Suresh and Slomp (2001) also developed a cross-training model and link team formation with the CFP. Bidanda, Ariyawongrat, LaScola Needy, Norman, and Tharmmaphornphilas (2005) reviewed a diverse range of human issues involved in CFP. They counted eight different and vital human-oriented concepts: worker assignment strategies, training, skill identification, autonomy, communication, reward/compensation systems, teamwork, and conflict management. Süer and Tummaluri (2008) studied manpower assignment issues in labor intensive cells, like: skills, learning and forgetting. Othman, Bhuiyan, and Gouw (2012) developed a workforce planning model with some human issues such as training, skills, and workers’ personalities and motivation. They introduced a multi-objective non-linear mathematical model toward minimizing the training, hiring, firing, and overtime costs and minimizing the number of fired most productive workers. Egilmez, Erenay, and Süer (2014) introduced the concept of processing time variations due to the different levels of workers’ skill in operating different machines as probabilistic values. Bootaki, Mahdavi, and Paydar (2014) presented a three-dimensional CFP with objective functions of minimizing the inter-cell movements and maximizing the total quality index of parts. In this paper, different worker skills are assumed through a cubic quality matrix. It is shown that different skills of workers in operating parts on different machines may change the design of cells in order to produce parts with a more skillful worker, however, it burdens the system with more inter-cell movements.
The reminder of the paper is organized as follows. In Section 2, the proposed problem and assumptions are described in detail. In Section 3, the mathematical formulation is developed. A survey of basic definitions and solution approaches toward multi-objective optimization problems (MOPs) is presented in Section 4. Section 5, describes the exact method of augmented ε-constraint (AUGMECON) method to solve the proposed bi-objective model. Section 6 deals with the proposed NSGAII algorithm and its implementation. In Section 7, some numerical examples of small and large size are solved to assess the performance of the AUGMECON method and NSGAII algorithm. Finally this paper is concluded in Section 8.
Section snippets
Problem description
In this paper, the relationship between workers and machines is represented by task matrix, which is a zero-one matrix of order (W is the number of workers and M is the number of machines). The task matrix exhibits the capability of workers in work with different machines. Also, interest matrix shows the mutual interest between each pair of workers, which is also a zero-one matrix of order . Table 1 shows a sample task matrix for 7 workers and 5 machines. This table demonstrates that,
Indices
: Number of workers
: Number of machines
: Number of cells
: Index of workers
: Index of machines
: Index of cells
Parameters
: 1 if worker can operate machine 0 otherwise
: 1 if worker and have mutual interest (); 0 otherwise
: The lower bound for workers to be assigned to each cell
: The lower bound for machines to be assigned to each cell
Decision variables
: 1 if worker is assigned to cell ; 0 otherwise
: 1 if machine is assigned to cell ; 0 otherwise
Multi-objective optimization
Multi-objective optimization is a part of optimization that has a huge practical importance in real world optimization problems. In multi-objective optimization problems (MOPs) there are more than one objective functions tried to be optimized; also there is no single optimal solution that optimizes all of the objective functions simultaneously. In other words, if a solution optimizes one of the objective functions alone, it has no guarantee that optimizes the other objective functions. In this
Exact augmented ε-constraint solution approach
Although MOEAs are suitable methods for MOPs, they cannot guarantee to provide the Pareto-optimal front. In this paper, to compare the performance of the proposed NSGAII algorithm with Pareto-optimal front in small-size problems, the authors apply the exact method of augmented ε-constraint (AUGMECON) developed by Mavrotas (2009). As AUGMECON is an enhanced type of ε-constraint method, so there is primarily a need for description of the ordinary ε-constraint method and then the mechanism of the
NSGAII algorithm
The Non-dominated Sorting Genetic Algorithm II (NSGAII) is an elitist, multi-objective evolutionary algorithm which is known by the concepts of non-dominated sorting and crowding distance (Deb, 2001). In NSGAII, non-dominated sorting is performed to select layers of non-dominated individuals to fill in the slots of the new population of a given population of parents and children. The crowding distance concept is used for valuing the more diverse solutions and increasing the chance of selecting
Computational results
In this section, two categories of small- and large-sized test problems are randomly generated and then some of the multi-objective performance metrics are computed in each instance. In small-size problems, the AUGMECON can reach the Pareto-optimal front, so it gains a favorable opportunity for us to make a direct comparison between NSGAII results and Pareto-optimal front. Before proceeding any further, the authors would like to define the applied performance metrics in the following section.
Conclusion
In this paper, a mathematical model has developed toward configuring the manufacturing cells with part mix variation assumption. Here, the part mix variation is meant as a periodically change in the part-machine incidence matrix, which is mostly assumed to be as a constant in the literature, even for the case of dynamic or stochastic CFP (in a dynamic/stochastic case the part-machine matrix is fixed but demands of these parts change periodically).
Part mix variation makes the manufacturing
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