Solving the generalized cubic cell formation problem using discrete flower pollination algorithm

Highlights

• A new mathematical model for generalized cubic cell formation problem is introduced.

• Discrete flower pollination algorithm is developed to solve the GCCFP problem.

• Our results outperform those of branch & bound and simulated annealing meta-heuristic.

Abstract

The manufacturing cell formation represents one of the important stages of the construction of cellular manufacturing systems. It focuses on grouping machines, parts and workers and assigning them to the corresponding cells. This assignment is guided by multiple objectives, and is subject to many constraints. In this paper, the focus is made on a variant of the cell formation problem, which is the Generalized Cubic Cell Formation Problem (GCCFP). In this study, a mathematical model is developed for this variant of the problem. Besides the multiple objectives considered in most research works, the quality index of the produced parts is also considered in this study. To solve the problem, a Discrete Flower Pollination Algorithm (DFPA) is developed. To validate the model and the DFPA, a set of randomly generated instances were solved using B&B under LINGO software, DFPA and Simulated Annealing (SA) algorithm. The performance of DFPA, from the standpoint of the considered objectives and the time of calculation, has been tested. The experiment results show the efficiency of the developed method.

Introduction

The problem of cell formation (CFP) in cellular manufacturing systems (CMSs) is an important problem in the operational research literature (Joines, King, & Culbreth, Nourie, Tang, Tuah, Ariffin, Samin, 2013). It consists on decomposing an entire production system into a set of manufacturing cells, assigning the machines and allocating the parts, to be produced, to these production cells. During this decomposition, some constraints and objectives must be considered to produce most manageable and independent cells.

This paper focuses on a variant of the CFP, known as the Generalized Cubic Cell Formation Problem (GCCFP). The basic CFP considers that each part has a single route. However, in real situations, a part may have more than one process routings (e.g. part p_i may be processed on machines m₁ and m₃ or it may be processed on m₁, m₂, and m₄). The CFP that considers several possible process routings is called Generalized Cell Formation Problem (GCFP) (Kusiak, 1987). Although the formation of cells by considering the parts and the machines is the essence of group technology, its full benefits cannot be reached without considering the human factor (MIN & SHIN, 1993). Due to the important role played by the workers, grouping together workers with similar expertise and skills to produce similar part families may enhance the quality of the CMS design. When the worker dimension is considered besides the part and the machine dimensions, the problem is transformed into a GCCFP.

In the most studies done on the cubic CFP, when assigning worker to cells, the focus was made on minimizing the inter-cellular movement of workers. But, in some cases, having inter-cellular movement of workers may result in building a high quality process, by calling for more skillful workers from outsider cells to execute an operation of a part on a machine in a given cell. Selecting the skillful worker to execute a given operation may enhance the quality index of the produced parts. In this situation, to get a better design of cellular manufacturing system, a trade-off between these two conflicted objectives (inter-cellular movement of worker and quality of the produced parts) must be defined. According to Bootaki, Mahdavi, and Paydar (2014), the concept of part quality emerges from the three possible relationships between the three dimensions of the cubic CFP (parts, machines, workers). The first connection links parts with machines, where the status of the machines (machine lifetime, old or new, machine accuracy, etc) may influence the part quality. The second connection concerns the workers and the machines, where the experience of a worker of working with different machines may have an impact on the parts produced by this worker on these machines. The last connection is between the parts and the workers, where some parts require very qualified workers to be produced correctly and with the recommended accuracy. By assigning the production of these parts to less skilled workers, the quality of the parts will be affected. Thus, expertise and experience of workers may define the quality of the produced parts. Bootaki et al. (2014) have adopted a three-dimensional matrix to represent the relation between these three factors affecting the part quality. The same structure is used in this study. The estimation of this matrix values may be done through historical data acquisition and worker errors.

In this paper, a mathematical model for GCCFP is introduced. However, the proposed mathematical model is inefficient to solve large-sized real life instances. To deal with this issue, a population-based algorithm is developed in this paper. The algorithm is called Discrete Flower Pollination Algorithm (DFPA), it is an adaptation of the Flower Pollination Algorithm (FPA) to the discrete GCCFP. The fast convergence and the simple computation of FPA make it a better choice to solve continuous and discrete problems a well. It has been extensively used in recent years to solve problems in many fields (Abdel-Basset & Shawky, 2019) such us computer science, bio-informatics, operational research, food industry, ophthalmology, engineering, etc. Zhou, Zhang, Luo, and Wen (2018) applied the FPA to shape matching problem. In the study, the authors investigates the convergence performances of FPA and other algorithms. Singh and Kaur (2019) used FPA to select the optimal features and the most critical attributes for anomaly detection in network classification problem. Wang, Xie, He, and Chan (2019) defined two new flower pollination algorithms to deploy heterogeneous node in the wireless sensor networks. The aim is to optimize the coverage rate, the node radiation overflow rate and the energy consumption rate. Li, He, Zheng, and Hu (2019) applied an improved FPA to the social networks. The study shows superior performance in practical applications of user identification across social networks. Mishra and Deb (2019) proposed a powerful FPA to solve the multi-modal optimization problem of assembly sequence optimization. Wang, Li, and Gao (2019) proposed a multi-objective discrete flower pollination algorithm based on Pareto dominance relations to solve the stochastic two-sided partial dis-assembly line balancing problem. Niu et al. (2019) combines the FPA and the wind driven optimization algorithm in order to optimize parameters of fast learning network to build thermal efficiency model of a 330 MW coal-fired boiler. Priya and Rajasekar (2019) applied the FPA to derive unknown model parameters of fuel cells having different characteristics and rating for enhanced proton exchange membrane fuel cell modeling. More applications of FPA may be found in a recent review (Abdel-Basset & Shawky, 2019).

The remainder of this paper is organized as follows: In Section 2, a literature review is presented. In Section 3, a description of the GCCFP is given. In Section 4, the mathematical formulation of the problem is provided. After, in Section 5, the adopted representation and evaluation of the solution are presented. In Section 6, a preliminary computation is performed. In Section 7, the solution approach containing a description of the proposed DFPA is detailed. Next, in Section 8, the computational results are exhibited. Finally, the conclusion and some perspectives are shown in Section 9.