Solving an extended multi-row facility layout problem with fuzzy clearances using GA

Accepted Manuscript Title: Solving an extended multi-row facility layout problem with fuzzy clearances using GA Authors: Soroush Safarzadeh, Hamidreza Koosha PII: DOI: Reference:

S1568-4946(17)30542-2 http://dx.doi.org/10.1016/j.asoc.2017.09.003 ASOC 4450

To appear in:

Applied Soft Computing

Received date: Revised date: Accepted date:

18-12-2015 3-8-2017 3-9-2017

Please cite this article as: Soroush Safarzadeh, Hamidreza Koosha, Solving an extended multi-row facility layout problem with fuzzy clearances using GA, Applied Soft Computing Journalhttp://dx.doi.org/10.1016/j.asoc.2017.09.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Solving an extended multi-row facility layout problem with fuzzy clearances using GA Soroush Safarzadeh a,*, Hamidreza Koosha b a

Ph.D. Student, Department of Industrial and Systems Engineering, Isfahan University of Technology, Isfahan, Iran, 84156-83111

b Assistant

Professor, Department of Industrial Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

* Corresponding author. Tel: +98 9156148684

E-mail address: [email protected]

Highlights 

We modeled an extended MRFLP, as a novel practical problem, with the following main assumptions: o the facilities are arranged in a two-dimensional area without splitter rows o multiple products are available (instead of single product) o minimum distances between facilities, due to inaccurate nature and according to comments from an expert panel are considered as fuzzy numbers o the objective function is considered to minimize material handling and lost opportunity costs, as a novel cost function.



We compared several crossover and mutation methods in the proposed GA.



We studied sensitivity analysis of the main parameters and presented their results.

Abstract Multi-row facility layout problem (MRFLP) is a class of facility layout problems, which decides upon the arrangement of facilities in some fixed numbers of rows in order to minimize material handling cost. Nowadays, according to the new layout requirements, the facility layout problems (FLPs) have many applications such as hospital layout, construction site layout planning and layout of logistics facilities. Therefore, we study an extended MRFLP, as a novel layout problem, with the following

1

main assumptions: 1) the facilities are arranged in a two-dimensional area and without splitter rows, 2) multiple products are available, 3) distance between each pair of facilities, due to inaccurate and flexible manufacturing processes and other limitations (such as WIPs, industrial instruments, transportation lines and etc.), is considered as fuzzy number, and 4) the objective function is considered as minimizing the material handling and lost opportunity costs. To model these assumptions, a nonlinear mixedinteger programming model with fuzzy constraints is presented and then converted to a linear mixed-integer programming model. Since the developed model is an NP-hard problem, a genetic algorithm approach is suggested to find the best solutions with a minimum cost function. Additionally, three different crossover methods are compared in the proposed genetic algorithm and finally, a sensitivity analysis is performed to discuss important parameters. Keywords: Facility layout, Multi-row facility layout problem (MRFLP), Genetic algorithm, Lost opportunity cost, Fuzzy sets

1. Introduction Layout design is an important and popular issue in the facility planning, which deals with the arrangement of facilities in a determined area. Facility layout problems have many applications in different fields. For example, recently, we can observe hospital layout [1-3], construction site layout planning [4-6] and layout of logistics facilities [7, 8], which are modeled by FLPs. In addition, according to the extent of layout design problems, researchers have conducted many studies on the FLPs’ definition, classification, and their related solving methods in recent decades. Regarding various arrangements in the plant layout area, according to the design restrictions, layout design problems can be divided into four distinct categories: single row facility layout (SRFLP), multi-row facility layout (MRFLP), loop layout and openfield layout [9]. Single row facility layout problem (SRFLP) is a classic problem that arranges rectangle departments, which all have the same height, in a single row to minimize the material handling cost. It is proven that this problem is an NP-hard combinatorial optimization problem [10]. On the other hand, multi-row facility layout problem (MRFLP) is a generalized problem of SRFLP, so which rectangular departments must be assigned to the multi-determined rows with the same height with the aim of minimizing the sum of the center-to-center distances between all pairs of facilities [11]. MRFLP, similar to others layout problems, has been used in different fields such as design computer and its accessories [12, 13] or production systems [14]. However, in MRFLP, the rectangular departments are placed in the multi-fixed rows as default, but facilities may need unusual or asymmetric layout, practically, in a two-dimensional area with considering other limitations (such as unequal areas, unequal departments and etc.). Thus, we propose a novel layout problem in a production environment that is an extension of MRFLP, as a two-dimensional planning with additional real assumptions. 2

Furthermore, based on the assigned feasible placement of facilities, a cost, usually called handling cost, is calculated in FLPs with considering the interactions between each pair of facilities. Although, handling cost is a significant cost in layout problems, to have a more realistic understanding of the problem and to enhance the accuracy of the objective, we add an important layout cost that is lost opportunity cost related to waste spaces, which remains after the facilities placement into the objective function of the proposed layout problem. Economically, any investment on an opportunity causes losing other economic opportunities. For example, in layout design problems, the inappropriate layout of facilities in an area leads to losing some opportunities in that area such as locating a new production line, making storage, renting the area and etc. Therefore, considering the cost of lost opportunities with higher utilities is reasonable for a layout problem. Unfortunately, this issue has not been mentioned in the extant literature yet. 1.1. Fuzzy approach in FLP Some important parameters such as demand or distance between the facilities have been considered in the literature of the FLPs, usually, as a crisp number to calculate the handling cost and it has made the proposed models impractical. Because of some limitations in the real world such as required space for instruments, work in process, transportation facilities and etc., we cannot put the facilities in the optimal location. Furthermore, diversity of products forces the manufacturer into the flexible production methods and industrial machines that may lead to a partial change in the facility layout. Thus, it is more realistic to consider the distance between each pair of facilities as fuzzy number as well as the recent studies in the FLPs [15-17]. By doing this, the existent inaccuracy of the distance is exerted in the model. Fuzzy set theory which first introduced by Zadeh [15] explains the inexact and ambiguous features of any measurable variable. In an uncertain environment, the fuzzy theory can help us to solve problems by considering inexact inputs for them and apply the special techniques, in accordance with the available limitations [18]. In this paper, we focus on an extension of MRFLP in a two-dimensional environment without splitter rows, to minimize lost opportunity cost and material handling cost of a multi-product system. In addition, we assume unequal departments and the fuzzy distance between each pair of facilities to improve the mathematical modeling of the extended problem. Thus far, however, no research has been found surveying these two assumptions with each other. Based on the problem definition, a linear mixed-integer programming model is proposed with fuzzy constraints. Since the proposed model belongs to NP-hard class, an effective GA, as a popular and useful algorithm in the FLPs’ literature, is developed to conclude the best layout solutions. The rest of this research is organized as follows. Section 2 consists of a literature review of the SRFLP and MRFLP and their solution methods. Then, in Section 3, the mathematical foundations of the proposed model are explained. In Section 4, a genetic algorithm is studied as the solution method for the defined 3

layout problem. Following, the numerical studies and its related computational results are described in Section 5. Furthermore, sensitivity analysis and related discussions are performed in Section 6. Finally, Section 7 presents conclusion, including the main understandings, limitations, and future research. 2. Literature Review In this section, we review the literature on SRFLP and MRFLP and their proposed mathematical models, which establish major part of the FLPs background, in the last decade. Moreover, we study the developed solving methods consisting of metaheuristics, heuristics, and exact techniques. Lastly, the summary of the reviewed literature is presented in Table 1. In most of the studied papers, the defined layout problems have similar assumptions and we can observe more innovation in solving methods. For example, Kothari and Ghosh [22] proposed the Lin–Kernighan method, as a novel heuristic technique, for SRFLP in which the lost opportunity cost had not been considered. In another study for SRFLP, Amaral and Letchford [23] provided a polyhedral approach based on the material handling cost. Moreover, a hybrid estimation of the distribution algorithm is proposed for SRFLP [24]. Researchers in the area of facility layout paid little attention to fuzzy layout environment [15, 16, 19]. For example, fuzzy techniques could help to solve the unequal facility layout problems (UA-FLPs) in a fuzzy random environment [17, 20]. Moreover, qualitative and quantitative criteria were incorporated into FLP model and then an integrated methodology, based on the synthetic value of fuzzy judgments and nonlinear programming, was provided [21]. Some of the main parameters of layout problems such as demand (related to the handling cost), and closeness of departments (in the multi-objective models) are considered as fuzzy numbers in these studies [6, 20]. However, the investigated studies of FLPs have not been dealt with the MRFLP in a fuzzy environment. Garey and Johnson [25] proved the high computational complexity of different layout design problems. Therefore, many heuristics [26, 27] and meta-heuristic algorithms have been proposed in the previous literature to solve single and multi-row layout design problems. For example, we can mention ant colony optimization (ACO) algorithm [28-30], tabu search algorithm (TS) [31-34], particle swarm optimization (PSO) algorithm and simulated annealing (SA) [35-37] as some metaheuristics to solve SRFLP [38]. However, some researchers used exact optimization methods; for example, Solimanpur and Jafari [39] provided a two-dimensional facility layout problem using a branch and bound algorithm. Nonetheless, genetic algorithm (GA) is the most popular approach used frequently in the literature of FLPs [40-45]. In most cases, it has been shown that GA concludes better solutions than other metaheuristics methods in FLPs. The genetic algorithm, which was introduced by Jon Holland, has been widely applied in solving optimization problems due to its capabilities. For example, in one of the first GA applications, 4

an FLP is solved by a genetic algorithm with a heuristic crossover method [46]. In addition, Wu, Chu, Wang and Yan [47] considered GA for cellular manufacturing design and layout problem. On the other hand, Sadrzadeh [48] used GA with a heuristic procedure to solve multi-line layout problem. In some papers, GA is proposed with novel features to improve the efficiency of algorithm. Datta, Amaral, and Figueira [44] proposed a permutation-based GA for single row facility layout problem. Aiello, Scalia and Enea [43] studied a multi-objective GA (MOGA) for the facility layout problem based upon slicing structure encoding. In addition, Kiaa, Khaksar-Haghanib, Javadianc and Tavakkoli-Moghaddam [42] solved a multi-floor FLP of a dynamic cellular manufacturing system by an efficient GA. Furthermore, Pourvaziri and Naderi [40] exhibited a hybrid multi-population GA for dynamic facility layout problem. Table 1 presents an overview of the studied publications in this paper as follows.

Reference Ficko, Brezocnik, and Balic [45] Solimanpur, Vrat and Shankar [30] Amaral [21] Wu, Chu, Wang and Yan [47] Hani, Amodeo, Yalaoui, and Chen [28] Samarghandi, Taabayan and Jahantigh [38] Datta, Amaral and Figueira [44] Sadrzadeh [48]

Table 1 Overview of studied literature Data Product Problem 2D Objective Crisp Fuzzy Single Multiple

Meta-heuristic Heuristic Exact

FMS

•

•

HC

GA

SRFLP

•

•

HC

ACO

SRFLP

•

•

HC

CMS

•

QAP

•

SRFLP

SRFLP MLFLP*

Aiello, Scalia and Enea [43] UA-FLP Amaral and Letchford [23]

Solution method

*

•

Novel MIP

HC

GA

•

HC

ACO

•

•

HC

PSO

•

•

HC

GA

HC

GA

•

HC

MOGA*

•

HC

•

HC

Interactive GA

Hybrid Alg.

•

•

•

SRFLP

•

UA-FLP

•

•

Polyhedral App.

García-Hernández, Pierreval, Salas-Morera and Arauzo-

•

Azofra [41] Ou-Yang and Utamima [24] SRFLP

•

•

HC

Kothari and Ghosh [22]

SRFLP

•

•

HC

Kothari and Ghosh [31]

SRFLP

•

•

HC

TS

•

HC+RC*

Fuzzy Apps.

Samarghandi, Taabayan and Behroozi [17]

•

FDFLP*

LKN*

DFLP

•

•

HC+RC

GA

SRFLP

•

•

HC

MOTS*

MFLP*

•

HC+RC+PC*+OC*

GA

Zuo, Murray, and Smith [33] DRLP*

•

•

HC+TAC*

MOTS

•

HC

•

HC

Pourvaziri and Naderi [40] Lenin, Siva Kumar, Ravindran and Islam [34] Kiaa, Khaksar-Haghanib, Javadianc and Tavakkoli-

•

•

Moghaddam [42]

Nematian [15] Niroomand, Mirzaei, Şahin

•

FRSRFLP* CLFLP

*

•

5

B&B SA

and Vizvári [37] Hungerländer and Anjos

Semidefinite

MRFLP

•

•

HC

Palubeckis [26]

SRFLP

•

•

HC

LS*

Palubeckis [36]

SRFLP

•

•

HC

SA

DRLP

•

•

HC+RC

SA

•

HC+RC

MOPSO

[11]

Wang, Zuo, Liu, Zhao and Li [35] Xu and Song [20] Rubio-Sánchez, Gallego, Gortázar and Duarte [27] Guan and Lin [29] The considered research

•

DFLP

App.

SRFLP

•

•

HC

GRASP*

SRFLP

•

•

HC

Hybrid Alg.

HC+LOC*

GA

Extended MRFLP

•

•

•

*Notes: HC—Handling Cost, LOC—Loss Opportunity Cost, MLFLP—Multi Line FLP, UA-FLP—Unequal Area FLP, MOGA—Multi-Objective GA, LKNS—Lin– Kernighan Neighborhood Search, FDFLP—Fuzzy Dynamic FLP, RC—Rearrangement Cost, MOTS—Multi-Objective TS, MFLP—Multi Floor Layout Problem, PC— Purchase Cost, OC—Operating and Overhead Cost, DRLP—Double Row FLP, TAC—Total Area Consumed by the resulting layout, FRSRFLP—Fuzzy Robust SRFLP, CLFLP—Closed Loop FLP, LS—Local Search, GRASP—Greedy Randomized Adaptive Search Procedure

Obviously, our proposed problem consists of multi-product with fuzzy distances [15, 17, 20] in a twodimensional area, is not observed in the studied FLPs’ literature. Moreover, we contributed to the studied cost function (lost opportunity cost) of the allocated spaces to the unequal departments, as a novel term, except for the handling cost. In the next section, firstly, a novel nonlinear programming model with fuzzy constraints is presented for the extended MRFLP along with the explained assumptions and notations. Then, the proposed mathematical model is converted to a linear mixed-integer programming model. 3. Mathematical modeling Firstly, in this section, the assumptions and notations of the proposed model are explained in two separate subsections. Next, we present a fuzzy nonlinear programming model as an extension of the ABSMODEL 3 for the defined problem. Finally, the primary model is converted to a linear mixed-integer programming model, due to the fuzzy nature of some right-hand side parameters. 3.1. Assumptions In this paper, the mathematical model is developed based on the following assumptions: i. Facilities are arranged in a two-dimensional planar area with determined dimensions, ii. The mathematical model is designed for multi-product, single period and flow shop process, iii. Demand value and processing route of each product are known and there is not any backflow, iv. Distance between each pair of facilities is calculated by the center-to-center method, v. Facilities have a predetermined rectangular shape, vi. Minimum horizontal and vertical distance between the machines are considered as fuzzy numbers, vii. The number of machines isn’t predetermined and is calculated according to the total demand of products, 6

viii. Origin is considered as the entrance in the manufacturing process. (This point is specified as a facility with a fixed location and zero dimensions for all of the products.) The related notations are illustrated in the next subsection. 3.2. Notations In this subsection, the used notations are divided into the indices, crisp and fuzzy parameters, and sequentially, the decision variables are defined for the mathematical model. Furthermore, a comprehensive introduction to the fuzzy theory and fuzzy optimization is provided from the fuzzy literature [49, 50]. Thus, at first, we describe the indexes and crisp parameters in Table 2 as follows. Table 2 The description of the used indices and crisp parameters

Indices

Description

𝑖, 𝑗 ∈ {1,2, ⋯ , 𝑁}

Machine index

𝑘 ∈ {1,2, ⋯ , 𝑃}

Product index

Crisp parameter

Description

𝑃

The number of products

𝐶

Length of one period

𝑁

The total number of facilities

𝐷𝑘

Product demand over one time period

𝑡𝑖𝑘

Processing time of product 𝑘 on the machine 𝑖

𝑁𝑖𝑘

The number of required machine 𝑖 for manufacturing product 𝑘

𝑏𝑖𝑗𝑘

Machine route requirement binary parameter, which is 1 if machine 𝑗 is needed after machine 𝑖 in the for product 𝑘, and zero otherwise.

𝑎𝑖𝑘

Assignment parameter, which is one if machine 𝑖 assign to product 𝑘, and zero, otherwise.

𝑓𝑖𝑗

The material flow between machine 𝑖 and 𝑗

𝑐𝑖𝑗𝑘

Cost per unit of material flow between machine 𝑖 and 𝑗 for product 𝑘

𝑐𝑖𝑗

Cost per unit of material flow between machine 𝑖 and 𝑗

𝑤𝑖

Width of machine 𝑖

𝑙𝑖

Length of machine 𝑖

𝑆0

Lost opportunity cost of each unit of area

𝑋

Length coordinate of available area

𝑌

Width coordinate of available area

Now, we present four mathematical relations between the introduced parameters to obtain the secondary parameters from the primary parameters as follows. It should be noted that; we will use the secondary parameters in modeling and the numerical studies in Section 5. 𝑁𝑖𝑘 = ⌈ 𝐽

𝐷𝑘 𝑡𝑖𝑘 ⌉ 𝐶

∀ 𝑖, 𝑘

(1)

𝐾

𝑁 = ∑ ∑ 𝑁𝑖𝑘

(2)

𝑖=1 𝑘=1

7

𝐾

𝐷𝑘 𝑁𝑗𝑘

∀ 𝑖, 𝑗

(3)

𝑐𝑖𝑗 = ∑ 𝑏𝑖𝑗𝑘 𝑎𝑖𝑘 𝑎𝑗𝑘 𝑐𝑖𝑗𝑘

∀ 𝑖, 𝑗

(4)

𝑓𝑖𝑗 = ∑ 𝑏𝑖𝑗𝑘 𝑎𝑖𝑘 𝑎𝑗𝑘 𝑘=1 𝐾

𝑘=1

Where Equations (1) and (2) address the number of total machines. In addition, Equations (3) and (4) refer to the material flow and handling cost between each pair of facilities, respectively. Furthermore, we have two fuzzy parameters and three types of variables, which are introduced in Table 3 as follows: Table 3 The description of the used fuzzy parameters and decision variables

Fuzzy parameter

Description

̃ 𝑖𝑗 𝑑ℎ

Minimum horizontal distance needed between machine 𝑖 and 𝑗

̃ 𝑖𝑗 𝑑𝑣

Minimum vertical distance needed between machine 𝑖 and 𝑗

Decision variable

Description

𝑥𝑖

length coordinates of machine 𝑖

𝑦𝑖

width coordinates of machine 𝑖

′

Maximum distance between x-axis and the last point of the facilities along x-axis

𝑌′

Maximum distance between y-axis and the last point of the facilities along y-axis

𝑅𝑖

Direction parameter, which is one if machine 𝑖 be horizontal, and zero, otherwise.

𝑋

In this paper, based on the fieldwork experiences and nature of fuzzy variables, we used the left trapezoidal numbers to show the minimum distances as the best possible choice. Generally, the left trapezoidal number denoted 𝐴𝑙 = (𝑎1 , 𝑏1 , 𝑏2 , 𝑏2 ) that has supporting interval [𝑎1 , 𝑏2 ] in theory [50]. Fig. 1 represents the general form of the left trapezoidal number as follows.

Fig. 1 The left trapezoidal number [50]

̃ 𝑖𝑗 and 𝑎1 = 𝑑𝑣𝑖𝑗 − 𝑣𝑖𝑗 , 𝑏1 = 𝑑𝑣𝑖𝑗 and Now, assume that 𝑎1 = 𝑑ℎ𝑖𝑗 − ℎ𝑖𝑗 , 𝑏1 = 𝑑ℎ𝑖𝑗 and 𝑏2 = ∞ for 𝑑ℎ ̃ 𝑖𝑗 in the proposed model. accordingly, the membership functions of the fuzzy parameters are 𝑏2 = ∞ for 𝑑𝑣 presented, based on the linearization methods in fuzzy sets theorem [51], in Equations (5) and (6) as follow:

8

1 ̃ 𝑖𝑗 𝑑ℎ

= (𝑑ℎ𝑖𝑗 − ℎ𝑖𝑗 , 𝑑ℎ𝑖𝑗 ),

𝜇𝑑ℎ𝑖𝑗 (𝑥) =

𝑥 − (𝑑ℎ𝑖𝑗 − ℎ𝑖𝑗 ) ℎ𝑖𝑗 0 {

𝑥 ≥ 𝑑ℎ𝑖𝑗 𝑑ℎ𝑖𝑗 − ℎ𝑖𝑗 ≤ 𝑥 ≤ 𝑑ℎ𝑖𝑗

= (𝑑𝑣𝑖𝑗 − 𝑣𝑖𝑗 , 𝑑𝑣𝑖𝑗 ),

𝜇𝑑𝑣𝑖𝑗 (𝑥) =

(5)

∀ 𝑖, 𝑗

(6)

𝑥 ≤ 𝑑ℎ𝑖𝑗 − ℎ𝑖𝑗

1 ̃ 𝑖𝑗 𝑑𝑣

∀ 𝑖, 𝑗

𝑥 ≥ 𝑑𝑣𝑖𝑗

𝑥 − (𝑑𝑣𝑖𝑗 − 𝑣𝑖𝑗 ) 𝑑𝑣𝑖𝑗 − 𝑣𝑖𝑗 ≤ 𝑥 ≤ 𝑑𝑣𝑖𝑗 𝑣𝑖𝑗 0 𝑥 ≤ 𝑑𝑣𝑖𝑗 − 𝑣𝑖𝑗 {

̃ 𝑖𝑗 and 𝑑𝑣 ̃ 𝑖𝑗 , Where (𝑑ℎ𝑖𝑗 − ℎ𝑖𝑗 , 𝑑ℎ𝑖𝑗 ) and (𝑑𝑣𝑖𝑗 − 𝑣𝑖𝑗 , 𝑑𝑣𝑖𝑗 ) are the expected intervals of 𝑑ℎ respectively. Note that, we have the indifferent mode between the two fuzzy parameters, if 𝜇𝑑ℎ𝑖𝑗 (𝑥) = 𝜇𝑑𝑣𝑖𝑗 (𝑥) = 0.5. Therefore, we present a novel non-linear mathematical model for the extended MRFLP in next subsection. 3.3. The Proposed Model The proposed mathematical model, in this subsection, is an extension of the ABSMODEL 3, which is presented by Heragu and Kusiak [14]. The objective function aims at minimizing the lost opportunity cost of the used rectangular area as well as the total distance traveled by the materials in the shop floor. It should be noted that calculating lost areas among the facilities changes the objective function to a fuzzy objective, which can be considered as future research. Based on the mentioned notation, the objective function is presented as follows in Eq. (7): 𝑁

𝑁

′ ′

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑆0 𝑋 𝑌 + ∑ ∑ 𝑐𝑖𝑗 𝑓𝑖𝑗 (|𝑥𝑖 − 𝑥𝑗 | + |𝑦𝑖 − 𝑦𝑗 |)

(7)

𝑖=1 𝑗=1 𝑗≠𝑖

Moreover, the constraints include the following: C1. Ensure that no two facilities in the layout overlap (Equations (8) and (9)): 𝑙

|𝑥𝑖 − 𝑥𝑗 | − [𝑅𝑖 2𝑖 + 𝑅𝑗 |𝑦𝑖 − 𝑦𝑗 | − [𝑅𝑖

𝑤𝑖 2

+

𝑙𝑗

+ (1 − 𝑅𝑖 )

2 𝑤 𝑅𝑗 2𝑗

𝑤𝑖 2

+ (1 − 𝑅𝑗 )

𝑤𝑗

̃ 𝑖𝑗 , ] + 𝑀𝛼𝑖𝑗 ≥ 𝑑ℎ

2 𝑙𝑗

𝑙 ̃ 𝑖𝑗 , + (1 − 𝑅𝑖 ) 2𝑖 + (1 − 𝑅𝑗 ) 2 ] + 𝑀(1 − 𝛼𝑖𝑗 ) ≥ 𝑑𝑣

∀ 𝑖, 𝑗

(8)

∀ 𝑖, 𝑗

(9)

C2. Ensure that the location of each machine is in the feasible area (Equations (10)-(16)):

9

𝑙

𝑤

∀𝑖

(10)

𝑙

∀𝑖

(11)

∀𝑖

(12)

∀𝑖

(13)

𝑥𝑖 − [𝑅𝑖 2𝑖 + (1 − 𝑅𝑖 ) 2𝑖 ] ≥ 0, 𝑦𝑖 − [𝑅𝑖

𝑤𝑖 2 𝑙𝑖

+ (1 − 𝑅𝑖 ) 𝑖 ] ≥ 0, 2 𝑤𝑖

𝑥𝑖 + [𝑅𝑖 2 + (1 − 𝑅𝑖 ) 2 ] ≤ 𝑋 ′ , 𝑦𝑖 + [𝑅𝑖

𝑤𝑖 2

𝑙

+ (1 − 𝑅𝑖 ) 2𝑖 ] ≤ 𝑌 ′ ,

𝑋′ ≤ 𝑋

(14)

𝑌′ ≤ 𝑌

(15) ∀ 𝑖, 𝑗

𝑅𝑖 , 𝛼𝑖𝑗 ∈ {0,1}.

(16)

To prepare and solve the proposed mathematical model, we convert the fuzzy constraints (Equations (8) and (9)) and the objective function to the linear mode and rewrite the model in the next subsection. 3.4. Modified Mathematical Model Considering the constraints (8) and (9), it can be observed that their right-hand side values are fuzzy numbers. By looking at the fuzzy literature [51, 52], to convert the fuzzy mathematical model to the deterministic model, we must define the membership functions of the objective function and fuzzy constraints, and then use the max-min method so that, the deterministic model can be obtained. Therefore, the membership function for the constraints set, including 𝑑ℎ parameter, can be presented in Eq. (17) as follows: 𝑙𝑗

𝑙

𝐻𝑙ℎ = 𝜇𝑑ℎ𝑖𝑗 (|𝑥𝑖 − 𝑥𝑗 | − [𝑅𝑖 2𝑖 + 𝑅𝑗 2 + (1 − 𝑅𝑖 )

𝑤𝑖 2

+ (1 − 𝑅𝑗 )

𝑤𝑗 2

] + 𝑀𝛼𝑖𝑗 ),

𝑙ℎ ∈ {1, ⋯ , 𝑁𝑑ℎ }

(17)

Where 𝑁𝑑ℎ is the number of constraints (including 𝑑ℎ parameter) in the proposed model. Similarly, the membership function for the constraints set, including 𝑑𝑣 parameter, is presented in Eq. (18) as follows: 𝐻𝑙𝑣 = 𝜇𝑑𝑣𝑖𝑗 (|𝑦𝑖 − 𝑦𝑗 | − [𝑅𝑖

𝑤𝑖 2

+ 𝑅𝑗

𝑤𝑗 2

𝑙𝑗

𝑙

+ (1 − 𝑅𝑖 ) 2𝑖 + (1 − 𝑅𝑗 ) 2 ] + 𝑀(1 − 𝛼𝑖𝑗 )),

𝑙𝑣 ∈ {1, ⋯ , 𝑁𝑑𝑣 }

(18)

Where 𝑁𝑑𝑣 is the number of constraints set (including 𝑑ℎ parameter) in the proposed model. Based on the fuzzy sets literature [51], the membership function for objective function is obtained similar to the relations (5) and (6) as follow: 1 𝑓(𝑠) ≤ 𝑓𝑙 𝑓𝑢 − 𝑓(𝑠) 𝐺𝑓 = 𝜇𝑓 (𝑠) = 𝑓𝑙 ≤ 𝑓(𝑠) ≤ 𝑓𝑢 𝑓𝑢 − 𝑓𝑙 { 0 𝑓𝑢 ≤ 𝑓(𝑠)

∀ 𝑖, 𝑗

(19)

Where 𝑓𝑙 is lower bound of the objective function which is obtained by placing lower bound of right values in the constraints and solving the mathematical model. In addition, the parameter 𝑓𝑢 indicates upper bound of the objective which is obtained by placing upper bound of right hand side values in the constraints and solving the mathematical model. Finally, the new objective function is obtained in Eq. (20) as follows: 10

𝑙ℎ=1

𝑙𝑣=1

𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 min ( ⋂ 𝐻𝑙ℎ , ⋂ 𝐻𝑙𝑣 , 𝐺𝑓 ) = 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝜆 𝑁𝑑ℎ

(20)

𝑁𝑑𝑣

To reduce the complexity of the model (due to absolute objective function and the constraints (5) and (6)), linearization is suggested. Thus, six new variables are defined as follows (Equations (21)-(26)): + 𝑥𝑖𝑗 ={

𝑥𝑖 − 𝑥𝑗 0

𝑥𝑖 ≥ 𝑥𝑗 , 𝑥𝑖 < 𝑥𝑗

−(𝑥𝑖 − 𝑥𝑗 ) 0 𝑦𝑖 − 𝑦𝑗 𝑦𝑖 + 𝑦𝑖𝑗 ={ 0 𝑦𝑖 − 𝑥𝑖𝑗 ={

∀ 𝑖, 𝑗

(21)

𝑥𝑖 ≤ 𝑥𝑗 , ∀ 𝑖, 𝑗 𝑥𝑖 > 𝑥𝑗 ≥ 𝑦𝑗 , ∀ 𝑖, 𝑗 < 𝑦𝑗

(22) (23)

−(𝑦𝑖 − 𝑦𝑗 ) 𝑦𝑖 ≤ 𝑦𝑗 , ∀ 𝑖, 𝑗 0 𝑦𝑖 > 𝑦𝑗 1 𝑥𝑖 < 𝑥𝑗 𝛽𝑖𝑗 = { , ∀ 𝑖, 𝑗 0 𝑥𝑖 ≥ 𝑥𝑗 1 𝑦𝑖 < 𝑦𝑗 𝛾𝑖𝑗 = { . ∀ 𝑖, 𝑗 0 𝑦𝑖 ≥ 𝑦𝑗 − 𝑦𝑖𝑗 ={

(24) (25) (26)

Now, by replacing the new variables, the mathematical model can be rewritten. In the modified mathematical model, only the fuzzy-related constraints (Equations (8) and (9)) could be extended to Equations (28) to (34). Moreover, the objective function is converted to a compatible form as follows (Equations (27)-(42)).

11

𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝜆

(27)

𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑁 ′)

𝜆(𝑓𝑢 − 𝑓𝑙 ) + 𝑆0 [𝑋(𝑌 − 𝑌 + 𝑌(𝑋 − 𝑋

′ )]

𝑁

+ − + ∑ ∑ 𝑐𝑖𝑗 𝑓𝑖𝑗 (𝑥𝑖𝑗 + 𝑥𝑖𝑗 + 𝑦𝑖𝑗+ + 𝑦𝑖𝑗− ) ≤ 𝑓𝑢

(28)

𝑖=1 𝑗=1 𝑗≠𝑖 𝑙

+ 𝑥𝑖𝑗 − [𝑅𝑖 2𝑖 + 𝑅𝑗

𝑙𝑗 2

𝑙

𝑙𝑗

2

2

− 𝑥𝑖𝑗 − [𝑅𝑖 𝑖 + 𝑅𝑗

+ (1 − 𝑅𝑖 )

𝑤𝑖

+ (1 − 𝑅𝑖 )

𝑤𝑖

𝑤𝑗

] + 𝑀(𝛼𝑖𝑗 + 𝛽𝑖𝑗 ) − 𝜆ℎ𝑖𝑗 ≥ 𝑑ℎ𝑖𝑗 − ℎ𝑖𝑗 ,

∀ 𝑖, 𝑗

(29)

] + 𝑀 (𝛼𝑖𝑗 + (1 − 𝛽𝑖𝑗 )) − 𝜆ℎ𝑖𝑗 ≥ 𝑑ℎ𝑖𝑗 − ℎ𝑖𝑗 ,

∀ 𝑖, 𝑗

(30)

∀ 𝑖, 𝑗

(31)

∀ 𝑖, 𝑗

(32)

∀ 𝑖, 𝑗

(33)

∀ 𝑖, 𝑗

(34)

𝑤

∀ 𝑖, 𝑗

(35)

𝑙

∀ 𝑖, 𝑗

(36)

𝑤

∀ 𝑖, 𝑗

(37)

𝑙

∀ 𝑖, 𝑗

(38)

2 2

+ (1 − 𝑅𝑗 ) + (1 − 𝑅𝑗 )

2 𝑤𝑗 2

+ − 𝑥𝑖 − 𝑥𝑗 = 𝑥𝑖𝑗 − 𝑥𝑖𝑗 ,

𝑦𝑖𝑗+ − [𝑅𝑖

𝑤𝑖

𝑦𝑖𝑗− − [𝑅𝑖

𝑤𝑖

2 2

+ 𝑅𝑗 + 𝑅𝑗

𝑤𝑗 2 𝑤𝑗 2

𝑙

𝑙

2

2

𝑙

𝑙𝑗

+ (1 − 𝑅𝑖 ) 𝑖 + (1 − 𝑅𝑗 ) 𝑗] + 𝑀 ((1 − 𝛼𝑖𝑗 ) + 𝛾𝑖𝑗 ) − 𝜆𝑣𝑖𝑗 ≥ 𝑑𝑣𝑖𝑗 − 𝑣𝑖𝑗 , + (1 − 𝑅𝑖 ) 2𝑖 + (1 − 𝑅𝑗 ) 2 ] + 𝑀 ((1 − 𝛼𝑖𝑗 ) + (1 − 𝛾𝑖𝑗 )) − 𝜆𝑣𝑖𝑗 ≥ 𝑑𝑣𝑖𝑗 − 𝑣𝑖𝑗 ,

𝑦𝑖 − 𝑦𝑗 = 𝑦𝑖𝑗+ − 𝑦𝑖𝑗− , 𝑙

𝑥𝑖 − [𝑅𝑖 2𝑖 + (1 − 𝑅𝑖 ) 2𝑖] ≥ 0 , 𝑦𝑖 − [𝑅𝑖

𝑤𝑖 2

+ (1 − 𝑅𝑖 ) 2𝑖 ] ≥ 0 ,

𝑙

𝑥𝑖 + [𝑅𝑖 2𝑖 + (1 − 𝑅𝑖 ) 2𝑖] ≤ 𝑋 ′ , 𝑦𝑖 + [𝑅𝑖

𝑤𝑖 2

+ (1 − 𝑅𝑖 ) 𝑖 ] ≤ 𝑌 ′ , 2

𝑋′ ≤ 𝑋 ,

(39)

′

𝑌 ≤𝑌,

(40)

𝜆≥0,

(41)

𝑅𝑖 , 𝛼𝑖𝑗 , 𝛽𝑖𝑗 , 𝛾𝑖𝑗 ∈ {0,1}.

∀ 𝑖, 𝑗

(42)

In the next section, we explain GA structure as the proposed solving method for the modified mixedinteger programming model, with considering the problem features, which influence the structure of GA. 4. The Proposed GA The GA is one of the most popular optimization methods proposed by Holland [53]. In this section, we explain the proposed GA and its structure for the extended MRFLP in detail. Therefore, we describe the structure of chromosomes and the initial population. Then, we focus on the improvement operators consisting of crossover and mutation operators by setting parameters that make GA as a strong metaheuristic algorithm. In the genetic algorithm, the general idea is based on the natural evolutionary systems. These systems help to improve breeding by fertilization and genetic mutations. The explained mechanism can be mimicked in the optimization problems to produce near optimal solutions. For this purpose, a set of solutions is created, which is called "population", and each solution is called "chromosome". Each chromosome is included in decision variables, which are called "genes". Based on a determined mechanism, some pairs of chromosomes are chosen (called parents) and crossed over with each other to generate next population (called children) [48]. Afterward, a fitness function is selected to evaluate the relative performance of 12

chromosomes. Finally, the solutions with better fitness value are selected (as usual) and reproduction operators (crossover and mutation) are applied in order to achieve acceptable solutions. This process continues until stop conditions are satisfied. As mentioned in the FLPs’ literature, many applications can be observed in GA with different improvement techniques. However, most of the investigated studies generate the initial population randomly (except ref.[40, 42]). Table 4 shows an overview of the proposed GAs in the studied publications as follows. Table 4 Overview of the GA studies Stop

Reference

Problem Fitness fun.

Selection

Crossover

Mutation

Mak, Wong, and Ghan [46]

JSSPFSSP

1⁄ * 𝜑

RWS*+Random

Heuristic + PMX* + OX* + CX*

Heuristic

G. No.*

Ficko, Brezocnik, and Balic [45]

FMS

𝜑

Tournament meth.

PMX

Reciprocal meth.

G. No.

Enea, Galante, and Panascia [19]

UA-FLP

1⁄ (𝜑 + 𝑝𝑒𝑛𝑎𝑙𝑡𝑦)

Heuristic

Heuristic

Heuristic

Wu, Chu, Wang and Yan [47]

CMS

Datta, Amaral and Figueira [44]

SRFLP

Sadrzadeh [48]

MLFLP*

Aiello, Scalia and Enea [43]

UA-FLP

𝜑

Heuristic

García-Hernández, Pierreval, SalasMorera and Arauzo-Azofra [41]

UA-FLP

𝜑

Pourvaziri and Naderi [40]

DFLP

𝜑

Tournament meth. The best solutions

Dynamic + Weighted sum meth. RWS + Dynamic PMX Tournament 𝜑 Heuristic meth. (max 𝜑 + 1 − 𝜑) Modified RWS Heuristic 𝑛 𝑝𝑜𝑝 × (max 𝜑 + 1) − 𝑠𝑢𝑚 𝜑

Nasab [16]

1⁄ 𝜑

RWS

Kiaa, Khaksar-Haghanib, Javadianc, MFLP* and Tavakkoli-Moghaddam [42]

𝜑

RWS

condition

Group meth.

G. No. + Conv.* G. No.

Swap meth.

G. No.

Heuristic

G. No.

Swap + Switch G. No. meth. DM* PMX + N-Point meth. Uniform meth. satisfaction G. No. + Heuristic Swap meth. Conv. Swap+ Single-point+ Heuristic G. No. Heuristic Uniformed based

Heuristic

Uniform meth. G. No.

PMX+2Uniform meth. G. No. Point+Uniform * 𝜑: Material handling cost; RWS: Roulette Wheel Selection; PMX: Partially Mapped Crossover; OX: Order Crossover; CX: Cycle Crossover; G. No.*: The The proposed model

MRFLP

𝜑

RWS

Number of Generations; Conv: Convergence; DM: Decision Maker

We can observe in Table 4 that roulette wheel selection is a popular selection method in the studied literature. In addition, the reproduction operators of GA consisting of PMX, N-point, and uniform methods have more application than those of others. On the other hand, the number of generations criterion is dominated stop condition in the proposed GAs. Accordingly, we illustrate the GA elements in detail as follows. 4.1. Structure of Chromosomes and Initial Population As mentioned, the proposed modified mixed-integer programming model contains both discrete (i.e. 𝑅𝑖 ) and continuous (i.e. 𝑥𝑖 , 𝑦𝑖 , 𝑋 ′ and 𝑌 ′ ) decision variables. In addition, due to the dependency of 𝑋 ′ and 𝑌 ′ on the coordinates of decision variables (i.e. 𝑥𝑖 , 𝑦𝑖 ), we can ignore them in the chromosome structure and keep them, only in the calculation of the fitness function. Therefore, the proposed structure of chromosome consists three parts, which the number of each part is equal to the number of facilities (N). The first part of 13

this structure is assigned to the facilities longitudinal coordinates (𝑥𝑖 ). The second part is assigned to the facilities transverse coordinates (𝑦𝑖 ), and the directions of facilities (𝑅𝑖 ) are allotted to the third part. Fig. 2 shows the proposed chromosome structure. 𝑥1

𝑥2

⋯

𝑥𝑛

𝑦1

𝑦2

⋯

𝑦𝑛

𝑅1

𝑅2

⋯

𝑅𝑛

Fig. 2. The structure of the proposed chromosome

Moreover, to create the initial population, according to the problem constraints, the values of the facilities central features and their directions are identified randomly and then, the flow value and distance between each pair of facilities are calculated for the solution. Afterward, the parents are selected by the roulette wheel method to create the next generation. It should be noted that the fitness value of each chromosome is the same as objective function calculated in the proposed model. 4.2. Crossover and Mutation Operators In this paper, to produce the new chromosomes and create the next population, both crossover and mutation operators are used. In the proposed GA, the PBX, 2-point and uniform crossover methods are applied to the improvement of future generations. Moreover, the random mutation method is used to create diversification in the population. 4.3. Choose Solutions and Stop Condition In this subsection, we clarify the selection method of the next generation and stop condition of the proposed GA. After reproduction step, the fitness function of the total population is calculated. After that, all of the solutions are sorted and the best solutions are separated (the number of solutions is equal to the number of initial population). Then, the remaining worst results are removed from the population. The GA algorithm continues to create generations and to move from one generation to another until the termination condition is satisfied. In this study, like many others studies, the number of iterations determines the termination of the algorithm. Fig. 3 presents the flowchart of the proposed GA.

14

1. Set Parameters

Enter Inputs

2. Population Initialization

3. Fitness Evaluation

4. Roulette Wheel Selection Single

Pair

5a. Mutation

5b. Crossover

9. Final Solutions

6. Fitness Evaluation

Yes No

8. Stop Condition

7. Merge Populations

Parameter: - Population size - Number of generations - Crossover rate - Mutation rate - Beta parameter

Fig. 3 The framework for proposed GA

The next section explains how we can implement the proposed GA for the MRFLP by the numerical examples. In addition, we show that the studied problem has a high time complexity, even in the small size problems. 5. Computational Results In this section, the computational results of the proposed algorithm are presented to study the efficiency of applied reproduction operators. Therefore, the number of variables and constraints is calculated by considering the proposed linear mathematical model. The results can be observed in Table 5 as follows. Table 5 The variant size of MRFLP

Problem No. of machines No. of variables No. of constraints 1

21

1,074

1,554

2

32

2,515

3,600

3

40

3,943

5,620

4

60

8,913

12,630

5

75

13,953

19,725

6

120

35,823

50,460

7

175

76,303

107,275

8

280

195,583

274,540

9

500

624,253

875,250

10

2,000

9,997,003

14,001,000

Clearly, the investigated layout design problem has a time complexity of the factorial order. Therefore, 15

solving the proposed FLP with a deterministic optimization method, in a reasonable time, is impossible. In such situations, application of the meta-heuristic optimization methods is strongly recommended. The proposed GA be encoded with the MATLAB programming language on a computer with 2.6 GHz processor and 4G RAM. The proposed GA pseudo-code for the presented layout design problem is presented as follows in Fig. 4. 1 Assume iter: index of generation number; M: the max of iter; n pop: The number of population 2 Input data 3 Calculate the number of required facilities 4 Calculate the flow matrix and unit-cost matrix 5 Set parameters 6 Generate population 7 Set iter ← 0 8 If (iter < M) 9 { 10 For i=1 to n pop 11 { 12 Fitness evaluation 13 Selection chromosomes based on the Rolette wheel 14 Crossover operation (based on crossover rate) 15 Mutation (based on mutation rate) 16 Fitness evaluation 17 Merge populations 18 Sort population 19 Remove worst solutions 20 } 21 Set iter ← iter+1 22 } 23 Report the best solution 24 Stop Fig. 4 The pseudo-code of the proposed GA

Before we start algorithm running, lower and upper bound of the objective function (𝑓𝑙 and 𝑓𝑢 ) were calculated and then, the best values were obtained for the main parameters. In the next subsection, we illustrate a numerical example in detail to clarify what kind of problems can be modeled and solved by the proposed method. 5.1. Numerical Example This section addresses a numerical example of the MRFLP with two products and five machines as the production system. Moreover, the length and width values of the area were considered 50 × 50, respectively. In each time period, the demand values of products A and B must be satisfied. In addition, the lost opportunity cost of each area unit is considered 10 units of currency. The specifications of the required equipment to manufacture the products and products are shown in Table 6 and Table 7, respectively. Moreover, Table 8 shows the processing times of products at each facility. Table 9 and Table 10 present the handling cost for products A and B. Finally, Table 11 andTable 12 denote the minimum horizontal and vertical distances needed among facilities as fuzzy numbers. Table 6 Dimensions and symbols of machines

16

Machine type

Symbol Length (m) Width (m)

Entrance

E

0

0

Cutting machine C

4

2

Milling machine M

3

2

Drilling machine D

2

1

Welding machine W

3

3

Table 7 The specification of products

Product Demand average (in each period) Processing Route (from left to right) A

35

E-C-M-W

B

20

E-C-D-W Table 8 Processing time of each product on each machine (min)

Product machine

A

B

Entrance

realmin*

realmin

Cutting

2

4

Milling

2

0

Drilling

0

3

Welding

5

3

*Is equals the smallest positive normalized floating-point number.

Table 9 Product A handling cost among the machines (currency per meter)

Entrance Cutting Milling Drilling Welding Entrance 0

3

1

5

2

Cutting

3

0

2

4

3

Milling

1

2

0

3

3

Drilling 5

4

3

0

5

Welding 2

3

3

5

0

Table 10 Product B handling cost among machines (currency per meter)

Entrance Cutting Milling Drilling Welding Entrance 0

5

2

1

2

Cutting

5

0

2

2

5

Milling

2

2

0

1

3

Drilling 1

2

1

0

1

Welding 2

5

3

1

0

Table 11 The minimum fuzzy horizontal distance needed among machines (m)

Entrance Cutting

Milling Drilling Welding

Entrance (0,0)

(0.1,0.2) (0.05,0.1) (0.1,0.15) (0.08,0.1)

Cutting

(0.1,0.2) (0.4,0.7)

(0.1,0.2)

17

(0.8,1.0)

(0.6,1.1)

Milling

(0.05,1.0) (0.4,0.7) (0.4,0.5)

(0.1,0.5)

(0.1,0.2)

Drilling (0.1,0.15) (0.8,1.0) (0.1,0.5) (0.8,0.9) (0.1,0.2) Welding (0.08,0.1) (0.6,1.1) (0.1,0.2) (0.1,0.2) (0.3,0.5) *format of each cell: (𝑑ℎ − ℎ,𝑑ℎ) Table 12 The minimum fuzzy vertical distance needed among machines (m)

Entrance Entrance (0,0)

Cutting

Milling Drilling Welding

(0.05,0.1) (0.1,0.2)

(0.05,0.1) (0.25,0.5)

Cutting

(0.05,0.1) (0.05,0.2) (0.55,0.9) (0.25,0.5) (0.55,0.8)

Milling

(0.1,0.2)

(0.55,0.9) (0.6,0.8)

(0.05,0.1) (0.4,0.5)

Drilling (0.05,0.1) (0.25,0.5) (0.95,1.0) (0.4,0.5) (0.2,0.3) Welding (0.25,0.5) (0.55,0.8) (0.4,0.5) (0.2,0.3) (0.3,0.4) *format of each cell: (𝑑𝑣 − 𝑣,𝑑𝑣)

According to the presented data in Table 7, Table 8 and available time in each time period (that is considered 70 minutes), the number of required equipment for each product is calculated by Eq. (1), and the results are demonstrated in Table 13 as follows: Table 13 The number of required machines

Machine

Product A Product B

Entrance

1

1

Cutting

1

2

Milling

1

0

Drilling

0

1

Welding

3

1

In the next step, 𝑓𝑙 and 𝑓𝑢 values (that were explained in subsection 3.4) obtain 10,000 and 14,000, respectively, by the optimal GA, which set its parameters in subsection Error! Reference source not found. (population size=300, iteration number=300). Thus, the objective value is obtained 0.6217 with the following arrangement, which is shown in Fig. 5:

18

Fig. 5. The best-obtained layout for the numerical example

As we can observe in Fig. 5, the facilities, with considering the existent constraints, have a reasonable and productive arrangement. In addition, we show the trend of fitness value improvement for the numerical example in Fig. 6. The main point of Fig. 6 is that the best and the mean fitness values have a dramatical ascendancy, in the entire process of optimization. This shows that the proposed GA is efficient and convergent.

Fig. 6. Iteration-to-fitness plot for the illustrated example

The next section investigates a comprehensive sensitivity analysis on the main parameters of the proposed GA and then, we define three levels for each parameter and determine the optimized level by the Taguchi method. Finally, according to the result of parameter settings, we compare the obtained results of different crossover operators and other GA parameters in detail. 6. Sensitivity Analysis In this section, the effects of changing the main parameters on the solutions are addressed in detail. In 19

the first step, the GA parameters are set by the Taguchi method. Next, the impact of crossover operator on the output of the proposed GA is examined by many experiments in subsection 6.2. Finally, we conduct a comprehensive sensitivity analysis on the size of the population and number of iterations, respectively. 6.1. Parameter Setting Taguchi method is one of the most prevalent methods to analyze the output of experiments. It is used where the number of experimental factors or factor levels are numerous and take considerable time to be optimized [54]. In this way, the experimental factors and their levels are identified and then the Taguchi method is used to determine the optimal output level of each factor. Table 14 presents the main parameters of the proposed GA and their ranges: Table 14 The GA parameter ranges and levels

Row Algorithm parameters Parameters range Low Medium High 1

Pcrossover

0.75-0.95

0.75 0.85

0.95

2

Pmutation

0.05-0.25

0.05 0.15

0.25

3

𝛽*

1-10

1

10

5

* beta is a GA parameter.

Based on Table 14, the optimal level of each factor is obtained according to the numerical experiments, which are determined by the Taguchi method with 9 experiments. The results can be observed in Table 15 in detail. In addition, Fig. 7 shows the main effects for means of factor levels. It should be noted that each level has been run 10-times. Table 15 Calibration process of GA

A B C R1

R2

R3

R4

S. D* Mean

1 3 1 15,044.5 13,327.2 12,053.1 12,053.1 1,227 13,119.5 1 2 2 16,225.2 15,569.4 15,302.4 14,077.0 778

15,293.5

1 1 3 16,802.0 14,123.3 16,991.0 16,032.8 1,134 15,987.3 2 3 2 12,001.8 13,144.1 14,174.9 19,380.0 2,823 14,675.2 2 2 3 13,620.3 15,013.5 16,311.1 15,224.5 957

15,042.4

2 1 1 16,697.8 15,961.3 17,084.3 19,168.7 1,191 17,228.0 3 3 3 14,728.2 13,804.2 14,708.3 15,008.8 454

14,562.4

3 2 1 14,767.1 14,565.7 13,990.7 15,940.1 709

14,815.9

3 1 2 15,842.2 15,634.4 15,113.0 19,464.5 1,724 16,513.5 * Standard deviation

20

Fig. 7. Taguchi means plot for the proposed GA

It can be observed in Fig. 7, which the optimal factor levels of crossover rate, mutation rate, and 𝛽 parameter are 0.75, 0.25 and 1, respectively. The results show that the proposed algorithm is sensitive to the mutation rate, and the 𝛽 parameter has insignificant effect on the output. In the next subsection, we apply the optimal GA to compare the crossover operators and the different size of MRFLP practically. 6.2. Comparisons and Numerical Results We compare between the crossover operators with the optimal GA, in this subsection. Thus, the proposed GA was run 100-times for 40 experiments. Furthermore, we apply uniform, 2-point and position based crossover (PBX) methods in GA to determine the best operator for the extended MRFLP. Table 16 shows the results as follows. Table 16 The experimental results for the proposed crossover operators PBX Problem

Best

Avg.

Uniform cr. Time Best

Avg.

2-P cr.

Time Best

Avg.

PBX Time

Problem

Best

Avg.

Time Best

Uniform cr.

2-P cr.

Avg.

Avg.

Time Best

Time

1

0.1347 0.1345 262

0.5475 0.5085 290

0.1592 0.1561 195

21

0.2470 0.2469 316 0.0855 0.0854 342 0.3632 0.2916 286

2

0.3792 0.2775 272

0.1590 0.1572 303

0.4519 0.4519 202

22

0.3995 0.3987 300 0.3073 0.3072 295 0.2709 0.2706 250

3

0.6544 0.6364 276

0.3710 0.3593 321

0.2323 0.2323 178

23

0.0799 0.0706 373 0.2552 0.2530 338 0.1726 0.1725 312

4

0.3517 0.3493 269

0.1590 0.1572 304

0.4411 0.4411 235

24

0.4629 0.3650 298 0.0478 0.0463 268 0.6322 0.4178 289

5

0.1545 0.1499 301

0.3710 0.3593 328

0.2304 0.2304 229

25

0.0318 0.0317 317 0.5231 0.5211 350 0.3473 0.3193 298

6

0.2716 0.2716 273

0.5692 0.3904 255

0.0623 0.0608 201

26

0.1223 0.1090 305 0.4214 0.4015 268 0.4732 0.3407 339

7

0.2865 0.2549 257

0.0608 0.0446 299

0.1258 0.0831 189

27

0.2518 0.1375 374 0.2330 0.2132 312 0.2915 0.2882 254

8

0.2800 0.2799 292

0.1590 0.1572 309

0.4996 0.4244 216

28

0.4030 0.3840 325 0.2660 0.2659 293 0.4268 0.3916 249

9

0.1855 0.1852 270

0.3710 0.3593 317

0.4062 0.3865 225

29

0.3551 0.3462 330 0.1296 0.0459 296 0.2950 0.2946 262

10

0.5897 0.5799 264

0.5692 0.3904 247

0.3128 0.2837 235

30

0.1098 0.0352 285 0.3532 0.3531 316 0.0485 0.1030 268

11

0.6720 0.6052 310

0.0253 0.0240 411

0.2274 0.2088 244

31

0.5339 0.5325 313 0.1775 0.1611 305 0.2274 0.2088 252

12

0.1408 0.1398 258

0.1590 0.1572 348

0.1894 0.1893 254

32

0.3727 0.2947 371 0.3391 0.3390 276 0.0754 0.0753 230

13

0.3292 0.3198 331

0.3710 0.3593 367

0.1587 0.1221 329

33

0.3792 0.3572 343 0.4803 0.1962 354 0.2274 0.2088 249

14

0.3452 0.3451 312

0.5692 0.3904 286

0.1942 0.1753 248

34

0.2640 0.2635 312 0.3994 0.3963 313 0.0754 0.0753 229

15

0.5011 0.4955 313

0.3729 0.3478 339

0.6150 0.5963 268

35

0.4255 0.4255 334 0.1618 0.1412 355 0.0335 0.0334 222

16

0.2092 0.1997 322

0.2277 0.2259 333

0.5466 0.5454 266

36

0.6001 0.6001 272 0.1514 0.1148 376 0.3632 0.2916 240

17

0.4696 0.4695 302

0.5146 0.4919 304

0.0203 0.0201 294

37

0.1159 0.1143 336 0.1590 0.1572 342 0.2709 0.2706 255

18

0.0165 0.0034 293

0.4605 0.4530 338

0.2274 0.2088 270

38

0.4352 0.4351 308 0.4083 0.4007 340 0.1726 0.1725 278

19

0.1285 0.0994 359

0.4291 0.2282 318

0.0754 0.0753 287

39

0.4263 0.4226 313 0.4711 0.4559 282 0.6322 0.4178 258

21

20

0.2220 0.2057 334

0.4203 0.1238 333

0.3350 0.3340 264

40

0.3480 0.3478 361 0.6010 0.5651 295 0.3473 0.3193 274

Average:

0.3171 0.2980 309 0.3214 0.2776 317 0.2814 0.2547 253

*The genetic parameters are: n pop=300, Pcrossover=0. 75, Pmutation=0. 25

Comparing the obtained values of the three different crossover operators shows minor differences among the results that make selecting a specific operator as the best one for the studied problem difficult. The mean of objective values for PBX solutions is better than that of other methods. However, the uniform operator provides the best results. Comparing the average run-time shows that 2-point crossover operator takes less run time compared with other ones. Moreover, another main point is that the minimum difference between the mean value and the best value can be observed in the PBX crossover method. In another study, the objective values are evaluated by changing the population size (n pop) and the number of iterations (n iter) at different levels. Table 17 summarizes the results as follows. It should be regarded that, each level has been run 10-times. Table 17 The experimental results of sensitivity analysis on the population and iteration size Fitness value Problem

n pop

Fitness value

n iter

Problem Best

Avg.

Time (s)

n pop

n iter Best

Avg.

Time (s)

1

40

100

0.1748

0.1656

5.92

21

150

100

0.2703

0.2621

22.49

2

40

200

0.2546

0.2475

10.47

22

150

200

0.3701

0.3701

33.02

3

40

300

0.2577

0.2577

13.99

23

150

300

0.4090

0.3923

43.56

4

40

400

0.2968

0.2965

16.32

24

150

400

0.4155

0.4155

60.60

5

40

500

0.3108

0.3108

21.59

25

150

500

0.4802

0.4802

78.97

6

50

100

0.1836

0.1762

7.50

26

200

100

0.2762

0.2661

28.00

7

50

200

0.2245

0.2241

13.09

27

200

200

0.3827

0.3689

49.36

8

50

300

0.2428

0.2425

18.06

28

200

300

0.4177

0.4177

63.98

9

50

400

0.2915

0.2915

21.69

29

200

400

0.4596

0.4596

77.30

10

50

500

0.3495

0.3491

27.86

30

200

500

0.4815

0.4813

104.73

11

75

100

0.1912

0.1909

11.34

31

300

100

0.2887

0.2419

39.34

12

75

200

0.1930

0.1836

20.33

32

300

200

0.4088

0.4064

65.73

13

75

300

0.2361

0.2350

26.81

33

300

300

0.4205

0.4103

88.12

14

75

400

0.3588

0.3588

32.17

34

300

400

0.4754

0.4665

112.80

15

75

500

0.3680

0.3680

42.11

35

300

500

0.4870

0.4756

139.01

16

100

100

0.2496

0.2288

14.21

36

500

100

0.3592

0.3293

67.66

17

100

200

0.2918

0.2851

24.26

37

500

200

0.4782

0.4643

105.93

18

100

300

0.3491

0.3488

33.06

38

500

300

0.4950

0.4921

134.75

19

100

400

0.3758

0.3757

42.16

39

500

400

0.6027

0.6018

168.47

20

100

500

0.3814

0.3814

53.91

40

500

500

0.6334

0.6334

208.72

*The GA parameters: Pcrossover=0. 75, Pmutation=0. 25 (2-point Crossover)

Table 17 shows that with the significant increase in the population size (n pop), the running time increases as well as the fitness function value. In addition, increasing the number of iterations not only includes higher run-time but also improves the objective value dramatically. For better data analysis, the results are presented in Fig. 8 as follows. Moreover, the coefficients and powers of exponential trend lines for the total population series are shown in Table 18 to improve the conclusion. 22

Avg. of best fitness values

0.7 0.6 0.5 0.4 0.3 0.2 0

50

100

150

200

Time (s) 40

50

75

100

150

200

300

500

Fig. 8. The fitness values change over time for population series Table 18 The coefficient and power of exponential trend lines Seri n pop Coefficient Power Normalized power* 1

40

0.1575

0.0350

1.0000

2

50

0.1397

0.0330

0.9429

3

75

0.1322

0.0252

0.7200

4

100

0.2239

0.0111

0.3171

5

150

0.2546

0.0085

0.2429

6

200

0.2520

0.0070

0.2000

7

300

0.2654

0.0049

0.1400

8

500

0.2927

0.0040

0.1143

* That is equal to 𝑃𝑜𝑤𝑒𝑟⁄𝑃𝑜𝑤𝑒𝑟(𝑆𝑒𝑟𝑖 40)

Table 18 determines that increasing the population size which is shown with different series in Fig. 8 generally leads to less power. In other words, with the growth of population size, the required time increases to achieve a significant best solution. Undoubtedly, spending considerable time with the exponential trend to achieve a mediocre solution in a large population size rather than a small population size, in terms of the objective value, is not reasonable. In addition, this results, justify using meta-heuristic for the proposed problem. 7. Conclusion and Future Research MRFLP is a popular and useful layout problem that can be applied in most industrial or non-industrial environments as a realistic layout problem. However, this problem restricts the facilities to multi-fixed rows, whereas we have some general problems such as unequal areas and unequal departments that cannot be arranged in this condition. Therefore, this paper proposed an extended MRFLP consisting of the lost opportunity cost per unit of space and the fuzzy distance between the facilities in a multi-product environment. To optimize the extended MRFLP, we propose an absolute linear programming model with fuzzy constraints. Next, we convert the proposed fuzzy model to the crisp linear mixed-integer 23

programming model, based on the principals of fuzzy model linearization. Since the extended model belongs to NP-hard class, an effective GA, as a popular solving method, was developed to solve the model. In addition, we provided the numerical studies to show how the proposed model works. Furthermore, the optimal GA parameters were obtained using the Taguchi method in the sensitivity analysis section. In the next step, three crossover methods and their results were compared. Accordingly, it was determined that the uniform crossover operator provides the best result compared to 2-point and PBX crossover operators. However, the average of the objective values in PBX solutions was better than the other crossover operators. Finally, we presented the results of changes of the population size, iterations, and their effects on the fitness value. Owing to the vast volume of required computation to reach the best solution, practically, it was difficult to compare various Metaheuristic methods in this paper. Research may be needed to examine the other meta-heuristics for possible improvements of layout solutions. In addition, we solved the problem with deterministic demand for products. However, considering it as a stochastic variable may be an innovation in the studied problem. Moreover, the investigation of the heuristic methods on the extended MRFLP can be recommended for future research.

24

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