MCDM Model for Evaluating and Selecting the Optimal Facility Layout Design: A Case Study on Railcar Manufacturing

Abstract

Facility layout in a manufacturing system is a complex production activity because decisions on layout design are influenced by numerous, ambiguous, and competing factors. This study proposes a method for determining and choosing an ideal layout using a hybridized Fuzzy Analytic Hierarchy Process (F-AHP) with the Fuzzy Technique for Order of Preference by Similarity to the Ideal Solution (F-TOPSIS). The F-AHP is used, in this case, because of its ability to generate design criteria weight. The railcar industrial case study results indicate that the developed model can effectively lead to selection of the most suitable facility layout design. The Discrete Event Simulation model is used to evaluate the performance of the suggested layout concepts with the purpose of determining quantitative criteria for use when selecting the most optimal concept by the proposed Fuzzy AHP-TOPSIS model. The proposed methodology demonstrated that a framework is a logical way to solve problems. The proposed Fuzzy AHP-TOPSIS methodology is capable of selecting the best layout concept based on the set decision criteria. Layout concept three was the best in terms of the closeness coefficient, which was more than 0.9 for both batching and non-batching processing.

1. Introduction

Facility layout design is a complex production process that influences layout choices based on a variety of multiple, conflicting, and uncertain factors. The assignment of facilities to various locations in an effort to reduce handling costs is known as the facility planning problem [1]. The facility layout selection is considered a Multi-Criteria Decision-Making (MCDM) problem, primarily involving evaluating alternative layouts designed to meet common criteria for choosing the best one [2]. Performance evaluation of a layout design concept plays a significant role in selecting the most optimal design.

Various MCDM and multi-criteria decision analysis (MCDA) techniques have been developed and applied in designing and selecting alternative layout designs. The Fuzzy Analytic Hierarch Process (F-AHP) approach, based on criteria, such as productivity, flexibility, initial investment cost, and ease of maintenance, has been used for the optimal selection of a manufacturing layout [3]. According to [4], fuzzy AHP methodology is more powerful as it can solve and support spatial reasoning problems in different contexts. Fuzzy AHP provides a hierarchical structure, facilitates the decompositions and pairwise comparisons, reduces inconsistencies, and generates priority vectors. The Analytic Hierarchy Process (AHP) coupled with the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) was used by [5] for the selection of a performant manual workshop layout based on criteria, such as flexibility, accessibility, noise, area utilisation as well as labour utilisation. The fuzzy TOPSIS technique, which deals with uncertainty of data, was applied to evaluate and rank the candidate cities for the placement of a wind farm [6]. A hybrid fuzzy MCDM method using combination weight, Delphi, fuzzy Analytic Network Process (ANP), Entropy, and fuzzy Preference Ranking Organization Method for Enrichment Evaluation (PROMETHEE) was applied in selecting the most suitable aircraft assembly layout based on criteria, such as space utilisation, transportation performance, personnel issues, and layout flexibility [7]. The AHP and Data Envelopment Analysis (DEA) approach was used by [8] to evaluate a manufacturing plant layout, with quantitative criteria of cost and distance and qualitative criteria of flexibility, safety, and utilisation. The AHP was able to assess layout design based on qualitative factors, such as routing flexibility, production area utilisation, and human issues [9]. Discrete Event Simulation (DES) and AHP were used to analyse the manufacturing facility layout design and the performance of a greenfield project based on criteria, such as average output, maintainability, safety, production pace, flexibility, and ease of implementation [10].

A considerable amount of research has been conducted on the use of MCDM techniques for selecting design alternatives; however, more attention needs to be given to comprehensively considering simulation results as criteria for choosing the best layout design. Most studies on the use of MCDM depend on the decision makers, as experts in criteria and alternative evaluation lacking objectivity in making pairwise comparisons. In MCDM, techniques such as AHP cannot provide deterministic preferences but perception-based judgment [3]. To deal with this kind of uncertainty, quantifiable criteria from DES and fuzzy set theory are used in modelling. Simulation-based MCDM is based on factors that could be easily quantifiable and removes biases in evaluating criteria and alternatives.

Recently, simulation-based MCDM has been conducted for planning and making decisions on infectious disease epidemics, such as COVID-19 in 2019 [11]; a combination of System Dynamics Model and Fuzzy MCDM was used as an approach for the assessment of sustainability in a transport sector company [12]; [13] developed a systematic simulation-based MCDM approach for the evaluation of semi-fully flexible machine system process parameters.

The research gap identified in the literature is that there is no comprehensive framework that exists to support decision making, especially in organisations that are in a production ramp-up phase. In particular, the use of simulation modelling with MCDM enhances decision making in that the output from the model could easily be scaled up for use in pairwise comparison with AHP; the minimization of subjectivity in decision making, and the use of group decision making, which requires an excessive amount of time.

This paper aims to develop an MCDM approach based on quantifiable criteria derived from a DES model to evaluate and select the most optimal facility layout concept for the underframe layout in the railcar manufacturing industry. The organisation is in a production ramp-up phase and experiencing low productivity in its operations. This article presents a model for selecting an optimal layout design by hybridising two MCDM methods. The results of this study showed that DES could provide the necessary criteria for evaluating the performance of layout design concepts based on the developed Fuzzy AHP-TOPSIS model.

To model the different processing scenarios and the performance of generated layout alternatives, the novel DES model was developed with the capability of producing outputs, such as production rate, productivity, and utilisation. This DES model is the primary contribution of this study. Furthermore, the outputs of the DES model were used as inputs for the decision matrix for the criteria and the selection of the best layout by the developed Fuzzy AHP-TOPSIS methodology.

2. Literature Review

Numerous terminologies have been developed to classify MCDM problems [14]. The first and most popular terms are MCDA or Multi-Attribute Decision Making (MADM). This category, in which decision makers (DMs) must choose from a finite number of explicitly available alternatives, is characterised by a set of multiple attributes or criteria. The second category is Multi-Objective Mathematical Programming (MOMP) or Multi-Objective Decision Making (MODM) that deals with decision problems characterised by various and conflicting objective functions that are to be optimised over a feasible set of decisions. One criticism of MCDM methods is that different results are obtained when applied to the same problem due to the differences in techniques [15]. The differences in MCDM algorithms include the use of different weights, attempting to scale the objectives, and introducing additional parameters that will ultimately affect the solution [15].

According to [16], various MCDM techniques have been widely used in different fields, such as healthcare systems, manufacturing systems, tourism management, strategic management, environmental analysis, as well as risk management. The DEA-based Malmquist productivity index, which is an MCDM method, was used for the assessment of cybersecurity in wireless communications for data security [17]. A combination of spherical fuzzy AHP (SF-AHP) and grey Complex Proportional Assessment (G-COPRAS) for the supplier selection was used in a study, where it was found that quality, cost/price, safety, health practices, as well as the wellbeing of suppliers were ranked as the five most important attributes [18].

MCDM, in general, follows six steps, including (1) problem formulation, (2) identification of the requirements, (3) setting goals, (4) identifying various alternatives, (5) establishing criteria, and (6) identifying and applying a decision-making technique [5].

Selecting the best facility layout among alternatives is a complicated MCDM problem [7]. The facility layout problem seeks to geographically locate the production units within a facility, subject to some design criteria and area limitations, with one or multiple objectives. A facility layout design (FLD) problem can be considered a multi-criteria problem due to the presence of qualitative criteria such as flexibility as well as quantitative criteria such as cost [19]. According to [20], the layout problem in a production system entails choosing the location of the equipment, workstations, and other departments to satisfy the following requirements:

(a)

Minimise the cost of moving finished goods, raw materials, parts, tools, and works in progress between departments.

(b)

Facilitate the flow of traffic.

(c)

Increase employee morale.

(d)

Minimise the risk of personnel injuries and damage to property.

(e)

Where necessary, provide supervision and face-to-face communication.

The FLD problems have been resolved using a variety of techniques, including the MCDM, heuristics, and intelligent approaches. Ref. [1] indicated that MCDM is classified into three categories, which include single MCDM, combining two or more MCDMs [21,22], and integration of an MCDM technique with another technique such as Genetic Algorithm (GA) [22]. A study by [5] found that AHP, TOPSIS, and ANP are the most common MCDM for evaluating layout design. The evaluation of layouts can be performed using the MCDM, DEA, Simulation, simple Criteria Comparison (s-CC), Fuzzy Constraint Theory (f-TOC), and Non-Linear Programming (NLP) techniques [23].

The VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) was developed to solve MCDM problems with conflicting and non-consumable (different units) criteria, assuming that compromising is acceptable for inconsistent resolutions so that a solution that is closest to the ideal solution is realized [19].

In the Best Worst Method (BWM), which is an MCDM technique, the decision maker selects the best and worst criteria and provides two pairwise comparison vectors for the best and worst criteria, resulting in a method that requires fewer comparisons [24]. In BWM, firstly, the most desirable, and then the worst, least desirable, or least essential, criteria are identified by the DM [25]. Secondly and finally, the best criterion is compared to the other criteria, and the other criteria to the worst criterion using a non-linear minimax model. The BWM, when compared to other pairwise-comparison-based MCDM techniques, such as AHP, requires fewer pairwise comparison data and it also produces more reliable results [26].

According to research conducted by [27], the methodologies used to design warehouse layouts using MCDM are AHP, Elimination and Choice Expressing Reality (ELECTRE), and PROMETHEE. Measuring Attractiveness by a Categorical-Based Evaluation Technique (MACBETH) is an MCDA approach motivated by the multi-attribute value theory [2]. The TOPSIS is an MCDA technique that is based on the principle that the chosen alternative should have the shortest geometric distance from the positive ideal solution (PIS) and the longest geometric distance from the negative ideal solution (NIS) [28]. The authors of [29], who corroborated [28], indicated that TOPSIS is founded on the notion that the ideal solution should be the one that is furthest from a negative ideal solution and the closest to a positive ideal solution.

Attributes or criteria are often of incongruous dimensions in multi-criteria problems, and this may create problems when evaluating alternatives. To avoid this problem, a fuzzy system is necessary [28]. The fuzzy AHP and Fuzzy TOPSIS are the preferred strategies when the criteria weights and performance ratings are ambiguous and inaccurate [30]. The Vector Measure Construction Method (VMCM) is similar to TOPSIS methodology; however, the VMCM is more sensitive to changes in data values in calculations.

It is becoming more and more vital to create hybrid and modular approaches, which are built on already existing techniques, such as the TOPSIS, AHP, ELECTRE, VIKOR, ANP, SAW, and their modification, using fuzzy and grey number techniques [31]. A new MCDM approach, called graph-based design language, was applied to evaluate assembly system design concepts [32].

Table 1. Summary literature analysis on MCDM techniques applied on FLD.

Author	Title	Criteria for Evaluation	Findings
[19]	An integrated AHP-VIKOR methodology for facility layout design	Flexibility, accessibility, maintenance, and material handling distance, adjacency score ratio	The fuzzy set comparisons show good reliability of the selection of optimal FLD alternatives
[33]	An AHP approach to the choice of manufacturing plant layout	Flexibility, Cost, and production volume	The AHP- KBS produced much more supporting evidence to justify the layout choice.
[34]	A hybrid method for the selection of facility layout experimental design and grey relational analysis: A case study	Flow distance, space utilisation, and shape ratio	The results of the study show that the DoE-GRA method is robust, calculations are quick and it is practical.
[8]	Selection of multi-criteria plant layout design by combining AHP and DEA	Distance, cost, flexibility, safety utilisation	The DEA model was useful in finding global priorities amongst the layout designs
[3]	A fuzzy AHP approach for optimal selection of manufacturing layout	Productivity, Initial investment, Flexibility, and Ease of maintenance	The criteria productivity was more preferred followed by the initial investment
[35]	Integrated SA-DEA-TOPSIS based solution approach for multi objective stochastic dynamic facility layout problem	Cost, Distance, Maintenance, Flow	This was found to be a unique method for ranking layouts were objective and subjective criteria were merged to find a solution
[36]	Optimal solution for multi-objective facility layout problem using a genetic algorithm	Material handling cost, and Closeness rating score	The algorithm developed was useful in finding the initial optimal solution
[37]	Evaluating Lean Facility Layout Designs Using a BWM-Based Fuzzy ELECTREI Methodology	Cycle Time, Cost, Productivity, WIP, Quality, and Distance	The proposed hybrid method could overcome the problem of missing information in numerical comparison
[38]	Optimization of plant layout in Jordan Light Vehicles Manufacturing Company	Travelling distance, Space usage, Activity relationship	AHP was successful in evaluating five layout designs with a consistency ratio of 8.33%
[39]	A New Optimization via Simulation Approach for Dynamic Facility Layout Problem with Budget Constraints	Cost and distance	The proposed model integrates computer simulation, ANN, and H-NSGA-II to overcome the limitations of traditional layout optimization methods
[40]	The evaluation of office layout design with MCDM techniques	Safety, Motion, and Comfort	AHP and Permutation methods yielded the same results
	Evaluating Facilities Layout Design using Fuzzy MCDM method	Investment, size, distance, and Quality	Subjective and Objective criteria are used to evaluate layouts
[29]	Layout Selection by AHP-TOPSIS and Performance Analysis by Computer Simulation	Flexibility, Distance, Route, Activity Relationship	AHP was used to determine the weights and TOPSIS was used to outrank the options
[41]	Comparison of MADM Methods for Layout Evaluation Selection	Volume, Distance, Utilisation, and Quality	Four methods resulted in the selection of the same layout
[42]	A multi-criteria decision-making framework for assessing the quality and cost of facility layout alternatives: a case study	Investment, Maintenance, Operational and Disposal Cost, and Quality	The developed framework accounts for quantitative and qualitative factors in selecting layouts
[43]	Selection of a layout configuration for reconfigurable manufacturing system using the AHP	Layout reconfiguration, Cost, Quality, and Reliability	The sensitivity analysis indicated that layout reconfigurability is the most important
[44]	Multi criteria decision making for the selection of a performant manual workshop layout: a case study	Accessibility, Flexibility, Area utilisation, Noise, Labour utilisation	AHP and TOPSIS were most applicable techniques for layout selection
[45,46]	Finding optimum Facility Layout by Developed Simulated Annealing Algorithm	Distance and Costs	Proper of layout facilities has a direct relationship with the cost goods manufactured

shows a summary of the literature review relevant to the domain of this research, which is facility layout design.

Based on Table 1, it is evident that the most popular MCDM technique used to solve layout problems is AHP followed by TOPSIS. A hybrid of fuzzy AHP and TOPSIS received some attention. Distance was found to be the most popular criterion for the evaluation of layout designs, followed by utilisation, flexibility, and cost. The research gap identified in the literature is that there are very few studies that rely on the output of a simulation model for computations in the MCDM model, which effectively means that they depend solely on judgements by experts when conducting pairwise comparisons.

3. Problem Statement

The economic development policy of South Africa emphasizes the need to embark on localization to resuscitate local manufacturing industries through the implementation of a comprehensive localization plan. South Africa is experiencing a 40-year technology gap in rail manufacturing, with a short supply of its engineering workforce.

Despite the increased use of simulation-based optimisation, the design of facility layout is challenged by high levels of uncertainty associated with new production processes [46]. Ref. [47] asserts that the integration of simulation models with optimisation techniques could handle the complexity of a manufacturing system and could help decision makers to take optimal decisions by considering the most influential factors. Together, simulation and optimisation are well-suited to improve complex manufacturing systems in which several events occur at the same time in unpredictable situations [48].

Research conducted on the use of MCDM ignores the use of simulation methodologies as a means to derive Key Performance Indicators (KPIs), such as utilisation and throughput, for use when conducting criteria evaluation. The following research questions are to be addressed by this paper:

(1)

How could the DES model be applied to provide the necessary KPIs to be used as criteria in selecting the best layout configuration for the underframe production line?

(2)

How could the quantitative KPIs from DES enhance pairwise comparison in the developed fuzzy AHP-TOPSIS methodology for selecting the best layout configuration?

4. Materials and Methods

4.1. Proposed Fuzzy AHP-TOPSIS Method to Evaluate Layout Design Concepts

The MCDM framework for evaluating and selecting a facility layout concept is customised to address the challenges faced by the railcar manufacturing organisation. A framework is a critical approach to establishing means to develop a model and the interpretation thereafter. A framework is a scientific procedure necessary to better understand the developed model, which renders the model easy to apply. The framework in Figure 1 aims to provide the methodology for the developed model that can be used for better decision making in facility layout design and selection problems. However, this model could be customised to suit other decision-making problems.

The proposed framework starts with the identification of the problem. The researcher has to take some time observing the underframe line with the aim of identifying problem areas and collecting production data. The main challenge in the production of railcars, specifically the underframe, is the increase in product variety and product families, which has led to the decision to assemble families of products in the same resource or jig, which leads to the concept of shared resources. This situation leads to production capacity challenges in terms of bottlenecks in the shared resource where there are many varieties of products. The underframe line has also been experiencing high lead times more, especially in workstation CBS1070, which created bottlenecks.

Problem identification was then followed by conceptualizing the different layout concepts with the aim of establishing alternatives that could lead to improved productivity. The alternative concepts are detailed in Section 5.2.

The proposed concepts were then subjected to a simulation model to test their effectiveness. The developed DES model was agile to simulate different processing strategies, such as batching and non-batching, for the underframe production line. The model was developed to enable the establishment of quantitative criteria to be used for the selection of the best layout concept by the developed fuzzy AHP-TOPSIS methodology.

The fuzzy AHP is used in this study to determine the criterion weights. The fuzzy AHP is more powerful and unique, as additional information from the DES model is available to aid decision making when computing pairwise comparisons.

The classical AHP needs to be revised to deal with the fuzziness and uncertainty due to the need for more information for decision makers. Therefore, this study takes advantage of DES and Fuzzy AHP-TOPSIS to advance the classical AHP and TOPSIS when making decisions. Finally, the fuzzy-TOPSIS techniques were then applied to evaluate and select the best layout concept based on five criteria.

4.2. Fuzzy AHP Technique

The Fuzzy AHP-TOPSIS methodology was chosen to evaluate the designed layout alternatives to select the most optimal design. The evaluation procedure for this study involves four main steps: identifying the evaluation criteria, utilising the DES model for criteria quantification, using Fuzzy AHP for weight determination, and using Fuzzy TOPSIS for final result ranking.

The AHP developed by Thomas Saaty in 1977 is a structured technique for organising and analysing complex decision problems involving subjective judgements. The procedure follows four basic steps, as suggested by [5]:

• The classification of the overall goal of the decision, criteria, and alternatives in a hierarchical structure;
• The construction of comparative judgment matrices by pairwise comparisons based on the decision makers’ preferences using the scale;
• Determination of the weights;
• Compute the consistency ratio.

Equation (1) shows a matrix (A) that attempts to compare a set of n attributes pairwise based on their respective weights, where the attributes are indicated as $a_1$,$a_2$,…,$a_n$ and the weights are $w_1$,$w_2$,…, $w_n$.

$$A=\left[\begin{array}{ccc}{a}_{l1}& \cdots & \begin{array}{ccc}{a}_{lj}& \cdots & a_{ln}\end{array}\\ ⋮& & \begin{array}{ccc}⋮& & ⋮\end{array}\\ \begin{array}{c}{a}_{i1}\\ ⋮\\ a_{n1}\end{array}& \begin{array}{c}\cdots \\ \\ \cdots \end{array}& \begin{array}{ccc}\begin{array}{c}{a}_{ij}\\ ⋮\\ a_{nj}\end{array}& \begin{array}{c}\cdots \\ \\ \cdots \end{array}& \begin{array}{c}{a}_{in}\\ ⋮\\ a_{nn}\end{array}\end{array}\end{array}\right]$$

(1)

where $a_{ij}=\frac{1}{a_{ji}}$ (positive reciprocal) and $a_{ij}=a_{ik}/a_{jk}$. In real situations, $w_i/w_j$ is usually unknown. Therefore, the problem is to find $a_{ij}$ such that $a_{ij}\cong w_i/w_j$, where $a_{ij}=\frac{1}{a_{ji}}$ (positive reciprocal) and $a_{ij}=w_i/w_j$, is usually unknown. Therefore, the problem is for the AHP is to find $a_{ij}$ such that $a_{ij}\cong w_i/w_j$.

The weight matrix can be represented as:

$$W=\begin{array}{c}\begin{array}{ccc}{w}_1& \cdots & \begin{array}{ccc}{w}_j& \cdots & w_n\end{array}\end{array}\\ \begin{array}{c}{w}_1\\ \begin{array}{c}⋮\\ \begin{array}{c}{w}_i\\ ⋮\\ w_n\end{array}\end{array}\end{array}\left[\begin{array}{ccc}w{}_1/w_1& \cdots & \begin{array}{ccc}w{}_1/w_j& \cdots & w{}_1/w_n\end{array}\\ ⋮& & \begin{array}{ccc}⋮& & ⋮\end{array}\\ \begin{array}{c}w{}_i/w_1\\ ⋮\\ w{}_n/w_1\end{array}& \begin{array}{c}\cdots \\ \\ \cdots \end{array}& \begin{array}{c}\begin{array}{ccc}w{}_i/w_j& \cdots & w{}_i/w_n\end{array}\\ \begin{array}{ccc}⋮& & ⋮\end{array}\\ \begin{array}{ccc}w{}_n/w_j& \cdots & w{}_n/w_n\end{array}\end{array}\end{array}\right]\end{array}$$

(2)

If W is multiplied by w, we obtain

$$W\times \omega =\begin{array}{c}\begin{array}{ccc}{w}_1& \cdots & \begin{array}{ccc}{w}_j& \cdots & w_n\end{array}\end{array}\\ \begin{array}{c}{w}_1\\ \begin{array}{c}⋮\\ \begin{array}{c}{w}_i\\ ⋮\\ w_n\end{array}\end{array}\end{array}\left[\begin{array}{ccc}w{}_1/w_1& \cdots & \begin{array}{ccc}w{}_1/w_j& \cdots & w{}_1/w_n\end{array}\\ ⋮& & \begin{array}{ccc}⋮& & ⋮\end{array}\\ \begin{array}{c}w{}_i/w_1\\ ⋮\\ w{}_n/w_1\end{array}& \begin{array}{c}\cdots \\ \\ \cdots \end{array}& \begin{array}{c}\begin{array}{ccc}w{}_i/w_j& \cdots & w{}_i/w_n\end{array}\\ \begin{array}{ccc}⋮& & ⋮\end{array}\\ \begin{array}{ccc}w{}_n/w_j& \cdots & w{}_n/w_n\end{array}\end{array}\end{array}\right]\left[\begin{array}{c}{w}_1\\ ⋮\\ \begin{array}{c}{w}_j\\ ⋮\\ w_n\end{array}\end{array}\right]=n\left[\begin{array}{c}{w}_1\\ ⋮\\ \begin{array}{c}{w}_j\\ ⋮\\ w_n\end{array}\end{array}\right]\end{array}$$

(3)

According to [49], a fuzzy set $\overline{a}$ in a universe of discourse X is characterised by a membership function $\mu \overline{a}\left(x\right)$ that maps each element x in X to a real number in the interval $\left[0, 1\right],$ as indicated in Figure 2. The function value $\mu \overline{a}\left(x\right)$ is the grade of membership of x in $\overline{a}$. The closer the value of x in $\overline{a}$, the higher the grade of membership of x in $\overline{a}$.

Triangular fuzzy numbers (TFNs) are used to establish the weighting scale at the beginning of the fuzzy AHP process. To improve the Saaty (1977) original 9-point scale based on linguistic characteristics, fuzzy weights for the pairwise comparison are determined using TFNs [50].

The following definition explains the calculation process for fuzzy numbers. A triangular fuzzy number is often represented as a triplet $\overline{a}=\left(l, m, u\right)$, where $l$ and $u$ are the lower and upper bounds of the fuzzy number $\overline{a}$, and m is the modal value for $\overline{a}$. The membership function $\mu \overline{a}\left(x\right)$ of triangular fuzzy number $\overline{a}$ is in Figure 2.

Based on the three parameters of $l, m, and u$, the membership function can be stated by Equation (4).

$$\mu \overline{a}\left(x\right)=\left\{\begin{array}{cc}x-l/m-l& if l \le x\le m\\ u-x/u-m& if m\le x\le u\\ 0& otherwise\end{array}\right.$$

(4)

Fuzzy numbers can be used to indicate the degrees of pairwise comparison of linguistic variables, as demonstrated in

Table 2. A pairwise comparison of linguistic variables using fuzzy numbers [52].

The Intensity of the Fuzzy Scale	Definition of Linguistic Variable	Fuzzy Number	User-Defined
$\tilde{1}$	Similar importance (SI)	$\left(l, m, u\right)$	(_,1,_)
$\tilde{3}$	Moderate importance (MI)	$\left(l, m, u\right)$	(_,3,_)
$\tilde{5}$	Intense importance (II)	$\left(l, m, u\right)$	(_,5,_)
$\tilde{7}$	Demonstrated importance (DI)	$\left(l, m, u\right)$	(_,7,_)
$\tilde{9}$	Extreme importance (EI)	$\left(l, m, u\right)$	(_,9,_)
$\tilde{2},\tilde{4},\tilde{6},\tilde{8}$	Intermediate values	$\left(l, m, u\right)$	(_,_,_)

.

Equation (5) presents a fuzzy positive reciprocal matrix, which is based on the information from pairwise comparison.

$$\tilde{A}=\left[\begin{array}{ccc}{\tilde{a}}_{l1}& \cdots & \begin{array}{ccc}{\tilde{a}}_{lj}& \cdots & {\tilde{a}}_{ln}\end{array}\\ ⋮& & \begin{array}{ccc}⋮& & ⋮\end{array}\\ \begin{array}{c}{\tilde{a}}_{i1}\\ ⋮\\ {\tilde{a}}_{n1}\end{array}& \begin{array}{c}\cdots \\ \\ \cdots \end{array}& \begin{array}{ccc}\begin{array}{c}{\tilde{a}}_{ij}\\ ⋮\\ {\tilde{a}}_{nj}\end{array}& \begin{array}{c}\cdots \\ \\ \cdots \end{array}& \begin{array}{c}{\tilde{a}}_{in}\\ ⋮\\ {\tilde{a}}_{nn}\end{array}\end{array}\end{array}\right]$$

(5)

where ${\tilde{a}}_{ij}\otimes a_{ji}\approx 1$ and ${\tilde{a}}_{ij}\cong w_i/w_j$.

The geometric mean value ($\widetilde{r_i})$ and the fuzzy weights for the criteria are computed using Equations (6) and (7), respectively.

$$\widetilde{r_i}={\left({\tilde{a}}_{i1}\otimes {\tilde{a}}_{12}\otimes \dots \otimes {\tilde{a}}_{in}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}$$

(6)

$${\tilde{w}}_i=\widetilde{r_i}\otimes \left(\widetilde{r_1}\oplus \widetilde{r_2}\oplus \dots \oplus \widetilde{ r_n}\right)$$

(7)

The following operations are also critical when using fuzzy numbers $\widetilde{A_1} \mathrm{and} \widetilde{A_2}$.

Multiplying fuzzy numbers

$$\widetilde{A_1}\otimes \widetilde{A_2}=\left(l_1,m_1,u_1\right)\otimes \left(l_2,m_2,u_2\right)=\left(l_1\ast l_2, m_1\ast m_2, u_1\ast u_2\right)$$

(8)

Adding fuzzy numbers

$$\widetilde{A_1}\oplus \widetilde{A_2}=\left(l_1, m_1, u_1\right)\oplus \left(l_2,m_2, u_2\right)=\left(l_1+l_2, m_1+m_2, u_1+u_2\right)$$

(9)

The reciprocal values are then calculated by using Equation (10)

$${\left(l_1, m_1, u_1\right)}^{-1}=\left(\frac{1}{u_1}, \frac{1}{m_1},\frac{1}{l_1}\right)$$

(10)

4.3. Fuzzy TOPSIS Technique

TOPSIS is the technique adopted for determining the order preference by similarity to obtain the ideal solution that maximises the criteria/attributes and minimises the cost criteria/attributes, whereas the negative ideal solution maximises the cost criteria and minimises the benefit criteria [51]. TOPSIS selects the most sustainable facility layout amongst the generated alternatives [52]. The classical TOPSIS method assumes that the rating of alternatives concerning the criteria as well as the criteria weights are expressed precisely by real numbers and it follows the steps described below [53]:

Let us assume a problem where a set of alternatives, given as $A=\left\{A_k\mathsf{{\rm I}} k=1,\dots ,n\right\}$, and a set of criteria, $C=\left\{C_j\mathsf{{\rm I}} j=1,\dots ,m\right\}$, where $X=\left\{X_{kj}\mathsf{{\rm I}} k=1,\dots ,n; j=1,\dots ,m\right\}$ denotes the set of performance ratings and $\omega =\left\{{\omega }_j\mathsf{{\rm I}} j=1,\dots ,m\right\}$, is the set of weights, the information for Alternatives (A), Criteria (C), Performance ratings (X), and Weights (W) can be represented, as shown in

Table 3. Information matrix for TOPSIS.

Alternatives	${\mathit{C}}_1$	${\mathit{C}}_2$	$\cdots$	${\mathit{C}}_{\mathit{m}}$
$A_1$	$x_{11}$	$x_{12}$	$\cdots$	$x_{1m}$
$A_2$	$x_{21}$	$x_{22}$	$\cdots$	$x_{2m}$
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
$A_n$	$x_{n1}$	$x_{n2}$	$\cdots$	$x_{nm}$
$W$	$w_1$	$w_2$	$\cdots$	$w_m$

.

The first step in the TOPSIS technique is to calculate the normalised ratings using Equation (11).

$$r_{kj}\left(x\right)=\frac{x_{kj}}{ \sqrt{{{\displaystyle \sum }}_{k=1}^n{x}_{kj}^2}}, k=1,\dots ,n; j=1,\dots ,m$$

(11)

For the benefit criteria (the larger the value the better), $r_{kj}\left(x\right)=\left(x_{kj}-x_j^{-}\right)/\left(x_j^{\ast }-x_j^{-}\right)$, where $x_j^{\ast }=ma{x}_k{x}_{kj}$ and $x_j^{-}=mi{n}_k{x}_{kj}$ or setting, $x_j^{\ast }$ is the desired level and $x_j^{-}$ is the worst level.

For the cost criteria (the smaller the value the better), $r_{kj}\left(x\right)=\left(x_j^{-}-x_{kj}\right)/\left(x_j^{-}-x_j^{\ast } \right)$, and then the weighted normal ratings are calculated by Equation (12).

$$v_{kj}\left(x\right)={\omega }_j{r}_{kj}\left(x\right), k=1,\dots ,n; j=1,\dots ,m$$

(12)

Next, the positive ideal solution (PIS) and the negative ideal solution (NIS) are derived as,

$$\begin{gathered} PIS=A^{+}=\left\{v_1^{+}\left(x\right),v_2^{+}\left(x\right),\dots ,v_j^{+}\left(x\right),\dots ,v_m^{+}\left(x\right) \right\} \\ =\left\{\left(ma{x}_k{v}_{kj}\left(x\right) \mathsf{{\rm I}} \mathrm{j}\in J_1\right), \left(mi{n}_k{v}_{kj}\left(x\right) \mathsf{{\rm I}} \mathrm{j}\in J_2\right) \mathsf{{\rm I}} k=1,\dots ,n\right\}, \end{gathered}$$

(13)

$$\begin{gathered} NIS=A^{-}=\left\{v_1^{-}\left(x\right),v_2^{-}\left(x\right),\dots ,v_j^{-}\left(x\right),\dots ,v_m^{-}\left(x\right) \right\} \\ =\left\{\left(mi{n}_k{v}_{kj}\left(x\right) \mathsf{{\rm I}} \mathrm{j}\in J_1\right), \left(ma{x}_k{v}_{kj}\left(x\right) \mathsf{{\rm I}} \mathrm{j}\in J_2\right) \mathsf{{\rm I}} k=1,\dots ,n\right\} \end{gathered}$$

(14)

where $J_1$ and $J_2$ are the benefit and the cost attributes, respectively.

The alternative step is to calculate the separation from the PIS and NIS between the alternatives. The separation values can be measured using the Euclidean distance given as:

$$D_k^{\ast }=\sqrt{{\displaystyle \sum }_{j=1}^m{\left[v_{kj}\left(x\right)-v_j^{+}\left(x\right)\right]}^2, } k=1,\dots ,n$$

(15)

and

$$D_k^{-}=\sqrt{{\displaystyle \sum }_{j=1}^m{\left[v_{kj}\left(x\right)-v_j^{-}\left(x\right)\right]}^2, } k=1,\dots ,n$$

(16)

The similarities to PIS can be derived as

$$C_k^{\ast }=D_k^{-}/\left(D_k^{\ast }+D_k^{-}\right), k=1,\dots ,n$$

(17)

where $C_k^{\ast }\in \left[0, 1\right] \forall k=1,\dots ,n$.

Finally, the preferred orders can be obtained according to the similarities to the PIS $\left(C_k^{\ast }\right)$ in descending order to choose the best alternatives. The methodology proposed for selecting the best layout mainly depends on the DES model for developing quantitative criteria, such as productivity and utilisation. The results of DES are then converted into a scale suitable for use in the fuzzy TOPSIS methodology.

5. Railcar Industrial Case Study

This section presents the application of the methodology to a railcar manufacturing case study.

5.1. Description of the Underframe Production System

The manufacturing company considered for this paper is a railcar manufacturing organisation that produces commuter railcars. The study focuses on the Car Body Shell section, which assembles underframes through the welding assembly operations. The line comprises four main workstations, which are CBS1080, CBS1070, CBS1140, and CBS1150, as indicated in brown in Figure 3.

The current layout of the underframe is divided into four main sections: the layout of the car body shell section. Figure 4 represents a detailed view of the two critical areas of the underframe line, which is the main focus of this paper. CBS 1080 has four jigs, jigs 1, 2, 3, and 4. CBS1070 has two jigs, which are CBS10701 and CBS10702.

5.2. Concept Generation for DES Simulation

To improve the performance of the current underframe layout, it was necessary to conceptualize the design and analyse layout configuration in line with the performance objectives of the railcar manufacturing organisation. Four concepts were generated based on the line’s available space and production performance. The first concept represents the AS IS (current production system) and is based on Figure 4. The criteria for configurations were based on factors, such as Line Balancing Efficiency (LBE), Flexibility, Cost, Utilisation of resources, and productivity. For simulation purposes, the following concepts were proposed:

Concept 2:

This proposal seeks to duplicate the bottleneck workstation CBS1070, with additional space requirements. In this concept, an effort is made to double the capacity of CBS1070 to eliminate the effect of this bottleneck station within the current space. This proposal will require additional space for the new jigs CBS20701 and CBS20702.

Concept 3:

The second proposal is the concept that proposes further changes to the ones detailed in layout alternative two. A dedicated welding gun is assigned for both jig one and three, as opposed to a shared welding gun. This proposal will minimise the waiting time experienced in jigs one and three as they were sharing a welding gun. Figure 5 is the concept where bottleneck station CBS1070 is duplicated.

Concept 4:

This concept represents a scenario that seeks to duplicate CBS1070, as in concepts two and three, then introduces a new jig that will replace jigs two and four. This layout configuration will create more space that will increase the efficient flow of products and increase the overall utilisation of the resources. This option will also mean that one welder will be assigned to the newly suggested merged jig, saving one welder and the welding gun.

5.3. Discrete Event Simulation Model

DES was identified as the ideal method for the evaluation of the performance of the proposed underframe layout concepts. The DES model was built as a representation of the physical underframe layout. Anylogic software was used to develop the model. This paper does not discuss the development process of the model; however, it uses the results from the model in the developed fuzzy AHP-TOPSIS model. Figure 6 is a picture of the DES model capturing some results.

5.4. Discrete Event Simulation Model Results

Table 4. Simulation results for generated concepts.

Alternatives	Concept 1	Concept 2	Concept 3	Concept 4
Operators	16	19	19	19
Production output	288	330	567	566
Productivity	18	17.37	29.84	29.79
Line Balancing Efficiency	72.85%	61.58%	61.58%	57.30%
Utilisation	52	46	69.4	75.7

shows the model performance results necessary for comparing criteria weight determination. The table indicates the results of three quantitative criteria of productivity, utilisation, and line balancing efficiency. The number of operators is an important parameter used to calculate productivity, which was the output divided by the proposed number of operators. The line balancing efficiency for the four layout alternatives was calculated based on the equation, which is dependent on the number of workstations, cycle time, and the work content.

The results in Table 4 show a significant improvement when concepts three and four are used. The simulation results indicate that the proposed underframe line will be able to produce just about 566 railcars, making 95 train sets per year, which is well over the production target of 72 train sets annually. Concepts three and four are capable of producing 567 underframes annually, which relates to productivity of about 30 underframes per welder. The utilisation of concepts three and four, on average, is 69.4% and 75.7%, respectively. The results of the simulation are an important input for the MCDM model developed and applied in Section 5.5.

5.5. Application of F-AHP and F-TOPSIS

The Fuzzy AHP-TOPSIS method is based on the following steps:

Step 1: Development of the problem hierarchy and the description of the criteria

In this step, the problem is to select the optimal concept that could improve productivity in an underframe line for railcars. Based on the relevant literature, five criteria are used for choosing the best concept. These criteria are layout cost, line efficiency, productivity, layout flexibility, and utilisation.

The MCDM approach begins with identifying appropriate criteria for evaluating the alternatives. The literature proposes several criteria; however, criteria could only be relevant to some environments whilst they might not be suitable in others. Facility layout performance criteria for the railcar manufacturing organisation are summarised in Figure 7.

The criteria presented in Figure 7 are described in the following section, emphasising how they are applied in the current study.

(1) 

Cost

In layout decisions, the most common factor used by facility designers to evaluate layout options, because it influences material handling costs, is distance. According to the average, material handling accounts for 50% of the total operating costs [54]. In many FLPs, the objective is to minimise the material handling cost, which is directly influenced by the travel distance. Cost is a measurable quantity that includes costs for new equipment (hardware components), material handling equipment (MHE), or overtime [55]. The cost of material handling is determined by the typical distance travelled of materials brought into or out of the underframe production line over the course of production. Cost also involves expenses incurred when relocating some of the workstations, which will be one-off expenditure and, in some instances, could be excessive. Implementing the new system requires purchasing new equipment, such as jigs and guns. Operational costs involve any expense incurred by the organisation in extra labour requirements and any other direct or indirect costs involved in running the operations.

(2) 

Flexibility

Flexibility is a qualitative factor defined as a company’s capacity to adjust to changes in its environment [44]. Flexibility is the capability built into the manufacturing system that allows it to cope with at least some of these changes and variations with minimal disruption in performance [56].

(3) 

Line balancing efficiency (LBE)

This is a quantitative measure that was derived from the line balancing exercise that was carried out earlier. Line efficiency takes into consideration the number of workstations, the total work content, and the service time or the highest cycle time.

(4) 

Utilisation of resources

The resources in the manufacturing systems, such as employees and equipment, impact the facility [7]. Labour utilisation is a measure of the ratio of average hours worked by an operator over the total working hours allocated to the resource within a given period. Ref. [44] found that labour utilisation was the most crucial criterion, with a weightage of 35% compared to flexibility (0.34), accessibility (0.073), and area utilisation (0.16). The utilisation of resources, such as workstations, machines, and workers, is of interest to management as this information will enable the reconfiguration of the layout to reduce bottlenecks or eliminate waste in the process [57]. Utilisation statistics were collected from the DES results.

(5) 

Productivity

Productivity is defined as the ratio between input and output. In this case, labour productivity was used and it constituted a measure of underframe output levels based on the number of labour inputs employed to run production for a period of one year.

Step 2: Determining the linguistic scale to be used for the assessment

In practice, various types of scales are used for the assessment of weights and alternatives. The literature is not prescriptive about which scale to use. Multiple scales, such as 5-point, 7-point, and 10-point Likert scales, have been used for different applications. This study uses a five-point Likert scale adopted by [28].

Table 5. Relative importance of factors by Saaty [58].

Crisp Scale	Fuzzy Scale	Meaning
1	(1,1,1)	Equally preferred
2	(1,2,3)	Equally to moderately preferred
3	(2,3,4)	Moderately preferred
4	(3,4,5)	Moderately to strongly preferred
5	(4,5,6)	Strongly preferred
6	(5,6,7)	Strongly to very strongly preferred
7	(6,7,8)	Very strongly preferred
8	(7,8,9)	Very to extremely strongly preferred
9	(9,9,9)	Very extremely preferred

presents the fuzzy set theory linguistic terms used to rate the criteria weights.

The ranking scale for criteria weights adopted from the conversion scales is applied to convert the linguistic terms into fuzzy numbers. These usually uses a scale of 1 to 9 for rating the criteria.

Step 3: Determination of AHP fuzzy weights

In this study, the Fuzzy AHP is used to determine the weights based on the performance criteria concerning layout design alternatives. Table 5 shows a pairwise comparison used to determine each criterion’s weight. The scale in Table 4 was used to develop the pairwise comparison in

Table 6. Pairwise comparison of criteria.

Criteria.	Flexibility	LBE	Utilisation	Cost	Productivity
Flexibility	(1,1,1)	(0.11,0,13,0.14)	(0.25,0.33,0.5)	(5,6,7)	(0.11,0.11,0.11)
LBE	(7,8,9)	(1,1,1)	(7,8,9)	(9,9,9)	(0.11,0.11,0.11)
Utilisation	(2,3,4)	(0.11,0.13,0.14)	(1,1,1)	(2,3,4)	(0.11,0.11,0.11)
Cost	(0.14,0.17,0.2)	(0.11,0.11,0.11)	(0.25,0.33,0.5)	(1,1,1)	(0.11,0.11,0.11)
Productivity	(9,9,9)	(9,9,9)	(9,9,9)	(9,9,9)	(1,1,1)

. To unpack how the pairwise comparison works, when flexibility is compared to cost, flexibility was rated as being strongly to very strongly preferred; hence, the fuzzy scale of (5, 6, 7) was used. The reciprocal of the fuzzy scale (5, 6, 7) was then determined for comparing cost and flexibility and it was found to be (0.14,0.17,0.2). Similarly, the rest of the reciprocal values were computed in this manner, and the rest of the shared values were also computed in this manner.

The geometric mean value for the criteria Flexibility ($\widetilde{r_F})$, Efficiency $(\widetilde{r_E)}$, Utilisation $\widetilde{(r_U}), \mathrm{Cos}\mathrm{t} (\widetilde{r_C)}$, and Productivity $(\widetilde{r_P)}$ was calculated by using the following equation:

$$\widetilde{r_i}={\left({\tilde{a}}_{i1}\otimes {\tilde{a}}_{12}\otimes \dots \otimes {\tilde{a}}_{in}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}.$$

(18)

$$\begin{gathered} \widetilde{r_F}=\left({\left(1\ast 0.11\ast 0.25\ast 5\ast 0.11\right)}^{\frac{1}{5}},{\left(1\ast 0.13\ast 0.33\ast 6\ast 0.11\right)}^{\frac{1}{5}}, {\left(1\ast 0.14\ast 0.5\ast 7\ast 0.11\right)}^{\frac{1}{5}}\right)= \\ \left(0.044331, 0.487281, 0.560977\right) \end{gathered}$$

$$\begin{gathered} \widetilde{r_E}=\left({\left(7\ast 1\ast 7\ast 9\ast 0.11\right)}^{\frac{1}{5}},{\left(8\ast 1\ast 8\ast 9\ast 0.11\right)}^{\frac{1}{5}},{\left(9\ast 1\ast 9\ast 9\ast 0.11\right)}^{\frac{1}{5}}\right) \\ = \left(2.17747067, 2.29693705, 2.40774285\right) \end{gathered}$$

$$\begin{gathered} \widetilde{r_U}=\left({\left(2\ast 0.11\ast 1\ast 2\ast 0.11\right)}^{\frac{1}{5}},{\left(3\ast 0.13\ast 1\ast 3\ast 0.11\right)}^{\frac{1}{5}}, {\left(4\ast 0.14\ast 1\ast 4\ast 0.11\right)}^{\frac{1}{5}}\right) \\ =\left(0.54769804, 0.65962195, 0.7602489\right) \end{gathered}$$

$$\begin{gathered} \widetilde{r_C}=\left({\left(0.14\ast 0.11\ast 0.25\ast 1\ast 0.11\right)}^{\frac{1}{5}},{\left(0.17\ast 0.11\ast 0.33\ast 1\ast 0.11\right)}^{\frac{1}{5}}, {\left(0.2\ast 0.11\ast 0.5\ast 1\ast 0.11\right)}^{\frac{1}{5}}\right)= \\ \left(0.21229648, 0.23372643, 0.2618962\right) \end{gathered}$$

$$\begin{gathered} \widetilde{r_P}=\left({\left(9\ast 9\ast 9\ast 9\ast 1\right)}^{\frac{1}{5}},{\left(9\ast 9\ast 9\ast 9\ast 1\right)}^{\frac{1}{5}}, {\left(\left(9\ast 9\ast 9\ast 9\ast 1\right)\right)}^{\frac{1}{5}}\right)= \\ \left(5.79954613, 5.79954613, 5.79954613\right) \end{gathered}$$

The fuzzy geometric mean values for the five criteria can be summarised in

Table 7. Geometric mean values.

Criteria	Fuzzy Geometric Mean
Flexibility	0.43402145	0.48728118	0.56097746
LBE	2.17747067	2.29693705	2.40774285
Utilisation	0.54769804	0.65962195	0.7602489
Cost	0.21229648	0.23372643	0.2618962
Productivity	5.79954613	5.79954613	5.79954613
Sum	9.17103278	9.47711274	9.79041154

.

Flexibility ($\widetilde{w_F})$, Efficiency $(\widetilde{w_E)}$, Utilisation $\widetilde{(w_U}), \mathrm{Cos}\mathrm{t} (\widetilde{w_C)}$, and Productivity $(\widetilde{w_P)}$ are then calculated. The final fuzzy weights are calculated by using the following equation:

$${\tilde{w}}_i=\widetilde{r_i}\oplus \left(\widetilde{r_1}\oplus \widetilde{r_2}\oplus \dots \oplus \widetilde{ r_n}\right)$$

(19)

Firstly, the summed reciprocal of the geometric mean value is determined following expression denoting its expression:

$$\begin{gathered} \left(\frac{1}{u_1}, \frac{1}{m_1},\frac{1}{l_1}\right)=\left(\frac{1}{9.790412}, \frac{1}{9.477113},\frac{1}{9.171033}\right) \\ =\left(0.10214075, 0.10551737, 0.10903897\right) \end{gathered}$$

(20)

To find the fuzzy weights, the geometric mean values $\widetilde{(r_i}$) for each criterion are multiplied by the reciprocal expression.

$$\begin{gathered} \widetilde{w_F}=\left(0.044331, 0.487281, 0.560977\right)\otimes \left(0.10214075, 0.10551737, 0.10903897\right) \\ =\left(0.04433128, 0.05141663, 0.06116841\right) \\ \widetilde{w_E}=\left(2.17747067,2.29693705, 2.40774285\right)\otimes \left(0.10214075, 0.10551737, 0.10903897\right) \\ =\left(0.22240849, 0.24236675, 0.26253781\right) \\ \widetilde{w_U}=\left(0.54769804, 0.65962195, 0.7602489\right)\otimes \left(0.10214075, 0.10551737, 0.10903897\right) \\ =\left(0.05594229, 0.06960157, 0.08289676\right) \\ \widetilde{w_C}=\left(0.21229648, 0.23372643,0.2618962\right)\otimes \left(0.10214075, 0.10551737, 0.10903897\right) \\ =\left(0.02168412, 0.0246622, 0.02855689\right) \\ \widetilde{w_P}=\left(5.79954613, 5.79954613, 5.79954613\right)\otimes \left(0.10214075, 0.10551737, 0.10903897\right) \\ =\left(0.59237001, 0.61195285, 0.63237656\right) \end{gathered}$$

Figure 8 depicts the geometric fuzzy weights as well as crisp numeric weights.

Step 4: Determine decision matrix for alternatives

Once the weights have been determined using the Fuzzy AHP technique, the next step will be to use Fuzzy TOPSIS to rank the generated alternative layouts. The linguistic terms used for this purpose were the fuzzy ratings in

Table 8. Fuzzy ratings for assessing alternatives.

Crisp Scale	Fuzzy Number	Linguistic Terms for Alternative Assessment
1	(1,1,3)	Very Poor (VP)
2	(1,3,5)	Poor (P)
3	(3,5,7)	Fair (F)
4	(5,7,9)	Good (G)
5	(7,9,9)	Very Good (VG)

adopted from [28].

In this study, the best layout concept has to be selected based on the layout performance of line balancing efficiency, flexibility, utilisation, productivity, and cost. The cost figures were based on the number of welders and the assumed price of new jigs and welding guns deployed in the underframe line. The labour cost for this calculation was assumed to be ZAR 10 000 per welder, whilst the costs of a welding gun and the new jig were ZAR 160 000 and ZAR 1000 000, respectively. Based on the changes proposed with the designed alternatives, the costs for layouts 1, 2, 3, and 4 were ZAR 1 280 000, ZAR 2 630 000, ZAR 2 790 000, and ZAR 3 630 00,0 respectively. The conversion scale used for the quantitative criteria of line balancing efficiency, utilisation, and productivity was derived from the simulation results in Table 4 and shown in

Table 9. Quantitative criteria conversion scales.

Criteria				Scale
Cost	Productivity	Utilisation	LBE
More than ZAR 3 million	15–20	40–50	50–60	1
ZAR 2.6 to 3 million	21–25	51–60	61–70	2
ZAR 2.1 to 2.5 million	26–30	61–70	71–80	3
ZAR 1.6 to 2 million	31–35	71–80	81–90	4
ZAR 1 to 1.5 million	More than 36	More than 81	More than 91	5

, whilst flexibility was a qualitative criterion guided by the literature.

The conversion scales in Table 8 were then used to derive the decision matrix for the four concepts, as shown in

Table 10. Decision matrix for alternatives.

Criteria	Alternative Concepts
	A1	A2	A3	A4
Cost	(7,9,9)	(1,3,5)	(1,3,5)	(1,1,3)
Flexibility	(1,1,3)	(5,7,9)	(5,7,9)	(3,5,7)
Line Balancing Efficiency (LBE)	(1,3,5)	(1,3,5)	(1,3,5)	(1,1,3)
Utilisation	(1,3,5)	(1,1,3)	(3,5,7)	(5,7,9)
Productivity	(1,1,3)	(1,1,3)	(3,5,7)	(3,5,7)

.

The Fuzzy TOPSIS approach selects the layout that maximises the benefit criteria while minimising the non-benefit criteria. The benefit criteria for this study were line balancing efficiency, flexibility, productivity, and utilisation, while the non-benefit criterion is the cost.

Step 5: Calculation of the normalised decision matrix R using vector normalisation

The normalised fuzzy decision values for the cost criterion $r_{kj}\left(C\right),$ which is a non-benefit criterion for layout alternative A, are calculated using Equation 20. These values are based on Table 10 for all the layout alternatives.

$$r_{kj}\left(C\right)=\left(\frac{l_1}{u_1}, \frac{l_1}{m_1},\frac{l_1}{l_1}\right)$$

(21)

The normalised fuzzy decision values for the benefit criteria Flexibility $r_{kj}\left(F\right)$, Line Balancing Efficiency $r_{kj}\left(E\right)$, Utilisation $r_{kj}\left(U\right)$, and Productivity $r_{kj}\left(P\right)$ are calculated by means of Equation (21) below. The results of this step are presented in

Table 11. Summarised normalised decision matrix.

Criteria	Alternative Concepts
	A1	A2	A3	A4
Cost	(0.11,0.11,0.14)	(0.20,0.33,0.10)	(0.20,0.33,0.10)	(0.33,1.00,1.00)
Flexibility	(0.11,0.11,0.33)	(0.56,0.78,1.00)	(0.56,0.78,1.00)	(0.33,0.56,0.78)
LBE	(0.43,0.71,1.00)	(0.14,0.43,0.71)	(0.14,0.43,0.71)	(0.14,1.00,0.43)
Utilisation	(0.11,0.33,0.56)	(0.11,0.11,0.33)	(0.33,0.56,0.78)	(0.56,0.78,1.00)
Productivity	(0.14,0.14,0.43)	(0.14,0.14,0.43)	(0,43,0.71,1.00)	(0.43,0.71,1.00)

.

$$r_{kj}\left(x\right)=\left(\frac{l}{u}, \frac{m}{u},\frac{u}{u}\right)$$

(22)

Table 12. Weighted normalised fuzzy decision matrix.

Criteria	Alternative Concepts
	A1	A2	A3	A4
Cost	(0.005,0.005,0.009)	(0.009,0.017,0.061)	(0.009,0.017,0.061)	(0.015,0.051,0.061)
Flexibility	(0.025,0.027,0.088)	(0.124,0.189,0.263)	(0.124,0.0189,0.263)	(0.124,0.189,0.263)
LBE	(0.024,0.050,0.083)	(0.008,0.030,0.060)	(0.030,0.030,0.060)	(0.080,0.007,0.040)
Utilisation	(0.002,0.008,0.016)	(0.002,0.003,0.010)	(0.007,0.014,0.022)	(0.012,0.019,0.029)
Productivity	(0.085,0.087,0.271)	(0.085,0.087,0.271)	(0.254,0.437,0.632)	(0.254,0.437,0.632)

shows the weighted normalised values for the developed alternatives. The weighted normalised values are a product of the normalised weights determined in Table 11 and the fuzzy weights determined in step 3. The following equation is used to multiply fuzzy numbers: $\widetilde{A_1}\otimes \widetilde{A_2}=\left(l_1,m_1,u_1\right)\otimes \left(l_2,m_2,u_2\right)=\left(l_1\ast l_2, m_1\ast m_2, u_1\ast u_2\right)$

Step 6: Determine the fuzzy positive ideal solution (FPIS) and the fuzzy negative ideal solution (FNIS)

Firstly, the maximum values from each row as A+ and the minimum value from each row as A- are determined, as shown in

Table 13. Maximum and minimum values for each criterion.

Criteria	Distance of Each Criteria to the FPIS and FNIS
	A+	A−
Cost	(0.015,0.051,0.061)	(0.005,0.006,0.009)
Flexibility	(0.124,0.189,0.263)	(0.025,0.027,0.089)
LBE	(0.024,0.070,0.083)	(0.008,0.030,0.036)
Utilisation	(0.012,0.019,0.029)	(0.002,0.003,0.010)
Productivity	(0.254,0.437,0.632	(0.085,0.085,0.271)

, followed by the calculations of FPIS and FNIS, which are depicted in

Table 14. Fuzzy positive ideal solution.

Criteria	Fuzzy Positive Ideal Solution for Alternative Concepts
	A1	A2	A3	A4
Cost	0.040557693	0.020082487	0.020082487	0
Flexibility	0.148898449	0	0	0.053997909
LBE	0.011481352	0.028274251	0.028274251	0.028863817
Utilisation	0.011167007	0.015552503	0.005583504	0
Productivity	0.306325637	0.306325637	0	0
di*	0.518430138	0.370234878	0.053940242	0.08286173

and

Table 15. Fuzzy negative ideal solution.

Criteria	Fuzzy Negative Ideal Solution for Alternative Concepts
	A1	A2	A3	A4
Cost	0	0.031064214	0.031064214	0.040557693
Flexibility	0	0.148898449	0.148898449	0.096022732
LBE	0.031063506	0.013674459	0.013674459	0.022962705
Utilisation	0.004841039	0	0.010073852	0.015552503
Productivity	0	0	0.306325637	0.306325637
d-	0.035904545	0.193637122	0.510036611	0.481421269

, respectively.

The FPIS for the four layout alternatives can be summarised in Table 14. According to Table 14, it is clear that layout design three is best (close to the ideal positive solution) when criteria flexibility, line balancing efficiency, utilisation, and productivity are considered, whilst layout four was not close in terms of the distance to the ideal solution when line balancing efficiency is considered.

The results presented in Table 15 indicate that layout concept one is very close to the negative ideal solution; hence, it is doubtful that it will be considered.

According to the coefficient closeness calculated in

Table 16. Closeness coefficient for the non-batching processing.

Alternatives	di*+di−	Cci	Rank
Concept 1	0.554334683	0.064770519	3
Concept 2	0.563872001	0.343406166	4
Concept 3	0.563976853	0.904357348	1
Concept 4	0.564282996	0.853155727	2

, alternative three ranked number one, with a closeness coefficient of 0.90, followed by layout four, with a closeness ranking of 0.85. Layout alternative two ranked last, with a closeness coefficient of 0.34, meaning that it is undesirable to use this layout design as it will yield poor production results. Layout three is recommended to ensure that the railcar organisation produces the products according to their targeted production.

Figure 9 is a graphical representation of Table 15, which confirms that layout 3 (A3), with a closeness coefficient of 0.904, is the recommended proposal because the alternative is very close to the ideal positive solution and the one which is furthest from the ideal negative solution. The graph shows the ranking order of preference as A3, A4, A2, and A1. The least preferred concept is A1, with a closeness coefficient of 0.0647.

Figure 10 presents the closeness coefficient for the proposed batching process model, which indicates that concept three (A3) is the best of all, with a coefficient closeness of 0.93, followed by concept 2 (A2), with a coefficient closeness of 0.35.

5.6. Validation of the Fuzzy AHP-TOPSIS Model

Model validation is an important part of any modelling and simulation process. To ensure the validity of the results, a traditional AHP model was developed based on the crisp numeric values found in Table 5.

Table 17. Criteria pairwise comparison for the AHP method and weights.

Criteria Weights	Flexibility	Efficiency	Utilisation	Cost	Productivity	Weights
Flexibility	1	0.13	0.33	6	0.11	0.07442
Efficiency	8	1	8	9	0.11	0.26168
Utilisation	3	0.13	1	3	0.11	0.078374
Cost	0.17	0.11	0.33	1	0.17	0.03074
Productivity	9	9	9	9	1	0.554785
Sum	21.16667	10.36111	18.66667	28	1.5

shows a comparison matrix for the problem based on crisp numeric values.

Table 17 depicts the weights of the five criteria used in selecting the best layout for the underframe production. Productivity has the highest weight, followed by efficiency.

According to

Table 18. Layout ranking.

	Layout Alternatives
Criteria	A1	A2	A3	A4
Cost	0.01	0.04	0.04	0.07
Flexibility	0.13	0.26168	0.26168	0.19626
Efficiency	0.052249	0.052249	0.052249	0.078374
Utilisation	0.020493	0.010247	0.03074	0.03074
Productivity	0.277393	0.277393	0.554785	0.554785
Performance Score	0.50	0.64	0.94	0.93
Ranking	4	3	1	2

, which depicts the ranking of the layout alternatives, layout alternative A3 has the highest performance score, followed by layout A4. This ranking confirms the performance scores achieved by the fuzzy AHP-TOPSIS; however, the scores are not exactly the same and this might be because of rounding off.

6. Conclusions

Facility layout selection is an important issue for an organisation that seeks to improve productivity, particularly due to its role in production activity. Layout decisions are crucial and must be carried out in a considerate manner since these are long-term decisions that are costly. The developed Fuzzy AHP TOPSIS methodology was used in this case as a tool to analytically perform the selection of an optimal alternative layout based on performance criteria. The DES model provides essential information when it comes to the performance analysis of a production line. This paper demonstrated that DES is an effective tool for deriving quantitative criteria that could be used in MCDM. A conversion scale was developed to derive quantitative criteria used in the proposed Fuzzy AHP-TOPSIS methodology to evaluate designed alternative layouts. The analysis results indicate that the company can implement layout three, irrespective of whether batching or non-batching processing is used; though it is the most expensive one to implement, it shows significant improvements in terms of productivity, efficiency, flexibility, and the utilisation of resources. Productivity ranked the highest, followed by flexibility in terms of geometric weights, which were (0.592, 0.612, 0.632) and (0.222, 0.242, 0.262), respectively. Layout alternative three was ranked the first option for both the batch and non-batching processing in terms of the coefficient closeness index, with 0.92 and 0.90, respectively. The simulated results indicate that the proposed layout is capable of producing 567 underframes annually, as compared to the 288 units produced by the current layout.

Future work is possible in this area due to the fact that, in manufacturing, there are many factors that contribute to improved efficiency and productivity, such as planning, skills, management, procurement, and quality management. This study focuses on the drivers of facility layout to improve productivity without looking at factors, such as people and management issues. The methodology presented in this research is just the foundation for facility layout design and the optimisation of performance for decision makers. The use of simulation-based MCDM is not a well-established phenomenon; therefore, there is a need for researchers to take advantage of the simulation and optimization tools for planning and decision making.

Although cost was used as a criterion in the MCDM evaluation of the layout concepts, when sufficient data are available, an exhaustive cost–benefit analysis could be performed to enhance decision making. This study is limited because the model was tested only in one railcar manufacturing setup. The designed model could be tested in different industrial cases for further validation.

Managerial implications in this study to be considered are for the adoption of approaches, such as simulation and optimization, to effectively decide on the policies and strategies for improving productivity across the organization. Application of the tools in this study shows the possibilities available for productivity improvement, which could effectively improve the organizations’ economic wellbeing.

The limitation of this study is that the research focused on only existing (brownfield) manufacturing facilities for the underframe production line, as opposed to designing new (greenfield) facility layouts from scratch. The focus was on using the current shape and size of the underframe production line. The design concept was based on the limitations and principles adopted by this start-up organisation, which focuses on employment creation. This means that there is a limitation in terms of the amount of automation that could be implemented.