Decision advisor based on uncertainty theories for the selection of rapid prototyping system

Abstract

A multitude of rapid prototyping (RP) systems and technologies have come up since the introduction of additive process. Owing to the enlarging number of these systems with distinctive efficacy, the problem of selecting an appropriate system for a particular requirement is a cumbersome task. Henceforth, this work comes up with a strategy based on multi-attribute decision making to select a most suitable RP system. The presence of subjectivity in decision making as well as the existence of imprecision from various sources emphasize the methods which must consider uncertainty and vagueness. A decision advisor based on uncertainty theories, including fuzzy analytical hierarchy process (FAHP) and grey relational analysis (GRA) has been introduced. It provides a comprehensive database comprising thirty nine commercially available RP systems. The evaluation attributes consisting of machine cost, accuracy, layer thickness, machine speed, material cost, net build size volume, machine weight, surface roughness, and material strength were utilized to characterize the different machines. The FAHP based on trapezoidal fuzzy number was implemented to determine the priority weights of various attributes, while the GRA was employed to realize the best RP system and technology. The authors believe that this system has the potential to transform into a fully developed RP selection system.

Keywords

Rapid prototyping fuzzy analytical hierarchy process trapezoidal fuzzy number grey relational analysis additive manufacturing sustainability

1 Introduction

The rapid prototyping (RP) (also known as Additive Manufacturing, AM) due to its innumerable benefits has revolutionized the manufacturing sector. Since the inception of Stereolithography (SLA) the first commercial AM printer in 1987, different AM technologies have emerged and evolved over the years. The AM is no longer used only for visualization purpose or prototyping, but are also being used to fabricate fully functional products. Some of the primary benefits of AM include reduced lead time (most often by 75%), reduced cost (many times, the cost can be minimized by 50%), fabrication of highly complex shapes as well as high design flexibility [1]. The AM process exhibit applications in various manufacturing industries such as aerospace, automobiles, home appliances, electrical, defense and medical applications. According to the Wohler’s report of 2018, there were more than 135 AM companies in 2017 which produced and sold industrial AM system worldwide [2]. Moreover, the polymer based AM machines occupies the largest share among the global AM Industry [3]. In 2017, the growth of AM worldwide was $7.36 billion, that was 21% higher when compared to $6.063 billion in 2016 [4].

The AM machines depending on the type of material used, can be categorized into three systems, namely liquid, solid (wire, sheet) and powder. In addition, the AM systems can also be classified into different technologies (processes) such as SLA, Selective Laser Sintering (SLS), Fused Deposition Modeling (FDM), etc. Among them, the SLA belongs to liquid based system, SLS comes under powder and FDM belongs to solid based AM systems. Each of these processes (or technologies) have their own strength, utilities, applications and limitations. Each AM machine have several attributes (machine characteristics) such as machine speed, material cost, layer thickness, accuracy etc., which drive their selection. The speed, cost, weight and size are not the only factors that should be considered while choosing a particular AM technology. The other factors consisting of surface finish, material strength, accuracy, etc., also varies for each process. Certainly, the strength and weakness are inherent in any manufacturing process and AM is no exception. It is crucial to realize that not one AM machine can do everything or can be used in all applications.

Due to the rapid growth of AM technology, the selection of the most appropriate AM system or machine has become increasingly important. A large number of new technologies are being introduced each year. With such a wide range of AM systems, technologies and machine attributes, selecting an AM machine for a user or a manufacturer is a challenging task and it takes a considerable amount of time and require greater expertise. There is no formal evaluation procedure which is generally followed in the selection of AM technique [2]. Indeed, the selection of optimum or most suitable system for AM process depends on many factors such as accuracy, material strength, build speed, surface finish and other system related parameters. To select the most suitable AM machine for a specific application, one must understand the strength and limitations of each AM machine. A number of factors (selection criteria) such as accuracy, speed, layer thickness, surface finish etc., which can be quantitative or qualitative must be synthetically considered while evaluating and selecting an AM machine. With so many factors involved and different AM machines available, it is really difficult for the potential customer or user to differentiate among them. In addition, the potential customer also confronts further complexity in choosing AM machine, as a result of the lack of knowledge of new AM system and its application. Consequently, to administer the selection of AM system efficiently, which is a multi-attribute problem, the decision makers should employ Multi-Attribute Decision Making (MADM) approaches. The MADM method represents a means of formulating fundamental information and providing judgements in situations involving multifarious objectives [5]. In the course of time, numerous approaches including Elimination and Choice Expressing Reality (ELECTRE) [6], Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) [7], Preference Ranking Organization Method For Enrichment of Evaluations (PROMETHEE) [8], Data Envelopment Analysis (DEA) [9], VIKOR [10], Full Consistency Method (FUCOM) [11, 12], Multi-Attributive Border Approximation Area Comparison (MABAC) [13 –15], Conjoint Analysis [16], Complex Proportional Assessment (COPRAS) [17, 18], etc., have been introduced. Henceforth, different MADM methods are available, which can be employed for various real decision problems. Certainly, no method is superior to the other because each possesses their own benefits and limitations in order to find an appropriate final solution [19]. It is generally difficult to differentiate between two MADM methods because they are based on different principles or they employ contrasting tactics, when dealing with diverse data sets [20, 21]. The selection of MADM is an extensive and a painstaking process which must take into regard the available information, decision maker’s role and all the decision process abilities, etc. [22]. For a given problem, the MADM approach can be selected either randomly or depending on the user’s knowledge and expertise with the particular technique or the ease of use of that technique [23, 24].

A number of studies were carried out based on MADM situations using different multicriteria decision support systems, benchmark studies and software tools for assisting AM users in selecting the most suitable AM machine. The first attempt in the selection of AM using a computer based system was done by Hornberger (1993) at Santa Clara University for education purpose [25]. In the other, relational databases were introduced by Campbell and Bernie (1996) [26]. Bauer et al introduced a software tool for AM selection that assisted the users in choosing the best combination of material and machines [27]. Philipson and Henderson (1997) from Arizona University introduced the AM advisor for selecting the best AM process based on relational database applying MS Access [28]. All these systems provided general and simple process selection information of AM process. Bibb et al. [29] and Masood and Soo [30] were the first to use the knowledge-based expert system for the selection of AM among availablecommercial AM machines, though this system lacked material, cost and build time information. Moreover, it was difficult to rank the most suitable AM among them using conditional statements. The web based AM selection techniques emerged in 2002. They consisted of a series of questions with a scaling factor for each question thus making it hard for the user in a single analysis [31]. Similarly, another web based AM selector was launched in 2005 with few questions and limitations in the ranking process [32]. Mahesh et al. (2005) [33] developed the first fuzzy decision making and benchmarking based on the five AM process by measuring several parts and building a database. The drawback of the system was that it did not adopt a generic approach and was applicable to specific household application. Byun and Lee [34] presented a modified TOPSIS method for AM selection process, but the fuzzy approach for assigning the machine attributes required relatively more computation. A total of six machine attributes comprising accuracy, surface roughness, part cost, strength, elongation and build time were used in their selection process. The Spanish Rapid Manufacturing Association (ASERM, 2006) launched an AM selector for the use in Additive prototype process and Rapid tooling methods. However the system was inadequate due to the absence of material information and costing [35]. In recent times, due to the prominence of grey theory in MADM, a number of approaches were introduced to assist in the RP selection process. For example, Mahapatra and Panda [36] proposed a benchmarking of AM system using Grey Relational Analysis (GRA) based on six AM machines and six machine characteristics such as dimensional accuracy, surface quality, part cost, material properties and build time. Similarly, Shende et al. [37] employed TOPSIS for ranking four AM processes: SLA, SLS, Three Dimensional Printing (3DP) and FDM based on seven attributes: accuracy, surface finish, tensile strength, elongation, part cost, process time, heat deflection temperature. In addition, Narayan panda et al. (2014) [38] developed an integrated Analytic Hierarchy Process (AHP) and fuzzy TOPSIS method for the selection of five AM processes: Laminated Object Manufacturing (LOM), SLS, 3DP, FDM and SLA based on five attributes: dimensional accuracy, surface quality, part cost, build time and material properties. Furthermore, Nitish et al. (2016) [39] compared two different AM system, namely FDM and an open source low cost AM machine for comparing dimensional performance using GRA method. Ivan peko et al. [40] also proposed an AHP, Fuzzy AHP (FAHP) and PROMETHEE method for AM selection based on four processes: 3DP, FDM, SLS, Photopolymer Jetting (PolyJet) and six attributes including dimensional accuracy, surface roughness, mechanical properties, process cost, process time and post-processing. Lately, different preference relations, including intuitionistic multiplicative preference relations [41], hesitant fuzzy preference relation [42, 43], etc., have also been utilized to derive priority weights in MADM. Although all the research works in literature utilized some logical process for AM selection. However, these works cannot be identified as the most comprehensive and best as far as the selection of AM process is concerned due to the following reasons.

Most of the existing methodologies used in the selection of AM were incomplete owing to limited number of attributes and processes.

All the methods were aimed at comparing the AM processes rather than selecting the best AM machine among the AM processes.

Some AM selection process also used benchmarking approach which is a costly and lengthy procedure. Additionally, all the vendors do not agree or conform to benchmarking of AM process.

Some authors also compared the polymer based system to metal based system which has different surface finish and accuracy.

Most of the MADM employed in earlier works assumed accurate judgement and crisp evaluation.

Henceforth, there is a need of a comprehensive but simple and logical approach involving more processes and attributes for the selection of AM. This work is different from previous works due to its exhaustive and extensive nature. This study has incorporated three AM systems, six AM technologies and thirty nine different AM machines with nine machine attributes (criteria). It intends to provide a most appropriate AM machine among the three AM based systems, different technologies and all the available AM machines in the database. Verily, it is imperative that the selection of a particular machine from an inclusive database always involve significant imprecision. This imprecision may arise from various sources such as unquantifiable data, deficient information, personal biases, inaccessible knowledge, and limited expertise, especially on the part of the users [44, 45]. Generally, most of the stakeholders or the system users are likely to depend on their knowledge and personal skill during machine selection, which lead to highly unorganized and uncertain decisions [46]. As a result of diverse sources of uncertainty, combination of fuzzy and crisp data as well as abundance of attributes and machines, a decision support methodology based on uncertainty theories, FAHP and GRA have been introduced in this work for the selection of a particular RP system. The FAHP (extent analysis based on trapezoidal fuzzy numbers) has been utilized to estimate the weights of various attributes, while the GRA was executed to achieve the best RP system and technology. The FAHP and GRA have been chosen owing to their ease of use, wide applicability as well as their robust nature and well-established outcomes in a number of real life decision problems. The FAHP and GRA have leverage over the other methods because they are adequate and competent in quantifying the uncertainty of information as well as the vagueness of judgements, can handle innumerable criteria and alternatives. Fuzzy set theory and GRA are very powerful mathematical tools for modeling and control of uncertain systems, while AHP represents a robust and a flexible framework for solving complex decision problems. This system can be identified as a user-friendly decision advisor which permit partial information as well as qualitative and quantitative data. As a result of growing complexity in decision-making, most of previous MADM approaches have only considered limited criteria and alternatives. However, in this work, most of the known criteria associated with AM selection have been analyzed. Furthermore, multi-level classification of alternatives have been considered for determining the AM process, technology and machine according to the requirement of the users. This approach is adept to minimize subjective or cognitive errors by streamlining, segregating and correlating multitude of attributes. The application of FAHP would shape up the opinions of the experts as well as enhance the quality of gathered data, without restricting them to distinct terminology or parameter. The FAHP and GRA based integrated approach can result in flexible and sustainable solutions as well as will expand the prospects of their future applications.

2 Methodology

This section discusses the different steps of the adopted methodology in order to reach out the best RP solution.

2.1 Generation of database

The foremost step in this methodology was the classification of different RP machines depending on the principles and raw material used. This work has only considered polymer based AM technologies. The different polymer based AM machines can be categorized into three systems depending on the build material (or the raw material) used. The different systems were liquid based system, solid based system and the powder based system. Each of these AM systems consisted of different technologies or processes as shown in Table 1. The six AM technologies, two each for each AM systems, have been considered in this research study. The different technologies were SLA, PolyJet technology, FDM, Multi-Jet Printing (MJP) or Multi-Jet Machining (MJM), SLS and Inkjet Printing (INKJET), etc. These technologies were chosen as they are widely used in industries and manufactured by five major manufacturing firms: 3D systems, Stratasys, Z-Corporation, Object geometrics Ltd. and Electrical optical systems (EOS). The number of machines (or different models) chosen for each technology was based on their commercial availability in the market as well their popularity among the masses.

Table 1
AM process classification

AM system Technology AM machines

Liquid SLA

PolyJet

Solid FDM

MJM

Powder INKJET

SLS

AM system	Technology	AM machines
Liquid	SLA
	PolyJet
Solid	FDM
	MJM
Powder	INKJET
	SLS

The selection of the most appropriate RP system becomes especially crucial because each machine or model possesses its benefits and limitations. For example, the dimensional accuracy and speed of SLA produced parts is good, but they require support structures and post processing operations [47]. Similarly, the PolyJet can print multiple materials simultaneously, however they also need support structures for overhanging parts and holes [48]. Moreover, the FDM produced parts are durable, functional and good in strength, while MJM offers the highest Z direction resolution and offers best combination of resolution and print speed [49, 50]. The build material used in SLS is in powder form and their fabricated parts are robust, durable and exhibits higher wear resistance as well as they do not require support material [51]. The INKJET originally developed by MIT in 1993 is capable of printing parts with multiple colors and offers faster speed when compared to other technologies of AM systems [52]. Therefore, it is very important to gather as much information as possible about the different machines and identify the best system which can satisfy most of the requirements of the users. The database was generated in four stages. The first stage involved the survey of different AM processes and technologies available in the market. This survey mainly focused on the high-end level, medium level and small machines of the major players in AM industry. The second stage consisted of the collection of data which took months as there were more than 2000 data (or specifications) files of AM machines on public domain. The third stage included the building of database using technical features (machine characteristics) of the AM machines available in the market. The database can be expanded when new AM technologies will be added with the passage of time. The database might not be exhaustive, but includes important players of AM Industry. The fourth stage comprised of the evaluation attributes needed in any MADM. The entire database can be summarized in a hierarchical structure with five levels as shown in Fig. 1. The level 1 represents the objective of selecting the best machine among thirty nine AM machines. The level 2 attributes consist of AM systems including liquid, solid and powder based material. The level 3 presents different AM technologies such as SLA, PolyJet, SLS, etc. The level 4 represents the machine characteristics for the thirty AM machines used in this study. The level 5 consist of thirty nine different AM machines assigned to these technologies.

Fig.1

Hierarchy of AM database.

The characteristic data (attributes) of thirty nine AM machines as shown in Table 2, was obtained from the Wohler’s reports, company catalogues that manufacture these AM machines, vendors and literature survey, thus making it highly reliable. The essential source of information was web-based or internet. A significant number of important attributes including the machine cost, error (or the accuracy), layer thickness, machine speed, material cost, net build size, machine weight, surface roughness and material strength were considered in this work.

Table 2

Machine characteristics (attributes) data

Characteristics	Unit	Description
Machine cost	US$	Cost of the AM machine
Error	mm	Dimensional error (or accuracy of produced parts)
Layer Thickness	mm	Layer height for each successive addition of material
Machine Speed	mm/hr	Time taken to build the part
Material cost	US$	Cost of the material
Net build size Vol	mm	Maximum build size
Machine Weight	kgs	Weight of the machine
Surface roughness	US$	Value of material roughness
Material Strength	–	Strength of the material before deformation

2.2 Acquisition of users requirements and expert opinions

A group of RP users or experts with different requirements were identified from academia and industries. Their requirements were gathered and a decision matrix was prepared using Saaty’s intensity scale as shown in Table 3. The different users or experts and their requirements can be described as follows.

Table 3
Saaty’s intensity importance scale [53]

Intensity of Definition

Importance

1 Equal Importance

3 Moderate Importance

5 Strong Importance

7 Very Strong Importance

9 Extreme Importance

2, 4, 6, 8 Comprises between the above

2 More than Equal Importance but less than Moderate Importance

4 More than Moderate Importance but less than Strong Importance

6 More than Strong Importance but less than Very Strong Importance

8 More than Very Strong Importance but less than Extreme Importance

Reciprocal of the above If an element i has any of the above non-zero numbers assigned to it in comparison to its element j, then j has the reciprocal value in comparison with i

Intensity of	Definition
1	Equal Importance
3	Moderate Importance
5	Strong Importance
7	Very Strong Importance
9	Extreme Importance
2, 4, 6, 8	Comprises between the above
2	More than Equal Importance but less than Moderate Importance
4	More than Moderate Importance but less than Strong Importance
6	More than Strong Importance but less than Very Strong Importance
8	More than Very Strong Importance but less than Extreme Importance
Reciprocal of the above	If an element i has any of the above non-zero numbers assigned to it in comparison to its element j, then j has the reciprocal value in comparison with i

User 1 – Industrial personnel who needed a solid based system, especially MJM. The crucial machine characteristics for this user were accuracy, machine speed, surface finish, material strength in the same order.

User 2 – Academician who required a solid based system, such as FDM for demonstration purposes. The machine cost, machine weight, surface finish and material strength were important attributes to accomplish his objectives.

User 3 – A user from a manufacturing company who was looking for a liquid based system, particularly SLA with higher surface finish, lower machine weight and enhanced material strength.

User 4 – A researcher who was keen on liquid based system (exclusively PolyJet) for experimentation. This user required a machine with lower material and machine cost, lesser machine weight and a higher net build size.

The three decision matrices for each level (machine characteristics, AM systems and AM technologies) were obtained as the outcome of this step. Three matrices included decisions regarding machine characteristics, AM technologies and AM systems. In order to model unbiased and logical behavior, the consistency analysis of the judgements was mandatory. Therefore, before processing further, the data obtained from different users was validated to conform their consistency [53]. To simplify the computation, the consistency test proposed by Saaty [54] which made use of crisp values was implemented. The Saaty’s consistency test was acceptable to ascertain that the judgements were rational and dependable to a reasonable extent, even when the fuzzy set theory was integrated with AHP. Indeed, the same procedure was also adopted by Amarasingha and Piantanakulchai [55], Radionovs and Užga-Rebrovs [56] as well as Meixner [57] in their analysis to verify the consistency of acquired judgements. A random consistency index (RI) as shown in Table 4 was employed to test the consistency of decision making. A consistency ratio, CR which compares between consistency index (CI) and RI has to be defined as follows.

Table 4

Random consistency index

n	RI
1	0
2	0
3	0.58
4	0.9
5	1.12
6	1.24
7	1.32
8	1.41
9	1.45
10	1.49

$CR = \frac{CI}{RI}, where CI = \frac{λ_{max} - n}{n - 1}$

The λ_max indicates the maximum eigenvalue of the decision matrix and n represents the number of attributes. If CR is less than or equal to 10%, the inconsistency is acceptable else the users have to revise their decision matrix.

2.3 Computation of weights using FAHP

In this research, the uncertainty and vagueness of expert’s opinions as well as of data acquisition have been managed through fuzzy sets theory proposed by Zadeh [58]. The FAHP based on the concepts of fuzzy set theory and hierarchical structure analysis can methodically deal with the RP machine selection problem [59, 60]. Indeed, the FAHP is an extension of a typical AHP technique into fuzzy environment through fuzzy numbers [61]. The original transformation of standard AHP into FAHP was proposed by Van Laarhoven and Pedrycz [62], Buckley [63] and Chang [64]. They represented the Saaty’s relative preferences by using fuzzy numbers with triangular membership functions. However, the crisp values in this work have been modeled using trapezoidal membership function owing to its superior performance as compared to triangular membership functions [65]. The different steps used in the estimation of fuzzy weights can be discussed as follows.

Step 1. Transformation of crisp values into fuzzy numbers

The decision matrices obtained for different users were transformed into trapezoidal fuzzy numbers in this step. The membership function of trapezoidal fuzzy number designated by F and defined as (l, m, n, u) can be described as follows [66]. $Membership function, μ_{F} (x) = {\begin{matrix} 0 & x ⩽ l \\ \frac{x - l}{m - l} & l ⩽ x ⩽ m \\ 1 & m ⩽ x ⩽ n \\ \frac{u - x}{u - n} & n ⩽ x ⩽ u \\ 0 & x ⩾ u \end{matrix}$

In order to achieve this transformation, the crisp values based on Saaty scale were first transformed into triangular fuzzy numbers and then to the trapezoidal fuzzy numbers. The triangular fuzzy numbers were converted into trapezoid numbers by keeping the upper and lower bounds of the fuzzy numbers constant and extending the center to a certain range [67]. For example, the crisp value of x was converted to triangular fuzzy number (a, b, c) as a = x - 1; b = x and c = x + 1. Subsequently, the triangular fuzzy numbers were reshaped into trapezoidal fuzzy number (l, m, n, u) as l = a; u = c; m = l + 0.5 and n = u - 0.5.

Step 2. Estimation of weights

In this study, priori weights of the performance measures were computed by using extent analysis of Chang [64]. The extent analysis for computation of priori weights can be described as follows [68, 69]. Let X = {x₁, x₂, x₃, …… , x_n} represent the set of objects and G = {g₁, g₂, g₃, …… , g_n} be an objective set. According to extent analysis approach, every object has to be subjected to extent analysis for each objective of the problem. As a result, m extent analysis values for each object can be obtained as follows. $\begin{matrix} M_{gi}^{1}, M_{gi}^{2}, M_{gi}^{3}, M_{gi}^{4}, \dots \dots, M_{gi}^{m}, \\ i = 1, 2, 3, 4, \dots, n \end{matrix}$

Where, $M_{gi}^{j}$ (j = 1, 2, 3, 4, …, m) represents trapezoidal fuzzy numbers. The Chang’s extent analysis can be implemented in the following manner.

The synthetic fuzzy values with respect to ith object can be computed by employing the Equation (1). $S_{i} = \sum_{j = 1}^{m} {M_{gi}^{j} \otimes [\sum_{i = 1}^{n} \sum_{j = 1}^{m} M_{gi}^{j}]}^{- 1}$ (1)

Since, FAHP utilizes four values (trapezoidal fuzzy number) for the evaluation of the particular attribute. Therefore, to compute $\sum_{j = 1}^{m} M_{gi}^{j}$ , the fuzzy addition operation for a particular matrix can be carried out using the Equation (2).

$\begin{matrix} \sum_{j = 1}^{m} M_{gi}^{j} \\ = (\sum_{j = 1}^{m} l_{j} \sum_{j = 1}^{m} m_{j}, \sum_{j = 1}^{m} n_{j}, \sum_{j = 1}^{m} u_{j}) \end{matrix}$ (2)

The value of ${[\sum_{i = 1}^{n} \sum_{j = 1}^{m} M_{gi}^{j}]}^{- 1}$ can be calculated through fuzzy addition operation of M_gi^j (j = 1, 2, 3, 4, …, m) values and by taking the inverse of its vector. Please refer Equations (3 and 4).

$\begin{matrix} \sum_{i = 1}^{n} \sum_{j = 1}^{m} M_{gi}^{j} \\ = (\sum_{i = 1}^{n} l_{i}, \sum_{i = 1}^{n} n_{i}, \sum_{i = 1}^{n} u_{i}) \end{matrix}$ (3)

$\begin{matrix} {[\sum_{i = 1}^{n} \sum_{j = 1}^{m} M_{gi}^{j}]}^{- 1} \\ = (\frac{1}{\sum_{i = 1}^{n} u_{i}}, \frac{1}{\sum_{i = 1}^{n} n_{i}}, \frac{1}{\sum_{i = 1}^{n} m_{i}}, \frac{1}{\sum_{i = 1}^{n} l_{i}}) \end{matrix}$ (4)

Suppose M₁ = (l₁, m₁, n₁, u₁) and M₂ = (l₂, m₂, n₂, u₂) represents two trapezoidal fuzzy numbers. The degree of possibility of M₂ = (l₂, m₂, n₂, u₂) ⩾ M₁ = (l₁, m₁, n₁, u₁) is dependent on conditions in Equation (5).

$V (M_{2} ⩾ M_{1}) = sup_{y ⩾ x} [min (μ_{M_{1}} (x), μ_{M_{2}} (y))]$ , can be defined as follows $\begin{matrix} V (M_{2} ⩾ M_{1}) = hgt (M_{1} \cap M_{2}) = μ_{M_{2}} (d) = \\ \begin{matrix} 1, & m_{1} ⩾ m_{2} \\ 0, & (m_{2} - n_{1}) > (u_{1} - l_{2}) \\ \frac{((n_{1} - m_{2}) + (u_{1} + l_{2}))}{(u_{1} + l_{2})}, & 0 < (m_{2} - n_{1}) < (u_{1} + l_{2}) \\ \frac{((m_{2} - n_{1}) + (u_{1} + l_{2}))}{(u_{1} + l_{2})}, & (m_{2} - n_{1}) < (u_{1} + l_{2}), where \\ m_{2} < n_{1} and m_{1} < m_{2} \end{matrix} \end{matrix}$ (5)

The degree of possibility of a fuzzy number greater than k fuzzy numbers can be defined as $\begin{matrix} V (M_{2} ⩾ M_{1}, M_{2}, M_{3}, M_{4} \dots \dots, M_{k}) = V [(M > M_{1})] and \\ V [(M ⩾ M_{2})] and V [(M ⩾ M_{3})] and V [(M ⩾ M_{4})] and \dots \\ and V [(M ⩾ M_{k})] = min V (M ⩾ M_{i}), i = 1, 2, 3, \dots, k . \end{matrix}$

Suppose that

$\begin{matrix} d^{'} (A_{i}) = \min V (S_{i} \geq S_{k}) f o r \\ k = 1, 2, 3, 4, \dots, n; k \neq i \end{matrix}$ (6)

The weight vector can be expressed as follows.

, where A_i (i = 1, 2 3 4, …, n) are n elements. To obtain the weight vectors for individual elements, the normalization has to be carried out. The weight vectors are normalized to get the normalized vectors.

w = (d (A₁) , d (A₂) , d (A₃) , d (A₄) , …, d (A_n)) ^T, where w is a vector of non-fuzzy numbers or crisp values.

In this work, three sets of weights (w₁, w₂ and w₃) were computed using the above discussed FAHP as shown in Fig. 2. The first set of weights (w₁) was computed using the decision matrix acquired for different attributes. These weights (represented by local weights 1) were used to determine the best machine within the respective technologies (SLA, PolyJet, FDM, MJM, INKJET and SLS). For example, the best SLA machine, the best MJM machine, etc. Consequently, the second set of weights (w₂) were determined by using the decision matrix of different technologies. In order to estimate the most suitable AM system (liquid, solid or powder based system) depending on the requirements, local weights 2 (=local weights 1 * w₂) were utilized as shown in Fig. 2. Finally, the decision matrix of AM systems was employed to compute the third set of weights (w₃). Then the global weights (=local weights 2* w₃) were determined and these were implemented to achieve the most appropriate RP machine.

Fig.2

Representation of weights at various levels of the hierarchy.

As the weights were computed for different users, therefore, the individual weights were combined using the geometric mean (GM) approach as follows in Equation (7). $Combined weights (GM) = {(\prod_{i = 1}^{d} w_{i})}^{\frac{1}{d}}$ (7)

Where, d is the number of users or experts.

2.4 Ranking of various alternatives

The GRA has been employed in this work to rank the various alternatives in the RP selection problem. The notion of grey theory was first introduced by Prof. Deng from the grey set in combination with theory of space and control theory [70]. The objective behind utilizing grey theory is its ability to take into account the ambiguity and uncertainty associated with user decisions and data collection. The execution of GRA can be carried out in the following manner [71, 72].

Suppose $x_{i}^{*} (k)$ represents the sequence after the data processing, $x_{i}^{\circ} (k)$ is the original sequence of responses, (where i = 1, 2, …, m and k = 1, 2, …, n), max $x_{i}^{\circ} (k)$ signifies the largest value of $x_{i}^{\circ} (k)$ and min $x_{i}^{\circ} (k)$ represents the smallest value of $x_{i}^{\circ} (k)$ . The data was normalized as follows. $x_{i}^{*} (k) = \frac{x_{i}^{(O)} (k) - min x_{i}^{(O)} (k)}{max x_{i}^{(O)} (k) - min x_{i}^{(O)} (k)} Larger - the - better$ $x_{i}^{*} (k) = \frac{max x_{i}^{(O)} (k) - x_{i}^{(O)} (k)}{max x_{i}^{(O)} (k) - min x_{i}^{(O)} (k)} Smaller - the - better$ $\begin{matrix} x_{i}^{*} (k) = \frac{| x_{i}^{(O)} (k) - IV |}{max {max x_{i}^{(O)} (k) - IV, IV - min x_{i}^{(O)} (k)}} \\ Norminal - the - better (define the response as intended \\ value (IV)) \end{matrix}$

Using the comparability sequences, a reference sequence was determined. After data processing was performed, grey relational coefficient (GRC) were estimated using the preprocessed sequences. The GRC was computed using the following equation Equation (8). $GRC (x_{o}^{*} (k), x_{i}^{*} (k)) = \frac{Δ_{min} + ɛ Δ_{max}}{Δ_{oi} (k) + \in Δ_{max}}$ (8)

Where Δ_oi (k) represented the deviation sequence of the reference sequence $x_{O}^{*} (k)$ and the comparability $x_{i}^{*} (k)$ , i.e. $Δ_{oi} (k) = | x_{O}^{*} (k) - x_{i}^{*} (k) |$ was the absolute value of the difference between $x_{O}^{*} (k)$ and $x_{i}^{*} (k)$ . $\begin{matrix} Δ_{min} = min_{\forall i} min_{\forall k} Δ_{oi} (k) and Δ_{max} = max_{\forall i} max_{\forall k} Δ_{Oi} (k), \\ where i = 1, 2, \dots, m and k = 1, 2, \dots, n \end{matrix}$ ∈: distinguishing coefficient, ∈[0, 1]. The value of ∈ was set as 0.5. The grey relational grade (GRG) can be defined as a weighting-sum of the GRC and it was computed using the following formulation Equation (9). $GRG (x_{o}^{*}, x_{i}^{*}) = \sum_{k = 1}^{n} β_{k} GRC (x_{o}^{*} (k), x_{i}^{*} (k))$ (9)

Where β_k represented the weighting value of the kth performance characteristic, and $\sum_{k = 1}^{n} β_{k} = 1$ .

The ranking was carried out for the individual users as well as for an imagined user whose requirements was the combination of the individual four users.

2.5 Sensitivity analysis

The objective of sensitivity analysis was to ascertain the robustness and stability of ranking acquired through the proposed approach. It can be defined as a technique to understand the variation of input values on the final model results [73]. Certainly, it was critical to examine the deviations in the outcome parameters of given model due to deviations in input parameter values [74, 75]. It suggests that the model is reliable and trustworthy if the model solution is not very sensitive to changes in input. The most commonly used sensitive analysis is accomplished by varying the weights of the performance attributes and analyzing the change in results [76]. In earlier studies also, it has been shown that the ranks of the alternatives significantly depend on the values of the weight coefficients of the attributes [77]. Therefore, in this work, sensitivity analysis depending on changes in weight coefficients was employed to validate the model and ensure the reliability of results.

In sensitivity analysis, the first step was the arbitrary selection of a criterion. It was followed by a percentage change in the selected criterion (more or less). Consequently, the computation of weights for remaining criteria was carried out using Equation (10) [80]. $w_{n}^{*} = \frac{w_{n} (1 - w_{i}^{*})}{(1 - w_{i})}$ (10)

Where,

w_i: Original weights for attribute i

$w_{i}^{*}$ : Weight obtained after changing the original

weight by a percentage change for attribute i

w_n: Original weight for attribute n

$w_{n}^{*}$ : Recomputed weight for attribute n

In order to establish the robustness of the developed approach and to examine the similarities of ranks obtained by varying the weights, Kendall’s coefficient (Z) of concordance was utilized [81, 82]. The Kendall’s coefficient depicts the similarities in the ordering of ranked quantities and its values ranges from 0 to 1. For example, the value of 1 represents a perfect match between the various ranking orders. It suggests that closer the value of Z to 1, the better will be the robustness. The value of Z can be computed by using Equations (11–13). $Z = \frac{12 R}{m^{2} (k^{3} - k)}$ (11) $R_{i} = \sum_{j = 1}^{m} r_{ij}$ (12) $R = \sum_{i = 1}^{k} (R_{i} - \bar{R})^{2}$ (13)

Where, r_ij= ranking scenario i provides to alternative j, m = number of scenarios and k = number of alternatives.

Altogether, four schemes depending on weight percentage (20%, 30%, 40% and 50%) were established for each user (decision maker). In each scheme, there were six scenarios based on variously selected attribute for percentage change. The different schemes and scenarios can be realized in Table 5. As an example, the scheme 2 and scenario 3 represents a 30% decrease in weight for surface finish (weights of remaining attributes were computed using Equation (10)), 30% increase in weight for SLS (weights of remaining technologies were calculated using Equation (10)) and 30% increase in weight for POWDER (weights of remaining systems were estimated using Equation (10)).

Table 5

Scenarios used in sensitivity analysis

Scenario	Description
1	Increase in weight for net build volume	Decrease in weight for MJM	Decrease in weight for LIQUID
2	Decrease in weight for accuracy	Decrease in weight for SLA	Decrease in weight for SOLID
3	Decrease in weight for surface finish	Increase in weight for SLS	Increase in weight for POWDER
4	Increase in weight for machine speed	Decrease in weight for POLYJET	Increase in weight for SOLID
5	Increase in weight for material strength	Increase in weight for 3D PRINT	Increase in weight for LIQUID
6	Decrease in weight for machine weight	Increase in weight for FDM	Decrease in weight for POWDER

3 RP system selection

The database established for the RP selection problem consisted of the type of AM system (liquid, solid, powder), principle or the technology (SLA, PolyJet, FDM, MJM, 3DP, SLS), manufacturer, machine name and the machine code. As shown in Table 6, the database also included information about specifications of various machines. In the last column of Table 6, the value of three represented weak material strength, six expressed the strong strength and nine was used to show the very strong material strength.

Table 6
Database developed for the selection of best RP machine

SYSTEM Technology Manufacturer Machine Name Machine Code Machine Cost ($) Error (mm) Layer Thickness (mm) Machine Speed (mm/hr) Material Cost /lb Net Build Size Vol (mm³) Machine weight (Kg) Surface Finis h (&mum) Material Strength

LIQUID SLA 3D SYSTEMS PROJECT 6000 LSLA3DS1M1 220,700 0.025 0.125 6 100 15625000 272 4 3

LIQUID SLA 3D SYSTEMS I PRO 8000 LSLA3DS2M2 405000 0.0508 0.05 7 100 68250000 1590 4 3

LIQUID SLA 3D SYSTEMS I PRO 9000 LSLA3DS3M3 556000 0.0508 0.05 7 100 268125000 2404 4 3

LIQUID SLA 3D SYSTEMS I PRO 9000 XL LSLA3DS4M4 945000 0.0508 0.05 7 100 618750000 2404 4 3

LIQUID SLA Z CORPORATION ZBUILDER ULTRA LSLAZCO1M5 34900 0.06 0.05 12.7 100 7904000 163 4 3

LIQUID POLY OBJECT GEOMETRICS 24 PERSONNEL 3D PRINTER LPOLOBJ1M6 19900 0.1 0.027 12.7 210 6697164.24 93 3 3

LIQUID POLY OBJECT GEOMETRICS 30 PERSONAL 3D PRINTER LPOLOBJ2M7 30000 0.1 0.027 12.7 210 8414385.84 93 3 3

LIQUID POLY OBJECT GEOMETRICS CONNEX 500 LPOLOBJ3M8 250000 0.02 0.015 12.7 210 38220000 580 3 3

LIQUID POLY OBJECT GEOMETRICS CONNEX 350 LPOLOBJ4M9 200000 0.02 0.015 12.7 210 23392800 580 3 3

LIQUID POLY OBJECT GEOMETRICS EDEN 500V LPOLOBJ5M10 170000 0.1 0.015 12.7 210 38220000 500 3 3

LIQUID POLY OBJECT GEOMETRICS EDEN 350V LPOLOBJ6M11 158000 0.1 0.015 12.7 210 23120000 500 3 3

LIQUID POLY OBJECT GEOMETRICS EDEN 260V LPOLOBJ7M12 60000 0.2 0.015 12.7 210 12852000 500 3 3

LIQUID POLY OBJECT GEOMETRICS EDEN 250V LPOLOBJ8M13 42000 0.03 0.015 12.7 210 12500000 330 3 3

SOLID FDM STRATASYS UPRINT PLUS SFDMSTR1M14 19900 0.2 0.254 6 180 6263768 98 18 9

SOLID FDM STRATASYS UPRINT SFDMSTR2M15 14900 0.2 0.254 6 180 4690112 80 18 9

SOLID FDM STRATASYS DIMESNION ELITE SFDMSTR3M16 29900 0.2 0.178 6 180 12568745 136 18 9

SOLID FDM STRATASYS DIMESNION 768 SERIES SFDMSTR4M17 25000 0.13 0.245 6 180 12568745 136 18 9

SOLID FDM STRATASYS DIMESNSION 1200 ES SERIES SFDMSTR5M18 33000 0.13 0.254 6 180 19677380 148 18 9

SOLID FDM STRATASYS FORTUS 360 MC SFDMSTR6M19 75000 0.127 0.127 6 180 22903180 687 18 9

SOLID FDM STRATASYS FORTUS 400MC SFDMSTR7M20 119000 0.127 0.127 6 180 22903180 687 18 9

SOLID FDM STRATASYS FORTUS 900MC SFDMSTR8M21 400000 0.089 0.178 6 180 509591560 3287 18 9

SOLID MJM 3D SYSTEMS PROJECT 1000 SMJM3DS1M22 10900 0.0254 0.101 12.7 200 6178914 68 7 6

SOLID MJM 3D SYSTEMS PROJECT 1500 SMJM3DS2M23 15000 0.0254 0.101 12.7 200 7914564 68 7 6

SOLID MJM 3D SYSTEMS PROJECT 3500 SMJM3DS3M24 67500 0.025 0.032 5 140 11191390 401 7 6

SOLID MJM 3D SYSTEMS PROJECT HD 3000 SMJM3DS4M25 70000 0.025 0.032 5 140 11191390 385 7 6

SOLID MJM 3D SYSTEMS PROJECT 5000 SMJM3DS5M26 140000 0.025 0.015 3.8 144 64845000 652 7 6

POWDER 3DP Z CORPORATION Z PRINTER 150 P3DPZCO1M27 14900 0.4 0.101 20.3 25 5544820 165 16 3

POWDER 3DP Z CORPORATION Z PRINTER 250 P3DPZCO2M28 24900 0.4 0.101 20.3 25 5544820 165 16 3

POWDER 3DP Z CORPORATION Z PRINTER 350 P3DPZCO3M29 29900 0.15 0.102 20.3 25 10467086 179 16 3

POWDER 3DP Z CORPORATION Z PRINTER 450 P3DPZCO4M30 39900 0.15 0.102 22.8 25 10467086 193 16 3

POWDER 3DP Z CORPORATION Z PRINTER 650 P3DPZCO5M31 59900 0.1 0.102 27.9 25 19645122 340 16 3

POWDER 3DP Z CORPORATION Z PRINTER 850 P3DPZCO6M32 109900 0.1 0.089 33 25 44322492 363 16 3

POWDER SLS ELECTRO OPTICAL SYSTEMS FORMIGA P100 PSLSELE1M33 60000 0.05 0.1 10 35 16500000 600 12 6

POWDER SLS ELECTRO OPTICAL SYSTEMS EOSINT P 395 PSLSELE2M34 200000 0.02 0.06 30.9 35 71672000 1060 12 6

POWDER SLS ELECTRO OPTICAL SYSTEMS EOSINT P 760 PSLSELE3M35 800000 0.02 0.06 31.7 35 154280000 2300 12 6

POWDER SLS ELECTRO OPTICAL SYSTEMS EOSINT P 800 PSLSELE4M36 1200000 0.02 0.12 33 35 154280000 2300 12 6

POWDER SLS 3D SYSTEMS S PRO 60 SD PSLS3DS1M37 178000 0.03 0.08 71.62 70 57458610 1950 12 6

POWDER SLS 3D SYSTEMS S PRO 140 PSLS3DS2M38 392000 0.03 0.08 71.62 70 139150000 1800 12 6

POWDER SLS 3D SYSTEMS S PRO 230 PSLS3DS3M39 485000 0.03 0.08 71.62 70 226875000 1900 12 6

SYSTEM	Technology	Manufacturer	Machine Name	Machine Code	Machine Cost ($)	Error (mm)	Layer Thickness (mm)	Machine Speed (mm/hr)	Material Cost /lb	Net Build Size Vol (mm³)	Machine weight (Kg)	Surface Finis h (&mum)	Material Strength
LIQUID	SLA	3D SYSTEMS	PROJECT 6000	LSLA3DS1M1	220,700	0.025	0.125	6	100	15625000	272	4	3
LIQUID	SLA	3D SYSTEMS	I PRO 8000	LSLA3DS2M2	405000	0.0508	0.05	7	100	68250000	1590	4	3
LIQUID	SLA	3D SYSTEMS	I PRO 9000	LSLA3DS3M3	556000	0.0508	0.05	7	100	268125000	2404	4	3
LIQUID	SLA	3D SYSTEMS	I PRO 9000 XL	LSLA3DS4M4	945000	0.0508	0.05	7	100	618750000	2404	4	3
LIQUID	SLA	Z CORPORATION	ZBUILDER ULTRA	LSLAZCO1M5	34900	0.06	0.05	12.7	100	7904000	163	4	3
LIQUID	POLY	OBJECT GEOMETRICS	24 PERSONNEL 3D PRINTER	LPOLOBJ1M6	19900	0.1	0.027	12.7	210	6697164.24	93	3	3
LIQUID	POLY	OBJECT GEOMETRICS	30 PERSONAL 3D PRINTER	LPOLOBJ2M7	30000	0.1	0.027	12.7	210	8414385.84	93	3	3
LIQUID	POLY	OBJECT GEOMETRICS	CONNEX 500	LPOLOBJ3M8	250000	0.02	0.015	12.7	210	38220000	580	3	3
LIQUID	POLY	OBJECT GEOMETRICS	CONNEX 350	LPOLOBJ4M9	200000	0.02	0.015	12.7	210	23392800	580	3	3
LIQUID	POLY	OBJECT GEOMETRICS	EDEN 500V	LPOLOBJ5M10	170000	0.1	0.015	12.7	210	38220000	500	3	3
LIQUID	POLY	OBJECT GEOMETRICS	EDEN 350V	LPOLOBJ6M11	158000	0.1	0.015	12.7	210	23120000	500	3	3
LIQUID	POLY	OBJECT GEOMETRICS	EDEN 260V	LPOLOBJ7M12	60000	0.2	0.015	12.7	210	12852000	500	3	3
LIQUID	POLY	OBJECT GEOMETRICS	EDEN 250V	LPOLOBJ8M13	42000	0.03	0.015	12.7	210	12500000	330	3	3
SOLID	FDM	STRATASYS	UPRINT PLUS	SFDMSTR1M14	19900	0.2	0.254	6	180	6263768	98	18	9
SOLID	FDM	STRATASYS	UPRINT	SFDMSTR2M15	14900	0.2	0.254	6	180	4690112	80	18	9
SOLID	FDM	STRATASYS	DIMESNION ELITE	SFDMSTR3M16	29900	0.2	0.178	6	180	12568745	136	18	9
SOLID	FDM	STRATASYS	DIMESNION 768 SERIES	SFDMSTR4M17	25000	0.13	0.245	6	180	12568745	136	18	9
SOLID	FDM	STRATASYS	DIMESNSION 1200 ES SERIES	SFDMSTR5M18	33000	0.13	0.254	6	180	19677380	148	18	9
SOLID	FDM	STRATASYS	FORTUS 360 MC	SFDMSTR6M19	75000	0.127	0.127	6	180	22903180	687	18	9
SOLID	FDM	STRATASYS	FORTUS 400MC	SFDMSTR7M20	119000	0.127	0.127	6	180	22903180	687	18	9
SOLID	FDM	STRATASYS	FORTUS 900MC	SFDMSTR8M21	400000	0.089	0.178	6	180	509591560	3287	18	9
SOLID	MJM	3D SYSTEMS	PROJECT 1000	SMJM3DS1M22	10900	0.0254	0.101	12.7	200	6178914	68	7	6
SOLID	MJM	3D SYSTEMS	PROJECT 1500	SMJM3DS2M23	15000	0.0254	0.101	12.7	200	7914564	68	7	6
SOLID	MJM	3D SYSTEMS	PROJECT 3500	SMJM3DS3M24	67500	0.025	0.032	5	140	11191390	401	7	6
SOLID	MJM	3D SYSTEMS	PROJECT HD 3000	SMJM3DS4M25	70000	0.025	0.032	5	140	11191390	385	7	6
SOLID	MJM	3D SYSTEMS	PROJECT 5000	SMJM3DS5M26	140000	0.025	0.015	3.8	144	64845000	652	7	6
POWDER	3DP	Z CORPORATION	Z PRINTER 150	P3DPZCO1M27	14900	0.4	0.101	20.3	25	5544820	165	16	3
POWDER	3DP	Z CORPORATION	Z PRINTER 250	P3DPZCO2M28	24900	0.4	0.101	20.3	25	5544820	165	16	3
POWDER	3DP	Z CORPORATION	Z PRINTER 350	P3DPZCO3M29	29900	0.15	0.102	20.3	25	10467086	179	16	3
POWDER	3DP	Z CORPORATION	Z PRINTER 450	P3DPZCO4M30	39900	0.15	0.102	22.8	25	10467086	193	16	3
POWDER	3DP	Z CORPORATION	Z PRINTER 650	P3DPZCO5M31	59900	0.1	0.102	27.9	25	19645122	340	16	3
POWDER	3DP	Z CORPORATION	Z PRINTER 850	P3DPZCO6M32	109900	0.1	0.089	33	25	44322492	363	16	3
POWDER	SLS	ELECTRO OPTICAL SYSTEMS	FORMIGA P100	PSLSELE1M33	60000	0.05	0.1	10	35	16500000	600	12	6
POWDER	SLS	ELECTRO OPTICAL SYSTEMS	EOSINT P 395	PSLSELE2M34	200000	0.02	0.06	30.9	35	71672000	1060	12	6
POWDER	SLS	ELECTRO OPTICAL SYSTEMS	EOSINT P 760	PSLSELE3M35	800000	0.02	0.06	31.7	35	154280000	2300	12	6
POWDER	SLS	ELECTRO OPTICAL SYSTEMS	EOSINT P 800	PSLSELE4M36	1200000	0.02	0.12	33	35	154280000	2300	12	6
POWDER	SLS	3D SYSTEMS	S PRO 60 SD	PSLS3DS1M37	178000	0.03	0.08	71.62	70	57458610	1950	12	6
POWDER	SLS	3D SYSTEMS	S PRO 140	PSLS3DS2M38	392000	0.03	0.08	71.62	70	139150000	1800	12	6
POWDER	SLS	3D SYSTEMS	S PRO 230	PSLS3DS3M39	485000	0.03	0.08	71.62	70	226875000	1900	12	6

In the consecutive step, the decision matrices were prepared for various users. The users were requested to measure their requirements on a scale of 1 to 9 (Saaty’s scale) needed in their particular applications. As result, three decision matrices were obtained for each user. The Table 7 represents the decision matrices for the user 1.

Table 7

Decision matrices for the user 1: (a) RP system; (b) RP technology; (c) Machine attributes

(a)
RP System	Liquid	Solid	Powder
Liquid	1	1/9	1/3
Solid	9	1	7
Powder	3	1/7	1
(b)
RP Technology	SLA	PolyJet	FDM	MJM	3D PRINT	SLS
PolyJet	1/2	1	1/5	1/9	1/2	1/5
FDM	6	5	1	1/5	4	5
MJM	8	9	5	1	6	7
3D PRINT	2	2	1/4	1/6	1	1/3
SLS	3	5	1/5	1/7	3	1
(c)
Machine	Machine Cost	Error (Accuracy)	Minimum Layer Thickness	Machine Speed	Material Cost	Net Build Size Vol	Machine Weight	Surface Finish	Material Strength
Error (Accuracy)	7	1	6	2	8	5	9	3	4
Minimum Layer Thickness	2	1/6	1	1/5	3	1/2	4	1/4	1/3
Machine Speed	6	1/2	5	1	7	4	8	2	3
Material Cost	1/2	1/8	1/3	1/7	1	1/4	2	1/6	1/5
Net Build Size Vol	3	1/5	2	1/4	4	1	5	1/3	1/2
Machine Weight	1/3	1/9	1/4	1/8	1/2	1/5	1	1/7	1/6
Surface Finish	5	1/3	4	1/2	6	3	7	1	2
Material Strength	4	1/4	3	1/3	5	2	6	1/2	1

The consistency was assessed for each user and the results are presented in Table 8. The outcome form consistency analysis implied that the value of the CR in each decision matrix did not exceed the maximum permissible value of 10%, therefore, these opinions from different users was acceptable in consequent computations.

Table 8

Results obtained using consistency analysis

User	Decision Matrix	CR (%)
User 1	RP system	6.92
	RP technology	8.38
	Machine attributes	3.45
User 2	RP system	2.51
	RP technology	8.40
	Machine attributes	3.45
User 3	RP system	6.92
	RP technology	9.05
	Machine attributes	3.48
User 4	RP system	6.34
	RP technology	9.45
	Machine attributes	3.45

Subsequently, the crisp values gathered within each decision matrix for every user were transformed into trapezoidal fuzzy numbers. The trapezoidal fuzzy numbers for different intensities of importance are shown in Table 9.

Table 9

Conversion of Saaty’s importance scale to trapezoidal fuzzy numbers

Saaty’s Scale		Conversion to Fuzzy number
Intensity of Importance	Definition	Triangular Fuzzy number	Trapezoidal Fuzzy
1	Equal Importance	(1, 1, 1)	(1,1,1,1)
3	Moderate Importance	(2, 3, 4)	(2, 2.5, 3.5, 4)
5	Strong Importance	(4, 5, 6)	(4, 4.5, 5.5, 6)
7	Very Strong Importance	(6, 7, 8)	(6, 6.5, 7.5, 8)
9	Extreme Importance	(9, 9, 9)	(9, 9, 9, 9)
2	Intermediate Values	(1, 2, 3)	(1, 1.5, 2.5, 3)
4		(3, 4, 5)	(3, 3.5, 4.5, 5)
6		(5, 6, 7)	(5, 5.5, 6.5, 7)
8		(7, 8, 9)	(7, 7.5, 8.5, 9)

After transformation into trapezoidal fuzzy numbers, Chang’s extent analysis was employed to compute the weights for individual users. The different calculation steps adopted in Chang’s method are discussed as follows. The decision matrix for machine attributes by user 1 is utilized as an example in this calculation.

Step 1. The foremost step was the generation of a fuzzy pairwise matrix as shown in Table 10.

Table 10

Fuzzy pairwise comparison matrix

Machine Attributes	Machine Cost	Accuracy	Min Layer Thickness	Machine Speed	Material Cost	Net Build Size Vol	Machine weight	Surface Finish	Material Strength
Machine Cost	(1,1,1,1)	(0.125, 0.133, 0.154, 0.167)	(0.333, 0.4, 0.667, 1)	(0.143, 0.154, 0.182, 0.2)	(1, 1.5, 2.5, 3)	(0.25, 0.286, 0.4, 0.5)	(2, 2.5, 3.5, 4)	(0.167, 0.182, 0.222, 0.25)	(0.2, 0.222, 0.286, 0.333)
Accuracy	(6, 6.5, 7.5, 8)	(1,1,1,1)	(5, 5.5, 6.5, 7)	(1, 1.5, 2.5, 3)	(7, 7.5, 8.5, 9)	(4, 4.5, 5.5, 6)	(9, 9, 9, 9)	(2, 2.5, 3.5, 4)	(3, 3.5, 4.5, 5)
Min Layer Thickness	(1, 1.5, 2.5, 3)	(0.143, 0.154, 0.182, 0.2)	(1,1,1,1)	(0.167, 0.182, 0.222, 0.25)	(2, 2.5, 3.5, 4)	(0.333, 0.4, 0.667, 1)	(3, 3.5, 4.5, 5)	(0.2, 0.222, 0.286, 0.333)	(0.25, 0.286, 0.4, 0.5)
Machine Speed	(5, 5.5, 6.5, 7)	(0.333, 0.4, 0.667, 1)	(4, 4.5, 5.5, 6)	(1,1,1,1)	(6, 6.5, 7.5, 8)	(3, 3.5, 4.5, 5)	(7, 7.5, 8.5, 9)	(1, 1.5, 2.5, 3)	(2, 2.5, 3.5, 4)
Material Cost	(0.333, 0.4, 0.667, 1)	(0.111, 0.118, 0.133, 0.143)	(0.25, 0.286, 0.4, 0.5)	(0.125, 0.133, 0.154, 0.167)	(1,1,1,1)	(0.2, 0.222, 0.286, 0.333)	(1, 1.5, 2.5, 3)	(0.143, 0.154, 0.182, 0.2)	(0.167, 0.182, 0.222, 0.25)
Net Build Vol	(2, 2.5, 3.5, 4)	(0.167, 0.182, 0.222, 0.25)	(1, 1.5, 2.5, 3)	(0.2, 0.222, 0.286, 0.333)	(3, 3.5, 4.5, 5)	(1,1,1,1)	(4, 4.5, 5.5, 6)	(0.25, 0.286, 0.4, 0.5)	(0.333, 0.4, 0.667, 1)
Machine Weight	(0.25, 0.286, 0.4, 0.5)	(0.111, 0.111, 0.111, 0.111)	(0.2, 0.222, 0.286, 0.333)	(0.111, 0.118, 0.133, 0.143)	(0.333, 0.4, 0.667, 1)	(0.167, 0.182, 0.222, 0.25)	(1,1,1,1)	(0.125, 0.133, 0.154, 0.167)	(0.143, 0.154, 0.182, 0.2)
Surface Finish	(4, 4.5, 5.5, 6)	(0.25, 0.286, 0.4, 0.5)	(3, 3.5, 4.5, 5)	(0.333, 0.4, 0.667, 1)	(5, 5.5, 6.5, 7)	(2, 2.5, 3.5, 4)	(6, 6.5, 7.5, 8)	(1,1,1,1)	(1, 1.5, 2.5, 3)
Material Strength	(3, 3.5, 4.5, 5)	(0.2, 0.222, 0.286, 0.333)	(2, 2.5, 3.5, 4)	(0.25, 0.286, 0.4, 0.5)	(4, 4.5, 5.5, 6)	(1, 1.5, 2.5, 3)	(5, 5.5, 6.5, 7)	(0.333, 0.4, 0.667, 1)	(1,1,1,1)

Step 2. Subsequently, the synthetic fuzzy values were computed using Equations (1–4).

S₁= (5.218, 6.3773, 8.911, 10.45) ⊗ (1/216.446, 1/195.029, 1/156.3072, 1/137.7299)

S₂= (38, 41.5, 48.5, 52) ⊗ (1/216.446, 1/195.029, 1/156.3072, 1/137.7299)

S₃= (8.0933, 9.744, 13.257, 15.283) ⊗ (1/216.446, 1/195.029, 1/156.3072, 1/137.7299)

S₄= (29.333, 32.9, 40.167, 44) ⊗ (1/216.446, 1/195.029, 1/156.3072, 1/137.7299)

S₅= (3.329, 3.9953, 5.544, 6.593) ⊗ (1/216.446, 1/195.029, 1/156.3072, 1/137.7299)

S₆= (11.9503, 14.09, 18.575, 21.083) ⊗ (1/216.446, 1/195.029, 1/156.3072, 1/137.7299)

S₇= (2.44, 2.6064, 3.155, 3.704) ⊗ (1/216.446, 1/195.029, 1/156.3072, 1/137.7299)

S₈= (22.583, 25.686, 32.067, 35.5) ⊗ (1/216.446, 1/195.029, 1/156.3072, 1/137.7299)

S₉= (16.7833, 19.4082, 24.853, 27.833) ⊗ (1/216.446, 1/195.029, 1/156.3072, 1/137.7299)

Step 3. These fuzzy values were then compared using Equation (5).

V (S₁ ⩾ S₂) = 0.3804, V (S₁ ⩾ S₃) = 0.9378, V (S₁ ⩾ S₄) = 0.4717, V (S₁ ⩾ S₅) = 1, V (S₁ ⩾ S₆) = 0.8838, V (S₁ ⩾ S_c7) = 1, V (S₁ ⩾ S₈) = 0.5855, V (S₁ ⩾ S₉) = 0.7229

V (S₂ ⩾ S₁) = 1, V (S₂ ⩾ S₃) = 1, V (S₂ ⩾ S₄) = 1, V (S₂ ⩾ S₅) = 1, V (S₂ ⩾ S₆) = 1, V (S₁ ⩾ S₇) = 1, V (S₂ ⩾ S₈) = 1, V (S₂ ⩾ S₉) = 1

V (S₃ ⩾ S₁) = 1, V (S₃ ⩾ S₂) = 0.5534, V (S₃ ⩾ S₄) = 0.6597, V (S₃ ⩾ S₅) = 1, V (S₃ ⩾ S₆) = 0.9244, V (S₃ ⩾ S₇) = 1, V (S₃ ⩾ S₈) = 0.7822, V (S₃ ⩾ S₉) = 0.922

V (S₄ ⩾ S₁) = 1, V (S₄ ⩾ S₂)=0.911, V (S₄ ⩾ S₃) = 1, V (S₄ ⩾ S₅) = 1, V (S₄ ⩾ S₆) = 1, V (S₄ ⩾ S₇) = 1, V (S₄ ⩾ S₈) = 1, V (S₄ ⩾ S₉) = 1

V (S₅ ⩾ S₁) = 1, V (S₅ ⩾ S₂) = 0.2064, V (S₅ ⩾ S₃) = 0.83, V (S₅ ⩾ S₄) = 0.2735, V (S₅ ⩾ S₆) = 0.6432, V (S₅ ⩾ S₇) = 1, V (S₅ ⩾ S₈) = 0.3677, V (S₅ ⩾ S₉) = 0.4893

V (S₆ ⩾ S₁) = 1, V (S₆ ⩾ S₂) = 0.7141, V (S₆ ⩾ S₃) = 1, V (S₆ ⩾ S₄) = 0.8272, V (S₆ ⩾ S₅) = 1, V (S₆ ⩾ S₇) = 1, V (S₆ ⩾ S₈) = 0.95, V (S₆ ⩾ S₉) = 0.9162

V (S₇ ⩾ S₁) = 0.755, V (S₇ ⩾ S₂) = 0.0487, V (S₇_geqslantS3) = 0.5368, V (S₇ ⩾ S₄) = 0.0856, V (S₇ ⩾ S₅) = 0.9929, V (S₇ ⩾ S₆) = 0.3659, V (S₇ ⩾ S₈) = 0.1502, V (S₇ ⩾ S₉) = 0.2404

V (S₈ ⩾ S₁) = 1, V (S₈ ⩾ S₂) = 0.9824, V (S₈ ⩾ S₃) = 1, V (S₈ ⩾ S₄) = 0.9037, V (S₈ ⩾ S₅) = 1, V (S₈ ⩾ S₆) = 1, V (S₈ ⩾ S₇) = 1, V (S₈ ⩾ S₉) = 1

V (S₉ ⩾ S₁) = 1, V (S₉ ⩾ S₂) = 0.8576, V (S₉ ⩾ S₃) = 1, V (S₉ ⩾ S₄) = 0.9713, V (S₉ ⩾ S₅) = 1, V (S₉ ⩾ S₆) = 1, V (S₉ ⩾ S₇) = 1, V (S₉ ⩾ S₈) = 0.9109

Step 3. The weights for various attributes can be estimated by using Equation (6).

d(Machine Cost) = min (0.3804, 0.9378, 0.4717,1, 0.8838, 1, 0.5855, 0.7229) = 0.3804

d(Accuracy) = min (1, 1, 1, 1, 1, 1, 1, 1) = 1

d(Min Layer Thickness) = min (1, 0.5534, 0.6597, 1, 0.9244, 1, 0.7822, 0.922) = 0.5534

d(Machine Speed) = min (1, 0.911, 1, 1, 1, 1, 1, 1) = 0.911

d(Material Cost) = min (1, 0.2064, 0.83, 0.2735, 0.6432, 1, 0.3677, 0.4893) = 0.2064

d(Net Build Vol.) = min (1, 0.7141, 1, 0.8272, 1, 1, 0.95, 0.9162) = 0.7141

d(Machine Weight) = min (0.755, 0.0487, 0.5368, 0.0856, 0.9929, 0.3659, 0.1502, 0.2404) = 0.0487

d(Surface finish) = min (1, 0.9824, 1, 0.9037, 1, 1, 1, 1) = 0.9037

d(Material Strength) = min (1, 0.8576, 1, 0.9713, 1, 1, 1, 0.9109) = 0.8576

The weight vector can now be expressed as w = (0.3806, 1, 0.5534, 0.911, 0.2064, 0.7141, 0.0487, 0.9037, 0.8576)

After normalization, the weight vectors can be represented as w = (0.0682, 0.1794, 0.0993, 0.1634, 0.0370, 0.1281, 0.0087, 0.1621, 0.1538)

Similarly, the remaining weights for other decision matrices were computed. The individual weights were also combined and normalized as shown in Table 10 using GM method. The local weights 1, local weights 2 and global weights were also estimated as discussed in the previous section.

Once the weights were computed, the GRA was implemented to determine the best alternative for different users as well for the imagined user whose requirement was the combination of all the individual users. The implementation of GRA commenced with data normalization to obtain comparability sequences, subsequently deviation sequences and GRC. As an example, the calculation steps of GRC for Machine Cost (LSLA3DS1M1) are shown below. Since, the objective is the minimization of machine cost, therefore, the smaller the better condition will be utilized in normalization.

Max = 1200000; Min = 10900; Given value = 220700

$x_{i}^{*} (k)$ = (1200000-220700)/ (1200000-10900)

$x_{i}^{*} (k)$ = 0.8236

Similarly, the remaining data were normalized depending on the conditions of maximization or minimization. Consequently, the deviation sequences were computed as follows.

Δ_oi (k) = |1 - 0.8236| $(Δ_{oi} (k) = | x_{O}^{*} (k) - x_{i}^{*} (k) |)$

Δ_oi (k) =0.1764

The GRC was computed using Equation (8) and by incorporating the following values.

∈=0.5; Δ_min = 0; Δ_max = 1; Δ_oi (k) =0.1764

GRC=(0 + 0.5*(1))/ (0.1764+(0.5*1)) GRC = 0.7392

For user 1, machine cost (machine attribute), SLA (technology) and Liquid (System), the following weights were obtained (please refer Fig. 2).

Local weight 1 = 0.0682

Local weight 2 = 0.0682*0.0768 = 0.005238

Local weight 3 = 0.005283*0.0838 = 0.0004389

Similarly, the GRC_s for other combinations of machines, technologies and systems were carried out. The Tables 11 –13 represent the comparability sequences, deviation sequences and GRC respectively.

Table 11

Weights computed using FAHP

		Weights
w ₁	RP System	User 1	User 2	User 3	User 4	Combined	Normalized
	LIQUID	0.0838	0.3120	0.6255	0.6585	0.3222	0.3809
	SOLID	0.6257	0.5950	0.2904	0.2325	0.3982	0.4708
	POWDER	0.2905	0.0930	0.0841	0.1090	0.1254	0.1483
w ₂	RP Technology	User 1	User 2	User 3	User 4	Combined	Normalized
	SLA	0.0768	0.1832	0.2767	0.2544	0.1774	0.2242
	POLYJET	0.0403	0.1198	0.2659	0.2692	0.1364	0.1723
	FDM	0.2537	0.2983	0.1439	0.0904	0.1771	0.2239
	MJM	0.2734	0.2922	0.2268	0.0136	0.1254	0.1585
	3D PRINT	0.1341	0.0264	0.0623	0.1593	0.0770	0.0973
	SLS	0.2217	0.0800	0.0244	0.2131	0.0980	0.1238
w ₃	Attributes	User 1	User 2	User 3	User 4	Combined	Normalized
	Machine Cost	0.0682	0.1793	0.0366	0.1632	0.0925	0.1172
	Accuracy	0.1794	0.0682	0.1588	0.0682	0.1073	0.1359
	Min Layer Thickness	0.0993	0.0370	0.1644	0.0370	0.0688	0.0871
	Machine Speed	0.1634	0.0087	0.1286	0.1280	0.0696	0.0882
	Material Cost	0.0370	0.1280	0.0088	0.1793	0.0523	0.0662
	Net Build Vol	0.1281	0.0992	0.1544	0.1537	0.1318	0.1670
	Machine Weight	0.0087	0.1632	0.0684	0.1626	0.0631	0.0800
	Surface Finish	0.1621	0.1626	0.1804	0.0087	0.0803	0.1017
	Material Strength	0.1538	0.1537	0.0996	0.0992	0.1236	0.1567

Table 12

Comparability after data normalization using GRA

Machine Code	Machine Cost	Error	Min Layer Thickness	Machine Speed	Material Cost	Net Build Size Vol	Machine weight	Surface Finish	Material Strength
LSLA3DS1M1	0.8236	0.9868	0.5397	0.0324	0.5946	0.0178	0.9366	0.9333	0.0000
LSLA3DS2M2	0.6686	0.9189	0.8536	0.0472	0.5946	0.1035	0.5272	0.9333	0.0000
LSLA3DS3M3	0.5416	0.9189	0.8536	0.0472	0.5946	0.4290	0.2743	0.9333	0.0000
LSLA3DS4M4	0.2144	0.9189	0.8536	0.0472	0.5946	1.0000	0.2743	0.9333	0.0000
LSLAZCO1M5	0.9798	0.8947	0.8536	0.1312	0.5946	0.0052	0.9705	0.9333	0.0000
LPOLOBJ1M6	0.9924	0.7895	0.9498	0.1312	0.0000	0.0033	0.9922	1.0000	0.0000
LPOLOBJ2M7	0.9839	0.7895	0.9498	0.1312	0.0000	0.0061	0.9922	1.0000	0.0000
LPOLOBJ3M8	0.7989	1.0000	1.0000	0.1312	0.0000	0.0546	0.8409	1.0000	0.0000
LPOLOBJ4M9	0.8410	1.0000	1.0000	0.1312	0.0000	0.0305	0.8409	1.0000	0.0000
LPOLOBJ5M10	0.8662	0.7895	1.0000	0.1312	0.0000	0.0546	0.8658	1.0000	0.0000
LPOLOBJ6M11	0.8763	0.7895	1.0000	0.1312	0.0000	0.0300	0.8658	1.0000	0.0000
LPOLOBJ7M12	0.9587	0.5263	1.0000	0.1312	0.0000	0.0133	0.8658	1.0000	0.0000
LPOLOBJ8M13	0.9738	0.9737	1.0000	0.1312	0.0000	0.0127	0.9186	1.0000	0.0000
SFDMSTR1M14	0.9924	0.5263	0.0000	0.0324	0.1622	0.0026	0.9907	0.0000	1.0000
SFDMSTR2M15	0.9966	0.5263	0.0000	0.0324	0.1622	0.0000	0.9963	0.0000	1.0000
SFDMSTR3M16	0.9840	0.5263	0.3180	0.0324	0.1622	0.0128	0.9789	0.0000	1.0000
SFDMSTR4M17	0.9881	0.7105	0.0377	0.0324	0.1622	0.0128	0.9789	0.0000	1.0000
SFDMSTR5M18	0.9814	0.7105	0.0000	0.0324	0.1622	0.0244	0.9751	0.0000	1.0000
SFDMSTR6M19	0.9461	0.7184	0.5314	0.0324	0.1622	0.0297	0.8077	0.0000	1.0000
SFDMSTR7M20	0.9091	0.7184	0.5314	0.0324	0.1622	0.0297	0.8077	0.0000	1.0000
SFDMSTR8M21	0.6728	0.8184	0.3180	0.0324	0.1622	0.8222	0.0000	0.0000	1.0000
SMJM3DS1M22	1.0000	0.9858	0.6402	0.1312	0.0541	0.0024	1.0000	0.7333	0.5000
SMJM3DS2M23	0.9966	0.9858	0.6402	0.1312	0.0541	0.0053	1.0000	0.7333	0.5000
SMJM3DS3M24	0.9524	0.9868	0.9289	0.0177	0.3784	0.0106	0.8966	0.7333	0.5000
SMJM3DS4M25	0.9503	0.9868	0.9289	0.0177	0.3784	0.0106	0.9015	0.7333	0.5000
SMJM3DS5M26	0.8914	0.9868	1.0000	0.0000	0.3568	0.0980	0.8186	0.7333	0.5000
P3DPZCO1M27	0.9966	0.0000	0.6402	0.2433	1.0000	0.0014	0.9699	0.1333	0.0000
P3DPZCO2M28	0.9882	0.0000	0.6402	0.2433	1.0000	0.0014	0.9699	0.1333	0.0000
P3DPZCO3M29	0.9840	0.6579	0.6360	0.2433	1.0000	0.0094	0.9655	0.1333	0.0000
P3DPZCO4M30	0.9756	0.6579	0.6360	0.2802	1.0000	0.0094	0.9612	0.1333	0.0000
P3DPZCO5M31	0.9588	0.7895	0.6360	0.3554	1.0000	0.0244	0.9155	0.1333	0.0000
P3DPZCO6M32	0.9167	0.7895	0.6904	0.4306	1.0000	0.0645	0.9084	0.1333	0.0000
PSLSELE1M33	0.9587	0.9211	0.6444	0.0914	0.9459	0.0192	0.8347	0.4000	0.5000
PSLSELE2M34	0.8410	1.0000	0.8117	0.3996	0.9459	0.1091	0.6918	0.4000	0.5000
PSLSELE3M35	0.3364	1.0000	0.8117	0.4114	0.9459	0.2436	0.3066	0.4000	0.5000
PSLSELE4M36	0.0000	1.0000	0.5607	0.4306	0.9459	0.2436	0.3066	0.4000	0.5000
PSLS3DS1M37	0.8595	0.9737	0.7280	1.0000	0.7568	0.0859	0.4153	0.4000	0.5000
PSLS3DS2M38	0.6795	0.9737	0.7280	1.0000	0.7568	0.2190	0.4619	0.4000	0.5000
PSLS3DS3M39	0.6013	0.9737	0.7280	1.0000	0.7568	0.3618	0.4309	0.4000	0.5000

Table 13

Deviation Sequences after data pre-processing

Machine Code	Machine Cost	Error	Min Layer Thickness	Machine Speed	Material Cost	Net Build Size Vol	Machine weight	Surface Finish	Material Strength
LSLA3DS1M1	0.1764	0.0132	0.4603	0.9676	0.4054	0.9822	0.0634	0.0667	1.0000
LSLA3DS2M2	0.3314	0.0811	0.1464	0.9528	0.4054	0.8965	0.4728	0.0667	1.0000
LSLA3DS3M3	0.4584	0.0811	0.1464	0.9528	0.4054	0.5710	0.7257	0.0667	1.0000
LSLA3DS4M4	0.7856	0.0811	0.1464	0.9528	0.4054	0.0000	0.7257	0.0667	1.0000
LSLAZCO1M5	0.0202	0.1053	0.1464	0.8688	0.4054	0.9948	0.0295	0.0667	1.0000
LPOLOBJ1M6	0.0076	0.2105	0.0502	0.8688	1.0000	0.9967	0.0078	0.0000	1.0000
LPOLOBJ2M7	0.0161	0.2105	0.0502	0.8688	1.0000	0.9939	0.0078	0.0000	1.0000
LPOLOBJ3M8	0.2011	0.0000	0.0000	0.8688	1.0000	0.9454	0.1591	0.0000	1.0000
LPOLOBJ4M9	0.1590	0.0000	0.0000	0.8688	1.0000	0.9695	0.1591	0.0000	1.0000
LPOLOBJ5M10	0.1338	0.2105	0.0000	0.8688	1.0000	0.9454	0.1342	0.0000	1.0000
LPOLOBJ6M11	0.1237	0.2105	0.0000	0.8688	1.0000	0.9700	0.1342	0.0000	1.0000
LPOLOBJ7M12	0.0413	0.4737	0.0000	0.8688	1.0000	0.9867	0.1342	0.0000	1.0000
LPOLOBJ8M13	0.0262	0.0263	0.0000	0.8688	1.0000	0.9873	0.0814	0.0000	1.0000
SFDMSTR1M14	0.0076	0.4737	1.0000	0.9676	0.8378	0.9974	0.0093	1.0000	0.0000
SFDMSTR2M15	0.0034	0.4737	1.0000	0.9676	0.8378	1.0000	0.0037	1.0000	0.0000
SFDMSTR3M16	0.0160	0.4737	0.6820	0.9676	0.8378	0.9872	0.0211	1.0000	0.0000
SFDMSTR4M17	0.0119	0.2895	0.9623	0.9676	0.8378	0.9872	0.0211	1.0000	0.0000
SFDMSTR5M18	0.0186	0.2895	1.0000	0.9676	0.8378	0.9756	0.0249	1.0000	0.0000
SFDMSTR6M19	0.0539	0.2816	0.4686	0.9676	0.8378	0.9703	0.1923	1.0000	0.0000
SFDMSTR7M20	0.0909	0.2816	0.4686	0.9676	0.8378	0.9703	0.1923	1.0000	0.0000
SFDMSTR8M21	0.3272	0.1816	0.6820	0.9676	0.8378	0.1778	1.0000	1.0000	0.0000
SMJM3DS1M22	0.0000	0.0142	0.3598	0.8688	0.9459	0.9976	0.0000	0.2667	0.5000
SMJM3DS2M23	0.0034	0.0142	0.3598	0.8688	0.9459	0.9947	0.0000	0.2667	0.5000
SMJM3DS3M24	0.0476	0.0132	0.0711	0.9823	0.6216	0.9894	0.1034	0.2667	0.5000
SMJM3DS4M25	0.0497	0.0132	0.0711	0.9823	0.6216	0.9894	0.0985	0.2667	0.5000
SMJM3DS5M26	0.1086	0.0132	0.0000	1.0000	0.6432	0.9020	0.1814	0.2667	0.5000
P3DPZCO1M27	0.0034	1.0000	0.3598	0.7567	0.0000	0.9986	0.0301	0.8667	1.0000
P3DPZCO2M28	0.0118	1.0000	0.3598	0.7567	0.0000	0.9986	0.0301	0.8667	1.0000
P3DPZCO3M29	0.0160	0.3421	0.3640	0.7567	0.0000	0.9906	0.0345	0.8667	1.0000
P3DPZCO4M30	0.0244	0.3421	0.3640	0.7198	0.0000	0.9906	0.0388	0.8667	1.0000
P3DPZCO5M31	0.0412	0.2105	0.3640	0.6446	0.0000	0.9756	0.0845	0.8667	1.0000
P3DPZCO6M32	0.0833	0.2105	0.3096	0.5694	0.0000	0.9355	0.0916	0.8667	1.0000
PSLSELE1M33	0.0413	0.0789	0.3556	0.9086	0.0541	0.9808	0.1653	0.6000	0.5000
PSLSELE2M34	0.1590	0.0000	0.1883	0.6004	0.0541	0.8909	0.3082	0.6000	0.5000
PSLSELE3M35	0.6636	0.0000	0.1883	0.5886	0.0541	0.7564	0.6934	0.6000	0.5000
PSLSELE4M36	1.0000	0.0000	0.4393	0.5694	0.0541	0.7564	0.6934	0.6000	0.5000
PSLS3DS1M37	0.1405	0.0263	0.2720	0.0000	0.2432	0.9141	0.5847	0.6000	0.5000
PSLS3DS2M38	0.3205	0.0263	0.2720	0.0000	0.2432	0.7810	0.5381	0.6000	0.5000
PSLS3DS3M39	0.3987	0.0263	0.2720	0.0000	0.2432	0.6382	0.5691	0.6000	0.5000

After estimating the GRC, the GRG and the respective ranking were obtained for different users and the combined user utilizing their respective weights. Using Equation (9), the values of GRG were computed to rank various machines, systems and technology. The GRG (k = 1) for machine cost, SLA and Liquid combination is as follows.

GRG₁ (based on different technologies)

=0.0682*0.7392

=0.0504

GRG₁ (based on build material)

=0.005238*0.7392

=0.003872

GRG₁ (overall)

=0.0004389*0.7392

=0.0003245

In order to rank the various alternatives, the inclusive GRG was estimated by summation of GRGs for different attributes (k = 1, 2, ... , n) using Equation (9). For example,

GRG (within AM technology) = (0.0504 + 0.1748 + 0.0517 + 0.0557 + 0.0204 + 0.0432 + 0.0078 + 0.1430 + 0.0513) = 0.5983.

GRG (within AM system) = (0.0039 + 0.0134 + 0.0040 + 0.0043 + 0.0016 + 0.0033 + 0.0006 + 0.0110 + 0.0039) = 0.0460.

GRG (overall machine) = (0.000325 + 0.001126 + 0.000333 + 0.000358 + 0.000132 + 0.000278 + 0.000050 + 0.000921 + 0.000330) = 0.003853.

Similarly, the GRG for other machines, technologies and systems were achieved. The Tables 14 –16 represents the GRG and the respective ranking of machines in a particular technology, AM systems and the most appropriate RP technology respectively for the user 1.

Table 14

GRC computed based on the deviation sequences

Machine Code	Machine Cost	Error	Min Layer Thickness	Machine Speed	Material Cost	Net Build Size Vol	Machine weight	Surface Finish	Material Strength
LSLA3DS1M1	0.7392	0.9744	0.5207	0.3407	0.5522	0.3373	0.8875	0.8824	0.3333
LSLA3DS2M2	0.6014	0.8605	0.7735	0.3442	0.5522	0.3580	0.5140	0.8824	0.3333
LSLA3DS3M3	0.5217	0.8605	0.7735	0.3442	0.5522	0.4669	0.4079	0.8824	0.3333
LSLA3DS4M4	0.3889	0.8605	0.7735	0.3442	0.5522	1.0000	0.4079	0.8824	0.3333
LSLAZCO1M5	0.9612	0.8261	0.7735	0.3653	0.5522	0.3345	0.9443	0.8824	0.3333
LPOLOBJ1M6	0.9851	0.7037	0.9087	0.3653	0.3333	0.3341	0.9847	1.0000	0.3333
LPOLOBJ2M7	0.9689	0.7037	0.9087	0.3653	0.3333	0.3347	0.9847	1.0000	0.3333
LPOLOBJ3M8	0.7132	1.0000	1.0000	0.3653	0.3333	0.3459	0.7587	1.0000	0.3333
LPOLOBJ4M9	0.7587	1.0000	1.0000	0.3653	0.3333	0.3402	0.7587	1.0000	0.3333
LPOLOBJ5M10	0.7889	0.7037	1.0000	0.3653	0.3333	0.3459	0.7884	1.0000	0.3333
LPOLOBJ6M11	0.8017	0.7037	1.0000	0.3653	0.3333	0.3401	0.7884	1.0000	0.3333
LPOLOBJ7M12	0.9237	0.5135	1.0000	0.3653	0.3333	0.3363	0.7884	1.0000	0.3333
LPOLOBJ8M13	0.9503	0.9500	1.0000	0.3653	0.3333	0.3362	0.8600	1.0000	0.3333
SFDMSTR1M14	0.9851	0.5135	0.3333	0.3407	0.3737	0.3339	0.9817	0.3333	1.0000
SFDMSTR2M15	0.9933	0.5135	0.3333	0.3407	0.3737	0.3333	0.9926	0.3333	1.0000
SFDMSTR3M16	0.9690	0.5135	0.4230	0.3407	0.3737	0.3362	0.9595	0.3333	1.0000
SFDMSTR4M17	0.9768	0.6333	0.3419	0.3407	0.3737	0.3362	0.9595	0.3333	1.0000
SFDMSTR5M18	0.9642	0.6333	0.3333	0.3407	0.3737	0.3388	0.9526	0.3333	1.0000
SFDMSTR6M19	0.9027	0.6397	0.5162	0.3407	0.3737	0.3401	0.7222	0.3333	1.0000
SFDMSTR7M20	0.8462	0.6397	0.5162	0.3407	0.3737	0.3401	0.7222	0.3333	1.0000
SFDMSTR8M21	0.6044	0.7336	0.4230	0.3407	0.3737	0.7377	0.3333	0.3333	1.0000
SMJM3DS1M22	1.0000	0.9724	0.5815	0.3653	0.3458	0.3339	1.0000	0.6522	0.5000
SMJM3DS2M23	0.9932	0.9724	0.5815	0.3653	0.3458	0.3345	1.0000	0.6522	0.5000
SMJM3DS3M24	0.9131	0.9744	0.8755	0.3373	0.4458	0.3357	0.8286	0.6522	0.5000
SMJM3DS4M25	0.9096	0.9744	0.8755	0.3373	0.4458	0.3357	0.8355	0.6522	0.5000
SMJM3DS5M26	0.8216	0.9744	1.0000	0.3333	0.4374	0.3566	0.7338	0.6522	0.5000
P3DPZCO1M27	0.9933	0.3333	0.5815	0.3979	1.0000	0.3336	0.9432	0.3659	0.3333
P3DPZCO2M28	0.9770	0.3333	0.5815	0.3979	1.0000	0.3336	0.9432	0.3659	0.3333
P3DPZCO3M29	0.9690	0.5938	0.5787	0.3979	1.0000	0.3354	0.9355	0.3659	0.3333
P3DPZCO4M30	0.9535	0.5938	0.5787	0.4099	1.0000	0.3354	0.9279	0.3659	0.3333
P3DPZCO5M31	0.9239	0.7037	0.5787	0.4368	1.0000	0.3388	0.8554	0.3659	0.3333
P3DPZCO6M32	0.8573	0.7037	0.6176	0.4675	1.0000	0.3483	0.8451	0.3659	0.3333
PSLSELE1M33	0.9237	0.8636	0.5844	0.3550	0.9024	0.3377	0.7516	0.4545	0.5000
PSLSELE2M34	0.7587	1.0000	0.7264	0.4544	0.9024	0.3595	0.6187	0.4545	0.5000
PSLSELE3M35	0.4297	1.0000	0.7264	0.4593	0.9024	0.3980	0.4190	0.4545	0.5000
PSLSELE4M36	0.3333	1.0000	0.5323	0.4675	0.9024	0.3980	0.4190	0.4545	0.5000
PSLS3DS1M37	0.7806	0.9500	0.6477	1.0000	0.6727	0.3536	0.4610	0.4545	0.5000
PSLS3DS2M38	0.6094	0.9500	0.6477	1.0000	0.6727	0.3903	0.4817	0.4545	0.5000
PSLS3DS3M39	0.5564	0.9500	0.6477	1.0000	0.6727	0.4393	0.4677	0.4545	0.5000

Table 15

GRG and ranking of the machines within various technologies

@@Machine Code	Machine Cost	Error	Layer Thickness	Machine Speed	Material Cost	Net Size Vol	Machine weight	Surface Finish	Material Strength	GRG	Rank
LSLA3DS1M1	0.0504	0.1748	0.0517	0.0557	0.0204	0.0432	0.0078	0.1430	0.0513	0.5983	4
LSLA3DS2M2	0.0410	0.1544	0.0768	0.0562	0.0204	0.0459	0.0045	0.1430	0.0513	0.5935	5
LSLA3DS3M3	0.0356	0.1544	0.0768	0.0562	0.0204	0.0598	0.0036	0.1430	0.0513	0.6011	3
LSLA3DS4M4	0.0265	0.1544	0.0768	0.0562	0.0204	0.1281	0.0036	0.1430	0.0513	0.6603	1
LSLAZCO1M5	0.0656	0.1482	0.0768	0.0597	0.0204	0.0428	0.0082	0.1430	0.0513	0.6161	2
LPOLOBJ1M6	0.0672	0.1262	0.0902	0.0597	0.0123	0.0428	0.0086	0.1621	0.0513	0.6204	4
LPOLOBJ2M7	0.0661	0.1262	0.0902	0.0597	0.0123	0.0429	0.0086	0.1621	0.0513	0.6194	5
LPOLOBJ3M8	0.0487	0.1794	0.0993	0.0597	0.0123	0.0443	0.0066	0.1621	0.0513	0.6636	3
LPOLOBJ4M9	0.0518	0.1794	0.0993	0.0597	0.0123	0.0436	0.0066	0.1621	0.0513	0.6660	2
LPOLOBJ5M10	0.0538	0.1262	0.0993	0.0597	0.0123	0.0443	0.0069	0.1621	0.0513	0.6159	7
LPOLOBJ6M11	0.0547	0.1262	0.0993	0.0597	0.0123	0.0436	0.0069	0.1621	0.0513	0.6160	6
LPOLOBJ7M12	0.0630	0.0921	0.0993	0.0597	0.0123	0.0431	0.0069	0.1621	0.0513	0.5898	8
LPOLOBJ8M13	0.0648	0.1704	0.0993	0.0597	0.0123	0.0431	0.0075	0.1621	0.0513	0.6705	1
SFDMSTR1M14	0.0672	0.0921	0.0331	0.0557	0.0138	0.0428	0.0086	0.0540	0.1538	0.5211	8
SFDMSTR2M15	0.0678	0.0921	0.0331	0.0557	0.0138	0.0427	0.0087	0.0540	0.1538	0.5217	7
SFDMSTR3M16	0.0661	0.0921	0.0420	0.0557	0.0138	0.0431	0.0084	0.0540	0.1538	0.5290	6
SFDMSTR4M17	0.0667	0.1136	0.0339	0.0557	0.0138	0.0431	0.0084	0.0540	0.1538	0.5430	4
SFDMSTR5M18	0.0658	0.1136	0.0331	0.0557	0.0138	0.0434	0.0083	0.0540	0.1538	0.5416	5
SFDMSTR6M19	0.0616	0.1147	0.0512	0.0557	0.0138	0.0436	0.0063	0.0540	0.1538	0.5548	2
SFDMSTR7M20	0.0577	0.1147	0.0512	0.0557	0.0138	0.0436	0.0063	0.0540	0.1538	0.5509	3
SFDMSTR8M21	0.0412	0.1316	0.0420	0.0557	0.0138	0.0945	0.0029	0.0540	0.1538	0.5896	1
SMJM3DS1M22	0.0682	0.1744	0.0577	0.0597	0.0128	0.0428	0.0087	0.1057	0.0769	0.6070	4
SMJM3DS2M23	0.0678	0.1744	0.0577	0.0597	0.0128	0.0428	0.0087	0.1057	0.0769	0.6066	5
SMJM3DS3M24	0.0623	0.1748	0.0869	0.0551	0.0165	0.0430	0.0072	0.1057	0.0769	0.6285	2
SMJM3DS4M25	0.0621	0.1748	0.0869	0.0551	0.0165	0.0430	0.0073	0.1057	0.0769	0.6283	3
SMJM3DS5M26	0.0561	0.1748	0.0993	0.0545	0.0162	0.0457	0.0064	0.1057	0.0769	0.6355	1
P3DPZCO1M27	0.0678	0.0598	0.0577	0.0650	0.0370	0.0427	0.0082	0.0593	0.0513	0.4489	5
P3DPZCO2M28	0.0667	0.0598	0.0577	0.0650	0.0370	0.0427	0.0082	0.0593	0.0513	0.4477	6
P3DPZCO3M29	0.0661	0.1065	0.0574	0.0650	0.0370	0.0430	0.0082	0.0593	0.0513	0.4938	4
P3DPZCO4M30	0.0651	0.1065	0.0574	0.0670	0.0370	0.0430	0.0081	0.0593	0.0513	0.4946	3
P3DPZCO5M31	0.0630	0.1262	0.0574	0.0714	0.0370	0.0434	0.0075	0.0593	0.0513	0.5165	2
P3DPZCO6M32	0.0585	0.1262	0.0613	0.0764	0.0370	0.0446	0.0074	0.0593	0.0513	0.5220	1
PSLSELE1M33	0.0630	0.1549	0.0580	0.0580	0.0334	0.0433	0.0066	0.0737	0.0769	0.5678	7
PSLSELE2M34	0.0518	0.1794	0.0721	0.0742	0.0334	0.0460	0.0054	0.0737	0.0769	0.6129	4
PSLSELE3M35	0.0293	0.1794	0.0721	0.0750	0.0334	0.0510	0.0037	0.0737	0.0769	0.5945	5
PSLSELE4M36	0.0227	0.1794	0.0528	0.0764	0.0334	0.0510	0.0037	0.0737	0.0769	0.5700	6
PSLS3DS1M37	0.0533	0.1704	0.0643	0.1634	0.0249	0.0453	0.0040	0.0737	0.0769	0.6761	1
PSLS3DS2M38	0.0416	0.1704	0.0643	0.1634	0.0249	0.0500	0.0042	0.0737	0.0769	0.6693	3
PSLS3DS3M39	0.0380	0.1704	0.0643	0.1634	0.0249	0.0563	0.0041	0.0737	0.0769	0.6719	2

Table 16

GRG and ranking of AM systems (build material classification)

Machine Code	Machine Cost	Error	Layer Thickness	Machine Speed	Material Cost	Net Build Size Vol	Machine weight	Surface Finish	Material Strength	GRG	Rank
LSLA3DS1M1	0.0039	0.0134	0.0040	0.0043	0.0016	0.0033	0.0006	0.0110	0.0039	0.0460	4
LSLA3DS2M2	0.0032	0.0119	0.0059	0.0043	0.0016	0.0035	0.0003	0.0110	0.0039	0.0456	5
LSLA3DS3M3	0.0027	0.0119	0.0059	0.0043	0.0016	0.0046	0.0003	0.0110	0.0039	0.0462	3
LSLA3DS4M4	0.0020	0.0119	0.0059	0.0043	0.0016	0.0098	0.0003	0.0110	0.0039	0.0507	1
LSLAZCO1M5	0.0050	0.0114	0.0059	0.0046	0.0016	0.0033	0.0006	0.0110	0.0039	0.0473	2
LPOLOBJ1M6	0.0027	0.0051	0.0036	0.0024	0.0005	0.0017	0.0003	0.0065	0.0021	0.0250	9
LPOLOBJ2M7	0.0027	0.0051	0.0036	0.0024	0.0005	0.0017	0.0003	0.0065	0.0021	0.0250	10
LPOLOBJ3M8	0.0020	0.0072	0.0040	0.0024	0.0005	0.0018	0.0003	0.0065	0.0021	0.0267	8
LPOLOBJ4M9	0.0021	0.0072	0.0040	0.0024	0.0005	0.0018	0.0003	0.0065	0.0021	0.0268	7
LPOLOBJ5M10	0.0022	0.0051	0.0040	0.0024	0.0005	0.0018	0.0003	0.0065	0.0021	0.0248	12
LPOLOBJ6M11	0.0022	0.0051	0.0040	0.0024	0.0005	0.0018	0.0003	0.0065	0.0021	0.0248	11
LPOLOBJ7M12	0.0025	0.0037	0.0040	0.0024	0.0005	0.0017	0.0003	0.0065	0.0021	0.0238	13
LPOLOBJ8M13	0.0026	0.0069	0.0040	0.0024	0.0005	0.0017	0.0003	0.0065	0.0021	0.0270	6
SFDMSTR1M14	0.0171	0.0234	0.0084	0.0141	0.0035	0.0109	0.0022	0.0137	0.0390	0.1322	13
SFDMSTR2M15	0.0172	0.0234	0.0084	0.0141	0.0035	0.0108	0.0022	0.0137	0.0390	0.1324	12
SFDMSTR3M16	0.0168	0.0234	0.0107	0.0141	0.0035	0.0109	0.0021	0.0137	0.0390	0.1342	11
SFDMSTR4M17	0.0169	0.0288	0.0086	0.0141	0.0035	0.0109	0.0021	0.0137	0.0390	0.1378	9
SFDMSTR5M18	0.0167	0.0288	0.0084	0.0141	0.0035	0.0110	0.0021	0.0137	0.0390	0.1374	10
SFDMSTR6M19	0.0156	0.0291	0.0130	0.0141	0.0035	0.0111	0.0016	0.0137	0.0390	0.1408	7
SFDMSTR7M20	0.0146	0.0291	0.0130	0.0141	0.0035	0.0111	0.0016	0.0137	0.0390	0.1398	8
SFDMSTR8M21	0.0105	0.0334	0.0107	0.0141	0.0035	0.0240	0.0007	0.0137	0.0390	0.1496	6
SMJM3DS1M22	0.0187	0.0477	0.0158	0.0163	0.0035	0.0117	0.0024	0.0289	0.0210	0.1660	4
SMJM3DS2M23	0.0185	0.0477	0.0158	0.0163	0.0035	0.0117	0.0024	0.0289	0.0210	0.1658	5
SMJM3DS3M24	0.0170	0.0478	0.0238	0.0151	0.0045	0.0118	0.0020	0.0289	0.0210	0.1718	2
SMJM3DS4M25	0.0170	0.0478	0.0238	0.0151	0.0045	0.0118	0.0020	0.0289	0.0210	0.1718	3
SMJM3DS5M26	0.0153	0.0478	0.0271	0.0149	0.0044	0.0125	0.0018	0.0289	0.0210	0.1737	1
P3DPZCO1M27	0.0091	0.0080	0.0077	0.0087	0.0050	0.0057	0.0011	0.0080	0.0069	0.0602	12
P3DPZCO2M28	0.0089	0.0080	0.0077	0.0087	0.0050	0.0057	0.0011	0.0080	0.0069	0.0600	13
P3DPZCO3M29	0.0089	0.0143	0.0077	0.0087	0.0050	0.0058	0.0011	0.0080	0.0069	0.0662	11
P3DPZCO4M30	0.0087	0.0143	0.0077	0.0090	0.0050	0.0058	0.0011	0.0080	0.0069	0.0663	10
P3DPZCO5M31	0.0085	0.0169	0.0077	0.0096	0.0050	0.0058	0.0010	0.0080	0.0069	0.0693	9
P3DPZCO6M32	0.0078	0.0169	0.0082	0.0102	0.0050	0.0060	0.0010	0.0080	0.0069	0.0700	8
PSLSELE1M33	0.0140	0.0343	0.0129	0.0129	0.0074	0.0096	0.0015	0.0163	0.0170	0.1258	7
PSLSELE2M34	0.0115	0.0398	0.0160	0.0165	0.0074	0.0102	0.0012	0.0163	0.0170	0.1359	4
PSLSELE3M35	0.0065	0.0398	0.0160	0.0166	0.0074	0.0113	0.0008	0.0163	0.0170	0.1318	5
PSLSELE4M36	0.0050	0.0398	0.0117	0.0169	0.0074	0.0113	0.0008	0.0163	0.0170	0.1263	6
PSLS3DS1M37	0.0118	0.0378	0.0143	0.0362	0.0055	0.0100	0.0009	0.0163	0.0170	0.1499	1
PSLS3DS2M38	0.0092	0.0378	0.0143	0.0362	0.0055	0.0111	0.0009	0.0163	0.0170	0.1484	3
PSLS3DS3M39	0.0084	0.0378	0.0143	0.0362	0.0055	0.0125	0.0009	0.0163	0.0170	0.1489	2

Based on the analysis using FAHP and GRA, the machines or systems as shown in Table 17 can be suggested to various users and the fictitious combined user. It can be realized that the integrated approach based on FAHP and GRA is able to satisfy all the users. For instance, PROJECT 5000 from 3D Systems, which is a solid based system based on MJM technology has been recommended to the User 1 by the RP selection system. Moreover, for a fictitious user with combined requirements, a machine with model DIMENSION 768 SERIES (solid based system and FDM technology) can be suggested.

Table 17

GRG and ranking to determine the most appropriate RP system

Machine Code	Machine Cost	Error	Min Layer Thickness	Machine Speed	Material Cost	Net Build Size Vol	Machine weight	Surface Finish	Material Strength	GRG	Rank
LSLA3DS1M1	0.0003	0.0011	0.0003	0.0004	0.0001	0.0003	0.0000	0.0009	0.0003	0.0039	30
LSLA3DS2M2	0.0003	0.0010	0.0005	0.0004	0.0001	0.0003	0.0000	0.0009	0.0003	0.0038	31
LSLA3DS3M3	0.0002	0.0010	0.0005	0.0004	0.0001	0.0004	0.0000	0.0009	0.0003	0.0039	29
LSLA3DS4M4	0.0002	0.0010	0.0005	0.0004	0.0001	0.0008	0.0000	0.0009	0.0003	0.0043	27
LSLAZCO1M5	0.0004	0.0010	0.0005	0.0004	0.0001	0.0003	0.0001	0.0009	0.0003	0.0040	28
LPOLOBJ1M6	0.0002	0.0004	0.0003	0.0002	0.0000	0.0001	0.0000	0.0005	0.0002	0.0021	35
LPOLOBJ2M7	0.0002	0.0004	0.0003	0.0002	0.0000	0.0001	0.0000	0.0005	0.0002	0.0021	36
LPOLOBJ3M8	0.0002	0.0006	0.0003	0.0002	0.0000	0.0001	0.0000	0.0005	0.0002	0.0022	34
LPOLOBJ4M9	0.0002	0.0006	0.0003	0.0002	0.0000	0.0001	0.0000	0.0005	0.0002	0.0023	33
LPOLOBJ5M10	0.0002	0.0004	0.0003	0.0002	0.0000	0.0001	0.0000	0.0005	0.0002	0.0021	38
LPOLOBJ6M11	0.0002	0.0004	0.0003	0.0002	0.0000	0.0001	0.0000	0.0005	0.0002	0.0021	37
LPOLOBJ7M12	0.0002	0.0003	0.0003	0.0002	0.0000	0.0001	0.0000	0.0005	0.0002	0.0020	39
LPOLOBJ8M13	0.0002	0.0006	0.0003	0.0002	0.0000	0.0001	0.0000	0.0005	0.0002	0.0023	32
SFDMSTR1M14	0.0107	0.0146	0.0053	0.0088	0.0022	0.0068	0.0014	0.0086	0.0244	0.0827	13
SFDMSTR2M15	0.0108	0.0146	0.0053	0.0088	0.0022	0.0068	0.0014	0.0086	0.0244	0.0828	12
SFDMSTR3M16	0.0105	0.0146	0.0067	0.0088	0.0022	0.0068	0.0013	0.0086	0.0244	0.0840	11
SFDMSTR4M17	0.0106	0.0180	0.0054	0.0088	0.0022	0.0068	0.0013	0.0086	0.0244	0.0862	9
SFDMSTR5M18	0.0104	0.0180	0.0053	0.0088	0.0022	0.0069	0.0013	0.0086	0.0244	0.0860	10
SFDMSTR6M19	0.0098	0.0182	0.0081	0.0088	0.0022	0.0069	0.0010	0.0086	0.0244	0.0881	7
SFDMSTR7M20	0.0092	0.0182	0.0081	0.0088	0.0022	0.0069	0.0010	0.0086	0.0244	0.0875	8
SFDMSTR8M21	0.0065	0.0209	0.0067	0.0088	0.0022	0.0150	0.0005	0.0086	0.0244	0.0936	6
SMJM3DS1M22	0.0117	0.0298	0.0099	0.0102	0.0022	0.0073	0.0015	0.0181	0.0132	0.1038	4
SMJM3DS2M23	0.0116	0.0298	0.0099	0.0102	0.0022	0.0073	0.0015	0.0181	0.0132	0.1038	5
SMJM3DS3M24	0.0107	0.0299	0.0149	0.0094	0.0028	0.0074	0.0012	0.0181	0.0132	0.1075	2
SMJM3DS4M25	0.0106	0.0299	0.0149	0.0094	0.0028	0.0074	0.0012	0.0181	0.0132	0.1075	3
SMJM3DS5M26	0.0096	0.0299	0.0170	0.0093	0.0028	0.0078	0.0011	0.0181	0.0132	0.1087	1
P3DPZCO1M27	0.0026	0.0023	0.0022	0.0025	0.0014	0.0017	0.0003	0.0023	0.0020	0.0175	25
P3DPZCO2M28	0.0026	0.0023	0.0022	0.0025	0.0014	0.0017	0.0003	0.0023	0.0020	0.0174	26
P3DPZCO3M29	0.0026	0.0041	0.0022	0.0025	0.0014	0.0017	0.0003	0.0023	0.0020	0.0192	24
P3DPZCO4M30	0.0025	0.0041	0.0022	0.0026	0.0014	0.0017	0.0003	0.0023	0.0020	0.0193	23
P3DPZCO5M31	0.0025	0.0049	0.0022	0.0028	0.0014	0.0017	0.0003	0.0023	0.0020	0.0201	22
P3DPZCO6M32	0.0023	0.0049	0.0024	0.0030	0.0014	0.0017	0.0003	0.0023	0.0020	0.0203	21
PSLSELE1M33	0.0041	0.0100	0.0037	0.0037	0.0022	0.0028	0.0004	0.0047	0.0050	0.0366	20
PSLSELE2M34	0.0033	0.0115	0.0046	0.0048	0.0022	0.0030	0.0003	0.0047	0.0050	0.0395	17
PSLSELE3M35	0.0019	0.0115	0.0046	0.0048	0.0022	0.0033	0.0002	0.0047	0.0050	0.0383	18
PSLSELE4M36	0.0015	0.0115	0.0034	0.0049	0.0022	0.0033	0.0002	0.0047	0.0050	0.0367	19
PSLS3DS1M37	0.0034	0.0110	0.0041	0.0105	0.0016	0.0029	0.0003	0.0047	0.0050	0.0435	14
PSLS3DS2M38	0.0027	0.0110	0.0041	0.0105	0.0016	0.0032	0.0003	0.0047	0.0050	0.0431	16
PSLS3DS3M39	0.0024	0.0110	0.0041	0.0105	0.0016	0.0036	0.0003	0.0047	0.0050	0.0433	15

Similarly, the DIMENSION 768 SERIES from STRATASYS (solid based FDM system), I PRO 9000 XL from 3D systems (liquid based system, SLA) and 24 PERSONNEL 3D PRINTER from OBJECT GEOMETRICS (liquid based system, PolyJet) were recommended to the user 2, user 3 and user 4 respectively.

The outcomes from sensitivity analysis as shown in Table 18 implies that the ranking order established using FAHP and GRA, is robust and stable. It can be attributed to the fact that the value Z for each user was closer to 1. To compute the values of Z, the ranking order obtained for six scenarios in each scheme and primarily established ranking order (in this work) were analyzed for their similarirties. The higher values of Z suggest that the ranking order established in this work is reliable and is not very sensitive to changes in weight.

Table 18

Appropriate outcomes of the RP selection system

Decision Makers/Experts	Requirements	Selection
	RP System	System/Category	Technology
User 1	Solid based system, MJM, Important Machine Characteristics (Accuracy, Machine Speed, Surface finish, Material Strength)	M26 (MJM) - PROJECT 5000	Liquid - I PRO 9000 XL (M4, SLA) Solid - PROJECT 5000 (M26, MJM) Powder - S PRO 60 SD (M37, SLS)	SLA - M4 (I PRO 9000 XL); PolyJet - M13 (EDEN 250 V); FDM - M21 (FORTUS 900MC); MJM - M26 (PROJECT 5000); 3DP - M32 (Z PRINTER 850); SLS - M37 (S PRO 60 SD)
User 2	Solid based system, FDM, Important Machine Characteristics (Machine Cost, Machine Weight, Surface finish, Material Strength)	M17 (FDM) - DIMESNION 768 SERIES	Liquid - ZBUILDER ULTRA (M5, SLA) Solid - DIMESNION 768 SERIES (M17, FDM) Powder - FORMIGA P100 (M33, SLS)	SLA - M5 (ZBUILDER ULTRA); PolyJet - M6 (24 PERSONNEL 3D PRINTER); FDM - M17 (DIMESNION 768 SERIES); MJM - M22 (PROJECT 1000); 3DP - M29 (Z PRINTER 350); SLS - M33 (FORMIGA P100)
User 3	Liquid based system, SLA, Important Machine Characteristics (Surface Finish, Machine Weight, Surface finish, Material Strength)	M4 (SLA) - I PRO 9000 XL	Liquid - I PRO 9000 XL (M4, SLA) Solid - PROJECT 5000 (M26, MJM) Powder - Z PRINTER 850 (M32, 3DP)	SLA - M4 (I PRO 9000 XL); PolyJet - M13 (EDEN 250 V); FDM - M21 (FORTUS 900MC); MJM - M26 (PROJECT 5000); 3DP - M32 (Z PRINTER 850); SLS - M39 (S PRO 230)
User 4	Liquid based system, PolyJet, Important Machine Characteristics (Material Cost, Machine Cost, Machine Weight, Net build size)	M6 (PolyJet) - 24 PERSONNEL 3D PRINTER	Liquid - 24 PERSONNEL 3D PRINTER (M6, Polyjet) Solid - DIMESNION 768 SERIES (M17, FDM) Powder - FORMIGA P100 (M33, SLS)	SLA - M5 (ZBUILDER ULTRA); PolyJet - M6 (24 PERSONNEL 3D PRINTER); FDM - M17 (DIMESNION 768 SERIES); MJM - M22 (PROJECT 1000); 3DP - M29 (Z PRINTER 350); SLS - M33 (FORMIGA P100)
Imagined User	Combined	M17 (FDM) - DIMESNION 768 SERIES	Liquid - I PRO 9000 XL (M4, SLA) Solid -DIMESNION 768 SERIES (M17, FDM) Powder - S PRO 60 SD (M37, SLS)Powder - S PRO 60 SD (M37, SLS)	SLA - M4 (I PRO 9000 XL); Polyjet - M13 (EDEN 250 V); FDM - M17 (DIMESNION 768 SERIES); MJM - M25 (PROJECT HD 3000); 3DP - M31 (Z PRINTER 650); SLS - M37 (S PRO 60 SD)

Table 19

Outcomes obtained in sensitivity analysis

	User 1	User 2	User 3	User 4	Imagined user
Z	0.9813	0.9908	0.9896	0.9025	0.9349
	0.9662	0.9849	0.9891	0.8855	0.8675
	0.9186	0.9793	0.9669	0.8841	0.8169
	0.8902	0.9788	0.9533	0.8832	0.8115

4 Conclusion

A decision advisor based on FAHP and GRA has been utilized to methodically deal with an unstructured and a complex RP selection problem. This method has the competence to consider the unclearness of human thinking and expertly solve MCDM problems. This approach has the potential and the capability to overcome any inconsistency and uncertainty with increase in the number of attributes. Certainly, the decisions made by the developed RP selection advisor successfully convinced every user and adequately satisfied all their requirements. This research is distinctive owing to its comprehensive database, categorization of RP selection procedure into three main levels, application of trapezoidal fuzzy numbers in extent analysis technique (FAHP), conversion of crisp values into trapezoidal fuzzy numbers and the utilization of GRA to rank various alternatives. As a result of its unique aspects, this method is easy to use and can yield rational, competent and sustainable results. Since, the outcomes in this work have been computed keeping in mind the available uncertainty in decision making and ambiguity in data collection. Therefore, it can be asserted through sensitivity analysis that the weights and the ranking were robust and effective. Moreover, many attributes have been considered in this investigation. Indeed, the complicated problems can be represented conveniently with this decision advisor which is simple and flexible in nature as well as that can exploit both objective and subjective information. The adaptability and resilience of the final outcome can also be investigated through sensitivity analysis. The utilization of FAHP weights and GRA ranking made the selection procedure adaptable, reliable and more realistic. This model is especially useful in RP selection or other similar problems, where the extent of uncertainty are expected to vary due to the dynamic and competitive market in conjunction with fickle customer demands. However, this selection approach may become tedious, computationally expensive and time consuming with increase in the number of hierarchical levels and the pairwise comparisons. Moreover, since the effectiveness of this decision approach depends on the judgement of users, it may sometime tends to overestimate the ranking process. In addition, the consistency of fuzzy pairwise comparison matrix with trapezoidal fuzzy numbers should be studied comprehensively. As the problem of RP selection has not been dealt with efficiently and effectively due to its complex nature, this work intends to aid RP users in selecting the most appropriate system. This work would inevitably help the vendors and industrial firms as a guideline in selecting the most appropriate RP system. Apart of RP selection, the proposed method can also be employed to other managerial applications such as project selection, appropriate supplier identification, hospitality industry, infrastructure management, etc. On account of continuously evolving RP market and dynamic customer demands, the database has to be updated periodically with addition of new technologies and machines. Therefore, in the future work, more attributes and alternatives will be considered to develop a more effective RP selection system. The authors also plan to solve the presented RP selection problem using other decision models, such as the method based on unbalanced hesitant fuzzy linguistic term sets [78], computational model based on extended linguistic hierarchies [79], etc.

Footnotes

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group number RG-1440-034.

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Intensity of	Definition
Importance
1	Equal Importance
3	Moderate Importance
5	Strong Importance
7	Very Strong Importance
9	Extreme Importance
2, 4, 6, 8	Comprises between the above
2	More than Equal Importance but less than Moderate Importance
4	More than Moderate Importance but less than Strong Importance
6	More than Strong Importance but less than Very Strong Importance
8	More than Very Strong Importance but less than Extreme Importance
Reciprocal of the above	If an element i has any of the above non-zero numbers assigned to it in comparison to its element j, then j has the reciprocal value in comparison with i

Decision advisor based on uncertainty theories for the selection of rapid prototyping system

Abstract

Keywords

1 Introduction

2 Methodology

2.1 Generation of database

Table 1 AM process classification AM system Technology AM machines Liquid SLA PolyJet Solid FDM MJM Powder INKJET SLS

Footnotes

Acknowledgments

References

Table 1
AM process classification

AM system Technology AM machines

Liquid SLA

PolyJet

Solid FDM

MJM

Powder INKJET

SLS