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

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.

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.

OPEN IN VIEWER

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.

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.

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.

CR = CI RI , where CI = 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.

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 ) = { 0 x ⩽ l x - l m - l l ⩽ x ⩽ m 1 m ⩽ x ⩽ n u - x u - n n ⩽ x ⩽ u 0 x ⩾ u

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. M gi 1 , M gi 2 , M gi 3 , M gi 4 , … … , M gi m , i = 1 , 2 , 3 , 4 , … , n

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 = ∑ j = 1 m M gi j ⊗ [ ∑ i = 1 n ∑ 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 ∑ j = 1 m M gi j , the fuzzy addition operation for a particular matrix can be carried out using the Equation (2).

∑ j = 1 m M gi j = ( ∑ j = 1 m l j ∑ j = 1 m m j , ∑ j = 1 m n j , ∑ j = 1 m u j )(2)

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

∑ i = 1 n ∑ j = 1 m M gi j = ( ∑ i = 1 n l i , ∑ i = 1 n n i , ∑ i = 1 n u i )(3)

[ ∑ i = 1 n ∑ j = 1 m M gi j ] - 1 = ( 1 ∑ i = 1 n u i , 1 ∑ i = 1 n n i , 1 ∑ i = 1 n m i , 1 ∑ i = 1 n l i )(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 V ( M 2 ⩾ M 1 ) = hgt ( M 1 ∩ M 2 ) = μ M 2 ( d ) = 1 , m 1 ⩾ m 2 0 , ( m 2 - n 1 ) > ( u 1 - l 2 ) ( ( n 1 - m 2 ) + ( u 1 + l 2 ) ) ( u 1 + l 2 ) , 0 < ( m 2 - n 1 ) < ( u 1 + l 2 ) ( ( 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(5)

The degree of possibility of a fuzzy number greater than k fuzzy numbers can be defined as V ( M 2 ⩾ M 1 , M 2 , M 3 , M 4 … … , M k ) = V [ ( M > M 1 ) ] and V [ ( M ⩾ M 2 ) ] and V [ ( M ⩾ M 3 ) ] and V [ ( M ⩾ M 4 ) ] and … and V [ ( M ⩾ M k ) ] = min V ( M ⩾ M i ) , i = 1 , 2 , 3 , … , k .

Suppose that

d ′ ( A i ) = min ⁡ V ( S i ≥ S k ) f o r k = 1 , 2 , 3 , 4 , … , n ; k ≠ i(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.

OPEN IN VIEWER

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 ) = ( ∏ i = 1 d w i ) 1 d(7)

Where, d is the number of users or experts.

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 ∘ ( k ) is the original sequence of responses, (where i = 1, 2, …, m and k = 1, 2, …, n), max x i ∘ ( k ) signifies the largest value of x i ∘ ( k ) and min x i ∘ ( k ) represents the smallest value of x i ∘ ( k ) . The data was normalized as follows. x i * ( k ) = 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 ) = max x i ( O ) ( k ) - x i ( O ) ( k ) max x i ( O ) ( k ) - min x i ( O ) ( k ) Smaller – the – better x i * ( k ) = | 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 ) )

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 ) ) = Δ min + ɛ Δ max Δ oi ( k ) + ∈ 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 ) . Δ min = min ∀ i min ∀ k Δ oi ( k ) and Δ max = max ∀ i max ∀ k Δ Oi ( k ) , where i = 1 , 2 , … , m and k = 1 , 2 , … , n ∈: 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 * ) = ∑ 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 ∑ 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.

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 * = 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 = 12 R m 2 ( k 3 - k )(11) R i = ∑ j = 1 m r ij(12) R = ∑ i = 1 k ( R i - R ¯ ) 2(13)

Where, r _i j= 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)).