Introduction

Background

Artificial intelligence (AI) technologies have been widely used in manufacturing [1]. However, AI applications are sometimes difficult to understand or communicate with. For example, applications of artificial neural networks (ANNs) [2], deep learning [2, 3], and fuzzy systems [4, 5] are often regarded as black boxes [2, 3, 6, 7]. The concept of explainable AI (XAI) has been proposed to solve this problem [8, 9]. XAI improves the understandability of an AI application by simulating its operation with simpler models, providing clues to enhance the effectiveness of the AI application [10].

XAI techniques have been applied to explain AI applications in manufacturing. According to the review by Ahmed et al. [11], XAI techniques have been applied to estimate the remaining life of a machine or tool [12, 13], predict product quality [12], identify failures [14, 15] and detect anomalies [15, 16]. XAI techniques applied in manufacturing include visualization (i.e., system diagrams or snapshots) [13], Shapley additive explanation (SHAP) [14, 15], “Explain Like I’m 5” (ELI5) [17], local interpretable model-agnostic explanation (LIME) [18], decision trees [16], random forests [13, 16, 19], association rules [20], and performance evaluation [13].

Research gaps, significant findings, and motivation

XAI applications in manufacturing have been limited to only a few topics [11,12,13,14,15,16,17,18,19,20]. Many manufacturing-related functions, such as job scheduling and decision-making, have been excluded. This study fills this gap by applying XAI techniques for a more comprehensive explanation of AI applications in three-dimensional (3D) printing-based ubiquitous manufacturing (UM) systems [21,22,23,24,25]: XAI applications for decision-making.

After reviewing relevant literature, the findings confirm that existing XAI applications in manufacturing have the following problems:

  • Most techniques to explain AI applications are outdated. These statistical and knowledge-mining techniques existed before the emergence of XAI but were not emphasized [11, 12].

  • Even with these technologies, AI applications remain difficult to understand or communicate with [13, 16].

  • Existing XAI techniques are heavily biased towards establishing input–output relationships [5, 7, 13, 16, 18, 20] or distinguishing the effects of different inputs [14, 15, 17].

The motivation for this research is as follows. Although similar topics appear to be overpublished, the increasing sophistication of applied AI techniques makes the decision-making process increasingly difficult to understand, as presented in Table 1. These AI applications include some difficult-to-understand black-box AI techniques, such as ANNs [26, 27], fuzzy mathematical programming [25, 28,29,30,31], agent technology [32, 33], and advanced fuzzy sets [30]. For example, it is difficult for a decision-maker to verify whether the derived criteria priorities match his or her judgment. Furthermore, if the derived priorities do not match, there is no easy way to adjust these priorities. This research proposes four new XAI techniques: (1) a gradient bar chart with baseline, (2) a group gradient bar chart, (3) a manually adjustable gradient bar chart, and (4) a bidirectional scatterplot. The gradient bar chart with baseline and manually adjustable gradient bar chart can solve these problems.

Table 1 AI technologies with applications in planning and controlling UM systems
Full size table

The proposed methodology starts with a review of existing XAI techniques and tools to explain AI applications in selecting 3D printing facilities in UM. Subsequently, this study proposes four new XAI techniques to address the deficiencies of existing XAI applications. The contributions of the proposed methodology are as follows:

  • XAI techniques are applied in the manufacturing domain. As described previously, the application of XAI in this domain has received less attention [11, 12, 35,36,37,38,39,40,41,42].

  • The application of XAI in decision-making, which has not been a focus in the past [29], is examined. Most past XAI research aims to support pattern recognition and analysis [2, 4, 10, 14,15,16, 37], anomaly detection [41, 42], and estimation (or prediction) [13, 17].

  • Four new XAI techniques (or tools) are proposed to improve the understandability of AI applications.

This study aims to improve the reasonableness and interpretability of existing AI applications in relevant selection processes. Reasonability means that the selection result of an AI application is reasonable. To be precise, the selection result should be consistent with the result of the user’s own selection [43]. First, most previous research in this field uses approximation techniques such as fuzzy geometric mean (FGM) [42], fuzzy extent analysis (FEA) [35], and others to derive the fuzzy criteria priorities. The derived fuzzy priorities may be inaccurate, misleading the selection process. This study addresses this issue by using alpha-cut operations (ACO) [36] to precisely derive the fuzzy priorities and designs an XAI tool to help decision-makers verify the derivation results subjectively. Furthermore, in previous studies, decision-makers often used methods such as fuzzy-weighted average (FWA) [40] and FGM [42] to aggregate their judgments or evaluations, producing an unacceptable aggregation result. Another XAI tool is designed to help decision-makers trace the aggregation process and solve this problem.

Organization of this research

The rest of this paper is organized as follows. “Methodology” reviews existing XAI tools and techniques to explain AI applications in selecting 3D printing facilities in UM. It also discusses how to evaluate the effectiveness of XAI applications. Subsequently, to improve the effectiveness, four new XAI techniques are proposed. A real-world case from the literature is used to demonstrate the applicability of the proposed methodology in “Case study”. Finally, “Conclusion and future research directions” summarizes this study and proposes directions for future research.

Methodology

UM is the application of ubiquitous computing in the manufacturing industry with the paradigm “design anytime, anywhere, manufacture anytime, anywhere, and sell anytime, anywhere” [22, 23]. A UM system consists of multiple 3D printing facilities that may not belong to but dynamically join the UM system [24, 25]. When choosing a suitable 3D printing facility to join, the UM system tends to consider 3D printing facilities that can fulfill orders quickly, inexpensively, and with high quality that have a good relationship with the UM system [23]. However, these requirements often conflict, making selecting a suitable 3D printing facility a complex multi-criteria decision-making (MCDM) problem [28, 29].

The proposed methodology starts by reviewing existing XAI techniques and tools, such as text descriptions, gradient bar charts [39], color management [39], annotated figures [39], common expressions [39], and traceable aggregation [39], to explain such AI applications. Subsequently, this study proposes four new XAI techniques—(1) a gradient bar chart with baseline, (2) a group gradient bar chart, (3) a manually adjustable gradient bar chart, and (4) a bidirectional scatterplot—to support the four goals of XAI: understandability, comprehensibility, interpretability, and transparency [8], as depicted in Fig. 1.

Fig. 1
figure 1

New tools to support the four goals of XAI

Full size image

Suitable 3D printing facility selection in UM

Selecting a suitable 3D printing facility starts with a pairwise comparison of the relative criteria priorities for assessing each 3D printing facility. The results are incorporated into fuzzy judgment matrix \({\tilde{\mathbf{A}}}(k) = [\tilde{a}_{ij} (k)]\). \(\tilde{a}_{ij} (k)\) is the relative priority of criterion i over criterion j to decision-maker k, which is usually interpreted as “the priority of criterion i is \(\tilde{a}_{ij} (k)\) times that of criterion j to decision-maker k”; \(\tilde{a}_{ij} (k) \in [0,\;9]\) [44]. Traditionally, all matrix elements are represented to facilitate their comparison. For the same purpose, this study proposes a new XAI tool, gradient bar charts with baselines, as depicted in Fig. 2:

  • Except for the baseline, gradient bars represent the uncertainty in pairwise comparison results [45].

  • The baseline is placed on the far left and compared with itself, so the relative priority is 1.

  • A dashed line is used to compare the relative priorities of other criteria.

Fig. 2
figure 2

Gradient bar chart with baseline to illustrate pairwise comparison results

Full size image

The fuzzy criteria priorities can be derived from the fuzzy judgment matrix using fuzzy eigenanalysis [46]:

$$ \det ({\tilde{\mathbf{A}}}(k)( - )\tilde{\lambda }(k){\mathbf{I}}) = 0 $$
(1)
$$ ({\tilde{\mathbf{A}}}(k)( - )\tilde{\lambda }(k))( \times ){\tilde{\mathbf{x}}}(k) = 0, $$
(2)

where \(\tilde{\lambda }(k)\) and \({\tilde{\mathbf{x}}}(k)\) are the fuzzy eigenvalue and eigenvector of \({\tilde{\mathbf{A}}}(k)\). \(( - )\) and \(( \times )\) are fuzzy subtraction and multiplication. Deriving the exact values of \(\tilde{\lambda }(k)\) and \({\tilde{\mathbf{x}}}(k)\) is a computationally intensive task [46]. The ACO method [47] is applied to address this issue.

ACO for deriving fuzzy criteria priorities

ACO derives the fuzzy criteria priorities by solving the following equations [46, 47]:

$$ \det ({\mathbf{A}}^{*} (k)(\alpha ) - \lambda^{!} (k)(\alpha ){\mathbf{I}}) = 0 $$
(3)
$$ ({\mathbf{A}}^{*} (k)(\alpha ) - \lambda^{!} (k)(\alpha )){\mathbf{x}}^{\& } (k)(\alpha ) = 0 $$
(4)

where *, !, and & can be L or R, indicating the left or right α cut. If α takes values from 0 to 1 every 0.1, Eqs. (3) and (4) need to be solved \(10 \cdot 2^{{C_{2}^{n} }} + 1\) times [46]. Then, the criteria priorities are derived as follows [44]:

$$ \tilde{w}_{i} (k) = \frac{{\tilde{x}_{i} (k)}}{{\sum\nolimits_{j = 1}^{n} {\tilde{x}_{j} (k)} }} $$
(5)

The results are usually illustrated with a group bar chart. Furthermore, the consistency ratio of \({\tilde{\mathbf{A}}}(k)\) is evaluated as follows [44]:

$$ \widetilde{CR}({\tilde{\mathbf{A}}}(k)) = \frac{{\tilde{\lambda }(k) - n}}{(n - 1)RI}, $$
(6)

If \(\widetilde{CR}({\tilde{\mathbf{A}}}(k))\) is less than 0.1 (for a small problem), \({\tilde{\mathbf{A}}}(k)\) is consistent [44]. Traditionally, a line chart is usually drawn to represent \(\widetilde{CR}({\tilde{\mathbf{A}}}(k))\). However, the range of \(\widetilde{CR}({\tilde{\mathbf{A}}}(k))\) may be extensive. Therefore, most previous studies only judged the consistency according to the core of \(\widetilde{CR}({\tilde{\mathbf{A}}}(k))\).

In the proposed methodology, a new XAI tool, group gradient bar charts, is proposed to validate the derivation process, as depicted in Fig. 3, in which

  • A criterion is used as the baseline. The relative priority between the criterion and another criterion is compared. In Fig. 3, criterion #2 is the baseline.

  • The relative priority between two criteria derived by ACO, i.e., \(\tilde{w}_{i} (k){(/)}\tilde{w}_{j} (k)\), is compared with the corresponding pairwise comparison result \(\tilde{a}_{ij} (k)\). If the deviation is too large, the derivation process may be questionable and must be modified.

Fig. 3
figure 3

Group gradient bar chart to validate derivation process

Full size image

Furthermore, the derivation result can be validated with another XAI tool, a manually adjustable gradient bar chart, as depicted in Fig. 4. The decision-maker compares the derived fuzzy priorities with his expectations. If any difference is significant, the derivation results are unacceptable. In this case, the decision-maker can manually adjust the priority of a criterion, as depicted in Fig. 4. However, raising the priority of one criterion will automatically lower the priorities of other criteria and vice versa.

Fig. 4
figure 4

Manually adjustable gradient bar chart to validate derivation results

Full size image

Theorem 1

If \(\tilde{w}_{i} (k) \to \tilde{w}_{i} (k)( + )\Delta \tilde{w}_{i} (k)\), then

$$ \tilde{w}_{j} (k) \to \tilde{w}_{j} (k)( - )\frac{{\tilde{w}_{j} (k)}}{{\sum\nolimits_{l \ne i} {\tilde{w}_{l} (k)} }}( \times )\Delta \tilde{w}_{i} (k) $$
(7)

Proof

Since \(\sum\nolimits_{j = 1}^{n} {\tilde{w}_{j} (k) = 1},\)

$$ \sum\limits_{j \ne i} {\tilde{w}_{j} (k)} = 1( - )\tilde{w}_{i} (k) $$
(8)

Therefore,

$$ \begin{aligned} & \tilde{w}_{i} (k)( + )\Delta \tilde{w}_{i} (k)( + )\sum\limits_{j \ne i} {(\tilde{w}_{j} (k)( - )\frac{{\tilde{w}_{j} (k)}}{{\sum\nolimits_{l \ne i} {\tilde{w}_{l} (k)} }}( \times )\Delta \tilde{w}_{i} (k)} ) \\ & \quad = \tilde{w}_{i} (k)( + )\Delta \tilde{w}_{i} (k)( + )\sum\limits_{j \ne i} {\tilde{w}_{j} (k)} ( - )\frac{{\sum\nolimits_{j \ne i} {\tilde{w}_{j} (k)} }}{{\sum\nolimits_{l \ne i} {\tilde{w}_{l} (k)} }}( \times )\Delta \tilde{w}_{i} (k) \\ & \quad = \tilde{w}_{i} (k)( + )\Delta \tilde{w}_{i} (k)( + )1( - )\tilde{w}_{i} (k)( - )\Delta \tilde{w}_{i} (k) \\ & \quad = 1 \\ \end{aligned} $$
(9)

Theorem 1 is proved.

Another XAI tool to validate the derivation results is an annotated line chart, as plotted in Fig. 5, a line chart enhanced by applying color management and common expressions [39]. A decision-maker unsatisfied with the derived fuzzy priorities can modify the pairwise comparison results and re-derive fuzzy priorities.

Fig. 5
figure 5

Annotated line chart to validate derived fuzzy priorities

Full size image

Furthermore, performance evaluation is widely applied to validate the derivation process and results. Accordingly, ACO minimizes the error in deriving the fuzzy criteria priorities, enhancing the interpretability of the derivation process and results [48]. In contrast, approximation methods, such as FGM [42] and FEA [49], may overestimate or underestimate fuzzy priorities [50]. Consequently, the explainability of the derivation results is low.

Fuzzy intersection for aggregating the derivation results

Subsequently, fuzzy intersection (FI) [51] is applied to aggregate the fuzzy priorities derived by all decision-makers to measure their consensus [52], as illustrated in Fig. 6:

$$ \;\mu_{{\widetilde{FI}(\{ \tilde{w}_{i} (k)|k = 1\sim K\} )}} (x) = \mathop {\min }\limits_{k} (\mu_{{\tilde{w}_{i} (k)}} (x)) $$
(10)
Fig. 6
figure 6

Aggregating the fuzzy priorities derived by all decision-makers using FI

Full size image

Theorem 2

$$ FI^{L} (\alpha {)} = \left\{ {\begin{array}{*{20}c} {\mathop {\max }\limits_{k} (w_{i}^{L} (k)(\alpha {)})} & {if\quad \mathop {\max }\limits_{k} (w_{i}^{L} (k)(\alpha {)}) \le \mathop {\min }\limits_{k} (w_{i}^{R} (k)(\alpha {)})} \\ \emptyset & {otherwise} \\ \end{array} } \right. $$
(11)
$$ FI^{R} (\alpha {)} = \left\{ {\begin{array}{*{20}c} {\mathop {\min }\limits_{k} (w_{i}^{R} (k)(\alpha {)})} & {if\quad \mathop {\min }\limits_{k} (w_{i}^{R} (k)(\alpha {)}) \ge \mathop {\max }\limits_{k} (w_{i}^{L} (k)(\alpha {)})} \\ \emptyset & {otherwise} \\ \end{array} } \right. . $$
(12)

The understandability of Fig. 6 can be enhanced by applying XAI techniques such as color management, common expressions, and annotated figures. The result is plotted in Fig. 7.

Fig. 7
figure 7

After applying XAI techniques to Fig. 6

Full size image

Furthermore, the aggregation process can be visualized using traceable aggregation proposed by Lin and Chen [39], as depicted in Fig. 8. If some decision-makers consider the priority of a criterion after aggregation as inappropriate (e.g., too large, too small, too wide, or too narrow), all decision-makers can discuss whether their judgments should be modified.

Fig. 8
figure 8

Traceable aggregation

Full size image

Assessing 3D printing facilities using FTOPSIS

Subsequently, fuzzy technique for order preference by similarity to ideal solution (FTOPSIS) [52,53,54] is applied to assess the overall performance of a 3D printing facility. First, the performance of a 3D printing facility in optimizing each criterion is normalized using fuzzy distributive normalization [53]:

$$ \begin{aligned} \tilde{\rho }_{qi} & = \frac{{\tilde{p}_{qi} }}{{\sqrt {\sum\nolimits_{\phi = 1}^{Q} {\tilde{p}_{\phi i}^{2} } } }} \\ & = \frac{1}{{\sqrt {1 + \sum\nolimits_{\phi \ne q} {\left( {\frac{{\tilde{p}_{\phi i} }}{{\tilde{p}_{qi} }}} \right)^{2} } } }} \\ \end{aligned} $$
(13)

where \(\tilde{p}_{qi}\) is the performance of the qth 3D printing facility in optimizing the ith criterion, and \(\tilde{\rho }_{qi}\) is the normalized performance, both independent of decision-makers. Replacing all variables in Eq. (13) with their α cuts produces

$$ \rho_{qi}^{L} (\alpha ) = \frac{1}{{\sqrt {1 + \sum\nolimits_{j \ne i} {\left( {\frac{{p_{qj}^{R} (\alpha )}}{{\tilde{p}_{qi}^{L} (\alpha )}}} \right)^{2} } } }} $$
(14)
$$ \rho_{qi}^{R} (\alpha ) = \frac{1}{{\sqrt {1 + \sum\nolimits_{j \ne i} {\left( {\frac{{p_{qj}^{L} (\alpha )}}{{\tilde{p}_{qi}^{R} (\alpha )}}} \right)^{2} } } }} $$
(15)

Subsequently, the fuzzy-weighted scores are calculated based on the aggregated fuzzy priorities [53]:

$$ \tilde{s}_{qi} = \tilde{w}_{i} ( \times )\tilde{\rho }_{qi} $$
(16)

Equivalently,

$$ s_{qi}^{L} = w_{i}^{L} (\alpha )\rho_{qi}^{L} (\alpha ) $$
(17)
$$ s_{qi}^{R} = w_{i}^{R} (\alpha )\rho_{qi}^{R} (\alpha ) $$
(18)

The fuzzy ideal (zenith) point and fuzzy anti-ideal (nadir) point are specified, as [53]

$$ \tilde{\Lambda }^{ + } = \{ \tilde{\Lambda }_{i}^{ + } \} = \{ \mathop {\max }\limits_{q} \tilde{s}_{qi} \} $$
(19)
$$ \tilde{\Lambda }^{ - } = \{ \tilde{\Lambda }_{i}^{ - } \} = \{ \mathop {\min }\limits_{q} \tilde{s}_{qi} \} $$
(20)

Their α cuts are

$$ [\Lambda^{ + L} (\alpha ),\;\Lambda^{ + R} (\alpha )] = \{ [\Lambda_{i}^{ + L} (\alpha ),\;\Lambda_{i}^{ + R} (\alpha )]\} , = \{ [\mathop {\max }\limits_{q} s_{qi}^{L} (\alpha ),\;\mathop {\max }\limits_{q} s_{qi}^{R} (\alpha )]\} $$
(21)
$$ [\Lambda^{ - L} (\alpha ),\;\Lambda^{ - R} (\alpha )] = \{ [\Lambda_{i}^{ - L} (\alpha ),\;\Lambda_{i}^{ - R} (\alpha )]\} , = \{ [\mathop {\min }\limits_{q} s_{qi}^{L} (\alpha ),\;\mathop {\min }\limits_{q} s_{qi}^{R} (\alpha )]\} $$
(22)

The fuzzy distances from each 3D printing facility to the two reference points are measured as [53]

$$ \tilde{d}_{q}^{ + } = \sqrt {\sum\limits_{i = 1}^{n} {(\tilde{\Lambda }_{i}^{ + } ( - )\tilde{s}_{qi} )^{2} } } $$
(23)
$$ \tilde{d}_{q}^{ - } = \sqrt {\sum\limits_{i = 1}^{n} {(\tilde{\Lambda }_{i}^{ - } ( - )\tilde{s}_{qi} )^{2} } } $$
(24)

Equivalently,

$$ d_{q}^{ + L} (\alpha ) = \sqrt {\sum\limits_{i = 1}^{n} {(\max (\Lambda_{i}^{ + L} (\alpha ) - s_{qi}^{R} (\alpha ),\;0))^{2} } } $$
(25)
$$ d_{q}^{ + R} (\alpha ) = \sqrt {\sum\limits_{i = 1}^{n} {(\Lambda_{i}^{ + R} (\alpha ) - s_{qi}^{L} (\alpha ))^{2} } } $$
(26)
$$ d_{q}^{ - L} (\alpha ) = \sqrt {\sum\limits_{i = 1}^{n} {(\min (\Lambda_{i}^{ - R} (\alpha ) - s_{qi}^{L} (\alpha ),\;0))^{2} } } $$
(27)
$$ d_{q}^{ - R} (\alpha ) = \sqrt {\sum\limits_{i = 1}^{n} {(\Lambda_{i}^{ - L} (\alpha ) - s_{qi}^{R} (\alpha ))^{2} } } $$
(28)

Accordingly, the fuzzy closeness of each 3D printing facility is derived as [53]

$$ \tilde{C}_{q} = \frac{{\tilde{d}_{q}^{ - } }}{{\tilde{d}_{q}^{ + } ( + )\tilde{d}_{q}^{ - } }} $$
(29)

The left and right α cuts of \(\tilde{C}_{q}\) can be calculated as

$$ C_{q}^{L} (\alpha ) = \min \left( {\frac{{d_{q}^{ - R} (\alpha )}}{{d_{q}^{ + R} (\alpha ){ + }d_{q}^{ - R} (\alpha )}},\;\frac{{d_{q}^{ - L} (\alpha )}}{{d_{q}^{ + R} (\alpha ){ + }d_{q}^{ - L} (\alpha )}}} \right) $$
(30)
$$ C_{q}^{R} (\alpha ) = \max \left( {\frac{{d_{q}^{ - R} (\alpha )}}{{d_{q}^{ + L} (\alpha ){ + }d_{q}^{ - R} (\alpha )}},\;\frac{{d_{q}^{ - L} (\alpha )}}{{d_{q}^{ + L} (\alpha ){ + }d_{q}^{ - L} (\alpha )}}} \right) $$
(31)

A 3D printing facility with a higher fuzzy closeness is more suitable. The assessment mechanism in FTOPSIS is to compare the distances from a 3D printing facility to the two reference points, which may not be very intuitive. Highlighting the difference in the two distances enhances the explainability of FTOPSIS. For this purpose, a new XAI tool, bidirectional scatterplots, is proposed in this study, as illustrated in Fig. 9:

  • The ideal solution is at the top, while the anti-ideal solution is at the bottom.

  • The ideal solution is white, while the anti-ideal solution is black.

  • All 3D printing facilities are placed anywhere in the bidirectional scatter plot if the following requirement is met. The distances from each 3D printing facility to the two reference points are measured using the FTOPSIS method, as illustrated in this figure.

  • The closer a 3D printing facility is to the ideal solution, the whiter (lighter) its color is. Conversely, a 3D printing facility becomes blacker if it is closer to the anti-ideal solution.

Fig. 9
figure 9

Bidirectional scatterplot to illustrate the assessment mechanism in FTOPSIS

Full size image

The complexity of the proposed methodology is determined by its most time-consuming step-deriving fuzzy priorities using fuzzy eigenanalysis. This step takes time proportional to \(2^{{\frac{n(n - 1)}{2}}}\). Therefore, its complexity is \(O(2^{{n^{2} }} )\), which can be improved by applying the approximating ACO method [46].

Evaluating the effectiveness of an XAI technique

In estimation and forecasting, a complex AI application is explained using a simpler XAI technique, and the average accuracy of the latter, when simulating the former, can be used to evaluate the effectiveness of the XAI technique [13]. However, in decision-making applications, there are no actual values to compare. Therefore, an XAI technique to explain AI applications to 3D printing facility selection is considered effective if it satisfies the following requirements:

  1. (i)

    Decision-makers have the required background knowledge: this can be verified by comparing the background knowledge of the XAI technology with that of the user.

  2. (ii)

    The XAI technique can process high-dimensional data: this requirement is met if the data processed using the XAI technique involves many dimensions (or criteria).

  3. (iii)

    The explanation formats are consistent in different applications: this can be verified if the explanations for different applications have the same format.

  4. (iv)

    The XAI technique is easy to communicate: this can be assessed subjectively by those responsible for communication using a numerical scale.

  5. (v)

    The XAI technique is easy to understand: this can be assessed subjectively by the communicated user using a numerical scale [55].

  6. (vi)

    The XAI technique embodies the uncertainty in the decision-making process: this requirement is met if the XAI technique uses probabilistic, stochastic, fuzzy, rough, or gray theory to deal with uncertainty.

  7. (vii)

    The XAI technique explains the derivation process and results: After the derivation process is over, any attempt to communicate, explain, or explain will count.

  8. (viii)

    The XAI technique visualizes the derivation process and results: The derivation process and results are explained using figures, tables, animations, or any other visualization tool.

  9. (ix)

    The XAI technique helps improve the derivation process and results: The XAI technique modifies the derivation process and results to improve the performance of the AI application.

  10. (x)

    The XAI technique explains the aggregation process and results: this requirement is considered when multiple users participate in the derivation process, and their results are aggregated. After the aggregation process is complete, any attempt to communicate, explain or interpret will be counted.

  11. (xi)

    The XAI technique visualizes the aggregation process and results: The aggregation process and results are explained using figures, tables, animations, or any other visualization tool.

  12. (xii)

    The XAI technique visualizes/compares the overall performance of 3D printing facilities: The overall performance of all alternatives is compared simultaneously using figures, tables, animations, or any other visualization tool.

These performance measures are chosen because they meet the requirements for understandability ((i) and (v)), comprehensibility ((ii) to (iv)), interpretability ((vi), (vii), (ix) and (x)), and transparency ((viii), (ix), (xi) and (xii)) [56, 57].

Case study

Background

The case discussed by Chen and Wang [29] was used to illustrate the applicability of the proposed methodology, where the administration of a UM system would like to select a suitable 3D printing facility to print an order. The UM system was in Taichung City, Taiwan, with a service area of 47 km2. The study included three decision-makers with different professional backgrounds and emphasized the performance of 3D printing facilities from different perspectives. Five criteria were used to evaluate the suitability of a 3D printing facility:

  • estimated completion time

  • estimated delivery time

  • relationship between the UM system and the 3D printing facility

  • average quality of 3D objects printed by the 3D printing facility

  • total costs, including printing and delivery costs.

Application of the proposed methodology

First, decision-makers compared the criteria priorities in pairs. The results are summarized by the following fuzzy judgment matrices:

$$ {\tilde{\mathbf{A}}}{(1)} = \left[ {\begin{array}{*{20}c} 1 & \quad {{(}3,\;5,\;7)} & \quad {(1,\;3,\;5)} & \quad{(1,\;3,\;5)} & \quad{(5,\;7,\;9)} \\ {1/{(}3,\;5,\;7)} & \quad1 &\quad {1/(1,\;3,\;5)} &\quad {1/(7,\;9,\;9)} & \quad{1/(5,\;7,\;9)} \\ {1/(1,\;3,\;5)} & \quad{(1,\;3,\;5)} & \quad1 & \quad{1/(1,\;3,\;5)} & \quad{(1,\;1,\;3)} \\ {1/(1,\;3,\;5)} & \quad{(7,\;9,\;9)} & \quad{(1,\;3,\;5)} &\quad 1 & \quad{(5,\;7,\;9)} \\ {1/(5,\;7,\;9)} & \quad{(5,\;7,\;9)} & \quad{1/(1,\;1,\;3)} & \quad{1/(5,\;7,\;9)} & \quad1 \\ \end{array} } \right] $$
$$ {\tilde{\mathbf{A}}}{(2)} = \left[ {\begin{array}{*{20}c} 1 & \quad{(2,\;4,\;6)} & \quad{1/(1,\;1,\;3)} &\quad {(3,\;5,\;7)} & \quad{(2,\;4,\;6)} \\ {1/(2,\;4,\;6)} &\quad 1 &\quad {1/(3,\;5,\;7)} & \quad{{1/}(1,\;3,\;5)} &\quad {1/(3,\;5,\;7)} \\ {(1,\;1,\;3)} &\quad {(3,\;5,\;7)} & \quad1 & \quad{(1,\;1,\;3)} &\quad {(2,\;4,\;6)} \\ {1/(3,\;5,\;7)} & \quad{(1,\;3,\;5)} & \quad{1/(1,\;1,\;3)} & \quad 1 &\quad {(2,\;4,\;6)} \\ {1/(2,\;4,\;6)} &\quad {(3,\;5,\;7)} & \quad {1/(2,\;4,\;6)} &\quad {1/(2,\;4,\;6)} & \quad1 \\ \end{array} } \right] $$
$$ {\tilde{\mathbf{A}}}{(3)} = \left[ {\begin{array}{*{20}c} 1 & \quad{(2,\;4,\;6)} &\quad {1/(1,\;3,\;5)} &\quad {(3,\;5,\;7)} &\quad {(1,\;3,\;5)} \\ {1/(2,\;4,\;6)} &\quad 1 &\quad {1/(2,\;4,\;6)} &\quad {1/(3,\;5,\;7)} &\quad {1/(2,\;4,\;6)} \\ {(1,\;3,\;5)} &\quad {(2,\;4,\;6)} &\quad 1 &\quad {(2,\;4,\;6)} &\quad {(1,\;3,\;5)} \\ {1/(3,\;5,\;7)} &\quad {(3,\;5,\;7)} &\quad {1/(2,\;4,\;6)} &\quad 1 &\quad {(3,\;5,\;7)} \\ {1/(1,\;3,\;5)} &\quad {(2,\;4,\;6)} &\quad {1/(1,\;3,\;5)} &\quad {1/(3,\;5,\;7)} & 1 \\ \end{array} } \right] $$

Gradient bar charts with baselines were drawn to confirm each decision-maker’s judgments visually. An example is given in Fig. 10.

Fig. 10
figure 10

Gradient bar chart with baseline to confirm decision-maker’s judgments

Full size image

Subsequently, the fuzzy criteria priorities were derived from fuzzy judgment matrices using the ACO method implemented with MATLAB R2021a on a computer having an i7-7700 central processing unit (CPU) 272 at 3.6 GHz and 8 GB of memory. The execution time for each decision-maker was less than 30 s. An annotated line chart was plotted (Fig. 11) so each decision-maker could validate the derivation results.

Fig. 11
figure 11

Annotated line charts to validate derivation results

Full size image

Manually adjustable gradient bar charts were drawn for decision-makers unsatisfied with the derivation results. In this case, decision-maker #3 manually lowered the fuzzy priority of “the relationship between the ubiquitous manufacturing system and the 3D printing facility,” as depicted in Fig. 12.

Fig. 12
figure 12

Decision-maker #3 lowered the fuzzy priority of “relationship” using a manually adjustable gradient bar chart

Full size image

Subsequently, FI was applied to aggregate the derivation results, visualized using traceable aggregation. The fuzzy priority of “the completion time” was used as an example. The result is plotted in Fig. 13.

Fig. 13
figure 13

Aggregation process using FI

Full size image

Based on the aggregated fuzzy priorities, nine 3D printing facilities were compared. The performance of these 3D printing facilities in optimizing various criteria is summarized in Table 2. FTOPSIS was applied to evaluate the overall performance of each 3D printing facility.

Table 2 Performance of nine 3D printing facilities
Full size table

After normalization, the aggregated fuzzy priorities were multiplied by the normalized performance to derive the fuzzy-weighted scores, based on which the two reference points were established, as presented in Table 3.

Table 3 Reference points in FTOPSIS
Full size table

The fuzzy distances between each 3D printing facility and the two references were measured and compared. The results were visualized using a bidirectional scatterplot (Fig. 14). The fuzzy proximities of these 3D printing facilities are summarized in Table 4.

Fig. 14
figure 14

Comparison of the fuzzy distances between each 3D printing facility and two references

Full size image
Table 4 Fuzzy proximity of 3D printing facilities
Full size table

Discussion

The following discussion is based on the experimental results:

  • From the bidirectional scatterplot, 3D printing facility #8 was the most suitable 3D printing facility, followed by 3D printing facilities #5, #9, and #4.

  • Typical features of the top two 3D printing facilities included short completion times and high product quality, ensuring that orders could be printed quickly and less likely to fail. Therefore, the production plan is practically feasible. Moreover, in the proposed methodology, only ACO requires time-consuming enumeration, but its logic is very simple. Therefore, the proposed methodology can be implemented using a worksheet with simple Visual Basic for Applications (VBA) coding. The proposed methodology is also computationally feasible.

  • Most 3D printing facilities were closer to the ideal than the anti-ideal solution.

  • In FTOPSIS, the suitability of a 3D printing facility was evaluated by its closeness, which could be easily compared after visualization with the bidirectional scatterplot.

  • The XAI techniques proposed in this study enhanced the understandability of an AI application (i.e., ACO-FTOPSIS) and provided a technique (i.e., manually adjustable gradient bar chart) to increase the effectiveness of the AI application for a decision-maker.

  • A sensitivity analysis was conducted to determine which decision-maker’s judgment had the most significant impact on the outcome. The results demonstrated that \(\tilde{a}_{{1{4}}} (1)\) (i.e., the relative priority of “completion time” over “quality” to decision-maker #1) was the most influential. Increasing this judgment by 1 changed the overall performance of 3D printing facilities by an average of 0.49%. The most suitable 3D printing facility also changed to 3D printing facility #5. Increasing \(\tilde{a}_{14} (1)\) further reinforced the advantage of 3D printing facility #5 over other 3D printing facilities.

  • The effectiveness of the proposed methodology was verified against the 12 requirements in Table 5. Accordingly, the person responsible for explaining the proposed methodology and the three decision-makers were asked to evaluate the effectiveness of the proposed methodology using a numerical scale. The evaluation results are shown in Table 6. Various XAI techniques to explain AI applications in selecting suitable 3D printing facilities in UM are compared in Table 7. An average score of 0.5 was used as the threshold for judging the effectiveness of a method. Most existing XAI techniques can only meet up to nine requirements, while the proposed methodology (XAI techniques V, VI, VIII to X) satisfies 11 requirements.

  • The new XAI tools proposed in this study are not intended to replace existing ones but to address their deficiencies. Applying multiple XAI tools, either existing or proposed in this study, simultaneously can achieve better synergy, especially when users come from different backgrounds.

Table 5 Verifying effectiveness of proposed methodology against 12 requirements
Full size table
Table 6 Evaluation results
Full size table
Table 7 Comparison of XAI techniques for AI applications to 3D printing facility selection
Full size table

Conclusion and future research directions

AI technologies have been widely used to assist in 3D printing facility selection in UM. However, these AI applications may not be easy to understand or communicate with, especially for decision-makers who lack a background in AI, which limits the usefulness of such applications. This study addresses this issue by reviewing existing XAI techniques for AI applications in selecting suitable 3D printing facilities in UM. Four XAI techniques are proposed in this study to address the weaknesses of existing techniques: (1) a gradient bar chart with baseline, (2) a group gradient bar chart, (3) a manually adjustable gradient bar chart, and (4) a bidirectional scatterplot.

A case from the literature was used to illustrate the applicability of the proposed methodology and compare it with existing XAI techniques. The key findings are as follows:

  1. 1.

    The proposed methodology can address the shortcomings of existing methods in modifying the derivation process and results and visually comparing 3D printing facilities.

  2. 2.

    Of the methods compared, only the proposed methodology can satisfy all the requirements of an effective XAI technique in explaining AI applications for selecting suitable 3D printing facilities in UM.

An advantage of the proposed methodology over existing methods is that it satisfies more requirements for an effective XAI tool. Moreover, the four proposed XAI tools are easily applied across domains. In contrast, a limitation of the proposed methodology is the inefficiency of ACO in deriving the fuzzy priority of criteria. Furthermore, the assessment of the effectiveness of the proposed methodology relies on subjective feedback from users, which may vary from case to case.

The proposed methodology can be applied to other decision-making problems in manufacturing [30, 58,59,60,61]. Furthermore, although the proposed methodology aims at explaining ACO and FTOPSIS applications in this study, the same treatments can be taken to explain other similar AI technology applications, such as those based on fuzzy VIseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR), the fuzzy measuring attractiveness by a categorical based evaluation technique (MACBETH), and the fuzzy ELimination Et Choix Traduisant la REalité (ELECTRE) method. These constitute some directions for future research.