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A novel collaborative decision-making method based on generalized abductive learning for resolving design conflicts

A novel collaborative decision-making method based on generalized abductive learning for... In complex product design, lots of time and resources are consumed to choose a preference-based compromise decision from non-inferior preliminary design models with multi-objective conflicts. However, since complex products involve intensive multi-domain knowledge, preference is not only a comprehensive representation of objective data and subjective knowledge but also characterized by fuzzy and uncertain. In recent years, enormous challenges are involved in the design process, within the increasing complexity of preference. This article mainly proposes a novel decision-making method based on generalized abductive learning (G-ABL) to achieve autonomous and efficient decision-making driven by data and knowledge collaboratively. The proposed G-ABL framework, containing three cores: classifier, abductive kernel, and abductive machine, supports preference integration from data and fuzzy knowledge. In particular, a subtle improvement is presented for WK-means based on the entropy weight method (EWM) to address the local static weight problem caused by the fixed data preferences as the decision set is locally invariant. Furthermore, fuzzy comprehensive evaluation (FCE) and Pearson correlation are adopted to quantify domain knowledge and obtain abducted labels. Multi-objective weighted calculations are utilized only to label and compare solutions in the final decision set. Finally, an engineering application is provided to verify the effectiveness of the proposed method, and the superiority of which is illustrated by comparative analysis. Keywords: Collaborative decision-making, Conflict resolution, Generalized abductive learning, EWM based WK-means, Fuzzy comprehensive evaluation 1 Introduction stemming from the difficulty of balancing both data and Preference-based decision-making with multi-objective knowledge for mutual benefit [1, 2]. Complex product design is typically categorized as conflicts is a widespread and critical task in the pre- multi-disciplinary design optimization (MDO) problem, liminary design of complex products. However, due to where multi-objective conflicts are widespread [3, 4]. the proliferation of product complexity and scale, prefer- However, there exists no single optimal solution that opti- ence determination, with uncertainty and fuzziness, is not mizes all objectives concurrently. Therefore, conflicts are only based on data-perceived features but also related to always weighed to obtain designer-interested “preferred” amounts of domain knowledge. Immense challenges re- solutions, formally known as the Pareto solution set [5–7]. main in the decision-making of complex product design, Further, designers are accustomed to selecting or devel- oping an appropriate product design in detail concern- Correspondence: 1152448@tongji.edu.cn ing “preferred” solutions. In recent years, research on the College of Electronics and Information Engineering, Tongji University, aided design of complex products based on MDO and Shanghai, 201804, China 2 multi-objective optimization (MOO) has attracted much TUM School of Engineering and Design, Technical University of Munich, Munich, 80333, Germany attention and obtained fruitful achievements, which are © The Author(s) 2023. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Cui et al. Autonomous Intelligent Systems (2023) 3:3 Page 2 of 13 applied in various engineering fields such as electromag- above two. However, existing researches generally follow a netics and space [8–10]. serial framework in which one model assists the other, so The Pareto solutions (PS) are unable to realize the en- the auxiliary model is obviously not fully exploited. There- hancement in one optimization objective without deteri- fore, autonomous and efficient decision-making, integrat- orating their performance in at least one of the rest [11]. ing preferences from data and knowledge in a mutually For complex product design problems with multiple ob- beneficial mode, becomes a novel problem for urgent at- jectives, variables, and constraints, it is indispensable to tention. keep the diversity of solutions so that solutions are suf- Recently, Zhou [28] proposed the abductive learning ficiently close to the Pareto front. As a result, the num- (ABL) method for bridging data-driven machine learning ber of preferred solutions recommended to designers re- and knowledge-driven logical reasoning. The ABL frame- mains large. Designers from different disciplines and de- work, originally created for classification, will undoubt- partments have to consume a lot of time and resources on edly help to address design decision-making problems co- collaborative actions (calculation, verification, discussion, driven by data and knowledge. However, to the best knowl- consensus, etc.) to obtain a compromise decision [12]. This edge of authors, no published related researches are avail- status fails to address the increasing engineering require- able. Simultaneously, ABL uses logical reasoning, which is ments of improving product design efficiency and shorten- helpless to deal with fuzzy domain knowledge in complex ing the research and development cycle. Hence, this article product design decision-making. This article offers the fol- seeks to propose a collaborative decision-making method lowing contributions. 1) The ABL framework is general- to help designers decide the most preferred solution more ized to fit MODM. In the G-ABL framework, it is possi- autonomously and efficiently. ble to realize the knowledge flow and data flow of deci- Multi-objective decision-making (MODM) is an effec- sion preference to develop in parallel and mutually bene- tive method to resolve multi-attribute conflicts to ob- ficial. 2) The G-ABL based collaborative decision-making tain a comprehensive solution [13]. The pivotal problem method is proposed to realize data and fuzzy knowledge of MODM is to acquire the appropriate decision pref- co-drivendesigndecision-making forcomplex products. erences, which are traditionally mapped into different 3) An engineering problem, the decision of horizontal tail objective weights [14–16]. Related researches, Entropy- control system design, is better addressed based on the based method [17], AHP [18], TOPSIS [19], and CCDS proposed method. [20], etc., contribute to measuring weights. Due to the The remainder of this article is arranged as follows. increasing scale and complexity of products, the uncer- Section 2 introduces some preliminaries. A novel G-ABL tainty, ambiguity, and disciplinary differences in prefer- framework is presented in Sect. 3.InSect. 4,the G-ABL- ences are prominent. Thus, collaborative decision-making based collaborative decision-making method is proposed, attracts research attention. Meaningful applications based and the “Classifier”, “Abductive Kernel”, and “Abductive on weighted least squares [21], distributed gradient [22], Machine” of the proposed method are established, respec- or weight estimation [23] are proposed in this area. Si- tively. Section 5 provides an engineering application to multaneously, various innovation methods are proposed. Zhang and Li [24] combined the fuzzy c-means method verify the effectiveness of the proposed method. Finally, (FCM) with the gray relational projection (GRP) to extract Sect. 6 and Sect. 7 discuss and conclude this article. the best compromise solution reflecting preferences in PS. A cross-entropy measure is presented to address the un- 2Preliminaries known attribute weights problem [25]. In addition, pref- This section introduces some propaedeutics of the pro- erence deviation and credibility of experts are concerned posed method. and discussed [26, 27]. Based on the above analysis, previous methods can be 2.1 Abductive learning divided into three categories: method based on expert A novel framework, Abductive Learning (ABL), bridges knowledge, method based on data feature analysis, and in- two independently evolving paradigms to artificial intelli- tegration method. However, there exist respective limita- gence for mutual benefit—Machine Learning and Logical tions. 1) The expert knowledge-based method has strong Reasoning [29]. Abduction differs from traditional deduc- subjective characteristics, and there still exist preference tion and induction, its essence is to correct data-perceived deviation and reliability risk, and even unknown knowl- facts in machine learning with prior knowledge. With it- edge problem. 2) The data-based method is relatively erative learning, the accuracy and reliability of machine easy to calculate. However, as for knowledge-intensive decision-making of complex product design, it is obviously learning models are significantly improved after consis- inadvisable to make decisions ignoring knowledge. 3) The tency optimization. The classical ABL framework is shown integrated method can combine the advantages of the in Fig. 1 [30]. Cui et al. Autonomous Intelligent Systems (2023) 3:3 Page 3 of 13 Figure 1 Classical ABL framework 2.2 EWM uation vector B. The entropy weight method is an objective weighting method based on data [31]. The information entropy is B = W ◦ R,(4) applied to measure the magnitude of objective variability and obtain objective weights. Suppose that the dimension- where W is the weight vector of the target, R is the fuzzy less data matrix X . n is the number of solutions, and m n×m evaluation matrix. “◦”isthe fuzzyoperator. is the number of objectives. The objective information en- Especially, after processing by the maximum member- tropy is given as ship principle or fuzzy scoring method, B can be regarded as a label corresponding to the alternative. The member- 1 ship function is obtained empirically and the fuzzy matrix e =– p ln p , j = 1,2,..., m,(1) j ij ij R is solved as a result. The process offers the possibility of ln n i=1 giving a wealth of domain knowledge to R.Itisthe basis n for introducing FCE into the proposed method. where p = X / X .If p =0, let e =0. The objective ij ij ij ij j i=1 entropy weight can be calculated by 3 G-ABL framework When AI is applied in knowledge-intensive complex sys- 1– e w = , j = 1,2,..., m.(2) tem engineering, the independence of the two main types 1– e j=1 of information, data and knowledge, needs to be broken. Decision-making that combines data and knowledge is al- 2.3 WK-means most certain. Thus, broadly, ABL can be regarded as an WK-means is an intelligent clustering algorithm with the intelligent framework integrating data and knowledge to autonomous calculation of objective weights [32]. The in- accommodate advances in engineering information. troduced iterative weight quantifies the importance of the The main challenge addressed by classical ABL is bridg- characterization objective in the clustering to reduce the ing numerical optimization and symbolic logic to evolve influence of the interference dimension. Its minimization artificial intelligence. Admittedly, data is the source of in- objective function is defined as formation for numerical optimization (machine learning model). However, due to the uncertainty and ambiguity k n m of complex product knowledge, the proportion of knowl- P(U, Z, W)= u w · d(x , z ), (3) il ij lj l=1 i=1 j=1 edge that can be expressed explicitly and with complete credibility decreases. Logical reasoning based on formal where U is the cluster assignment matrix, Z is the cluster symbols is not a single method for exploiting knowledge, center matrix and W is the weight vector. k is the number much less a single criterion for correcting facts inducted of classes. β is a dynamic parameter, varying with iteration, from data. Hence, we presented a novel G-ABL frame- for objective weight w . work, shown in Fig. 2. Simultaneously, the consistency op- timization in classical ABL should also be transformed. 2.4 FCE The G-ABL framework contains three primary blocks: FCE is a method based on fuzzy criteria, comprehensively Classifier, Abductive Kernel, and Abductive Machine. All evaluating alternatives [33].Thecoreistogiveafuzzyeval- of them are general terms for a class of methods, algo- Cui et al. Autonomous Intelligent Systems (2023) 3:3 Page 4 of 13 Figure 2 G-ABL framework rithms, or principles with similar functions, whose details direction is to select data that better match the preferences are explained as follows. and eliminate the rest, the G-ABL framework works for 1) Classifier solving MODM problems by combining data and knowl- The classifier outputs labels based on numerical edge. feature analysis. The initial classifier can be obtained by clustering, transferring, etc. Since the training 4 Collaborative decision-making method based on data with real labels are sparse, pseudo labels (red G-ABL labels in Fig. 2), even a large number, may exist in the This section first provides an overview of the collaborative previous generations. decision-making method, followed by the internal details. 2) Abductive Kernel The abductive kernel (a generalization of logical 4.1 Methodology reasoning) can output labels based on In complex product design, the collaborative decision- knowledge-based methods, such as reasoning, and making task is to select a “most preferred” design from evaluation. Incompletely reliable grey labels may a set of “preferred” preliminary design models with con- exist. However, aided by prior knowledge, grey labels flicting objectives, autonomously and efficiently based on are fewer and more credible than pseudo labels. decision preferences. Formally, the input of the proposed 3) Abductive Machine method is a Pareto solution set constituted by various non- The abductive machine (a generalization of dominated solutions, which is described as consistency optimization) can correct data-based labels by referring to knowledge-based labels based Ps =(X , X ,..., X ),(5) 1 2 N on established principles, and obtain abducted labels. The determination of abducted labels combines two where X =(x , x ,..., x ), i = 1,2,..., N, x , l = 1,2,..., L i i1 i2 iL il are optimization variables. Correspondingly, the Pareto information sources, data and knowledge. Generally, front is represented as there are more correct labels in abducted labels, whose reliability is superior to the above two. As the abductive process iterates, the continuously aug- Pf = f (Ps)=(Y , Y ,..., Y ),(6) 1 2 N mented supervised data optimize the classifier perfor- mance to excel in addressing the classification problem. It where Y = f (X )=(f (X ), f (X ),..., f (X )) = (y , y ,..., i i 1 i 2 i M i i1 i2 falls within the scope of classical ABL. However, the pos- y ), f (·) is combinatorial mapping of multi-objective iM sibilities for data updates go far beyond that. If the update functions, f (·), m = 1,2,..., M are objective functions. m Cui et al. Autonomous Intelligent Systems (2023) 3:3 Page 5 of 13 Figure 3 Collaborative decision-making method based on G-ABL In terms of the G-ABL framework, the process of the where (X , Y ), i = 1,2,..., N are solutions of the decision i i collaborative decision-making method can be depicted in set. The input Pf is expanded as Fig. 3. EWM based WK-means is proposed as a classifier ⎡ ⎤ to address the classification problem with iterative variable y y ··· y 11 12 1M weights that symbolize data preference. FCE is imported ⎢ ⎥ y y ··· y 21 22 2M ⎢ ⎥ as an abductive kernel to evaluate knowledge-based labels. Pf = ,(8) ⎢ ⎥ . . . . . . . ⎣ ⎦ . . . The membership function generated from prior domain y y ··· y knowledge and fuzzy logic can effectively measure com- N1 N2 NM plex knowledge preferences. The label update principle is defined as an abductive machine to obtain abductive labels where N is the number of Pareto solutions in the current based on Pearson correlation. decision set and M is the number of objectives. All objec- Three points should be interpreted. tives are assumed to be minimization objectives (subse- 1) Since WK-means is a clustering method, labeling quent calculations are based on the premise, and if other is necessary. We calculate the multi-objective weighted kinds of objectives exist, they are transferred to mini- mean within a class, while the later inter-class comparison mize.). Then, Pf is forward-oriented. For a single y in Pf , ij is made to label. The multi-objective weighted sums in the we have last generation are also used to select the “most preferred” solution. y = max(y )– y + σ,(9) ij j ij 2) Data update means removing the Pareto solutions la- beledasthe currentworst classinthe decision set. where σ is a minimal positive number and y >0 is guar- ij 3) There are two iteration termination conditions listed anteed. Then, y is nondimensionalized as ij below. One of them is satisfied to stop the iteration. • Condition 1 ij The number of Pareto solutions in the abducted ε = . (10) ij decision set is less than the class number. max(y ) • Condition 2 Only the best class has elements and the rest are The information entropy of jth objective is calculated by empty. 1 ε ε ij ij 4.2 Classifier: improved EWM based WK-means e =– · ln . (11) N N ln N ε ε Formally, the decision set is represented as ij ij i=1 i=1 i=1 ⎛ ⎞ (X , Y ) 1 1 The entropy weight of jth objective is calculated by ⎜ ⎟ (X , Y ) 2 2 ⎜ ⎟ Ds =(Ps, Pf )= ,(7) ⎜ ⎟ 1– e ⎝ ⎠ j w = . (12) 1– e (X , Y ) j N N j=1 Cui et al. Autonomous Intelligent Systems (2023) 3:3 Page 6 of 13 Entropy weight vector is obtained as Extending to the entire decision set, all solutions are given the data-based label Dlb(i), i = 1,2,..., N, i.e., W =(w , w ,..., w ). (13) 1 2 M if : (X , Y ) ∈ Class – k,then: Dlb(i)= lb . (19) i i k The entropy weight vector W is introduced to quantify Hereto, labeling is completed, resulting in the data preferences of the decision set, so that weights are closely related to the decision set. In the global decision- ∀(X , Y ) ∈ Ds, ∃Dlb(i), ¬Dlb(i) ← C (X , Y ), (20) i i i i making process, W should be constantly changing with the decision set. However, for a single abductive deduc- where C (·) denotes classified mapping. tion process, the decision set is locally invariant. Hence, the weights in WK-means also should not change with 4.3 Abductive kernel: FCE the iteration of the clustering algorithm. We describe the For all Solutions in the decision set, the index set of the above situation as a local static weight problem. The orig- FCEisconstructed as inal WK-means model, shown in Eq. (3), can not im- plement static weights in algorithm iterations. There- U =(u , u ,..., u ), i = 1,2,..., N, (21) i 1 2 M fore, a subtle improvement is presented, and the ob- jective function of the WK-means model is modified where u , j = 1,2,..., M is the jth dimensional optimization to objective of the Pareto front. The remark set is constructed as K N M P(U, Z, W)= u w · d(y , z ), (14) il j ij kj V =(v , v ,..., v ) = (1,2,..., K), i 1 2 K (22) k=1 i=1 j=1 i = 1,2,..., N. where K is the predetermined number of classes. The Especially, the number of remark levels is equal to the detailed clustering process of WK-means is described in number of classes. Then, the domain knowledge induced [32]. Here, it is simplified to a clustering mapping C(·), by prior expertise needs to be transferred to the fuzzy ma- as trix R to form knowledge preferences. However, the pro- posed method emphasizes the autonomy and efficiency of pre lb ← C(X , Y ), i = 1,2,..., N, (15) i i i decision-making. For each solution, there must be an R corresponding to it, and diversity and iteration-varying ex- pre where lb is the clustering label. However, clustering la- ist. Obviously, fuzzy statistics and expert scoring methods bels can only represent classes but cannot judge the fea- are not applicable. Therefore, the membership function is tures of classes. For example, assuming that the cluster- constructed for each index in a hierarchy based on domain ing results are “Class-1”and “Class-2”. But which class knowledge, represented as of solutions is more consistent with the current data preferences on the Pareto front? In response, a multi- f (·), j1 objective weighted mean vector is established. Suppose f (·), j2 “Class-k”contains Num solutions, whose correspond- A F (·)= j = 1,2,..., M. (23) ing Pareto front is represented as Y =(y , y ,..., y ), j = j j1 j2 jM ⎪ 1,2,..., Num . The multi-objective weighted mean Fm is k k A f (·), jK calculated by Then, there is a fuzzy matrix R for any solution (X , Y ). i i i Num W · ξ j=1 Fm = , k = 1,2,..., K, (16) R (X , Y ) i i i Num ⎡ ⎤ A A A f (y ) f (y ) ··· f (y ) i1 i1 i1 11 12 1K where ξ =(ε , ε ,..., ε ), by Eq. (10). Arrange Fm in de- j j1 j2 jM k A A A ⎢ ⎥ f (y ) f (y ) ··· f (y ) i2 i2 i2 scending order to obtain Fm,as 21 22 2K ⎢ ⎥ = ⎢ ⎥ . (24) . . . . . . . ⎣ ⎦ . . . Fm =(Fm ,Fm ,Fm ,...,Fm ) . (17) A A A 1 k 3 2 1×K f (y ) f (y ) ··· f (y ) iM iM iM M1 M2 MK Then, all solutions in “Class-k”obtainthe same label lb , The fuzzy comprehensive evaluation of the solution is as calculated as lb = (0,1,0,...,0) , k = 1,2,..., K. (18) B (X , Y )= W ◦ R =(b , b ,..., b ) , (25) k 1×K i i i i i1 i2 iK 1×K Cui et al. Autonomous Intelligent Systems (2023) 3:3 Page 7 of 13 Algorithm 1 Label Update Principle where W is the current objective entropy weight (by Input: Dlb, Klb =(b , b , ··· , b ), class number K. 1 2 K Eq. (13)). Extrapolating to the entire decision set, all Output: Alb. solutions will obtain the knowledge-based label Klb(i), 1: corr = r(Dlb, Klb). as 2: if corr >2/3 then 3: Alb = Dlb. Klb(i)= B , i = 1,2,..., N. (26) 4: else if 1/3 < corr <2/3 then T T Dlb·V +Klb·V 5: tag = ceil( ). ceil(·) is the round-up Hereto, the task of abductive kernel is achieved, resulting 2 function in 6: while i ≤ K do 7: if i == tag then ∀(X , Y ) ∈ Ds, ∃Klb(i), ¬Klb(i) ← K (X , Y ), (27) i i i i 8: Bool(i)=1. 9: else where K (·) is abductive kernel mapping. 10: Bool(i)=0. 11: end if 4.4 Abductive machine: label update principle based on 12: Alb =(Bool(1), Bool(2), ··· , Bool(K)). Pearson correlation 13: i = i +1. Hereinbefore, the data-based label Dlb and the knowledge- 14: end while based label Klb of solutions are obtained, both of which 15: else are K-dimension vectors. Combining information about 16: max = Max(Klb). Max(·) means to select the thetwo typesoflabelsisessential to form appropriateab- largest element in a vector ducted labels. Pearson correlation allows to analyze the de- 17: while i ≤ K do gree of linear correlation and it is a valid measure of the 18: if b == max then correlation between Dlb and Klb.The correlationcoeffi- 19: Bool(i)=1. cient is calculated as 20: else 21: Bool(i)=0. r(X, Y ) 22: end if n n n n x y – x · y i i i i i=1 i=1 i=1 23: Alb =(Bool(1), Bool(2), ··· , Bool(K)). =   . (28) 2 2 n n n n 2 2 24: i = i +1. n x –( x ) · n y –( x ) i i i i i=1 i=1 i=1 i=1 25: end while 26: end if Then, the resulting correlation coefficients are divided 27: return Alb. into three levels, with different principles for obtaining the abducted label Alb. The details of the label update princi- pleare showninAlgorithm 1. • Level 1 ∗ ∗ ∗ (w , w ,..., w ), is calculated again. The multi-objective 1 2 M r(Dlb, Klb)>2/3, implies similar data preferences weighted sums are calculated by and knowledge preferences. Dlb is used as Alb. • Level 2 1/3 ≤ r(Dlb, Klb) ≤ 2/3,implies uncertaintyexists, F = w ε , i = 1,2,..., N , (29) i j ij f Alb is co-determined by Dlb and Klb. Especially, to j=1 prevent the removal of solutions with a high degree of uncertainty, Alb should be assigned to a better class where N is the number of the solutions in Ds . Finally, the f f whenever possible. solution with the minimal multi-objective weighted sum • Level 3 will be the “most preferred” one. r(Dlb, Klb)<1/3, implies the existence of a significant conflict. In this case, the reliability of Klb is 5 Experiment usually high. The maximum membership method is In this section, an engineering application, the decision used to correct Klb to obtain Alb. problem of the horizontal tail control system design, is Hereto, the abducted label Alb is obtained. According used to verify the effectiveness of the proposed method. to the methodology proposed in Subsect. 4.1, when the Data required for this article are derived from our previ- iteration is terminated, the final decision set Ds is de- termined. If Ds contains a single solution, it is consid- ous research [12], a multi-objective optimization design of ered to be the “most preferred” solution recommended the horizontal tail control system. Finally, simulation and to designers. Otherwise, the entropy weight vector, W = comparison analysis are involved. f Cui et al. Autonomous Intelligent Systems (2023) 3:3 Page 8 of 13 Figure 4 Initial Pareto front 5.1 Data and problem description The horizontal tail is one of the key components to con- trol the pitch stability of an aircraft [34]. Its design process involves mechanical, electronic, hydraulic and control dis- ciplines, making it a typical aviation complex product. Be- sides, the core of horizontal tail control is a hydraulic servo Figure 5 Membership functions of three objectives system, which is also widely used in various engineering products [35]. In [12], the MOO-based preliminary design is completed for the horizontal tail control system. 43 Pareto solutions 5.2 Details are obtained, the Pareto front is shown in Fig. 4.Each To present the experimental results concisely, 3 classes are Pareto solution represents a non-inferior equipment selec- set in the classifier, respectively named “Excellent Class”, tion in the preliminary design of the horizontal tail control “Medium Class” and “Inferior Class”. Correspondingly, the system. remark set of the abductive kernel is The optimization model consists of five optimization variables and three optimization objectives. The 5 vari- V = (1, 2, 3), i = 1,2,...,43. (31) ables correspond to the models of control valve, hydraulic cylinder, hydraulic oil, transmission components, and hor- Based on the knowledge preferences provided by the do- izontal tail. The 3 objectives are control performance (f ), main experts, the membership functions of the objectives weight (f ), and price(f ), and they are all minimization ob- 2 3 are constructed, shown in Fig. 5. jectives. By Fig. 4, obviously, the optimization objectives Based on the raw data (initial decision set described in are in conflict with each other and non-consistent change Eq. (30)and Fig. 4) and the fuzzy knowledge input (shown trends. in Fig. 5), the G-ABL based collaborative decision-making Excessive recommended preliminary designs are not de- method can be generated as signer friendly and do not have a targeted reference value. • Step 1: data-based labeling Thus, the collaborative decision problem is defined as ob- 1) Input the decision set Ds, calculate the entropy taining a “most preferred” design from these preliminary weight vector W based on Eqs. (8)-(13). designs to push to designers. The initial decision set is con- 2) Input the obtained W to Eq. (14) and perform structed as WK-means to cluster Ds into 3 classes. ⎛ ⎞ 3) Labeling data-based classes (Dlb)for Ds (X , Y ) 1 1 according to Eqs. (16)-(19). ⎜ ⎟ (X , Y ) 2 2 ⎜ ⎟ • Step 2: knowledge-based labeling Ds = ⎜ ⎟ , (30) ⎝ ⎠ 1) Based on the membership function (as Fig. 5), (X , Y ) 43 43 calculate the fuzzy matrix R for Ds 2) Bring the calculated W and R into Eqs. (25)-(26), where X =(x , x ,..., x ), Y =(y , y , y ), i = 1,2,...,43. and obtain the knowledge-based label (Klb)for Ds. i i1 i2 i5 i i1 i2 i3 Cui et al. Autonomous Intelligent Systems (2023) 3:3 Page 9 of 13 Figure 6 Evolutionary process of abducted decision set • Step 3: abduction 1) Input Dlb and Klb,the abducted labelof Ds is obtained based on Algorithm 1. 2) Remove solutions with the worst class label to update Ds. • Step 4: iteration termination judgment If Condition 1 or Condition 2 is satisfied? Yes: obtain the final decision set Ds ;No: return Step 1. After 4 abductive iterations, Condition 2 is satisfied and the iteration terminates. The evolution of the abducted decision set is shown in Fig. 6. The iterative variation of decision set size and objective entropy weight is shown in Fig. 7.Meanwhile,todemonstrate thetrend of conver- gence of the abductive iterations towards the decision pref- erences, the variations in the mean objective function val- Figure 7 Iterative variation of decision set size and objective entropy ues of the excellent solutions are shown in Fig. 8. weight In Fig. 6, it is clear from the Ds that the number of solu- tions is greater than 1, and they all belong to the “Excellent Class”. When W = (0.35974584, 0.40878349, 0.23147067), the multi-objective weighted sums of Ds need to be cal- 5.3 Result analysis culated by Eq. (29), which are listed in Table 1. Horizontal tail control system is military, with a prefer- Based on the description at the end of Subsect. 4.4,solu- ence for performance metrics (including control perfor- tion 8, which has the minimum multi-objective weighted mance and weight) significantly over cost. The weight re- sum in Table 1, is decided as the “most preferred” design, duction demand also serves the improvement of control pushing to designers. The details of solution 8, described performance. Correspondingly, the trend of decreasing in Table 2, illustrate the model selections and the objective cost weight in Fig. 7 is consistent with the decision pref- performances of the horizontal tail control system. erences. Besides, Pareto solutions, the input of the pro- Cui et al. Autonomous Intelligent Systems (2023) 3:3 Page 10 of 13 MPO and FCE are knowledge-driven decision-making methods, and EWM is data-driven. All solutions are non- dominated, and no optimal decision occurs. However, they consider only a single preference and are suitable for deal- ing with simple decision-making problems. Moreover, the classical ABL method is not applicable to address such knowledge-fuzzy decision-making problems in complex product design. In comparison, praiseworthily, the G-ABL method selects the design with the excellent control per- formance and the lightest weight (in Table 3). Further, by Fig. 9, it is evident that the solution decided with the G-ABL method has a shorter rise time and almost no overshoot and oscillation. The outstanding control perfor- mance of the horizontal tail significantly ensures the agility and safety of the aircraft. To sum up, the G-ABL method provides excellent in- sight into the above data and knowledge preferences. The proposed method is advanced because it enables collabo- rative decision-making co-driven by data and knowledge, and gives a novel framework for intelligent autonomous decision-making with strong generalization capability. Especially for complex, knowledge-intensive decision- making tasks, the proposed method is more comprehen- sive and reliable. Figure 8 Mean values of the objective functions for solutions in the “excellent class” 6 Discussion Two aspects need to be further discussed, especially con- cerning the limitations. 1) The proposed method does not Table 1 Multi-objective weighted sums of Ds conform to the convergence of the general heuristic clas- sification learning models. 2) The decided preliminary de- Solution Multi-objective weighted sum sign does not have absolute optimality, merely conforms to 1 0.90726041 2 0.68434845 the decision preferences. 3 0.67019106 Convergence of algorithms is a prerequisite for heuris- 4 0.66471741 tic learning. In classification tasks, the convergence of the 5 0.7296347 classical ABL framework is demonstrated by the improve- 6 0.95559967 ment of the classification accuracy of the classifier with 7 0.65668525 8* 0.643856923* iterations. Where, the classifier can be chosen from var- ious classification learning models, such as SVM, CNN, and RNN. However, essentially, the convergence should be attributed to the classifier itself rather than to the posed method, are mutually non-dominated. It is difficult ABL framework. The significant contribution of the ABL to continue the simultaneous convergence of all objectives. framework is to constantly correct the training data labels Thus, Fig. 8 shows that price is compromised in exchange of the learning model based on prior knowledge during for the reduction of control performance and weight. The iterative training to improve the classifier performance. convergence of the abductive learning process to the deci- In the design decision problem addressed in this paper, sion preferences is proved. the purpose is to achieve the classification evaluation of The decisions of the proposed method are compared designs in the Pareto solution set co-driven by data and with the maximum priority objective method (MPO, pro- knowledge, so that the preliminary design is decided by re- posed by [12]), EWM, and FCE in this engineering prob- moving the “inferior” class solutions with integrated con- lem, shown in Table 3. The control performance is an sideration of data and knowledge preferences. Data-based implicit objective. The surrogate model (established by and knowledge-based classification evaluation in the pro- [12]) measures its performance but is difficult to visualize. posed decision-making method respectively relies on the For better model comparison and validation, a simulation labeling of entropy-based multi-objective weighted sum analysis is provided in Fig. 9. and FCE. The learning process of the classification model Cui et al. Autonomous Intelligent Systems (2023) 3:3 Page 11 of 13 Table 2 Details of the “most preferred” design Control valve Hydraulic cylinder Hydraulic oil Transmission components Horizontal tail Model selection Model 1 Model 1 Model 3 Model 4 Model 1 Control performance Weight (Kg) Price (Thousand CNY) Objective value 0.0394 167 155 Table 3 Decision results based on various methods Type Method Control performance Weight Price Knowledge-Driven MPO 0.0391 171 148 FCE 0.0994 169 140 Data-Driven EWM 0.0994 169 140 Knowledge and Data Co-Driven ABL Not Applicable G-ABL 0.0394 167 155 Figure 9 Simulation Comparison about training and testing is not involved, and its conver- fective structure for the integration of data and knowledge gence is not applicable. By Fig. 7 and Fig. 8,the conver- in decision-making, making possible the mutual benefit of gence of the proposed decision-making method is demon- data and knowledge. strated by the fact that the characteristics of the decision The input of the proposed method is the Pareto solution set such as the changing trends of target values and tar- set obtained by multi-objective optimization, in which the get weights can keep conforming to the prior expertise design solutions are mutually non-dominated. It means stated in Subsect. 5.3. The proposed method may not be that there does not exist an absolutely optimal design the general usage of the G-ABL framework, which has not whose all minimization objectives are the smallest in the been used in typical classification tasks. Thus, its con- Pareto solution set. Simultaneously, it is impossible to con- vergence, when combined with a classification learning tinue the simultaneous convergence of all objectives. This model, remains to be demonstrated, where the limitation research fails to address the above questions. Therefore, a lies. However, the prominent contribution of the proposed preference-based decision has to be made to resolve the G-ABL framework to this paper is that it provides an ef- design conflict in a sub-optimal way. In terms of data pref- Cui et al. Autonomous Intelligent Systems (2023) 3:3 Page 12 of 13 erence, the constraint criterion is set as the design with and method should be further verified by the expanded ap- the smallest entropy-based multi-objective weighted sum plications and adaptive improvements in different specific is relatively optimal. However, prior expertise is ignored. andcustomizedtasks. For example, the cost may be incorrectly treated as the most important target in a military product and given a Acknowledgements large weight. As for knowledge preference, the result of This research is supported by the National Key Research and Development FCE is considered the criterion for judging relative opti- Program of China (Grant No. 2018YFB1700900). mality. However, trends in data are not taken seriously. Funding For example, it is entirely possible to expend significant This work was funded by the National Key R&D Program of China costs for a slight performance improvement, which is re- (2018YFB1700900). dundant and unworthy. Further, a correlation-based algo- Availability of data and materials rithm is proposed in Subsect. 4.4 to complement the rel- Not applicable. ative optimality evaluation criterion when the results ob- tained from the above two criteria are inconsistent. The Declarations essential role of the G-ABL framework is to provide a pat- tern to help achieve a mutually beneficial complementar- Competing interests ity of the two types of preferences, rather than to provide The authors declare that they have no competing interests. a learning model used to surrogate optimality classifica- Author contributions tion criteria. Finally, we illustrate through simulations and All authors contributed to the research’s conception. ZC wrote the paper; QX, comparisons that the preliminary design obtained by the JY, and ZC provided the research idea; JY provided the funding acquisition; ZC proposed method has superior control performance and and WT contributed to the algorithm implementation; CW contributed to the organization, language, writing, and revision of the manuscript. All authors is more consistent with prior preferences. Although the read and approved the final manuscript. evaluation criteria are provided in both conditions and the simulation verification is valid, it is still not enough to sup- Publisher’s Note port the complete and formal demonstration of the opti- Springer Nature remains neutral with regard to jurisdictional claims in mality of the decided design, which needs further research. published maps and institutional affiliations. 7Conclusion Received: 28 August 2022 Revised: 10 December 2022 Accepted: 7 February 2023 This article proposes a G-ABL-based collaborative decision-making method for resolving multi-objective References conflicts during the design process of complex prod- 1. S. Zhou, Y. Cao, Z. Zhang, Y. Liu, System design and simulation integration for complex mechatronic products based on SysML and modelica. J. ucts. A G-ABL framework is first presented, providing Comput.-Aided Des. Comput. 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A novel collaborative decision-making method based on generalized abductive learning for resolving design conflicts

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Abstract

In complex product design, lots of time and resources are consumed to choose a preference-based compromise decision from non-inferior preliminary design models with multi-objective conflicts. However, since complex products involve intensive multi-domain knowledge, preference is not only a comprehensive representation of objective data and subjective knowledge but also characterized by fuzzy and uncertain. In recent years, enormous challenges are involved in the design process, within the increasing complexity of preference. This article mainly proposes a novel decision-making method based on generalized abductive learning (G-ABL) to achieve autonomous and efficient decision-making driven by data and knowledge collaboratively. The proposed G-ABL framework, containing three cores: classifier, abductive kernel, and abductive machine, supports preference integration from data and fuzzy knowledge. In particular, a subtle improvement is presented for WK-means based on the entropy weight method (EWM) to address the local static weight problem caused by the fixed data preferences as the decision set is locally invariant. Furthermore, fuzzy comprehensive evaluation (FCE) and Pearson correlation are adopted to quantify domain knowledge and obtain abducted labels. Multi-objective weighted calculations are utilized only to label and compare solutions in the final decision set. Finally, an engineering application is provided to verify the effectiveness of the proposed method, and the superiority of which is illustrated by comparative analysis. Keywords: Collaborative decision-making, Conflict resolution, Generalized abductive learning, EWM based WK-means, Fuzzy comprehensive evaluation 1 Introduction stemming from the difficulty of balancing both data and Preference-based decision-making with multi-objective knowledge for mutual benefit [1, 2]. Complex product design is typically categorized as conflicts is a widespread and critical task in the pre- multi-disciplinary design optimization (MDO) problem, liminary design of complex products. However, due to where multi-objective conflicts are widespread [3, 4]. the proliferation of product complexity and scale, prefer- However, there exists no single optimal solution that opti- ence determination, with uncertainty and fuzziness, is not mizes all objectives concurrently. Therefore, conflicts are only based on data-perceived features but also related to always weighed to obtain designer-interested “preferred” amounts of domain knowledge. Immense challenges re- solutions, formally known as the Pareto solution set [5–7]. main in the decision-making of complex product design, Further, designers are accustomed to selecting or devel- oping an appropriate product design in detail concern- Correspondence: 1152448@tongji.edu.cn ing “preferred” solutions. In recent years, research on the College of Electronics and Information Engineering, Tongji University, aided design of complex products based on MDO and Shanghai, 201804, China 2 multi-objective optimization (MOO) has attracted much TUM School of Engineering and Design, Technical University of Munich, Munich, 80333, Germany attention and obtained fruitful achievements, which are © The Author(s) 2023. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Cui et al. Autonomous Intelligent Systems (2023) 3:3 Page 2 of 13 applied in various engineering fields such as electromag- above two. However, existing researches generally follow a netics and space [8–10]. serial framework in which one model assists the other, so The Pareto solutions (PS) are unable to realize the en- the auxiliary model is obviously not fully exploited. There- hancement in one optimization objective without deteri- fore, autonomous and efficient decision-making, integrat- orating their performance in at least one of the rest [11]. ing preferences from data and knowledge in a mutually For complex product design problems with multiple ob- beneficial mode, becomes a novel problem for urgent at- jectives, variables, and constraints, it is indispensable to tention. keep the diversity of solutions so that solutions are suf- Recently, Zhou [28] proposed the abductive learning ficiently close to the Pareto front. As a result, the num- (ABL) method for bridging data-driven machine learning ber of preferred solutions recommended to designers re- and knowledge-driven logical reasoning. The ABL frame- mains large. Designers from different disciplines and de- work, originally created for classification, will undoubt- partments have to consume a lot of time and resources on edly help to address design decision-making problems co- collaborative actions (calculation, verification, discussion, driven by data and knowledge. However, to the best knowl- consensus, etc.) to obtain a compromise decision [12]. This edge of authors, no published related researches are avail- status fails to address the increasing engineering require- able. Simultaneously, ABL uses logical reasoning, which is ments of improving product design efficiency and shorten- helpless to deal with fuzzy domain knowledge in complex ing the research and development cycle. Hence, this article product design decision-making. This article offers the fol- seeks to propose a collaborative decision-making method lowing contributions. 1) The ABL framework is general- to help designers decide the most preferred solution more ized to fit MODM. In the G-ABL framework, it is possi- autonomously and efficiently. ble to realize the knowledge flow and data flow of deci- Multi-objective decision-making (MODM) is an effec- sion preference to develop in parallel and mutually bene- tive method to resolve multi-attribute conflicts to ob- ficial. 2) The G-ABL based collaborative decision-making tain a comprehensive solution [13]. The pivotal problem method is proposed to realize data and fuzzy knowledge of MODM is to acquire the appropriate decision pref- co-drivendesigndecision-making forcomplex products. erences, which are traditionally mapped into different 3) An engineering problem, the decision of horizontal tail objective weights [14–16]. Related researches, Entropy- control system design, is better addressed based on the based method [17], AHP [18], TOPSIS [19], and CCDS proposed method. [20], etc., contribute to measuring weights. Due to the The remainder of this article is arranged as follows. increasing scale and complexity of products, the uncer- Section 2 introduces some preliminaries. A novel G-ABL tainty, ambiguity, and disciplinary differences in prefer- framework is presented in Sect. 3.InSect. 4,the G-ABL- ences are prominent. Thus, collaborative decision-making based collaborative decision-making method is proposed, attracts research attention. Meaningful applications based and the “Classifier”, “Abductive Kernel”, and “Abductive on weighted least squares [21], distributed gradient [22], Machine” of the proposed method are established, respec- or weight estimation [23] are proposed in this area. Si- tively. Section 5 provides an engineering application to multaneously, various innovation methods are proposed. Zhang and Li [24] combined the fuzzy c-means method verify the effectiveness of the proposed method. Finally, (FCM) with the gray relational projection (GRP) to extract Sect. 6 and Sect. 7 discuss and conclude this article. the best compromise solution reflecting preferences in PS. A cross-entropy measure is presented to address the un- 2Preliminaries known attribute weights problem [25]. In addition, pref- This section introduces some propaedeutics of the pro- erence deviation and credibility of experts are concerned posed method. and discussed [26, 27]. Based on the above analysis, previous methods can be 2.1 Abductive learning divided into three categories: method based on expert A novel framework, Abductive Learning (ABL), bridges knowledge, method based on data feature analysis, and in- two independently evolving paradigms to artificial intelli- tegration method. However, there exist respective limita- gence for mutual benefit—Machine Learning and Logical tions. 1) The expert knowledge-based method has strong Reasoning [29]. Abduction differs from traditional deduc- subjective characteristics, and there still exist preference tion and induction, its essence is to correct data-perceived deviation and reliability risk, and even unknown knowl- facts in machine learning with prior knowledge. With it- edge problem. 2) The data-based method is relatively erative learning, the accuracy and reliability of machine easy to calculate. However, as for knowledge-intensive decision-making of complex product design, it is obviously learning models are significantly improved after consis- inadvisable to make decisions ignoring knowledge. 3) The tency optimization. The classical ABL framework is shown integrated method can combine the advantages of the in Fig. 1 [30]. Cui et al. Autonomous Intelligent Systems (2023) 3:3 Page 3 of 13 Figure 1 Classical ABL framework 2.2 EWM uation vector B. The entropy weight method is an objective weighting method based on data [31]. The information entropy is B = W ◦ R,(4) applied to measure the magnitude of objective variability and obtain objective weights. Suppose that the dimension- where W is the weight vector of the target, R is the fuzzy less data matrix X . n is the number of solutions, and m n×m evaluation matrix. “◦”isthe fuzzyoperator. is the number of objectives. The objective information en- Especially, after processing by the maximum member- tropy is given as ship principle or fuzzy scoring method, B can be regarded as a label corresponding to the alternative. The member- 1 ship function is obtained empirically and the fuzzy matrix e =– p ln p , j = 1,2,..., m,(1) j ij ij R is solved as a result. The process offers the possibility of ln n i=1 giving a wealth of domain knowledge to R.Itisthe basis n for introducing FCE into the proposed method. where p = X / X .If p =0, let e =0. The objective ij ij ij ij j i=1 entropy weight can be calculated by 3 G-ABL framework When AI is applied in knowledge-intensive complex sys- 1– e w = , j = 1,2,..., m.(2) tem engineering, the independence of the two main types 1– e j=1 of information, data and knowledge, needs to be broken. Decision-making that combines data and knowledge is al- 2.3 WK-means most certain. Thus, broadly, ABL can be regarded as an WK-means is an intelligent clustering algorithm with the intelligent framework integrating data and knowledge to autonomous calculation of objective weights [32]. The in- accommodate advances in engineering information. troduced iterative weight quantifies the importance of the The main challenge addressed by classical ABL is bridg- characterization objective in the clustering to reduce the ing numerical optimization and symbolic logic to evolve influence of the interference dimension. Its minimization artificial intelligence. Admittedly, data is the source of in- objective function is defined as formation for numerical optimization (machine learning model). However, due to the uncertainty and ambiguity k n m of complex product knowledge, the proportion of knowl- P(U, Z, W)= u w · d(x , z ), (3) il ij lj l=1 i=1 j=1 edge that can be expressed explicitly and with complete credibility decreases. Logical reasoning based on formal where U is the cluster assignment matrix, Z is the cluster symbols is not a single method for exploiting knowledge, center matrix and W is the weight vector. k is the number much less a single criterion for correcting facts inducted of classes. β is a dynamic parameter, varying with iteration, from data. Hence, we presented a novel G-ABL frame- for objective weight w . work, shown in Fig. 2. Simultaneously, the consistency op- timization in classical ABL should also be transformed. 2.4 FCE The G-ABL framework contains three primary blocks: FCE is a method based on fuzzy criteria, comprehensively Classifier, Abductive Kernel, and Abductive Machine. All evaluating alternatives [33].Thecoreistogiveafuzzyeval- of them are general terms for a class of methods, algo- Cui et al. Autonomous Intelligent Systems (2023) 3:3 Page 4 of 13 Figure 2 G-ABL framework rithms, or principles with similar functions, whose details direction is to select data that better match the preferences are explained as follows. and eliminate the rest, the G-ABL framework works for 1) Classifier solving MODM problems by combining data and knowl- The classifier outputs labels based on numerical edge. feature analysis. The initial classifier can be obtained by clustering, transferring, etc. Since the training 4 Collaborative decision-making method based on data with real labels are sparse, pseudo labels (red G-ABL labels in Fig. 2), even a large number, may exist in the This section first provides an overview of the collaborative previous generations. decision-making method, followed by the internal details. 2) Abductive Kernel The abductive kernel (a generalization of logical 4.1 Methodology reasoning) can output labels based on In complex product design, the collaborative decision- knowledge-based methods, such as reasoning, and making task is to select a “most preferred” design from evaluation. Incompletely reliable grey labels may a set of “preferred” preliminary design models with con- exist. However, aided by prior knowledge, grey labels flicting objectives, autonomously and efficiently based on are fewer and more credible than pseudo labels. decision preferences. Formally, the input of the proposed 3) Abductive Machine method is a Pareto solution set constituted by various non- The abductive machine (a generalization of dominated solutions, which is described as consistency optimization) can correct data-based labels by referring to knowledge-based labels based Ps =(X , X ,..., X ),(5) 1 2 N on established principles, and obtain abducted labels. The determination of abducted labels combines two where X =(x , x ,..., x ), i = 1,2,..., N, x , l = 1,2,..., L i i1 i2 iL il are optimization variables. Correspondingly, the Pareto information sources, data and knowledge. Generally, front is represented as there are more correct labels in abducted labels, whose reliability is superior to the above two. As the abductive process iterates, the continuously aug- Pf = f (Ps)=(Y , Y ,..., Y ),(6) 1 2 N mented supervised data optimize the classifier perfor- mance to excel in addressing the classification problem. It where Y = f (X )=(f (X ), f (X ),..., f (X )) = (y , y ,..., i i 1 i 2 i M i i1 i2 falls within the scope of classical ABL. However, the pos- y ), f (·) is combinatorial mapping of multi-objective iM sibilities for data updates go far beyond that. If the update functions, f (·), m = 1,2,..., M are objective functions. m Cui et al. Autonomous Intelligent Systems (2023) 3:3 Page 5 of 13 Figure 3 Collaborative decision-making method based on G-ABL In terms of the G-ABL framework, the process of the where (X , Y ), i = 1,2,..., N are solutions of the decision i i collaborative decision-making method can be depicted in set. The input Pf is expanded as Fig. 3. EWM based WK-means is proposed as a classifier ⎡ ⎤ to address the classification problem with iterative variable y y ··· y 11 12 1M weights that symbolize data preference. FCE is imported ⎢ ⎥ y y ··· y 21 22 2M ⎢ ⎥ as an abductive kernel to evaluate knowledge-based labels. Pf = ,(8) ⎢ ⎥ . . . . . . . ⎣ ⎦ . . . The membership function generated from prior domain y y ··· y knowledge and fuzzy logic can effectively measure com- N1 N2 NM plex knowledge preferences. The label update principle is defined as an abductive machine to obtain abductive labels where N is the number of Pareto solutions in the current based on Pearson correlation. decision set and M is the number of objectives. All objec- Three points should be interpreted. tives are assumed to be minimization objectives (subse- 1) Since WK-means is a clustering method, labeling quent calculations are based on the premise, and if other is necessary. We calculate the multi-objective weighted kinds of objectives exist, they are transferred to mini- mean within a class, while the later inter-class comparison mize.). Then, Pf is forward-oriented. For a single y in Pf , ij is made to label. The multi-objective weighted sums in the we have last generation are also used to select the “most preferred” solution. y = max(y )– y + σ,(9) ij j ij 2) Data update means removing the Pareto solutions la- beledasthe currentworst classinthe decision set. where σ is a minimal positive number and y >0 is guar- ij 3) There are two iteration termination conditions listed anteed. Then, y is nondimensionalized as ij below. One of them is satisfied to stop the iteration. • Condition 1 ij The number of Pareto solutions in the abducted ε = . (10) ij decision set is less than the class number. max(y ) • Condition 2 Only the best class has elements and the rest are The information entropy of jth objective is calculated by empty. 1 ε ε ij ij 4.2 Classifier: improved EWM based WK-means e =– · ln . (11) N N ln N ε ε Formally, the decision set is represented as ij ij i=1 i=1 i=1 ⎛ ⎞ (X , Y ) 1 1 The entropy weight of jth objective is calculated by ⎜ ⎟ (X , Y ) 2 2 ⎜ ⎟ Ds =(Ps, Pf )= ,(7) ⎜ ⎟ 1– e ⎝ ⎠ j w = . (12) 1– e (X , Y ) j N N j=1 Cui et al. Autonomous Intelligent Systems (2023) 3:3 Page 6 of 13 Entropy weight vector is obtained as Extending to the entire decision set, all solutions are given the data-based label Dlb(i), i = 1,2,..., N, i.e., W =(w , w ,..., w ). (13) 1 2 M if : (X , Y ) ∈ Class – k,then: Dlb(i)= lb . (19) i i k The entropy weight vector W is introduced to quantify Hereto, labeling is completed, resulting in the data preferences of the decision set, so that weights are closely related to the decision set. In the global decision- ∀(X , Y ) ∈ Ds, ∃Dlb(i), ¬Dlb(i) ← C (X , Y ), (20) i i i i making process, W should be constantly changing with the decision set. However, for a single abductive deduc- where C (·) denotes classified mapping. tion process, the decision set is locally invariant. Hence, the weights in WK-means also should not change with 4.3 Abductive kernel: FCE the iteration of the clustering algorithm. We describe the For all Solutions in the decision set, the index set of the above situation as a local static weight problem. The orig- FCEisconstructed as inal WK-means model, shown in Eq. (3), can not im- plement static weights in algorithm iterations. There- U =(u , u ,..., u ), i = 1,2,..., N, (21) i 1 2 M fore, a subtle improvement is presented, and the ob- jective function of the WK-means model is modified where u , j = 1,2,..., M is the jth dimensional optimization to objective of the Pareto front. The remark set is constructed as K N M P(U, Z, W)= u w · d(y , z ), (14) il j ij kj V =(v , v ,..., v ) = (1,2,..., K), i 1 2 K (22) k=1 i=1 j=1 i = 1,2,..., N. where K is the predetermined number of classes. The Especially, the number of remark levels is equal to the detailed clustering process of WK-means is described in number of classes. Then, the domain knowledge induced [32]. Here, it is simplified to a clustering mapping C(·), by prior expertise needs to be transferred to the fuzzy ma- as trix R to form knowledge preferences. However, the pro- posed method emphasizes the autonomy and efficiency of pre lb ← C(X , Y ), i = 1,2,..., N, (15) i i i decision-making. For each solution, there must be an R corresponding to it, and diversity and iteration-varying ex- pre where lb is the clustering label. However, clustering la- ist. Obviously, fuzzy statistics and expert scoring methods bels can only represent classes but cannot judge the fea- are not applicable. Therefore, the membership function is tures of classes. For example, assuming that the cluster- constructed for each index in a hierarchy based on domain ing results are “Class-1”and “Class-2”. But which class knowledge, represented as of solutions is more consistent with the current data preferences on the Pareto front? In response, a multi- f (·), j1 objective weighted mean vector is established. Suppose f (·), j2 “Class-k”contains Num solutions, whose correspond- A F (·)= j = 1,2,..., M. (23) ing Pareto front is represented as Y =(y , y ,..., y ), j = j j1 j2 jM ⎪ 1,2,..., Num . The multi-objective weighted mean Fm is k k A f (·), jK calculated by Then, there is a fuzzy matrix R for any solution (X , Y ). i i i Num W · ξ j=1 Fm = , k = 1,2,..., K, (16) R (X , Y ) i i i Num ⎡ ⎤ A A A f (y ) f (y ) ··· f (y ) i1 i1 i1 11 12 1K where ξ =(ε , ε ,..., ε ), by Eq. (10). Arrange Fm in de- j j1 j2 jM k A A A ⎢ ⎥ f (y ) f (y ) ··· f (y ) i2 i2 i2 scending order to obtain Fm,as 21 22 2K ⎢ ⎥ = ⎢ ⎥ . (24) . . . . . . . ⎣ ⎦ . . . Fm =(Fm ,Fm ,Fm ,...,Fm ) . (17) A A A 1 k 3 2 1×K f (y ) f (y ) ··· f (y ) iM iM iM M1 M2 MK Then, all solutions in “Class-k”obtainthe same label lb , The fuzzy comprehensive evaluation of the solution is as calculated as lb = (0,1,0,...,0) , k = 1,2,..., K. (18) B (X , Y )= W ◦ R =(b , b ,..., b ) , (25) k 1×K i i i i i1 i2 iK 1×K Cui et al. Autonomous Intelligent Systems (2023) 3:3 Page 7 of 13 Algorithm 1 Label Update Principle where W is the current objective entropy weight (by Input: Dlb, Klb =(b , b , ··· , b ), class number K. 1 2 K Eq. (13)). Extrapolating to the entire decision set, all Output: Alb. solutions will obtain the knowledge-based label Klb(i), 1: corr = r(Dlb, Klb). as 2: if corr >2/3 then 3: Alb = Dlb. Klb(i)= B , i = 1,2,..., N. (26) 4: else if 1/3 < corr <2/3 then T T Dlb·V +Klb·V 5: tag = ceil( ). ceil(·) is the round-up Hereto, the task of abductive kernel is achieved, resulting 2 function in 6: while i ≤ K do 7: if i == tag then ∀(X , Y ) ∈ Ds, ∃Klb(i), ¬Klb(i) ← K (X , Y ), (27) i i i i 8: Bool(i)=1. 9: else where K (·) is abductive kernel mapping. 10: Bool(i)=0. 11: end if 4.4 Abductive machine: label update principle based on 12: Alb =(Bool(1), Bool(2), ··· , Bool(K)). Pearson correlation 13: i = i +1. Hereinbefore, the data-based label Dlb and the knowledge- 14: end while based label Klb of solutions are obtained, both of which 15: else are K-dimension vectors. Combining information about 16: max = Max(Klb). Max(·) means to select the thetwo typesoflabelsisessential to form appropriateab- largest element in a vector ducted labels. Pearson correlation allows to analyze the de- 17: while i ≤ K do gree of linear correlation and it is a valid measure of the 18: if b == max then correlation between Dlb and Klb.The correlationcoeffi- 19: Bool(i)=1. cient is calculated as 20: else 21: Bool(i)=0. r(X, Y ) 22: end if n n n n x y – x · y i i i i i=1 i=1 i=1 23: Alb =(Bool(1), Bool(2), ··· , Bool(K)). =   . (28) 2 2 n n n n 2 2 24: i = i +1. n x –( x ) · n y –( x ) i i i i i=1 i=1 i=1 i=1 25: end while 26: end if Then, the resulting correlation coefficients are divided 27: return Alb. into three levels, with different principles for obtaining the abducted label Alb. The details of the label update princi- pleare showninAlgorithm 1. • Level 1 ∗ ∗ ∗ (w , w ,..., w ), is calculated again. The multi-objective 1 2 M r(Dlb, Klb)>2/3, implies similar data preferences weighted sums are calculated by and knowledge preferences. Dlb is used as Alb. • Level 2 1/3 ≤ r(Dlb, Klb) ≤ 2/3,implies uncertaintyexists, F = w ε , i = 1,2,..., N , (29) i j ij f Alb is co-determined by Dlb and Klb. Especially, to j=1 prevent the removal of solutions with a high degree of uncertainty, Alb should be assigned to a better class where N is the number of the solutions in Ds . Finally, the f f whenever possible. solution with the minimal multi-objective weighted sum • Level 3 will be the “most preferred” one. r(Dlb, Klb)<1/3, implies the existence of a significant conflict. In this case, the reliability of Klb is 5 Experiment usually high. The maximum membership method is In this section, an engineering application, the decision used to correct Klb to obtain Alb. problem of the horizontal tail control system design, is Hereto, the abducted label Alb is obtained. According used to verify the effectiveness of the proposed method. to the methodology proposed in Subsect. 4.1, when the Data required for this article are derived from our previ- iteration is terminated, the final decision set Ds is de- termined. If Ds contains a single solution, it is consid- ous research [12], a multi-objective optimization design of ered to be the “most preferred” solution recommended the horizontal tail control system. Finally, simulation and to designers. Otherwise, the entropy weight vector, W = comparison analysis are involved. f Cui et al. Autonomous Intelligent Systems (2023) 3:3 Page 8 of 13 Figure 4 Initial Pareto front 5.1 Data and problem description The horizontal tail is one of the key components to con- trol the pitch stability of an aircraft [34]. Its design process involves mechanical, electronic, hydraulic and control dis- ciplines, making it a typical aviation complex product. Be- sides, the core of horizontal tail control is a hydraulic servo Figure 5 Membership functions of three objectives system, which is also widely used in various engineering products [35]. In [12], the MOO-based preliminary design is completed for the horizontal tail control system. 43 Pareto solutions 5.2 Details are obtained, the Pareto front is shown in Fig. 4.Each To present the experimental results concisely, 3 classes are Pareto solution represents a non-inferior equipment selec- set in the classifier, respectively named “Excellent Class”, tion in the preliminary design of the horizontal tail control “Medium Class” and “Inferior Class”. Correspondingly, the system. remark set of the abductive kernel is The optimization model consists of five optimization variables and three optimization objectives. The 5 vari- V = (1, 2, 3), i = 1,2,...,43. (31) ables correspond to the models of control valve, hydraulic cylinder, hydraulic oil, transmission components, and hor- Based on the knowledge preferences provided by the do- izontal tail. The 3 objectives are control performance (f ), main experts, the membership functions of the objectives weight (f ), and price(f ), and they are all minimization ob- 2 3 are constructed, shown in Fig. 5. jectives. By Fig. 4, obviously, the optimization objectives Based on the raw data (initial decision set described in are in conflict with each other and non-consistent change Eq. (30)and Fig. 4) and the fuzzy knowledge input (shown trends. in Fig. 5), the G-ABL based collaborative decision-making Excessive recommended preliminary designs are not de- method can be generated as signer friendly and do not have a targeted reference value. • Step 1: data-based labeling Thus, the collaborative decision problem is defined as ob- 1) Input the decision set Ds, calculate the entropy taining a “most preferred” design from these preliminary weight vector W based on Eqs. (8)-(13). designs to push to designers. The initial decision set is con- 2) Input the obtained W to Eq. (14) and perform structed as WK-means to cluster Ds into 3 classes. ⎛ ⎞ 3) Labeling data-based classes (Dlb)for Ds (X , Y ) 1 1 according to Eqs. (16)-(19). ⎜ ⎟ (X , Y ) 2 2 ⎜ ⎟ • Step 2: knowledge-based labeling Ds = ⎜ ⎟ , (30) ⎝ ⎠ 1) Based on the membership function (as Fig. 5), (X , Y ) 43 43 calculate the fuzzy matrix R for Ds 2) Bring the calculated W and R into Eqs. (25)-(26), where X =(x , x ,..., x ), Y =(y , y , y ), i = 1,2,...,43. and obtain the knowledge-based label (Klb)for Ds. i i1 i2 i5 i i1 i2 i3 Cui et al. Autonomous Intelligent Systems (2023) 3:3 Page 9 of 13 Figure 6 Evolutionary process of abducted decision set • Step 3: abduction 1) Input Dlb and Klb,the abducted labelof Ds is obtained based on Algorithm 1. 2) Remove solutions with the worst class label to update Ds. • Step 4: iteration termination judgment If Condition 1 or Condition 2 is satisfied? Yes: obtain the final decision set Ds ;No: return Step 1. After 4 abductive iterations, Condition 2 is satisfied and the iteration terminates. The evolution of the abducted decision set is shown in Fig. 6. The iterative variation of decision set size and objective entropy weight is shown in Fig. 7.Meanwhile,todemonstrate thetrend of conver- gence of the abductive iterations towards the decision pref- erences, the variations in the mean objective function val- Figure 7 Iterative variation of decision set size and objective entropy ues of the excellent solutions are shown in Fig. 8. weight In Fig. 6, it is clear from the Ds that the number of solu- tions is greater than 1, and they all belong to the “Excellent Class”. When W = (0.35974584, 0.40878349, 0.23147067), the multi-objective weighted sums of Ds need to be cal- 5.3 Result analysis culated by Eq. (29), which are listed in Table 1. Horizontal tail control system is military, with a prefer- Based on the description at the end of Subsect. 4.4,solu- ence for performance metrics (including control perfor- tion 8, which has the minimum multi-objective weighted mance and weight) significantly over cost. The weight re- sum in Table 1, is decided as the “most preferred” design, duction demand also serves the improvement of control pushing to designers. The details of solution 8, described performance. Correspondingly, the trend of decreasing in Table 2, illustrate the model selections and the objective cost weight in Fig. 7 is consistent with the decision pref- performances of the horizontal tail control system. erences. Besides, Pareto solutions, the input of the pro- Cui et al. Autonomous Intelligent Systems (2023) 3:3 Page 10 of 13 MPO and FCE are knowledge-driven decision-making methods, and EWM is data-driven. All solutions are non- dominated, and no optimal decision occurs. However, they consider only a single preference and are suitable for deal- ing with simple decision-making problems. Moreover, the classical ABL method is not applicable to address such knowledge-fuzzy decision-making problems in complex product design. In comparison, praiseworthily, the G-ABL method selects the design with the excellent control per- formance and the lightest weight (in Table 3). Further, by Fig. 9, it is evident that the solution decided with the G-ABL method has a shorter rise time and almost no overshoot and oscillation. The outstanding control perfor- mance of the horizontal tail significantly ensures the agility and safety of the aircraft. To sum up, the G-ABL method provides excellent in- sight into the above data and knowledge preferences. The proposed method is advanced because it enables collabo- rative decision-making co-driven by data and knowledge, and gives a novel framework for intelligent autonomous decision-making with strong generalization capability. Especially for complex, knowledge-intensive decision- making tasks, the proposed method is more comprehen- sive and reliable. Figure 8 Mean values of the objective functions for solutions in the “excellent class” 6 Discussion Two aspects need to be further discussed, especially con- cerning the limitations. 1) The proposed method does not Table 1 Multi-objective weighted sums of Ds conform to the convergence of the general heuristic clas- sification learning models. 2) The decided preliminary de- Solution Multi-objective weighted sum sign does not have absolute optimality, merely conforms to 1 0.90726041 2 0.68434845 the decision preferences. 3 0.67019106 Convergence of algorithms is a prerequisite for heuris- 4 0.66471741 tic learning. In classification tasks, the convergence of the 5 0.7296347 classical ABL framework is demonstrated by the improve- 6 0.95559967 ment of the classification accuracy of the classifier with 7 0.65668525 8* 0.643856923* iterations. Where, the classifier can be chosen from var- ious classification learning models, such as SVM, CNN, and RNN. However, essentially, the convergence should be attributed to the classifier itself rather than to the posed method, are mutually non-dominated. It is difficult ABL framework. The significant contribution of the ABL to continue the simultaneous convergence of all objectives. framework is to constantly correct the training data labels Thus, Fig. 8 shows that price is compromised in exchange of the learning model based on prior knowledge during for the reduction of control performance and weight. The iterative training to improve the classifier performance. convergence of the abductive learning process to the deci- In the design decision problem addressed in this paper, sion preferences is proved. the purpose is to achieve the classification evaluation of The decisions of the proposed method are compared designs in the Pareto solution set co-driven by data and with the maximum priority objective method (MPO, pro- knowledge, so that the preliminary design is decided by re- posed by [12]), EWM, and FCE in this engineering prob- moving the “inferior” class solutions with integrated con- lem, shown in Table 3. The control performance is an sideration of data and knowledge preferences. Data-based implicit objective. The surrogate model (established by and knowledge-based classification evaluation in the pro- [12]) measures its performance but is difficult to visualize. posed decision-making method respectively relies on the For better model comparison and validation, a simulation labeling of entropy-based multi-objective weighted sum analysis is provided in Fig. 9. and FCE. The learning process of the classification model Cui et al. Autonomous Intelligent Systems (2023) 3:3 Page 11 of 13 Table 2 Details of the “most preferred” design Control valve Hydraulic cylinder Hydraulic oil Transmission components Horizontal tail Model selection Model 1 Model 1 Model 3 Model 4 Model 1 Control performance Weight (Kg) Price (Thousand CNY) Objective value 0.0394 167 155 Table 3 Decision results based on various methods Type Method Control performance Weight Price Knowledge-Driven MPO 0.0391 171 148 FCE 0.0994 169 140 Data-Driven EWM 0.0994 169 140 Knowledge and Data Co-Driven ABL Not Applicable G-ABL 0.0394 167 155 Figure 9 Simulation Comparison about training and testing is not involved, and its conver- fective structure for the integration of data and knowledge gence is not applicable. By Fig. 7 and Fig. 8,the conver- in decision-making, making possible the mutual benefit of gence of the proposed decision-making method is demon- data and knowledge. strated by the fact that the characteristics of the decision The input of the proposed method is the Pareto solution set such as the changing trends of target values and tar- set obtained by multi-objective optimization, in which the get weights can keep conforming to the prior expertise design solutions are mutually non-dominated. It means stated in Subsect. 5.3. The proposed method may not be that there does not exist an absolutely optimal design the general usage of the G-ABL framework, which has not whose all minimization objectives are the smallest in the been used in typical classification tasks. Thus, its con- Pareto solution set. Simultaneously, it is impossible to con- vergence, when combined with a classification learning tinue the simultaneous convergence of all objectives. This model, remains to be demonstrated, where the limitation research fails to address the above questions. Therefore, a lies. However, the prominent contribution of the proposed preference-based decision has to be made to resolve the G-ABL framework to this paper is that it provides an ef- design conflict in a sub-optimal way. In terms of data pref- Cui et al. Autonomous Intelligent Systems (2023) 3:3 Page 12 of 13 erence, the constraint criterion is set as the design with and method should be further verified by the expanded ap- the smallest entropy-based multi-objective weighted sum plications and adaptive improvements in different specific is relatively optimal. However, prior expertise is ignored. andcustomizedtasks. For example, the cost may be incorrectly treated as the most important target in a military product and given a Acknowledgements large weight. As for knowledge preference, the result of This research is supported by the National Key Research and Development FCE is considered the criterion for judging relative opti- Program of China (Grant No. 2018YFB1700900). mality. However, trends in data are not taken seriously. Funding For example, it is entirely possible to expend significant This work was funded by the National Key R&D Program of China costs for a slight performance improvement, which is re- (2018YFB1700900). dundant and unworthy. Further, a correlation-based algo- Availability of data and materials rithm is proposed in Subsect. 4.4 to complement the rel- Not applicable. ative optimality evaluation criterion when the results ob- tained from the above two criteria are inconsistent. The Declarations essential role of the G-ABL framework is to provide a pat- tern to help achieve a mutually beneficial complementar- Competing interests ity of the two types of preferences, rather than to provide The authors declare that they have no competing interests. a learning model used to surrogate optimality classifica- Author contributions tion criteria. Finally, we illustrate through simulations and All authors contributed to the research’s conception. ZC wrote the paper; QX, comparisons that the preliminary design obtained by the JY, and ZC provided the research idea; JY provided the funding acquisition; ZC proposed method has superior control performance and and WT contributed to the algorithm implementation; CW contributed to the organization, language, writing, and revision of the manuscript. All authors is more consistent with prior preferences. Although the read and approved the final manuscript. evaluation criteria are provided in both conditions and the simulation verification is valid, it is still not enough to sup- Publisher’s Note port the complete and formal demonstration of the opti- Springer Nature remains neutral with regard to jurisdictional claims in mality of the decided design, which needs further research. published maps and institutional affiliations. 7Conclusion Received: 28 August 2022 Revised: 10 December 2022 Accepted: 7 February 2023 This article proposes a G-ABL-based collaborative decision-making method for resolving multi-objective References conflicts during the design process of complex prod- 1. S. Zhou, Y. Cao, Z. Zhang, Y. Liu, System design and simulation integration for complex mechatronic products based on SysML and modelica. J. ucts. A G-ABL framework is first presented, providing Comput.-Aided Des. Comput. 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Journal

Autonomous Intelligent Systems – Springer Journals

Published: Feb 28, 2023

Keywords: Collaborative decision-making; Conflict resolution; Generalized abductive learning; EWM based WK-means; Fuzzy comprehensive evaluation

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