
==== Front
ScientificWorldJournal
ScientificWorldJournal
tswj
The Scientific World Journal
2356-6140
1537-744X
Wiley

10.1155/2024/9545184
Research Article
Multiobjective Binary Differential Approach with Parameter Tuning for Discovering Business Process Models: MoD-ProM
https://orcid.org/0000-0003-2528-6677
Deshmukh A. Sonia 1
https://orcid.org/0000-0002-0121-572X
Gupta B. Shikha shikhagupta@sscbsdu.ac.in
2
https://orcid.org/0000-0003-3401-6927
Kumar C. Naveen 3
1 Department of Computer Science and Information Technology KIET Group of Institutions, Ghaziabad, Uttar Pradesh, India
2 Department of Computer Science S.S. College of Business Studies University of Delhi, New Delhi, Delhi, India
3 Department of Computer Science University of Delhi, New Delhi, Delhi, India
Academic Editor: Ho SoonMin

2024
27 8 2024
2024 954518431 12 2023
19 6 2024
25 6 2024
Copyright © 2024 A. Sonia Deshmukh et al.
2024
https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Process discovery approaches analyze the business data to automatically uncover structured information, known as a process model. The quality of a process model is measured using quality dimensions, completeness (replay fitness), preciseness, simplicity, and generalization. Traditional process discovery algorithms usually output a single process model. A single model may not accurately capture the observed behavior and overfit the training data. We have formed the process discovery problem in a multiobjective framework that yields several candidate solutions for the end user who can pick a suitable model based on the local environmental constraints (possibly varying). We consider the Binary Differential Evolution approach in a multiobjective framework for the task of process discovery. The proposed method employs dichotomous crossover/mutation operators. The parameters are tuned using grey relational analysis combined with the Taguchi approach. We have compared the proposed approach with the well-known single-objective algorithms and state-of-the-art multiobjective evolutionary algorithm—Nondominated Sorting Genetic Algorithm (NSGA-II). Additional comparison via computing a weighted average of the quality dimensions is also undertaken. Results show that the proposed algorithm is computationally efficient and produces diversified candidate solutions that score high on the fitness functions. It is shown that the process models generated by the proposed approach are superior to or at least as good as those generated by the state-of-the-art algorithms.
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pmc1. Introduction

Processes are ubiquitous in any organization. An efficient organization is built on processes that run in a symphony to achieve growth and customer/employee satisfaction. In the present digital era, organizations maintain the process execution information in the form of transaction logs that are amenable to analyses. However, amidst routine activities, an organization may not analyze the effectiveness of the processes being followed. Process mining aims to extract nontrivial knowledge and exciting insights from data recorded by the information systems, stored in the form of an event log. In the past decade, process mining adoption has expanded considerably, evidenced by numerous industry and academic use cases, especially in auditing and healthcare, with the field maturing through enhanced tools and techniques [1, 2]. The prominent process mining challenges include process discovery, conformance checking, and enhancement. Process discovery algorithms build a process model from the given event log [3–5]. Conformance checking verifies the goodness of the discovered process models. Enhancement techniques extend or improve existing processes by identifying and removing bottlenecks, finding deviations, recommending adjustments, and repairing processes using the information in an event log [3, 4]. The present work is focused on the challenge of the process discovery.

Process discovery concerns itself with extracting information on existing processes to recognize the bottlenecks, deviations, and inefficiencies in the day-to-day process workflows, providing concrete steps towards business process improvement. The last decade has seen several process discovery techniques that optimize one or more quality metrics, namely, completeness (also known as replay fitness [3]), preciseness, simplicity, and generalization or their weighted function. Typically, process discovery algorithms output a single model. However, a single process model may not always describe the recorded behavior of the log effectively and may be a consequence of overfitting the training data.

In this paper, we present multiobjective differential approach in process mining (MoD-ProM), a process discovery algorithm that generates several competing process models, representing different trade-offs in the quality dimensions. The present work formulates process discovery as a multicriterion problem. The proposed approach applies the differential evolution algorithm and optimizes Completeness and Generalization quality metrics to output several candidate process models. Subsequently, the solutions may either be evaluated by a domain expert to best suit the situation at hand or be chosen by the user based on his/her preference.

The contributions of this proposal areA novel application of differential evolution approach for discovering a Pareto-front of the process models.

We adapted a binary version of the multiobjective differential evolution algorithm and used dichotomous operators [6].

The proposed algorithm (MoD-ProM) is evaluated on ten synthetic and four real-life event logs, and results are compared with the state-of-the-art algorithms.

The parameters are tuned using grey relational analysis combined with the Taguchi approach [7, 8].

The computation of fitness functions (completeness and generalization) has been reformulated in terms of the causality relation matrix.

The results reveal that the proposed approach (MoD-ProM) outperforms the compared algorithms regarding the quality of the process model. Compared to Nondominated Sorting Genetic Algorithm II (NSGA-II) [9], the proposed algorithm exhibits a lower computational cost. The competing solutions (Pareto set) generated by the proposed approach are better than the nondominated solutions generated by NSGA-II.

The remainder of this paper is organized as follows: Section 2 outlines the basic concepts related to process discovery and the related work. Section 3 describes the solution strategy, and Section 4 presents the results of the experiments. Finally, Section 5 gives the conclusion of the paper.

2. Background and Related Work

2.1. Process Discovery

Process discovery is an evolving domain that leverages event logs to analyze business processes and present factual insights. An event log is the starting point for process discovery algorithms and represents a business process using the case notation to correlate events. A “case” in this notation refers to an instance of a process and is also known as a trace. Each case is assigned a unique ID, called the Case ID. An instance of a process may involve multiple activities or tasks over many days. An occurrence of a task in the context of a particular process instance (case), along with its timestamp, is called an event. Table 1 gives an example of an event log. In this example, 101, 102, and 103 represent the Case ID of three process instances, and T1, T2,…, and T7 represent the various tasks carried out in the system.

2.1.1. Visualisation of a Process Model

A process model can be discovered from the given event log and may be visualized in various forms such as Business Process Modelling Notation (BPMN models), Petri nets, and Data Flow Graphs (DFGs). In this paper, the discovered process model is graphically represented as a Petri net, a popular method for representation. A Petri net is a bipartite graph, composed of nodes, tokens, and directed arcs. A node could be a place (denoted by a circle) or a transition (denoted by a square). The places and the transitions are joined by directed arcs. For example, in the following figure, p1 and p2 are places and t1 is a transition.

A transition is also called a task. The token is the information that needs to be processed. Each place can hold zero or more tokens. In the above figure, the place p1 holds a single token. The directed arcs can transfer one token. Transitions cannot store tokens. Arcs connect (input) places to transitions and transitions to (output) places. The state of a Petri net is given by its assignment of tokens to places.

A transition is said to be enabled if each input place holds at least one token. In the following figure, t1 transition is enabled.

An enabled transition may fire at any time. When fired, the tokens in the input places are moved to the output places of the transition. Firing of a transition results in a new state of the Petri net. The following figure shows the change in the above Petri net after transition t1 fires.

A transition cannot be enabled if a token is absent (missing token) at any input place. For example, in the following figure, transition t1 cannot be enabled.

Figure 1 depicts a Petri net that conforms to the example event log in Table 1.

2.1.2. Well-Known Algorithms for Process Model Discovery

State-of-the-art process discovery techniques include α [10], α+ [11], multiphase miner [12, 13], heuristics miner [14], genetic process mining (GPM) [15], α++ [16], α# [17], α∗ [18], fuzzy miner [19], and Inductive Logic Programming (ILP) [20] algorithms [21]. Other algorithms in the domain of process discovery include evolutionary tree miner (ETM) [22], inductive miner [23], and multiparadigm miner [24]. Reference [25] proposed a hybrid process mining approach that integrates the GPM, particle swarm optimization (PSO), and discrete differential evolution (DE) techniques to extract process models from event logs. Reference [26] proposed the Fodina algorithm, an extension of the heuristic miner algorithm [26].

Reference [27] proposed an extension of the ETM algorithm [22] that discovers a collection of mutually nondominating process trees using NSGA-II [28]. This algorithm optimizes replay fitness, precision, generalization, simplicity, and the number of low-level edits.

2.1.3. Motivation for the Proposed Algorithm

Usually, state-of-the-art process discovery algorithms output a single process model that may overfit the training data. To capture the observed behavior more accurately, we propose a multiobjective algorithm for process discovery. The proposed approach yields several candidate solutions. Subsequently, the solutions may either be evaluated by a domain expert to best suit the situation at hand or be chosen by the user based on the local environmental constraints (possibly varying).

The proposed algorithm formulates the problem of process model discovery in a multiobjective framework using the differential evolution approach [29]. Differential evolution (DE) is a versatile and stable evolutionary algorithm. It evolves individuals through the perturbation of population members with scaled differences of distinct population members. DE algorithm has consistent robust performance and is suitable for solving various numerical optimization problems [30].

2.2. Multiobjective Binary Differential Evolution

The proposed algorithm employs a binary version of the differential evolution approach to suit the process mining domain. While DE was initially designed to operate in continuous space, the authors in [6] proposed a binary DE (BDE) algorithm based on dichotomous mutation and crossover operators. The authors in [6] verified that compared to other notable BDE variants, the dichotomous BDE improves the diversity of the population and enhances the exploration ability in the binary search space. Also, it has been shown that as compared to other BDE variants, the dichotomous algorithm does not involve any additional computation cost and is faster than other variants of BDE [6].

The past decade has seen the application of the DE approach to problems where the optimization of multiple objectives is required. Reference [31] first proposed a DE-based approach for multiobjective real-coded optimization problems. According to [32], in the case of binary-coded optimization problems, multiobjective BDE algorithms explore the decision space more efficiently than other multiobjective evolutionary algorithms. Subsequently, multiobjective BDE algorithms were also proposed [33–37].

3. Materials and Methods

A process discovery algorithm is a function that maps an event log onto a process model that best represents the behavior seen in the event log. In the present work, a process model is represented by a causality relation matrix C = (ct1,t2), where t1, t2 ∈ [1, n] represent the tasks, ct1,t2 ∈ {0, 1}, and n is the number of tasks in the given event log. That is, an individual in the population is binary-coded. We, therefore, adapted a binary version of the multiobjective differential evolution algorithm using dichotomous operators [6]. The steps for the proposed multiobjective differential approach for process mining (MoD-ProM) are outlined in Algorithm 9. These steps are explained in the following subsections.

3.1. Initialization

The given event log E with n tasks is first consolidated into a dependency measure matrix D indicating the degree of dependencies between tasks [15]. Considering the example in Figure 1, where T1, T2,…, and T7 represent the tasks. A dependency exists between activities T1 and T2 if, in a trace, either T1 directly precedes T2 or vice versa. This is indicated by the presence of either the strings T1 T2 or T2 T1 in a process instance (trace) of the event log. The strength of dependency is proportional to the frequency of occurrence of these strings. In the example log (Figure 1), T1 directly precedes T2 whereas the string T2 T1 does not occur at all. That is, in the given system, task T1 is more likely to be the cause of task T2 than vice versa. Dependency measure is computed by counting the length-one-loops (for example, T1 T2), self-loops (for example, T1 T1), length-two-loops (for example, T1 T2 T1), and parallel tasks (for example, T1 T2 and T2 T1 occur an equal number of times). In the example log (Figure 1), T5 and T6 are parallel tasks and T1 T2 is a length-one-loop [38].

As proposed by [39], the present work represents a process model as a causality relation matrix. To represent a process model, [39] have favored a causality relation matrix over the more popular Petri net since representing the population individual as a causality relation matrix makes it easier to initialize the population and define the genetic operators. While a causality relation matrix can be directly derived from the information in the event log, in Petri nets, there are places whose existence cannot be derived directly from the event log [39]. The mapping between a Petri net and a causality relation matrix is detailed in [39]. We have graphically depicted both representations for an example event log in the section titled “Process Model Representation.”

3.2. Objective Functions and Fitness Evaluation

In the proposed algorithm, we use a novel combination of completeness and generalization as objective functions. Completeness is an important quality dimension because a discovered process model is expected to describe the behavior stored in the log. Completeness is the process of computation of all the parsed tasks while replaying the traces of the log in the model. The missing tokens in a trace and the extra ones left behind during parsing (unconsumed tokens) contribute to the penalty value. Generalization shows whether the process model accurately represents the system as it is and is not “overfitting” to the behavior observed in the event log [40]. Completeness [15] and generalization [41] are computed as in Algorithms 1 and 2, respectively. The algorithms make use of the following function for a given event log E:(1) followst1,t2,E=1,if t1 t2 is length−one−loop in E,0,otherwise,

(2) follow skt1,t2,E=1,if t2 is the kth task after t1 in E,0,otherwise.

The present proposal performs an additional analysis of the discovered process models by evaluating their preciseness and simplicity values. The preciseness value of a model is relative to an event log and quantifies the behavior existing in the model but not observed in the event log [42]. A process model with a high precision value is not expected to show behavior not observed in the event log [43]. Completeness and preciseness only consider the relationship between the event log and the process model. However, just a portion of all potential behavior that the system permits is recorded in the event log. Simplicity, instead of telling about the behavior observed in the event log, shows the internal structure of the discovered model. Preciseness [22] and simplicity [44] values are computed as in Algorithms 3 and 4, respectively.

3.3. Constraints and Decision Variables

For a given event log E, dependency measure matrix D=(D(t1, t2)) is used to generate causality relation matrices Ci = (ct1,t2i) where ∈[1, N] , t1,  t2 ∈ [1, n], and N is the population size (Algorithm 5). The dependency measure matrix and the causality matrix correspondingly represent the constraints and the decision variables for the problem. Each causality relation matrix represents an individual of the initial population and is computed as [15](3) ct1,t2i=1,if r<Dt1,t2,0,otherwise,

r ∈ [0, 1) is a random number.

3.4. Mutation

For a population member Ci = (ct1,t2i), i ∈ [1, N] , t1,  t2 ∈ [1, n], two other causal matrices Cr1, Cr2, r1 ≠ r2 ≠ i, r1, r2 ∈ [1, N] are chosen randomly from the current population. A mutant individual Vi = (vt1,t2i) is then created using the following dichotomous mutation scheme [6].(4) vt1,t2i=ct1,t2r1⊕ct1,t2r2∧rand∨¬ct1,t2r1⊕ct1,t2r2∧ct1,t2r1

where rand ∈{0, 1}, ∧ denotes the AND operator, ∨ denotes the OR operator, ¬ denotes the NOT operator, and ⊕ denotes the XOR operator. (4) can also be expressed as(5) vt1,t2i=rand,if ct1,t2r1⊕ct1,t2r2=1,ct1,t2r1,if ct1,t2r1⊕ct1,t2r2=0.

That is, if ct1,t2r1 and ct1,t2r2 are distinct, then the corresponding bit of the mutant individual vt1,t2i is randomly chosen as “0” or “1”; otherwise, vt1,t2i is set as ct1,t2r1.

3.5. Crossover

The dichotomous crossover operator [6] starts from the mutant individual Vi = vt1,t2i, obtained after application of the dichotomous mutation operator. In this step, the original individual Ci = (ct1,t2i) and the mutated individual Vi are used to generate a candidate individual Ui = ut1,t2i using the following equation:(6) ut1,t2i=vt1,t2i, if randt1,t2<CRt1,t2,ct1,t2i, otherwise,

where randt1,t2∈[0, 1],(7) CRt1,t2=CR1, if ct1,t2r1⊕ct1,t2r2=0,CR2, if ct1,t2r1⊕ct1,t2r2=1.

This operation uses two crossover probabilities CR1 and CR2 based on dichotomous psychological thinking or “black and white” thinking, with a proclivity for only seeing extremes. After mutation, to generate a candidate individual, if the bits in the randomly chosen individuals from the original population are the same (distinct), then crossover probability CR1(CR2) is used. This approach induces diversity in the population and enhances the exploration ability of the proposed approach [6].

3.6. Selection

In this section, we outline the selection procedure (Algorithm 6) used to determine the individuals to be preserved from the current population Pop = {C1, C2,…, CN}, and the candidate population 1 = {U1, U2,…, UN} generated after the crossover operation. The process involves identifying the nondominated individuals.

The ith individual from the current population (parent) (Ci) is said to dominate (≺) the corresponding ith individual in the candidate population (child) (Ui) if the parent is superior for both the objectives of completeness and generalization, that is,(8) Ci≺Ui=1 , if fcUi≥fcCi && fgUi≥fgCi,0, otherwise,

where fc and fg denote the completeness and generalization values, respectively.

If the parent (child) dominates the child (parent), then the parent (child) is preserved, while the child (parent) is discarded. When neither parent nor child is superior to each other, both the parent and the child are retained.

After eliminating dominated individuals, the number of remaining nondominated individuals will be between N and 2 ∗ N. Since the population size to be carried for the next generation is N, a truncation procedure based on nondominated sorting (Algorithm 7) and crowding distance (Algorithm 8) is applied [31].

Nondominated sorting algorithm (Algorithm 7) involves finding rank 1 individuals of the population that are not dominated by any other individual. Rank 2 is assigned to those individuals of the population that are dominated by rank 1 individuals and so on.

If the number of nondominated solutions is greater than the population size N, Euclidean distance is used to truncate individuals from the most crowded region (Algorithm 8). If the rank 1 individuals are less than N, then rank 2 individuals are added and so on.

4. Results and Discussion

4.1. Experimentation

The proposed algorithm is tested on both synthetic and real-world datasets (Table 2). Over the last decade, BPI challenge event logs have become important real-world benchmarks in the data-driven research area of process mining. The proposed algorithm is tested for three BPI event logs, namely, BPI 2012 [45], BPI 2013 [46], and BPI 2018 [47], varying in the number of tasks, number of traces, and their domain. BPI 2012 is one of the most studied datasets in process mining. This dataset contains 13,087 traces and 23 tasks and is derived from a structured real-life loan application procedure released to the community by a Dutch financial institute. The BPI 2013 dataset is from the IT incident management system of Volvo Belgium with 7554 traces and 13 tasks. BPI 2018 covers the handling of applications for EU direct payments for German farmers from the European Agricultural Guarantee Fund. BPI 2018-reference dataset contains 43802 traces and 6 tasks. The proposed algorithm is also tested on a real-life medical event log containing events of sepsis cases from a hospital with 1000 traces and 16 tasks [48]. The proposed algorithm is also run for synthetic logs (ETM, g2-g10 [15, 38, 44]).

The proposed approach is compared with state-of-the-art algorithms, α++ [16], Heuristic Miner [14], Genetic Miner [15], ILP [20], and Inductive Miner [23] algorithms. For the compared algorithms, the completeness, preciseness, and simplicity values for the synthetic datasets are taken as reported by [44]. However, the authors in [44] do not report the value of generalization for these datasets. For the models generated using the Prom tool, α++, Heuristic Miner, Genetic Miner, and ILP algorithms, the generalization value is computed using the Cobefra tool [44, 49]. We have also compared the proposed strategy with the NSGA-II algorithm for process discovery.

In the proposed multiobjective differential approach for process mining (MoD-ProM), the population size is set to 100 and the value of control parameters CR1 and CR2 is tuned using grey relational analysis combined with the Taguchi approach (Section).

The algorithm is run for a maximum of 100 iterations as the proposed algorithm converges before 100 iterations for most datasets. The total number of runs is fixed at 30.

4.2. Parameter Tuning

To find values of the crossover probabilities, CR1 and CR2 are suitable for the domain of process discovery and the grey relational analysis combined with the Taguchi approach is used [7, 8]. The Taguchi method efficiently determines optimal settings of numerous process variables with a minimal set of experiments. The Taguchi method suggests replication of the experiment to achieve improved accuracy of the results. Taguchi L16 orthogonal array (OA) design containing 16 experimental runs is used. The results for completeness and generalization are shown in Table 3 and Figure 2. Dr. Taguchi's Signal-to-Noise ratios (S/N), which are log functions of the desired output, serve as objective functions for optimization [50]. The optimization of numerous performance variables requires a comprehensive assessment of the S/N ratio. The grey relational analysis is used in the study to solve this issue [8].

In the grey relational analysis combined with the Taguchi approach, the experimental data are normalized using equation (9) to avoid different units and to reduce the variability as presented in Table 4.(9) xi∗k=xik− minxikmaxxik− minxik

where i = 1,…, m; k = 1,…, n, m is the number of experimental data and n is the number of responses. xi (k) denotes the original value of kth response for ith experimental run, xi∗ (k) denotes the normalized value after the data preprocessing, max (xi(k)) denotes the largest value of xi(k), and min (xi(k)) denotes the smallest value of xi(k). The next step is to calculate the grey relational coefficient, ξi(k), from the normalized values by using the following equation (Table 5):(10) ξik=Δmin−ξΔmaxΔ0ik−ξΔmax

where Δ0i is the deviation value obtained from the reference value (x0(k)) and the comparability value (xi(k)).(11) Δ0i=x0k−xik

Δmin and Δmax are the minimum and maximum values of the absolute difference (Δ0i). ξ is the distinguishing coefficient, where ξ∈[0, 1] and value 0.5 is used for experimentation [7]. The next step is to find out the grey relational grade (GRG) using the following equation (Table 5):(12) γi=1n∑k=1nξik

where γi is the required grey relational grade for the ith experiment. The results are utilized for optimizing the multiresponses as they are converted into a single grade.

From the value of GRG, the effects of each process parameter at different levels are plotted and shown in Figure 3. Using these results optimal settings for the parameters CR1 and CR2 are derived as 0.2 and 0.5, respectively.

4.3. Analysis of the Results

The proposed algorithm (MoD-ProM) is run for the real life and for the synthetic datasets and the values for quality dimensions, namely, completeness (fc), preciseness (fp), simplicity (fs), and generalization (fg), for the discovered nondominated solutions are shown in Tables 6 and 7, respectively.

The proposed approach is compared with the NSGA-II algorithm for process discovery. Tables 7 and 8 present the values for the quality dimensions for the discovered nondominated solutions for real-life and synthetic datasets, respectively.

Pareto-curves for the nondominated solutions of NSGA-II and the proposed multiobjective differential evolution for process mining (MoD-ProM) are plotted for comparison (Figures 4 and 5). The Pareto-curves show that in 12 out of 14 datasets, the results of the proposed algorithm are superior to the NSGA-II algorithm.

We also compute the convergence rate and per iteration computation time for NSGA-II and the proposed MoD-ProM, over 30 runs (Figures 6, 7, 8). While in 2 datasets, the algorithms (NSGA-II, MoD-ProM) show a similar convergence rate, and in 8 out of 14 datasets, the proposed MoD-ProM converges faster than NSGA-II, demonstrating superior exploration of the proposed approach. Figure 8 shows that in all cases, the proposed algorithm is superior to NSGA-II in terms of running time per iteration. It is evident from the results that NSGA-II is computationally more expensive than the proposed MoD-ProM algorithm.

The proposed algorithm is also compared with Genetic Miner, Heuristic Miner, α++, ILP, and Inductive Miner. To rank the proposed approach and the traditional algorithms, additional comparison based on a weighted average [22] of the quality dimensions is made (Table 9). Reference [22] proposed a weighted average computation methodology suitable to the process mining domain as follows:(13) Weighted Sum=10∗fc+1∗fp+1∗fs+1∗fg13

where for a given process model, fc, fp, fs, and fg denote the completeness, preciseness, simplicity, and generalization values, respectively. A higher weight is assigned to completeness as the process model should be able to reproduce the behavior expressed in the event log.

Table 10 shows the quality dimensions for the process model discovered by the state-of-the-art algorithms. The results (Table 9) show that the proposed algorithm produces superior-quality process models for all the datasets in terms of the weighted average.

It is also observed that the models generated through the optimization of a combination of completeness and generalization exhibit superior values for the other quality dimensions.

4.3.1. Process Model Representation

As discussed earlier (Section on Initialization), the proposed approach represents a process model as a causality relation matrix [39]. However, many state-of-the-art approaches use other semantics, such as Petri net, BPMN models, and DFGs. Petri net is possibly the more popular technique for visualizing the discovered process model. We apply the methodology given by [39] to map between a Petri net and a causality relation matrix.

To better explain our results, we have graphically depicted the discovered models (causality relation matrices) as Petri nets for the ETM event log. ETM is a popular dataset in the literature comprising seven tasks. Being a small dataset, it is feasible to show (Figure 9) the causality relation matrices and the corresponding Petri nets of the four models discovered by the proposed algorithm (MoD-ProM).

For the ETM dataset, the ProM tool generated Petri nets for the state-of-the-art algorithms is shown in Figure 10. To compare with the proposed approach, the Petri net of the model with the highest completeness value, discovered by the proposed MoD-ProM, is also drawn in Figure 10.

The completeness or replay fitness [3] quantifies the ability to replay the trace from an event log onto the Petri net [15]. That is, a process model (Petri net) will exhibit a perfect completeness value if every process instance in the given event log can be replayed (simulated) in the Petri net. For the ETM dataset (Figure 10), it is observed that the proposed MoD-ProM algorithm and the ILP algorithm can replay every process instance in the event log. It is observed that the process model discovered by inductive miner and α++ does not replay some of the traces, such as (a, b, c, f, g) and (a, c, d, f, g). Traces (a, b, c, d, e, g) and (a, c, b, d, f, g) are not replayed by the model generated by the heuristic miner algorithm. Similarly, the process model generated by the genetic miner algorithm does not replay (a, c, d, b, f, g) and (a, d, c, b, f, g).

5. Conclusions and Future Works

While conventional process mining algorithms generate a unique process model from the event logs, multiobjective evolutionary algorithms generate several candidate models. The goodness of the generated process models is measured based on quality dimensions such as completeness, generalization, simplicity, and preciseness. A practitioner in the field of process mining may select the most appropriate process model based on the domain requirement. For example, if a user requires a model that replays the maximum number of traces, he/she may pick the model with a better value of completeness [5].

In this paper, we are using the idea of differential evolution towards generating a Pareto-front in the domain of process discovery, a first attempt in this direction. The proposed algorithm performs optimization using completeness and generalization as objective functions. These two quality dimensions make a good pair, as a model with high generalization value can help in improving the current system and can be used for designing future improved processes. Completeness is an important quality dimension because a discovered process model is expected to describe the behavior stored in the log.

The experiments were run for ten synthetic and four real-life datasets and are repeated 30 times for each dataset. The results are compared with state-of-the-art process discovery algorithms such as α++, heuristic miner, genetic miner, ILP, and inductive miner, and also with NSGA-II for process discovery.

Results show that the models generated by the proposed approach vis-a-vis the compared approaches exhibit a higher value for all the quality dimensions indicating the discovery of “good” process models. The nondominated solutions generated by the proposed approach (MoD-ProM) are better than those generated by the NSGA-II algorithm for process discovery. The Pareto-curve shows that the results of the proposed algorithm are superior or at least as good as those of the NSGA-II algorithm. In terms of computational time requirement, the MoD-ProM algorithm performs consistently better for all datasets as compared to the NSGA-II algorithm.

In summary, we present a novel proposal for process model discovery. The approach employs a multiobjective differential evolution method to optimize the novel combination of completeness and generalization. Results show that the proposed approach is computationally efficient in discovering good-quality process models. However, the proposed approach is limited by the hardware availability.

In the future, we plan to evaluate the applicability of recent multiobjective algorithms [51, 52] in the domain of process discovery and study their computational complexity. In addition, to address the computational intensity and time consumption of process discovery for large event logs, we can explore parallel implementations (multicore processors, GPU-based processing, and distributed computing environments) for the proposed algorithm.

Acknowledgments

The authors Dr. Sonia Deshmukh, Dr. Shikha Gupta, and Prof. Naveen Kumar are correspondingly affiliated with the Department of Computer Science and Information Technology, KIET Group of Institutions; Shaheed Sukhdev College of Business Studies (University of Delhi); and Department of Computer Science, University of Delhi.

Data Availability

Previously reported data (event logs) were used to support this study and are described in the section on Experimentation. These prior studies (and datasets) are cited at relevant places within the text (References [15, 44–47]).

Disclosure

A preprint has been previously published “[53].”

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Figure 1 Petri net for the example event log of Table 1.

Figure 2 Main effects plot of completeness and generalization.

Figure 3 Main effect plot of grey relational grade.

Figure 4 Pareto-curve for the nondominated solutions for NSGA-II and MoD-ProM for synthetic datasets, namely (a) ETM dataset (b) G2 dataset (c) G3 dataset (d) G4 dataset (e) G5 dataset (f) G6 dataset (g) G7 dataset (h) G8 dataset (i) G9 dataset (j) G10 dataset.

Figure 5 Pareto-curve for the nondominated solutions for NSGA-II and MoD-ProM for real-life datasets, namely (a) BPI 2012 dataset (b) BPI 2013 dataset (c) BPI 2018 dataset (d) Sepsis dataset.

Figure 6 Convergence rate for NSGA-II and MoD-ProM for synthetic datasets, namely (a) ETM dataset (b) G2 dataset (c) G3 dataset (d) G4 dataset (e) G5 dataset (f) G6 dataset (g) G7 dataset (h) G8 dataset (i) G9 dataset (j) G10 dataset.

Figure 7 Convergence rate for NSGA-II and MoD-ProM for real-life datasets, namely (a) BPI 2012 dataset (b) BPI 2013 dataset (c) BPI 2018 dataset (d) Sepsis dataset.

Figure 8 Comparison of per iteration running time for NSGA-II/MoD-ProM, over 30 runs. (a) Synthetic dataset. (b) Real-life dataset.

Figure 9 Petri net and causality relation matrix of nondominated process models of the proposed algorithm for the ETM dataset.

Figure 10 Petri net of the process models discovered by the compared algorithms for the ETM dataset.

Algorithm 1 Completeness (C, E).

Algorithm 2 Generalization computation: Generalization (C, E).

Algorithm 3 Preciseness (C, E).

Algorithm 4 Simplicity (C, E).

Algorithm 5 Computation of the initial population of N individuals, each of size n × n: Initialize Population (n, N, D).

Algorithm 6 Selection of the fittest individuals from current (Pop) and candidate (PopU) populations, each of size N: Selection (PopU, Pop).

Algorithm 7 Non-dominated sorting algorithm: Nondominated Sorting (Pop, N).

Algorithm 8 Pseudo-code for crowding distance algorithm: Crowding distance (Cs, E), Cs: Nondominated solutions.

Algorithm 9 Pseudo code for multiobjective differential approach for process mining (MoD-ProM).

Table 1 An event log with three process instances.

Case ID	Process instance	
101	T 1 T2 T3	
102	T 1 T2 T4 T6 T5 T7	
103	T 1 T2 T4 T5 T6 T7	

Table 2 Details of datasets used in experiments.

Type	Event log	Tasks	Traces	Events	Source	
Synthetic event logs	ETM	7	100	790	[15, 44]	
g2	22	300	4501	
g3	29	300	14599	
g4	29	300	5975	
g5	20	300	6172	
g6	23	300	5419	
g7	29	300	14451	
g8	30	300	5133	
g9	26	300	5679	
g10	23	300	4117	
	
Real-life event logs	BPI 2012	23	13087	262200	[45]	
BPI 2013-incident	13	7554	65533	[46]	
BPI 2018-reference	6	43802	128554	[47]	
Sepsis	16	1050	150000	[48]	

Table 3 Experimental results for completeness and generalization as per Taguchi L16 Orthogonal Array.

Run No.	Process parameters	Experimental results	
CR1	CR2	Completeness	Generalization	
1	0.2	0.3	0.9959	0.8931	
2	0.2	0.5	0.9996	0.8927	
3	0.2	0.7	0.99531	0.8926	
4	0.2	0.9	0.9943	0.8919	
5	0.4	0.3	0.9945	0.8917	
6	0.4	0.5	0.998	0.8928	
7	0.4	0.7	0.9971	0.8922	
8	0.4	0.9	0.9973	0.8929	
9	0.6	0.3	0.9958	0.892	
10	0.6	0.5	0.9961	0.893	
11	0.6	0.7	0.9994	0.8926	
12	0.6	0.9	0.9965	0.8925	
13	0.8	0.3	0.9984	0.8929	
14	0.8	0.5	0.995	0.8915	
15	0.8	0.7	0.994	0.8923	
16	0.8	0.9	0.9993	0.8924	

Table 4 Grey relational generation values.

Run No.	Completeness	Generalization	
1	0.339	1.000	
2	1.000	0.750	
3	0.234	0.687	
4	0.054	0.250	
5	0.089	0.125	
6	0.714	0.813	
7	0.554	0.438	
8	0.589	0.875	
9	0.321	0.313	
10	0.375	0.938	
11	0.964	0.687	
12	0.446	0.625	
13	0.786	0.875	
14	0.179	0.000	
15	0.000	0.500	
16	0.946	0.562	

Table 5 Grey relational coefficient and grey relational grade values.

Run No	Evaluation of Δ0i	Grey relational coefficient	GRG	Rank	
Completeness	Generalization	Completeness	Generalization	
Ideal sequence	1	1	1	1	 	 	
1	0.661	0.000	0.431	1.000	0.715	5	
2	0.000	0.250	1.000	0.667	0.833	1	
3	0.766	0.313	0.395	0.615	0.505	10	
4	0.946	0.750	0.346	0.400	0.373	14	
5	0.911	0.875	0.354	0.364	0.359	15	
6	0.286	0.187	0.636	0.727	0.682	6	
7	0.446	0.562	0.528	0.471	0.499	11	
8	0.411	0.125	0.549	0.800	0.675	7	
9	0.679	0.687	0.424	0.421	0.423	12	
10	0.625	0.062	0.444	0.889	0.667	8	
11	0.036	0.313	0.933	0.615	0.774	2	
12	0.554	0.375	0.475	0.571	0.523	9	
13	0.214	0.125	0.700	0.800	0.750	3	
14	0.821	1.000	0.378	0.333	0.356	16	
15	1.000	0.500	0.333	0.500	0.417	13	
16	0.054	0.438	0.903	0.533	0.718	4	

Table 6 Quality dimensions of nondominated solutions obtained from the MoD-ProM algorithm for synthetic datasets.

ETM	g2	g3	g4	g5	
f C	f G	f S	f P	f C	f G	f S	f P	f C	f G	f S	f P	f C	f G	f S	f P	f C	f G	f S	f P	
0.625	0.961	0.967	0.435	0.896	0.946	0.996	0.824	0.818	0.97	1	0.85	0.927	0.969	0.998	0.867	0.916	0.9695	1	0.873	
0.813	0.9419	0.984	0.621	0.954	0.934	0.998	0.915	0.906	0.96	1	0.854	0.947	0.957	0.998	0.916	0.946	0.946	1	0.916	
0.965	0.911	0.993	0.814	0.972	0.929	0.998	0.941	0.927	0.958	1	0.872	0.948	0.936	0.999	0.944	0.974	0.942	1	0.953	
1	0.79	0.994	1	0.988	0.923	0.999	0.979	0.947	0.956	1	0.893	0.96	0.932	1	0.953	1	0.938	1	1	
 	 	 	 	1	0.916	1	1	0.966	0.953	1	0.915	0.971	0.927	1	0.96	 	 	 	 	
 	 	 	 	 	 	 	 	0.978	0.951	1	0.956	0.982	0.923	1	0.972	 	 	 	 	
 	 	 	 	 	 	 	 	0.99	0.948	1	0.978	0.988	0.917	1	0.984	 	 	 	 	
 	 	 	 	 	 	 	 	1	0.945	1	1	0.994	0.911	1	0.992	 	 	 	 	
 	 	 	 	 	 	 	 	 	 	 	 	1	0.904	1	1	 	 	 	 	
	
g6	g7	g8	g9	g10	
f C	f G	f S	f P	f C	f G	f S	f P	f C	f G	f S	f P	f C	f G	f S	f P	f C	f G	f S	f P	
	
0.621	0.9623	1	0.685	0.92	0.979	1	0.902	0.933	0.945	1	0.893	0.842	0.947	1	0.754	0.697	0.938	1	0.767	
0.691	0.959	1	0.667	0.947	0.977	1	0.945	0.948	0.935	1	0.932	0.871	0.942	1	0.795	0.787	0.936	1	0.835	
0.781	0.951	1	0.743	0.967	0.965	1	0.975	0.96	0.93	1	0.943	0.907	0.939	1	0.83	0.872	0.933	1	0.86	
0.844	0.948	1	0.772	0.979	0.963	1	0.981	0.968	0.925	1	0.963	0.937	0.936	1	0.83	0.937	0.932	1	0.799	
0.895	0.947	1	0.766	0.989	0.95	1	0.994	0.976	0.919	1	0.983	0.965	0.933	1	0.829	0.959	0.927	1	0.875	
0.928	0.943	1	0.798	1	0.947	1	1	0.984	0.914	1	0.987	0.984	0.928	1	0.945	0.976	0.921	1	0.935	
0.96	0.94	1	0.906	 	 	 	 	0.989	0.907	1	0.989	1	0.924	1	1	0.988	0.914	1	0.967	
0.973	0.934	1	0.945	 	 	 	 	0.995	0.9	1	0.993	 	 	 	 	1	0.907	1	1	
0.986	0.929	1	0.993	 	 	 	 	0.999	0.893	1	1	 	 	 	 	 	 	 	 	
1	0.923	1	1	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	
f C : Completeness, fP: Preciseness, fS: Simplicity, fG: Generalization.

Table 7 Quality dimensions of nondominated solutions obtained by NSGA-II for process mining and MoD-ProM algorithm for real-life datasets.

Nondominated solutions obtained from NSGA-II for process mining algorithm	
BPI 2012	BPI 2013	BPI 2018	Sepsis	
f C	f G	f S	f P	f C	f G	f S	f P	f C	f G	f S	f P	f C	f G	f S	f P	
0.794	0.9839	0.9984	0.6945	0.6637	0.9834	0.9964	0.6956	0.9671	0.9857	0.9461	0.406	0.7509	0.9529	0.9955	0.499	
0.815	0.9838	0.9986	0.6756	0.8453	0.9758	0.9953	0.6433	0.9873	0.9824	0.9568	0.5989	0.8198	0.9524	0.9958	0.538	
0.892	0.9836	0.9987	0.6819	0.9749	0.975	0.997	0.7879	0.9886	0.9792	0.9848	0.7978	0.8872	0.9448	0.9979	0.664	
0.899	0.9835	0.9989	0.7151	0.9816	0.9749	0.9971	0.805	0.99	0.979	0.9768	0.7922	0.9033	0.943	0.9974	0.649	
0.907	0.9834	0.999	0.6688	0.9832	0.9699	0.9973	0.8308	0.9998	0.9236	0.9907	0.9855	0.9158	0.9345	0.9981	0.704	
0.914	0.9832	0.9991	0.6874	0.9871	0.9685	0.9985	0.8818	 	 	 	 	0.9165	0.931	0.9977	0.670	
0.929	0.9831	0.9992	0.7271	0.9916	0.9683	0.9988	0.8915	 	 	 	 	0.9434	0.9299	0.998	0.714	
0.931	0.982	0.9993	0.7056	0.9966	0.9241	0.9983	0.869	 	 	 	 	 	 	 	 	
 	 	 	 	0.9974	0.9239	0.9987	0.8458	 	 	 	 	 	 	 	 	
	
Non-dominated solutions obtained from MoD-ProM algorithm	
BPI 2012	BPI 2013	BPI 2018	Sepsis	
	
f C	f G	f S	f P	f C	f G	f S	f P	f C	f G	f S	f P	f C	f G	f S	f P	
0.9146	0.9853	0.9995	0.9229	0.9942	0.9768	0.998	0.7398	0.9965	0.9793	0.985	0.806	0.994	0.961	0.9994	0.849	
0.9957	0.985	0.9996	0.9294	0.9946	0.97	0.9979	0.8078	0.9986	0.9792	0.9848	0.7978	0.9957	0.949	0.9995	0.892	
0.9965	0.9849	0.9997	0.930	0.9948	0.9696	0.9985	0.8312	0.999	0.979	0.981	0.792	0.9973	0.936	0.9997	0.941	
0.9974	0.9828	0.9998	0.9310	0.996	0.9687	0.9989	0.8548	0.9998	0.9236	0.993	0.9858	0.9977	0.908	0.9998	0.972	
 	 	 	 	0.9961	0.9686	0.999	0.8543	 	 	 	 	 	 	 	 	
 	 	 	 	0.9981	0.9618	0.9996	0.9524	 	 	 	 	 	 	 	 	
 	 	 	 	0.9987	0.9617	0.9994	0.8672	 	 	 	 	 	 	 	 	
 	 	 	 	0.9988	0.9245	0.9995	0.8791	 	 	 	 	 	 	 	 	
f C : Completeness, fP: Preciseness, fS: Simplicity, fG: Generalization.

Table 8 Quality dimensions of nondominated solutions obtained by NSGA-II in process mining domain for synthetic datasets.

ETM	g2	g3	g4	g5	
f C	f G	f S	f P	f C	f G	f S	f P	f C	f G	f S	f P	f C	f G	f S	f P	f C	f G	f S	f P	
0.505	0.957	0.948	0.371	0.515	0.947	0.987	0.497	0.514	0.969	0.996	0.515	0.48	0.97	0.984	0.372	0.529	0.965	0.986	0.525	
0.67	0.942	0.959	0.371	0.775	0.947	0.991	0.671	0.698	0.966	1	0.663	0.532	0.97	0.978	0.396	0.643	0.964	0.992	0.673	
0.83	0.927	0.984	0.628	0.858	0.94	0.994	0.786	0.791	0.963	0.998	0.716	0.568	0.969	0.982	0.412	0.755	0.959	0.996	0.753	
0.965	0.911	0.993	0.814	0.936	0.933	0.997	0.888	0.863	0.956	1	0.799	0.666	0.963	0.985	0.532	0.809	0.956	0.822	0.796	
1	0.79	0.994	1	0.97	0.928	0.998	0.953	0.899	0.956	1	0.845	0.774	0.956	0.992	0.674	0.862	0.953	0.999	0.836	
 	 	 	 	1	0.916	1	1	0.95	0.952	1	0.862	0.856	0.943	0.994	0.733	0.892	0.949	1	0.879	
 	 	 	 	 	 	 	 	0.956	0.949	1	0.87	0.893	0.943	0.996	0.784	1	0.938	1	1	
 	 	 	 	 	 	 	 	0.98	0.947	1	0.91	0.915	0.937	0.997	0.857	 	 	 	 	
 	 	 	 	 	 	 	 	0.995	0.945	1	0.966	0.931	0.929	0.998	0.926	 	 	 	 	
	
g6	g7	g8	g9	g10	
f C	f G	f S	f P	f C	f G	f S	f P	f C	f G	f S	f P	f C	f G	f S	f P	f C	f G	f S	f P	
	
0.654	0.961	0.997	0.504	0.36	0.982	0.989	0.433	0.634	0.949	0.993	0.59	0.565	0.949	1	0.548	0.714	0.931	1	0.583	
0.712	0.954	1	0.576	0.464	0.98	0.989	0.468	0.666	0.945	0.994	0.604	0.599	0.948	1	0.641	0.781	0.925	1	0.535	
0.765	0.95	1	0.647	0.559	0.979	0.994	0.577	0.792	0.941	0.996	0.677	0.661	0.946	1	0.555	0.868	0.925	1	0.647	
0.799	0.946	1	0.72	0.601	0.977	0.992	0.588	0.876	0.939	0.997	0.759	0.693	0.945	1	0.561	0.898	0.922	1	0.647	
0.884	0.94	1	0.721	0.698	0.977	0.996	0.678	0.89	0.926	0.998	0.802	0.726	0.943	1	0.643	0.916	0.917	1	0.676	
0.926	0.937	1	0.751	0.704	0.974	0.996	0.642	0.912	0.918	0.999	0.819	0.787	0.94	1	0.61	0.917	0.914	1	0.739	
0.978	0.928	1	0.831	0.745	0.97	0.996	0.672	0.965	0.903	1	0.92	0.846	0.934	1	0.717	0.942	0.911	1	0.667	
0.994	0.923	1	0.871	0.896	0.958	1	0.854	0.989	0.899	1	0.945	0.924	0.929	1	0.754	0.971	0.905	1	0.788	
 	 	 	 	0.937	0.954	1	0.872	0.995	0.892	1	0.945	0.966	0.927	1	0.787	0.976	0.905	1	0.805	
 	 	 	 	 	 	 	 	 	 	 	 	0.984	0.923	1	0.89	 	 	 	 	
f C : Completeness, fP: Preciseness, fS: Simplicity, fG: Generalization.

Table 9 Weighted sum (equation (13)) of the quality dimensions [22] and the resulting algorithm ranks for each dataset. Rank 7 indicates the best solution.

Event log	Weighted sum	Rank	
GM	HM	α ++	ILP	Inductive miner	NSGA-II	MoD-ProM	GM	HM	α ++	ILP	Inductive miner	NSGA-II	MoD-ProM	
ETM	0.423	0.485	0.881	0.978	0.881	0.983	0.983	1	2	3.5	5	3.5	6.5	6.5	
g2	0.993	0.993	0.435	0.991	0.94	0.994	0.994	4.5	4.5	1	3	2	6.5	6.5	
g3	0.429	0.991	0.132	0.986	0.77	0.989	0.996	2	6	1	4	3	5	7	
g4	0.675	0.816	0.991	0.99	0.71	0.993	0.993	1	3	5	4	2	6.5	6.5	
g5	0.994	0.994	0.994	0.994	0.81	0.995	0.995	3.5	3.5	3.5	3.5	1	6.5	6.5	
g6	0.984	0.722	0.546	0.963	0.64	0.98	0.994	6	3	1	4	2	5	7	
g7	0.993	0.984	0.143	0.988	0.756	0.99	0.996	6	3	1	4	2	5	7	
g8	0.334	0.618	0.473	0.966	0.795	0.984	0.992	1	3	2	5	4	6	7	
g9	0.578	0.776	0.552	0.972	0.679	0.974	0.994	2	4	1	5	3	6	7	
g10	0.561	0.8	0.602	0.941	0.6	0.959	0.993	1	4	3	5	2	6	7	

Table 10 Quality dimensions of the models using the state-of-the-art algorithms.

 	ETM	g2	g3	g4	g5	g6	g7	g8	g9	g10	
Genetic miner	f C	0.3	1	0.31	0.59	1	1	1	0.26	0.48	0.48	
f P	0.94	1	0.6	1	1	1	1	0.15	1	1	
f S	1	1	1	0.97	1	1	1	0.72	0.96	0.88	
f G	0.56	0.91	0.88	0.90	0.921	0.80	0.91	0.88	0.75	0.61	
	
Heuristic miner	f C	0.37	1	1	0.78	1	0.66	1	0.52	0.74	0.78	
f P	0.98	1	1	1	1	0.99	1	1	1	1	
f S	1	1	1	1	1	0.99	0.98	0.93	0.96	1	
f G	0.62	0.913	0.89	0.81	0.92	0.80	0.81	0.90	0.73	0.60	
	
α ++	f C	0.89	0.33	0	1	1	0.45	0	0.35	0.48	0.563	
f P	1	0.96	0.18	0.97	1	1	0.12	1	1	1	
f S	1	0.78	0.79	1	1	0.76	0.93	0.74	0.79	0.76	
f G	0.56	0.62	0.74	0.91	0.92	0.84	0.81	0.91	0.59	0.43	
	
ILP	f C	1	1	1	1	1	1	1	1	1	1	
f P	1	0.97	0.97	1	1	0.99	1	0.98	0.98	0.95	
f S	0.93	0.93	0.92	0.96	1	0.74	0.93	0.667	0.9	0.68	
f G	0.79	0.99	0.93	0.91	0.921	0.79	0.91	0.92	0.76	0.61	
	
Inductive miner	f C	0.89	0.958	0.757	0.70	0.80	0.63	0.74	0.79	0.668	0.61	
f P	1	0.89	0.73	0.56	0.75	0.41	0.64	0.637	0.423	0.26	
f S	1	0.9	0.82	0.81	0.9	0.7	0.85	0.63	0.84	0.65	
f G	0.56	0.91	0.94	0.91	0.94	0.91	0.95	0.91	0.88	0.9	
f C : Completeness, fP: Preciseness, fS: Simplicity, fG: Generalization.
==== Refs
1 Mamudu A. Bandara W. Wynn M. T. Leemans S. J. J. Process mining success factors and their interrelationships Business & Information Systems Engineering 2024 1 20 10.1007/s12599-024-00860-z
2 Hejazi S. M. Zandieh M. Mirmozaffari M. A multi-objective medical process mining model using event log and causal matrix Healthcare Analytics 2023 3 100188 10.1016/j.health.2023.100188
3 Van der Aalst W. M. P. Process Mining: Data Science in Action 2016 Berlin, Germany Springer
4 Mining P. Discovery, conformance and enhancement of business processes Springer-Verlag 2011 8 18
5 Kumar N. Agarwal M. Deshmukh S. Gupta S. Moea for discovering pareto-optimal process models: an experimental comparison International Journal of Computational Science and Engineering 2020 21 3 446 456 10.1504/ijcse.2020.10027621
6 Peng H. Wu Z. Shao P. Deng C. Dichotomous binary differential evolution for knapsack problems Mathematical Problems in Engineering 2016 2016 12 5732489 10.1155/2016/5732489 2-s2.0-85008939523
7 Panda A. Sahoo A. Rout A. K. Multi-attribute decision making parametric optimization and modeling in hard turning using ceramic insert through grey relational analysis: a case study Decision Science Letters 2016 5 4 581 592 10.5267/j.dsl.2016.3.001 2-s2.0-84974715528
8 Lin C. L. Use of the taguchi method and grey relational analysis to optimize turning operations with multiple performance characteristics Materials and Manufacturing Processes 2004 19 2 209 220 10.1081/amp-120029852 2-s2.0-2442559377
9 Deb K. Pratap A. Agarwal S. Meyarivan T. A. M. T. A fast and elitist multiobjective genetic algorithm: nsga-ii IEEE Transactions on Evolutionary Computation 2002 6 2 182 197 10.1109/4235.996017 2-s2.0-0036530772
10 Van der Aalst W. Weijters T. Maruster L. Workflow mining: discovering process models from event logs IEEE Transactions on Knowledge and Data Engineering 2004 16 9 1128 1142 10.1109/tkde.2004.47 2-s2.0-4544353101
11 Alves de Medeiros A. K. Van Dongen B. F. Van Der Aalst W. M. P. Weijters A. J. M. M. Process mining: extending the α-algorithm to mine short loops Technical report 2004
12 Boudewijn F. V. D. Van der Aalst W. M. P. Multi-phase process mining: building instance graphs International Conference on Conceptual Modeling November 2004 Shanghai, China 362 376
13 Boudewijn F. V. D. Van der Aalst W. M. P. Multi-phase process mining: aggregating instance graphs into epcs and petri nets PNCWB 2005 Workshop January 2005 Miami, FL 35 58
14 Weijters A. J. M. M. van Der Aalst W. M. P. Alves De Medeiros A. K. Process mining with the heuristics miner-algorithm Technische Universiteit Eindhoven 2006 166 1 34
15 Ana Karla Alves de Medeiros Genetic process mining 2006 Eindhoven, Netherlands Department of Technology Management, Technical University Eindhoven
16 Wen L. van der Aalst W. M. P. Wang J. Sun J. Mining process models with non-free-choice constructs Data Mining and Knowledge Discovery 2007 15 2 145 180 10.1007/s10618-007-0065-y 2-s2.0-34548089470
17 Wen L. Wang J. Sun J. Mining invisible tasks from event logs Lecture Notes in Computer Science 2007 4505 358 365 10.1007/978-3-540-72524-4_38
18 Li J. Liu D. Yang B. Process mining: extending α-algorithm to mine duplicate tasks in process logs Advances in Web and Network Technologies, and Information Management June 2007 Huang Shan, China 396 407
19 Van der Aalst W. M. P. Gunther C. W. Finding structure in unstructured processes: the case for process mining Seventh International Conference on Application of Concurrency to System Design (ACSD 2007) July 2007 Bratislava, Slovakia IEEE 3 12
20 Goedertier S. Martens D. Vanthienen J. Baesens B. Robust process discovery with artificial negative events Journal of Machine Learning Research 2009 10 Jun 1305 1340
21 Deshmukh S. Gupta S. Kumar N. Ga-prom: a genetic algorithm for discovery of complete process models from unbalanced logs International Conference on Big Data Analytics December 2021 Orlando, FL, USA Springer 209 218
22 Buijs J. C. A. M. Van Dongen B. F. van der Aalst W. M. P. On the role of fitness, precision, generalization and simplicity in process discovery Lecture Notes in Computer Science 2012 7565 305 322 10.1007/978-3-642-33606-5_19 2-s2.0-84872786846
23 Sander J. J. L. Fahland D. van der Aalst W. M. P. Discovering block-structured process models from event logs-a constructive approach International Conference on Applications and Theory of Petri Nets and Concurrency June 2013 Milan, Italy Springer 311 329
24 De Smedt J. De Weerdt J. Vanthienen J. Multi-paradigm process mining: retrieving better models by combining rules and sequences OTM Confederated International Conferences” on the Move to Meaningful Internet Systems October 2014 Rhodes, Greece Springer 446 453
25 Cheng H.-J. Ou-Yang C. Juan Y.-C. A hybrid approach to extract business process models with high fitness and precision Journal of Industrial and Production Engineering 2015 32 6 351 359 10.1080/21681015.2015.1065519 2-s2.0-84942194328
26 vanden Broucke S. K. De Weerdt J. Fodina: a robust and flexible heuristic process discovery technique Decision Support Systems 2017 100 109 118 10.1016/j.dss.2017.04.005 2-s2.0-85018714239
27 Joos C. A. M. B. van Dongen B. F. van der Aalst W. M. P. Discovering and navigating a collection of process models using multiple quality dimensions International Conference on Business Process Management August 2013 Beijing, China Springer 3 14
28 Srinivas N. Deb K. Muiltiobjective optimization using nondominated sorting in genetic algorithms Evolutionary Computation 1994 2 3 221 248 10.1162/evco.1994.2.3.221
29 Storn R. Price K. Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces Journal of Global Optimization 1997 11 4 341 359 10.1023/a:1008202821328 2-s2.0-0142000477
30 Huang Z. Chen Y. An improved differential evolution algorithm based on adaptive parameter Journal of Control Science and Engineering 2013 2013 5 462706 10.1155/2013/462706 2-s2.0-84885447902
31 RobicT F. I. L. I. P. IcB. Demo: differentialevolutionfor multiobjectiveoptimization Proceedingsofthe3rd InternationalconferenceonEvolutionaryMulti- criterion Optimization (EMO2005) March 2005 Guanajuato, Mexico
32 Tušar T. Filipič B. Differential evolution versus genetic algorithms in multiobjective optimization International Conference on Evolutionary Multi-Criterion Optimization March 2007 Leiden, Netherlands Springer 257 271
33 Li Y. Sansavini G. Zio E. Non-dominated sorting binary differential evolution for the multi-objective optimization of cascading failures protection in complex networks Reliability Engineering and System Safety 2013 111 195 205 10.1016/j.ress.2012.11.002 2-s2.0-84871110815
34 Xue B. Fu W. Zhang M. Differential evolution (de) for multi-objective feature selection in classification Proceedings of the Companion Publication of the 2014 Annual Conference on Genetic and Evolutionary Computation July 2014 Berlin, Germany 83 84
35 Sikdar U. K. Ekbal A. Saha S. Mode: multiobjective differential evolution for feature selection and classifier ensemble Soft Computing 2015 19 12 3529 3549 10.1007/s00500-014-1565-5 2-s2.0-84947080353
36 Asilian Bidgoli A. Ebrahimpour-Komleh H. Rahnamayan S. A novel multi-objective binary differential evolution algorithm for multi-label feature selection 2019 IEEE Congress on Evolutionary Computation (CEC) June 2019 Wellington, New Zealand IEEE 1588 1595
37 Zhang Y. Gong D.-W. Gao X.-Z. Tian T. Sun X.-Y. Binary differential evolution with self-learning for multi-objective feature selection Information Sciences 2020 507 67 85 10.1016/j.ins.2019.08.040
38 Gupta S. Deshmukh S. Kumar N. Mantaray-prom: an efficient process model discovery algorithm AI Communications 2024 1 20 10.3233/aic-220219
39 de Medeiros A. K. A. Weijters A. J. M. M. van der Aalst W. M. P. Genetic process mining: an experimental evaluation Data Mining and Knowledge Discovery 2007 14 2 245 304 10.1007/s10618-006-0061-7 2-s2.0-33947207063
40 Buijs J. C. A. M. van Dongen B. F. van der Aalst W. M. P. Quality dimensions in process discovery: the importance of fitness, precision, generalization and simplicity International Journal of Cooperative Information Systems 2014 23 01 1440001 10.1142/s0218843014400012 2-s2.0-84900485338
41 Van Eck M. L. Alignment-based Process Model Repair and its Application to the Evolutionary Tree Miner 2013 Eindhoven, Netherlands Technische Universiteit Eindhoven
42 van Dongen B. F. Carmona J. Chatain T. A unified approach for measuring precision and generalization based on anti-alignments Business Process Management: 14th International Conference, BPM 2016 September 2016 Rio de Janeiro, Brazil Springer 39 56
43 Mannhardt F. De Leoni M. Reijers H. A. Van Der Aalst W. M. P. Measuring the precision of multi-perspective process models Business Process Management Workshops: BPM 2015, 13th International Workshops August 2015 Innsbruck, Austria Springer 113 125
44 Vázquez-Barreiros B. Mucientes M. Lama M. Prodigen: mining complete, precise and minimal structure process models with a genetic algorithm Information Sciences 2015 294 315 333 10.1016/j.ins.2014.09.057 2-s2.0-84961287965
45 Bf Van Dongen and Bpi Challenge Event Log of a Loan Application Process 2012 Eindhoven, Netherlands Eindhoven University of Technology
46 Ward S. Bpi challenge 2013, incidents 2013 10.4121/uuid:500573e6-accc-4b0c-9576-aa5468b10cee
47 Boudewijn Van Bpi challenge 2018 2018 https://data.4tu.nl/articles/dataset/BPI_Challenge_2018/12688355/1
48 Mannhardt F. Sepsis cases–event log 2016 https://data.4tu.nl/articles/dataset/Sepsis_Cases_-_Event_Log/12707639
49 Broucke S. K. L. M. V. De Weerdt J. Vanthienen J. Baesens B. A comprehensive benchmarking framework (cobefra) for conformance analysis between procedural process models and event logs in prom 2013 IEEE Symposium on Computational Intelligence and Data Mining (CIDM) September 2013 Singapore IEEE 254 261
50 Freddi A. Salmon M. Introduction to the taguchi method Design Principles and Methodologies July 2019 Berlin, Germany Springer 159 180
51 Abed-alguni B. H. Al-Jarah S. H. Ibja: an improved binary djaya algorithm for feature selection Journal of Computational Science 2024 75 102201 10.1016/j.jocs.2023.102201
52 Alawad N. A. Abed-alguni B. H. Al-Betar M. A. Jaradat A. Binary improved white shark algorithm for intrusion detection systems Neural Computing & Applications 2023 35 26 19427 19451 10.1007/s00521-023-08772-x
53 Deshmukh S. Gupta S. Kumar N. Multi-objective binary differential approach with parameter tuning for discovering business process models: mod-prom 2024 https://arxiv.org/abs/2406.17713
