Interactive data-driven multiobjective optimization of metallurgical properties of microalloyed steels using the DESDEO framework

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Interactive data-driven multiobjective optimization of metallurgical properties of microalloyed steels using the DESDEO framework © 2023 the Authors Published version Saini, Bhupinder Singh; Chakrabarti, Debalay; Chakraborti, Nirupam; Shavazipour, Babooshka; Miettinen, Kaisa Saini, B. S., Chakrabarti, D., Chakraborti, N., Shavazipour, B., & Miettinen, K. (2023). Interactive data-driven multiobjective optimization of metallurgical properties of microalloyed steels using the DESDEO framework. Engineering Applications of Artificial Intelligence, 120, Article 105918. https://doi.org/10.1016/j.engappai.2023.105918 2023 Engineering Applications of Artificial Intelligence 120 (2023) 105918 Contents lists available at ScienceDirect Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai Interactive data-driven multiobjective optimization of metallurgical properties of microalloyed steels using the DESDEO framework Bhupinder Singh Sainia,∗, Debalay Chakrabartib, Nirupam Chakrabortic,b, Babooshka Shavazipoura, Kaisa Miettinen a aUniversity of Jyvaskyla, Faculty of Information Technology, University of Jyvaskyla, P.O. Box 35 (Agora), FI-40014, Finland bDepartment of Metallurgical and Materials Engineering, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal, 721302, India cFaculty of Mechanical Engineering, Czech Technical University, Prague, Czech Republic ARTICLE INFO Keywords: Data-driven evolutionary computation Multiple criteria optimization Surrogate-assisted optimization Multiple decision makers Interactive optimization Open-source software ABSTRACT Solving real-life data-driven multiobjective optimization problems involves many complicated challenges. These challenges include preprocessing the data, modelling the objective functions, getting a meaningful formulation of the problem, and supporting decision makers to find preferred solutions in the existence of conflicting objective functions. In this paper, we tackle the problem of optimizing the composition of microalloyed steels to get good mechanical properties such as yield strength, percentage elongation, and Charpy energy. We formulate a problem with six objective functions based on data available and support two decision makers in finding a solution that satisfies them both. To enable two decision makers to make meaningful decisions for a problem with many objectives, we create the so-called MultiDM/IOPIS algorithm, which combines multiobjective evolutionary algorithms and scalarization functions from interactive multiobjective optimization methods in novel ways. We use the software framework called DESDEO, an open-source Python framework for interactively solving multiobjective optimization problems, to create the MultiDM/IOPIS algorithm. We provide a detailed account of all the challenges faced while formulating and solving the problem. We discuss and use many strategies to overcome those challenges. Overall, we propose a methodology to solve real-life data-driven problems with multiple objective functions and decision makers. With this methodology, we successfully obtained microalloyed steel compositions with mechanical properties that satisfied both decision makers. 1. Introduction Digitalization sheds light on new data collection and information sharing methods, leading the world towards a data-centered era and opens up many opportunities for novel data-based methodology developments in data analytics and decision-making. However, various elements and challenges are involved in any decision-making process, starting from data. Decision makers (DMs), in many real-life problems, often need to consider multiple objectives functions (or objectives, in brief), simultaneously when making decisions. In a decision-making process involving data, they first need to identify the objectives to be optimized with the help of the data available and the independent variables that control them. This process may require the data to be preprocessed. A multiobjective optimization problem (MOP) that considers the objectives important to the DMs can then be formulated. A DM is expected to ∗Corresponding author. E-mail address: [email protected] (B.S. Saini). 1Note that optimization is required in problems where solution alternatives are not known. Instead, we optimize objective functions which depend on the independent, so-called decision variables to find the solutions. be a domain expert. An analyst, who is an expert in multiobjective optimization,1typically coordinates the formulation of the MOP with the DMs and supports in solving it. One of the ways to solve data-driven MOPs is to use surrogateassisted optimization algorithms (Chugh et al.,2019;Jin et al.,2021). They use regression models, called surrogate models, to mimic the behaviour of the objectives as recorded in the data. In this, we assume that the data has been obtained from phenomena that can be treated as objectives to be optimized. The choice of surrogate modelling techniques to model the objectives and the methods of training and validating the models significantly impact the solutions found by the optimization method. MOPs generally do not have a single optimal solution. Instead, due to potentially conflicting objectives, there exist many so-called Pareto optimal solutions that reflect the trade-offs between the various objectives (Miettinen,1999). Many optimization algorithms aim to find https://doi.org/10.1016/j.engappai.2023.105918 Received 4 July 2022; Received in revised form 15 November 2022; Accepted 24 January 2023 Available online xxxx 0952-1976/©2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). B.S. Saini, D. Chakrabarti, N. Chakraborti et al. Engineering Applications of Artificial Intelligence 120 (2023) 105918 Fig. 1. A schematic of the seamless chain from data to decision-making. Source: Adapted from http://jyu.fi/demo. a representative approximation of the Pareto optimal solutions, see, e.g. Branke et al. (2008), Miettinen (1999). However, such algorithms can return upwards of a thousand solutions to the DMs. Comparing many solutions, especially in problems with many objectives, can be a cognitively challenging task. Thus, without assistance, DMs may find it difficult to make decisions in such problems. One way to tackle this challenge is to use interactive methods for multiobjective optimization (Miettinen,1999). Such methods incorporate the preferences of a DM during the optimization process to focus on Pareto optimal solutions that the DM prefers. Interactive methods search for new solutions iteratively, i.e., once the DM sees the solutions discovered using their preferences, they can learn about the tradeoffs and interdependencies among the objective functions as well as about the feasibility of preferences. If satisfied, the DM can select a solution they like as the final solution. Alternatively, if they do not like the solutions, they can interact with the method by providing new preferences to the method, signifying interest in a different set of tradeoffs. The interactive method then provides them with new solutions that reflect the new preferences. Focusing the search in a smaller area helps interactive methods find Pareto optimal solutions quicker and reduces the number of alternative solutions that a DM must consider at a time. This process enables the DM to learn about the possible tradeoffs in the MOP in manageable and iterative steps, making the task of finding preferable solutions easier. The process of problem formulation, optimization and decisionmaking is called a seamless chain (Heikkinen et al.). We show a simple schematic of the steps involved in data-based decision-making in Fig. 1. Similar schematics for data-driven MOPs solved using non-interactive methods have also been proposed (Jin et al.,2021). The figure presents a simple, linear pathway from data and modelling to multiobjective optimization and decision-making. However, solving real-life MOPs can be a more complicated process. It must involve lengthy deliberations between the DM and the analyst to formulate a meaningful MOP. The data may introduce constraints limiting which objectives the DMs can consider in their MOP and how the analyst can model such objectives. The optimization and decisionmaking steps may reveal significant issues in the problem formulation of modelling phases. This would require going back and fixing the issues in the earlier steps and conducting optimization and decisionmaking again. The presence of more than one DM can also complicate the consideration, as most interactive methods are designed for a single DM. Although various data-driven optimization applications have been introduced in the literature, most of them are limited to explaining the results and miss reporting practical challenges and how they overcome those issues. Indeed a suitable framework or guidelines are lacking. To fill this gap, in this paper, we propose a more realistic representation of the seamless chain that will benefit both analysts and DMs by providing them with a structured guideline to tackle challenges associated with real-life data-driven MOPs. The major contributions of this paper are as follows: 1. Microalloyed steel design problem: We formulate and solve a data-driven MOP with six objectives to obtain alloy compositions that optimize multiple mechanical properties of microalloyed steel. To achieve this, we follow the seamless chain structure to make the most efficient use of the data available. We discuss in detail the challenges we faced during the modelling and the solution process, and the steps we took to overcome them. 2. Novel interactive method: We developed the interactive MultiDM/IOPIS (multiple decision makers supported using IOPIS) algorithm, an extension of the IOPIS algorithm (Saini et al., 2020), to support multiple DMs to find a solution that meets their different preferences. The new method can be applied for group decision making. In this paper, we use the MultiDM/IOPIS algorithm to support two DMs simultaneously to solve the microalloyed steel design problem. 3. An updated seamless chain: We provide a detailed explanation of all the steps we took to solve the microalloyed steel design problem. We used many established techniques, and created new and novel techniques. We provide an updated schematic of the seamless chain to reflect the challenges faced while solving real data-driven MOPs. By providing our detailed observations of the tools and techniques used, we present a guideline to solve data-driven MOPs interactively. In Section 2, we establish the core background concepts. In Sections 3and 4, we describe the various steps involved in formulating and solving the MOP, respectively. Section 3covers the first two steps of the seamless chain, whereas Section 4covers the last two steps. We start in Section 3by describing the dataset which contains information about alloys of microalloyed steels, training surrogate models, and formulating the MOP. More specifically, the dataset contains the compositions and the corresponding values of various metallurgical properties of the alloys. We then discuss the tools and strategies used to preprocess the data to make it suitable for use in MOPs. Following this, we test a large number of surrogate modelling techniques to find the ones that work best with the data. Finally, using the results of the previous steps, and with the help of the DMs, we formulate meaningful MOPs to be solved. In Section 4we use the open-source software framework called DESDEO2(Misitano et al.,2021) to implement the MultiDM/IOPIS algorithm and solve the MOPs interactively with two DMs. We also use two non-interactive optimization methods and compare the results. It should be noted that the process of problem formulation and solution is usually an iterative one: we update the methodologies used in earlier steps based on the results obtained in the later steps. However, in Sections 3and 4we provide a linear narrative of the steps involved for the benefit of the reader. In Section 5, we discuss the effectiveness of the various steps we took to solve the MOP, including steps that did not succeed. We also discuss the insights gained via the seamless chain process. In doing so, we provide a general framework (i.e., not limited to the application considered) and a guideline to solve data-driven MOPs. Finally, in Section 6, we conclude and mention some future research directions. The aim of the article is not just to solve a challenging MOP. Instead, we use the MOP to showcase how to tackle various challenges encountered in data-driven MOPs and decision making with multiple DMs. We describe the tools we used during various steps of the seamless chain, and also discuss why we use them. We also implemented the MultiDM/IOPIS algorithm to enable multiple DMs to control the interactive optimization process and find a mutually satisfactory solution. 2https://desdeo.it.jyu.fi 2 B.S. Saini, D. Chakrabarti, N. Chakraborti et al. Engineering Applications of Artificial Intelligence 120 (2023) 105918 We make all algorithms proposed in the article openly available to facilitate future research in data-driven optimization and group decision making. 2. Background 2.1. Multiobjective optimization This paper considers the following form of a multiobjective optimization problem: minimize {𝑓1(𝐱),…, 𝑓𝑘(𝐱)} subject to 𝐱∈𝑆 ⊂ R𝑛,(1) where 𝐱= (𝑥1,…, 𝑥𝑛)𝑇represents decision vectors (vectors of decision variables) in the feasible region 𝑆of the decision space R𝑛. The number of objective functions is 𝑘and the vectors of objective function values, denoted by 𝐟(𝐱)=(𝑓1(𝐱),…, 𝑓𝑘(𝐱))𝑇, are defined as objective vectors belong to the objective space R𝑘. Because the analytical form of functions is not available in datadriven optimization problems, the decision variables and their corresponding objective function values are often collected from simulators, real-life processes, or experiments and are only available in a dataset format. In such problems, approximation models, also called surrogates (or metamodels), are created, using the available data to approximate objective function values. Then, these surrogates are utilized to perform the optimization process. Surrogates may also be utilized as replacements for computationally expensive functions or simulators to reduce the evaluation time and save computations resources (Chugh et al.,2016;Tabatabaei et al.,2015). When collecting new data is not possible during the optimization process, as in the case of this paper, the problem is called an offline data-driven optimization (Wang et al., 2018) problem. It means that the constructed surrogates cannot be updated with some new data. Typically, in MOPs, identifying a single optimum is not possible because of the existing conflict between the objective functions. Instead, multiple (can be infinitely many) so-called Pareto optimal solutions exist, where improving any objective function value is impossible without impairment in at least one of the other objective function values. The set of Pareto optimal objective vectors in the objective space is called a Pareto front. DM needs to compare Pareto optimal solutions, study the existing trade-offs between objectives, and choose the most preferred one based on their preferences. Besides the DM, an analyst (or a group of analysts) is responsible for performing the analyses, computations, and supporting the interactive decision-making process. Generally, an analyst can be a human or a computer program (Miettinen et al.,2008). Based on when preference information is incorporated, multiobjective optimization methods can be classified as a priori, a posteriori, and interactive methods (Miettinen,1999). DM provides their preferences before and after the solution process, respectively, in a priori and a posteriori methods. Providing unrealistic preferences is the major shortcoming of the a priori method, particularly if the DMs do not have prior deep insight into the problem. On the other hand, computational cost of generating a representative set of Pareto optimal solutions and heavy cognitive loads of many comparisons are the main difficulties in using the a posteriori methods, especially when there are many objectives (Deb and Saxena,2006;Ishibuchi et al.,2008). When the DM actively directs the solution process by providing preferences iteratively, the multiobjective optimization method is called interactive (Miettinen,1999;Miettinen et al.,2008). In this way, the DM can learn about the interdependencies of the objectives in the problem and the feasibility of their preferences. Furthermore, these methods limit both cognitive and computational load since only a limited amount of information needs to be analyzed at a time, and only solutions reflecting the DM’s preferences need to be generated, respectively. The DM can pursue the interactions by adjusting the preferences until they get satisfied and converge to the most preferred solution (see Miettinen et al.,2008 for more details). There are different ways to solve MOPs (see, e.g., Branke et al., 2008;Miettinen,1999;Miettinen et al.,2008); among them, one widely used approach is to transform the MOP into an equivalent single-objective problem utilizing a so-called scalarization function while incorporating DM’s preferences (Miettinen,1999;Miettinen and Mäkelä,2002;Ruiz et al.,2009). The scalarized MOP can then be solved using an appropriate single objective optimizer, resulting in one or more Pareto optimal solutions that reflect the preferences to satisfy the DM. Another way to solve MOPs is to use multiobjective evolutionary algorithms (MOEAs) (Branke et al.,2008;Ishibuchi et al.,2008). They are metaheuristic approaches which use a ‘‘population’’ of solutions simultaneously to mimic the process of evolution. Popular MOEAs such as RVEA (Cheng et al.,2016) and NSGA-III (Deb and Jain,2014) have proven to be successful at generating a representative set of Pareto optimal solutions for MOPs with many objectives. However, MOPs become exponentially more difficult to solve with an increasing number of objectives (Deb and Saxena,2006). The DESDEO framework provides open-source and modular Python implemenetations of RVEA and NSGA-III (and many other MOEAs), as well as many scalarization functions and interactive methods (Misitano et al.,2021). The DESDEO framework is the only open-source framework designed for optimization using interactive methods (Misitano et al.,2021). Its modularity enables easy creation of new interactive methods using components of other interactive methods implemented in the framework. The IOPIS algorithm (Saini et al.,2020) provides a middle ground between using scalarization functions (which optimize in a single dimension) and MOEAs (which generally optimize in the objective space with many dimensions). The IOPIS algorithm incorporates a DM’s preferences using multiple scalarization functions (typically fewer than the number of objectives in the original problem). Together, these functions form a new space called a preference incorporated space (PIS). An appropriate MOEA is then used to optimize in this new space, making the MOEA interactive (since preferences are incorporated). An analyst can therefore control the number of dimensions in which the MOEA optimizes by changing the number of scalarization functions, which form the PIS. The IOPIS algorithm is of note as it enables easy and modular creation of interactive MOEAs. Moreover, as shown in Saini et al. (2020), it guarantees that the interactive MOEAs will have the beneficial properties of optimality (the solutions found by the MOEA are Pareto optimal), preferability (the solutions found by the MOEA follow the preferences of the DM), and searchability (the MOEA enables the DM to find any Pareto optimal solution by changing the preferences). However, the IOPIS algorithm was originally designed for solving MOPs with a single DM. 2.2. Microalloyed steels As mentioned, we consider a data-driven problem of microalloyed steel. Steels used for structural, linepipe and naval applications must meet strict performance standards and withstand mechanical stresses imposed in such applications without failure. Microalloyed steels3ex- hibit the required high strength, toughness, ductility, and weldability capacity (Kim,1983). Entities such as the British and European standards (BS EN standards), the American petroleum institute (API standards), and the US naval sea systems command (military or MIL standards) have published standards for usage of microalloyed steels in various domains. Each published standard includes one or more grades of microalloyed steel which set the minimum requirements and maximum bounds that should be met by the steel for specific applications. Yield strength (YS) and ultimate tensile strength (UTS) are two important measures of the strength of materials. The YS measures the 3A subcategory of high-strength low-alloy steels. 3 B.S. Saini, D. Chakrabarti, N. Chakraborti et al. Engineering Applications of Artificial Intelligence 120 (2023) 105918 maximum stress (force per unit area) a material can sustain before undergoing permanent (plastic) deformation. Below the YS, the material deforms elastically, i.e., it reverts to its original shape and size after removing the external force. The UTS measures the maximum stress a material can sustain before undergoing a fracture. Percentage elongation (ELON) measures the ductility as the fractional increase in the length of a specimen that undergoes fracture upon reaching the UTS. We can use Charpy impact tests to measure the toughness of materials at various temperatures. The test measures the energy required to fracture a standard specimen made from the material using an impact. The Charpy impact energy generally decreases at lower temperatures as ductile materials (such as steels) start showing brittle behaviour at such temperatures. Good toughness at low temperatures is required for steels used at such temperatures. The result of these tests can be reported as Charpy energy values at specific temperatures or impact transition temperature (ITT) values at specific Charpy energy values. The aforementioned mechanical properties of microalloyed steels depend strongly on the composition of the steel alloy. For example, in minute quantities, alloying elements such as titanium, molybdenum, and vanadium generally positively affect the strength, ductility, and toughness of microalloyed steels. However, they can hinder the ability of structures made from such steels to be welded. The weldability of steels is correlated with the carbon equivalent (𝐶𝑒𝑞), which can be calculated as (Lancaster,1999): 𝐶𝑒𝑞 = %𝐶+%𝑀𝑛 + %𝑆𝑖 6+%𝐶𝑟 + %𝑀𝑜 + %𝑉 5+%𝐶𝑢 + %𝑁𝑖 15 .(2) The terms on the right-hand side of (2),%𝐶, %𝑀𝑛, %𝑆𝑖, %𝐶𝑟, %𝑀𝑜, %𝑉 , %𝐶𝑢, and %𝑁𝑖 measure the concentration of the alloying elements carbon, manganese, silicon, chromium, molybdenum, vanadium, copper, and nickel as a mass percentage respectively. Various steel grades set different upper limits to the acceptable levels of 𝐶𝑒𝑞 to account for weldability needs. For different applications, different compositions of steels are needed which satisfy the various standards mentioned previously. Therefore, there is a need to formulate and solve MOPs to design microalloyed steel compositions. Past studies (Roy et al.,2020;Chakraborti, 2022) have considered MOPs with up to three objectives to design microalloyed steels. These studies solved the MOPs using non-interactive MOEAs. 3. Preprocessing, surrogate modelling and problem formulation In this section, we follow the first two steps of the seamless chain to formulate an MOP. We first describe the characteristics of the raw data we used for our study in the text below. Then in Section 3.1, we preprocess the raw data to use it in later steps. In Sections 3.2 and 3.3, we train and validate surrogate models using the processed data. Finally, in Section 3.4, we formulate two MOPs using the surrogate models for the microalloyed steel composition design problem. The raw dataset, compiled from a database and available in the DESDEO framework, contains details about the metallurgical properties of microalloyed steels. There are 736 rows and 51 columns in the dataset. Each row denotes information related to microalloyed steel of a specific composition, which we later refer to as a sample. The first twenty columns contain information about the concentration of various alloying elements. The rest of the columns contain information about various metallurgical properties of the samples. The first three among these are YS, UTS, and ELON. The next ten columns denote the ITT at ten different Charpy impact energy levels (ranging from 13 J to 80 J). The next six columns measure the Charpy impact energy value at different temperatures (ranging from −80 ◦C to 19 ◦C). The remaining columns contain properties such as fracture toughness, pearlite content and hardness. As in may cases with real data, there are many issues with the raw dataset. As the raw dataset is a compilation of data from various sources, the data quality is not consistent across the rows. For example, some cells have exact numbers, whereas others have a range. Moreover, measurements of properties such as the Charpy energy are very noisy. While the raw dataset contains 736 rows of samples, many cells are empty in various columns. Consequently, many rows do not completely contain information about the samples’ composition, grain size, and metallurgical properties. For example, there are only 599 samples that measure YS, 537 UTS measurements, and 296 ELON measurements. The number of rows that contain information about other metallurgical properties is much lower. Moreover, these measurements are spread across the data points such that the overlap in rows that measure two different metallurgical properties is very small. This means that the different properties are measured for entirely different alloy compositions, with minimal overlap. 3.1. Preprocessing the data The first step of the seamless chain is preprocessing of the data. We cleaned and divided the raw dataset into multiple sets to be used in later steps. Firstly, all empty cells in the alloy composition columns were assigned a value of zero. As the dataset was collected from various studies, empty cells signify that those alloying elements were not a part of the study, and hence did not exist in the alloy. Secondly, we merged the columns named ‘‘Nb’’ (niobium) and ‘‘Cb’’ (columbium) as they refer to the same alloying element. Finally, we replaced the cells in the alloy composition columns that were represented as a range of values by the average value of the range. At this step, all cells in the alloy composition columns had numerical values. We then divided the dataset into multiple sets such that each set contained all the alloy composition columns but only one metallurgical property. For this, we only considered rows which documented the respective metallurgical property. This led to, for example, a YS dataset with 599 samples. We repeated the process for UTS and ELON. None of the columns documenting ITT or the Charpy energy had more than 300 samples. Hence, instead of breaking these columns into multiple small sets, we combined the 10 ITT columns and the 6 Charpy energy columns into just two columns documenting the temperature and the corresponding Charpy energy.4This process led to a Charpy dataset with 781 samples.5We used the Pandas Python package to carry out the aforementioned tasks (Reback et al.,2020). 3.2. Model selection and training The second step of the seamless chain is training surrrogate models. To begin the modelling process, we first analysed the dataset to identify which alloying elements significantly impacted the various metallurgical properties. We compared the results of the feature importance analysis against metallurgical literature to confirm the reliability of the data. In brief, we conducted the following tests to identify significant alloying elements for the YS, UTS, Elongation, and Charpy datasets using the scikit-learn Python package (Pedregosa et al.,2011): •Principal component analysis (PCA): This method can help us find ‘‘features’’ (alloying elements in our case) that have the most variance in the dataset. •Cross decomposition: We use the PLSCanonical, PLSSVD, PLSRegression, and Canonical Correlation Analysis (CCA) algorithms to find out which features lead to the most variance in the various metallurgical properties. 4This process is known is data ‘‘melting’’ and converts ‘‘wide’’ data (more columns, fewer rows) to ‘‘tall’’ data (fewer columns, more rows) 5Note that, at this step, we have datasets for four mechanical properties of microalloyed steels already. Therefore, we can create an MOP with four objectives, which is more than the three objectives considered in earlier studies. However, in later subsections, we will add even more objectives to our MOP formulation. 4 B.S. Saini, D. Chakrabarti, N. Chakraborti et al. Engineering Applications of Artificial Intelligence 120 (2023) 105918 Fig. 2. Relative ranking of the features calculated for all datasets. A lower rank value (lighter colour in the heatmap) represents higher importance. •Random forest regression: We calculate feature importance (FeatImp) and permutation importance (PermImp) using random forest models trained on the dataset. •ANOVA tests: We rank various features according to their F- values. •Mutual importance (MI): We rank the various features according to their MI values. The results of the tests are presented in Fig. 2 as heatmaps. The 𝑥-axis in each of the sub-figures denotes the aforementioned tests, whereas the 𝑦-axis represents the alloying elements (and temperature values for the Charpy dataset). The colour of the heatmap represents the relative importance (calculated as a rank) of the alloying element as measured by each test. Elements with lighter hues achieved a better rank and are considered more important by the tests. There are various reasons why a test may consider a certain alloying element unimportant for a given mechanical property. The alloying element may have no effect, or a mixed effect on the mechanical property. For example, nitrogen can form nitrides with titanium, which can lead to grain refinement, which leads to better YS. However, in the presence of aluminium, it can form AlN, which can lead to embrittlement of the steel, which lowers the YS. Hence, nitrogen can have a mixed effect on YS, and is ranked low by the tests. The dataset may also have a very skewed distribution of the values of alloying element concentrations. In the UTS dataset, only 25 (out of 537) samples contained Zirconium. The effect of such elements may not be adequately represented in the dataset, which can lead to a low rank. Finally, certain alloying elements may show a mixed effect on the mechanical properties because of noise in the dataset. Based on these tests, we removed unimportant or noisy columns from the datasets. This can increase the overlap in the ranges of alloying element compositions for which the various properties are measured and lead to better modelling and optimization results. We discuss this further in Section 3.4. We trained surrogate models for YS, UTS, ELON, and Charpy energy based on their respective datasets using many surrogate modelling algorithms and compared their training accuracy using the 𝑅2 value. We considered the following surrogate modelling algorithms: neural networks (Gardner and Dorling,1998), support vector machines (Steinwart and Christmann,2008), Gaussian process regression (Matheron,1963;Emmerich,2005), and various ensemble modelling techniques. These surrogate modelling algorithms have been used extensively and successfully in data-driven multiobjective optimization (Jin et al.,2021). We conducted K-fold cross-validation to choose the best performing surrogate modelling technique for each metallurgical property. Details of the test, along with a Python implementation, can be found in the DESDEO framework. The results of the test are shown in Table 1. The full names and details of the surrogate modelling techniques are presented in Table 6 in Appendix B. In general, ensemble techniques worked better 5 B.S. Saini, D. Chakrabarti, N. Chakraborti et al. Engineering Applications of Artificial Intelligence 120 (2023) 105918 Table 1 The K-fold cross-validation performance (𝑅2score) of various surrogate modelling techniques on YS, UTS, Elongation, and Charpy datasets. Surrogate modelling techniques YS UTS Elongation Charpy energy Median 𝑅2Standard deviation Median 𝑅2Standard deviation Median 𝑅2Standard deviation Median 𝑅2Standard deviation ExTR 0.746 0.059 0.8380 0.0709 0.673 0.124 0.166 0.129 Ada 0.604 0.0818 0.7403 0.154 0.575 0.126 0.293 0.0724 Bagging 0.716 0.0492 0.8358 0.0839 0.612 0.108 0.275 0.109 GradBoost 0.712 0.0619 0.8437 0.0629 0.666 0.177 0.436 0.0843 XGBoost 0.742 0.0694 0.8440 0.0781 0.58 0.128 0.178 0.117 XGBRF 0.642 0.0571 0.7678 0.0932 0.651 0.136 0.423 0.0765 LightGBM 0.732 0.0511 0.7833 0.0587 0.642 0.141 0.305 0.103 RandomForest 0.726 0.037 0.8307 0.0679 0.64 0.12 0.317 0.0994 Kriging −204 1920 −85.18 2030 −509 835 −6.5 3.6 SVM1 −0.0253 0.0435 −0.03087 0.0352 0.172 0.0707 −0.0803 0.11 SVM2 −0.232 0.14 −0.3843 0.143 0.0344 0.134 −0.0244 0.184 NN −6.79 1.12 −11.35 7.29 −0.452 0.259 −0.0519 0.113 Table 2 Effect of alloying element composition on mechanical properties calculated using ICE plots. The alloying elements with no reported effects were not considered for the model. Alloying element Effect on YS Effect on UTS Effect on ELON Effect on Charpy energy Carbon positive positive negative mixed Silicon mixed positive negative – Manganese positive positive mixed positive Phosphorus negative negative negative negative Sulphur mixed negative negative negative Molybdenum positive positive negative positive Nickel positive positive negative mixed Aluminium negative negative positive positive Nitrogen – – mixed negative Niobium positive positive positive positive Vanadium positive positive negative negative Boron – – mixed – Titanium positive positive negative mixed Chromium positive positive negative negative Cerium positive positive – – Copper negative positive negative – Zirconium negative negative mixed – than other surrogate modelling techniques tested. The extra trees regression (Geurts et al.,2006), gradient boost regression (Friedman) and random forest regression (Breiman,2001) (from Python package scikit-learn), XGBoost (Chen and Guestrin,2016) (from Python package xgboost, and LightGBM (Ke et al.,2017) (from Python package lightgbm) performed the best. It should be noted that we use the default hyperparameter settings, as provided in their respective Python packages, for training the surrogate models. As the Charpy dataset was much noisier than the other datasets, the surrogate models for Charpy energy performed much worse. The model with the highest median 𝑅2 value was chosen for each mechanical property, as highlighted in bold in Table 1. 3.3. Model validation In the absence of the ability to create and test new alloys, we can validate the models by comparing the effect of changing the alloying element compositions on the models’ responses. We can do so using individual conditional expectation (ICE) plots (Goldstein et al.,2015). They plot the changes in the output of a surrogate model based on changes in one of the input variables, which are the alloying element concentrations in our problem. The effect of alloying elements on the mechanical properties is presented in Table 2. In the table, we divide the effect of changing alloying element concentrations into four categories. A positive effect means that the alloying element benefits the property modelled by the surrogate model, whereas a negative effect implies a negative correlation. A mixed effect implies that the alloying element has a complex relationship with the modelled property. It can have a beneficial or a detrimental effect based on other criteria, for example, the concentration of other alloying elements. In contrast, a nil effect (signified with ‘‘-’’) means that the alloying element has no effect on the surrogate model, or its effect could not be detected. We present a complete discussion of the effect of all alloying elements in Appendix A. The discussion is based on an extensive review (Abbaschian and Reed-Hill,2009;Bhadeshia and Hansraj,2019;Bhadeshia and Honeycombe,2017;Gladman,1997;Bae et al.,2002;Chen et al.,2004;Li et al.,2001;Rozhkova et al.,1981; Mesquita and Kestenbach,2011;Erhart and Grabke,1981;Vervynckt et al.,2012;Kim et al.,2001;Norström and Vingsbo,1979;DeArdo, 2003;Morrison,2009;Yan et al.,2006;Wilson and Gladman,1988; Uggowitzer et al.,1996;Baker,2009;Ghosh et al.,2014;Isheim et al.,2006;Ye et al.,2012;Adabavazeh et al.,2017;Guo et al., 2008) covering the effects of alloying elements on strengths (YS and UTS), ductility (ELON), and impact toughness (Charpy impact energy absorption at different test temperatures) in a variety of low-carbon steels used for structural, linepipe, automotive, naval, and pressure vessel applications. The DMs concluded that the metallurgical theory and literature back a majority of ICE plot results. The models can be deemed valid and be used to study steels’ mechanical behaviour. We can see in Table 2 that many elements have conflicting effects on YS, UTS, Elongation, and Charpy energy. This is ultimately the source of trade-offs in MOPs that consider such objectives. 3.4. Problem formulation Based on the consideration described in the previous subsection, the selected surrogate models (ExTR for YS and ELON, XGBoost for UTS, and GradBoost for Charpy energy), can predict the values of YS, UTS, ELON, and Charpy energy value of vanadium microalloyed steels 6 B.S. Saini, D. Chakrabarti, N. Chakraborti et al. Engineering Applications of Artificial Intelligence 120 (2023) 105918 Table 3 The upper and lower bound for the concentrations (in percentage weight) of the alloying elements and their cost as used in (MOP-I) and MOP-II. Alloying element Lower bound (MOP-I) Upper bound (MOP-I) Lower bound (MOP-II) Upper bound (MOP-II) Cost (USD per Kg) Carbon 0.009 0.34 0.006 0.4 0 Silicon 0.01 0.55 0 0.6 0.122 Manganese 0.28 1.63 0.28 1.63 1.7 Phosphorus 0 0.035 0 0.035 1.82 Sulphur 0 0.035 0 0.042 2.69 Molybdenum 0 0.53 0 1.0 0.0926 Nickel 0 3.12 0 3.12 40.1 Aluminium 0 0.06 0 0.06 13.9 Nitrogen 0 0.024 0 0.024 1.79 Niobium 0 0.086 0 0.45 0.140 Vanadium 0 0.2 0 0.2 72 Boron 0 0.001 0 0.001 3.68 Titanium 0 0.18 0 0.31 11.5 Chromium 0 1.24 0 1.5 9.4 Cerium 0 0.015 0 0.015 4.6 Copper 0 0.312 0 0.312 6 Zirconium 0 0.008 0 0.008 237 as functions of their compositions (and additionally the temperature for the Charpy energy). Thus, the metallurgical properties form the objectives of an MOP to be optimized with the alloy composition as the decision variables. Besides the surrogate models, two more objectives were added to the problem formulation: carbon equivalent value and the cost of materials. These objectives can be numerically calculated as functions of the steel composition. Eq. (2) is used to calculate the carbon equivalent value. The cost of materials is calculated as a linear combination of the alloy composition values weighted by the cost of the elemental components as stated in Table 3. We formally define the MOP (MOP-I) as: maximize 𝚈𝚂(𝐜𝐨𝐦𝐩) maximize 𝚄𝚃𝚂(𝐜𝐨𝐦𝐩) maximize 𝙴𝙻𝙾𝙽(𝐜𝐨𝐦𝐩) maximize 𝙲𝚑𝚊𝚛𝚙𝚢(𝐜𝐨𝐦𝐩, 𝑡𝑒𝑚𝑝) minimize 𝐶𝑒𝑞(𝐜𝐨𝐦𝐩) minimize 𝐶𝑜𝑠𝑡(𝐜𝐨𝐦𝐩) subject to 𝐋𝐁 ≤𝐜𝐨𝐦𝐩 ≤𝐔𝐁, (MOP-I) where YS,UTS,ELON, and Charpy are the surrogate models for YS, UTS, ELON, and the Charpy energy, respectively. The decision variable 𝐜𝐨𝐦𝐩 is the vector of the concentration of 17 alloying elements in the steel bounded by lower and upper bounds 𝐋𝐁 and 𝐔𝐁. The function 𝐶𝑒𝑞 refers to the carbon equivalent value, and the 𝐶𝑜𝑠𝑡 function refers to the cost of materials. The temperature 𝑡𝑒𝑚𝑝 input for the 𝙲𝚑𝚊𝚛𝚙𝚢 surrogate model is kept constant at −80 ◦𝐶. The bounds 𝐋𝐁 and 𝐔𝐁 are calculated as the bounds of the intersection of the datasets used for all four surrogate models and the values are given in Table 3. Expanding this range makes the search space larger, which increases the likelihood of finding suitable alloy compositions. This is why removing unimportant or noisy columns was a crucial step in surrogate modelling, as described in Section 3.2. We create a simpler version of (MOP-I) (named MOP-II) by removing the Charpy energy objective from (MOP-I). We consider this version since, as mentioned, the Charpy surrogate model had a worse accuracy than the other surrogate models. Removing this objective also led to an expansion of the bounds of the decision variables (calculated as the bounds of the intersection of the remaining datasets). The upper and lower bound values for the various alloying elements can be seen in Table 3. 4. Optimization and decision making The third and fourth steps of the seamless chain are multiobjective optimization and decision making. The open-source software framework DESDEO (Misitano et al.,2021) provides a variety of interactive and some non-interactive methods to solve multiobjective optimization problems. We begin by solving the two MOPs formulated in the previous section with two non-interactive evolutionary algorithms: RVEA and NSGA-III, to generate an approximate representation of Pareto optimal solutions. These method have been reported to work well with MOPs with more than four objectives. The details of the optimization and the results are discussed in Section 4.1. We conducted optimization using non-interactive MOEAs and presented visualizations of the results to the DMs to help them form their initial preferences. In Section 4.2, we outline the details of the interactive MOEA called MultiDM/IOPIS, a method developed to solve the MOP interactively with more than one DM. We then describe the interactive optimization process itself and the results obtained in Section 4.3. 4.1. Non-interactive optimization We ran the RVEA and NSGA-III algorithms for 150 generations each to solve MOP-I and MOP-II. There was no improvement in the objective values attained by the solutions after around 120 generations. The other parameters of the algorithms were otherwise unchanged from the values suggested in original publications which proposed RVEA and NSGA-III. We combined the final solutions from both RVEA and NSGAIII and removed the ones where some objective function value was worse and other values not better than in some other solution. Figs. 3 and 4show the remaining solutions for MOP-I and MOP-II, respectively, in parallel coordinates plots. Interactive versions of these plots can be seen at https://desdeo.it.jyu.fi. The two methods found 1055 solutions for MOP-I and 1599 solutions for MOP-II. The ideal, i.e., best possible, values for the objectives of MOP-I were (726 MPa, 1577 MPa, 34%, 103 J, 0.087, 43 USD per kg). The ideal values for the objectives of MOP-II were (752 MPa, 1593 MPa, 34%, 0.055, 42.9 USD per kg). The methods were able to find slightly better solutions for MOP-II than for MOP-I. This may be because optimization becomes exponentially more difficult with an increasing number of objectives. As the parameters of the methods were the same for solving MOP-I and MOP-II, the results were slightly worse in the version with one more objective. Despite the differing performance, certain similarities can be visually observed in the solutions of MOP-I and MOP-II. For example, in both cases, there is a cluster of solutions with very high UTS values, followed by a discontinuity, and then a different cluster of medium and low UTS values. There are also three clusters in the 𝐶𝑜𝑠𝑡 objective. In both problem formulations, 𝐶𝑜𝑠𝑡 did not have significant trade-offs with the ELON objective. The solutions represented high values for ELON for both low 𝐶𝑜𝑠𝑡 and high 𝐶𝑜𝑠𝑡 alloys. However, 𝐶𝑜𝑠𝑡 was very strongly correlated with UTS. Low 𝐶𝑜𝑠𝑡 alloys could only achieve UTS values of up to 1200 MPa. Only the highest 𝐶𝑜𝑠𝑡 alloys could achieve the best UTS values. Increasing 𝐶𝑜𝑠𝑡 also generally increased the YS values. 7 B.S. Saini, D. Chakrabarti, N. Chakraborti et al. Engineering Applications of Artificial Intelligence 120 (2023) 105918 Fig. 3. Solutions of MOP-I presented as a parallel coordinates plot. The solution traces are coloured based on the Carbon equivalent value. Fig. 4. Solutions of MOP-II presented as a parallel coordinates plot. The solution traces are coloured based on the Carbon equivalent value. However, the low 𝐶𝑜𝑠𝑡 alloys could still achieve the best YS values. Among the solutions of MOP-I specifically, the Charpy objective was weakly conflicting with all other objectives. We compared the solutions of MOP-I and MOP-II against structural and pipeline steel grades. We enumerate the solutions that match various steel grades in Table 4. The MOP-II formulation resulted in many more feasible alloy compositions for structural steels than the MOP-I formulation. This is because, as mentioned earlier, MOP-II is easier to optimize than MOP-I. However, for linepipe steels, the MOP- I formulation discovered alloy compositions to satisfy a more diverse range of grades than MOP-II. The inclusion of the Charpy energy objective in MOP-I allowed a more diverse search space. Hence, the formulation was able to satisfy more grades. Neither formulation was able to discover solutions that matched the naval steel grades. This 8 B.S. Saini, D. Chakrabarti, N. Chakraborti et al. 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