Computer-Supported Strategic Decision Making for Ecosystems Creation

Citation: Rodriguez-Garcia, P.; Carracedo, P.; Lopez-Lopez, D.; Juan, A.A.; Martin, J.A. ComputerSupported Strategic Decision Making for Ecosystems Creation. Computers 2024,13, 322. https://doi.org/ 10.3390/computers13120322 Academic Editor: Paolo Bellavista Received: 27 October 2024 Revised: 23 November 2024 Accepted: 2 December 2024 Published: 4 December 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Article Computer-Supported Strategic Decision Making for Ecosystems Creation Patricia Rodriguez-Garcia 1, Patricia Carracedo 2, David Lopez-Lopez 3, Angel A. Juan 2,* and Jon A. Martin 1 1Department of Computer Science, Universitat Oberta de Catalunya, Rambla Poblenou, 08018 Barcelona, Spain; patr[email protected] (P.R.-G.); [email protected] (J.A.M.) 2Research Center on Production Management and Engineering, Universitat Politècnica de València, Plaza Ferrandiz-Carbonell, 03801 Alcoy, Spain; [email protected].es 3ESADE Business School, Universitat Ramon Llul, Av. Torre Blanca, 08172 Sant Cugat, Spain; [email protected] *Correspondence: [email protected] Abstract: In the corporate strategy arena, the concept of ecosystems has emerged as a transformative approach to promote competitive advantage, growth, and innovation. Corporate ecosystems enable companies to benefit from interconnections among diverse partners, products, and services to deliver enhanced value to customers. However, the process of ecosystem creation represents a significant challenge for CEOs, as they must analyze a wide number of alternative sectors, partners, business cases, and other critical elements. Particularly, as it is a strategic decision, it lies beyond the traditional approach of risk-return by incorporating other factors, e.g.: the feasibility, desirability and sustainability of each alternative. This paper investigates how computer-supported optimization algorithms can help to solve the complex problem faced by CEOs when making these factors to create a successful and sustainable ecosystem. The paper shows how a CEO can make informed strategic decisions by identifying the best projects to include in the ecosystem portfolio, balancing financial risk and return with technical feasibility, customer appeal, and technical considerations. Keywords: strategic decision making; corporate ecosystems; computers in industry; optimization algorithms 1. Introduction Corporate ecosystems refer to networks of organizations—including businesses, suppliers, customers, competitors, and other stakeholders—that collaborate and interact within a shared environment to achieve common goals or create value [ 1 ]. Becoming a relevant player in such a corporate ecosystem is one of the top priorities of the next CEO agenda [2] . Among the various strategies to achieve corporate growth, CEOs seek to bring a more complete offer to their customers that evolve from pure industry players to an integral value proposition by encouraging different forms of collaboration with companies of different industries. In this ecosystem context, CEOs face the constant challenge of allocating their limited resources efficiently to a portfolio of strategic projects that reinforces the competitive advantage of the company and supports a sustainable and profitable growth. The CEO’s decision on which projects to invest in is considered strategic, as it will significantly impact the organization’s success in the medium and long term [ 3 ]. Knowing how to tackle this strategic decision implies that CEOs understand the portfolio optimization problem (POP) [4–6] , where they will select the project that meets the strategic objectives from a wide range of candidate projects. In traditional portfolio optimization, the goal is either to maximize the returns for a given level of risk or minimize the risk for a given level of returns. In the rich version of the POP that models ecosystem construction, other strategic metrics need to be considered Computers 2024,13, 322. https://doi.org/10.3390/computers13120322 https://www.mdpi.com/journal/computers Computers 2024,13, 322 2 of 15 Menold et al. [7] . As Urli and Terrien [8] elaborated in their paper, objectives and constraints can be of quantitative nature (such as return or risk), or pertain to qualitative measures (such as the desirability for the customer or the feasibility of the solution). In the dynamic and interconnected environment of corporate strategy, CEOs must consider a variety of both quantitative and qualitative factors that influence long-term sustainability and alignment with the company’s overarching goals [ 9 ]. The classical Markowitz model focuses primarily on minimizing risk while keeping the expected return above a user-defined threshold [ 10 ]. While the expected return, as financial viability, remains a crucial aspect, the decisionmaking process of creating a corporate ecosystem demands a broader consideration of factors to ensure alignment with the company’s vision and long-term strategy [ 11 , 12 ]. Drawing insights from various researches, we present a structured overview of the main factors that guide CEOs in this strategic decision. Key focal points include customer perception as well as feasibility alignment with sustainability objectives. A visual example of corporate ecosystems factors is represented in Figure 1. Figure 1. Corporate Ecosystems Factors. In particular, we have considered four key factors related to the creation of a corporate ecosystem according to [7,13]: • Viability (V) is the financial return. It is generally measured as the expected net present value of cash flows or the percentage of return according to the investment amount. Milhomem and Dantas [14] provide an overview of current developments in methodologies for the project portfolio selection problem (PPSP) to measure riskreturn performance. • Desirability (D) measures how attractive a project or partnership is to potential customers or stakeholders. Several authors are working on measuring this factor, e.g., Lopez and Castillo [15], Barnum and Palmer [16] and Benedek and Miner [17]. • Feasibility (F) is the technical consideration of the solution, assessing whether the ecosystem can realistically be implemented given the current technological, logistical, and expertise-related resources. Feasibility requires aligning technological capabilities with strategic objectives, considering logistical constraints, and evaluating the compatibility of potential partnerships or technologies. Menold et al. [7] elaborates on the importance of measuring feasibility, especially in supply chain management. • Sustainability (S) encompasses the economical, environmental and social dimensions (including ethical and governance aspects). Sustainability ensures that the ecosystem adheres to corporate social responsibility standards and aligns with long-term environmental and social goals. This factor is increasingly crucial in strategic decision making due to growing regulatory pressures and stakeholder expectations about sustainable practices. This is a hot topic where Haessler [18] and Chernev and Blair [19] sets Computers 2024,13, 322 3 of 15 the importance of considering environmental, social, and governance factors of the corporate sustainability strategy. Hence, the main goals of this paper are: (i) to map the major factors of CEOs when making strategic decisions for the creation of ecosystems; (ii) to explore the benefits of computer-supported optimization algorithms for creating ecosystems; (iii) to analyze the results of a case study regarding the application of such algorithms to support decision making; and (iv) to determine limitations and future areas of study regarding the implementation of these algorithms in strategic decisions on creating ecosystems. The remaining of the paper is structured as follows: Section 2presents a review of the literature to investigate and synthesize studies on the creation of ecosystems and the use of computer-supported algorithms in decision-making processes. Section 3contains the mathematical formulation of the problem. Section 4explains the solution approach. Then, the computational experiments and their associated results are presented in Section 5. Lastly, we present our main findings and further research lines in Section 6. 2. Related Work This section focuses on achieving insights from the scientific community regarding the major factors influencing CEOs’ strategic decision making in the creation of corporate ecosystems alongside an exploration of the current state of artificial intelligence (AI) in strategic decision-making processes, particularly focusing on the PPSP and POP. To achieve this, an exhaustive search strategy was implemented, including publications from Elsevier, Google Scholar, Scopus and Web of Science. Keywords such as: ‘CEO decision making on corporate ecosystems’, ‘ecosystem creation’, ‘strategic decision-making processes’, ‘measurement of feasibility, desirability and sustainability in strategic projects’, ‘project portfolio selection problem’, and ‘multi-objective portfolio optimisation’ were employed. To ensure currency and relevance, we restricted the period to the last five years (from 2019 to 2023). As a result, 123 publications were initially identified, comprising 10 book chapters, 26 conference proceedings, and 87 articles. Following a rigorous screening process, 47 papers were selected based on their alignment with the objectives of this study. 2.1. CEOs Deciding the Creation of Ecosystems Strategic decision making in ecosystem creation requires significant investment in resources and affects long-term profitability and survival of the firm. As defined by Adner [20], Vera et al. [21] and Shepherd and Rudd [22] , this process encompasses a series of rational, comprehensive, and political tasks, including information gathering, alternative creation, and selection. While much of the existing literature provides a qualitative review of ecosystem strategies, focusing on the rationale behind CEOs’ shift towards ecosystem models, our analysis extends deeper into both theoretical and empirical dimensions. Discussions around ecosystems have often been based on Markowitz portfolio theory, emphasizing the optimal balance of return and risk. However, Adner and Kapoor [23] , Adner [24] has shifted the focus toward how ecosystems can create collective value beyond mere profitability, enhancing product depth and customer complementarities. Scholars like Clarysse et al. [25] , Jacobides et al. [26] , and Wei et al. [27] further detail the mechanics of ecosystem construction, examining strategic decisions that emphasize the importance of high-quality partnerships and selective promotion strategies. Talmar et al. [28] introduced the Ecosystem Pie Model (EPM), which helps to visualize the ecosystem’s components including value propositions, user segments, and the roles of various actors and resources. Furthermore, Rodriguez-Garcia et al. [29] have mapped the integration of AI and the internet of things (IoT) into these strategic processes, pinpointing a gap in understanding the evolutionary dynamics of ecosystems, including partner networks, internal capabilities, and governance structures. The literature also covers strategic portfolio matrices, like the Boston Consulting Group Matrix and the GE Matrix of McKinsey, which serve as tools for analyzing and selecting business strategies based on market position and industry attractiveness. Discussions around non-financial factors such as desirability, particularly concerning user experience, Computers 2024,13, 322 4 of 15 are found in works by Griffin [30] and Adikari et al. [31] . Menold et al. [7] and Najmi and Makui [32] take a look at the feasibility aspects, considering internal business processes and the environment, highlighting the importance of flexibility, reliability, and responsiveness in ecosystem management. Sustainability has become a significant focus, integrating environmental and social governance into strategic frameworks as discussed by Husted and Allen [33] and Kuhlman and Farrington [34] . This shift towards sustainability is reflected in modern strategic decisions, incorporating principles of the triple bottom line and corporate social responsibility to ensure ecosystems are not only profitable, but also ethically and environmentally sound. This enriched understanding of ecosystem creation underscores the complexity of CEO decision making in today’s business environment, where strategic agility and comprehensive analysis are relevant. 2.2. AI in Strategic Decision Making for Ecosystems The strategic application of AI in ecosystem management, particularly through the PPSP and POP, is a rapidly evolving field with extensive literature exploring its complexities Trunk et al. [35] , Adesina et al. [36] . The methodologies for these problems are generally categorized into two main types: standard methods, which include tools like multi-criteria decision trees and the analytical hierarchy process, and advanced quantitative methods. These sophisticated approaches are designed to address the difficulties of multi-objective decision making in complex environments. For example, Mehrez and Sinuany-Stern [37] described PPSP as a search for utility functions that encapsulate various organizational goals. Rao [38] utilized goal programming in conjunction with a DELPHI process to map decision-maker preferences, highlighting the effort required to align strategic objectives with practical outcomes. The introduction of decision support systems like PROSEL by R˘adulescu and R˘adulescu [6] further aimed to improve the quality of the selection of the project portfolio. Innovations continued with the analytic network process [ 39 ] and the strategic portfolio management tool [ 40 ], which incorporate complex analytical models to refine decisionmaking processes. Techniques such as fuzzy logic [ 41 ] and mathematical programming models [ 42 ] integrate stochastic elements and simulations to manage complexity effectively. Recent developments have seen the rise of simheuristic algorithms, as discussed by Chica et al. [43] . These algorithms combine metaheuristics with simulation techniques, allowing decision makers to handle uncertainties and achieve high-quality solutions in unpredictable environments. Metaheuristics, as described by Glover and Kochenberger [44] , provide feasible solutions within acceptable time frames for complex problems, demonstrating their utility in strategic ecosystem management. Applications of metaheuristics in finance are considered in and Doering et al. [45] . This diverse array of models and methods underscores a significant trend towards integrating more intricate, data-driven approaches to optimize ecosystem creation and management, reflecting a shift towards more adaptive, robust strategic planning tools that cater to the dynamic needs of modern businesses. 2.3. Gaps on the Application of AI in Strategic Decision Making Despite the scientific literature identifies and utilizes various strategic decision models and tools for ecosystem creation, significant gaps remain. Advances in AI for strategic decision making in ecosystem creation could be applied with particular models and visualizations. The EPM, inspired by Ron Adner’s work [ 20 ], highlights the need for decision models that dynamically adapt to changing market conditions, regulatory changes, and technological disruptions. Furthermore, existing AI applications lack comprehensive metrics to accurately measure the effectiveness of ecosystem strategies and often fail to account for complex interdependencies [ 46 ]. Additionally, the complexity of AI tools poses barriers for non-technical decision makers, suggesting a need for more user-friendly designs to make AI more accessible [ 47 ]. Simplifying user interfaces and improving the interpretability of AI outputs are essential steps toward making advanced AI tools more accessible and actionable for strategic decision making. Computers 2024,13, 322 5 of 15 Based on the reviewed literature, Table 1highlights the contribution of our article and how our approach differs from previous related papers. Table 1. Comparative Analysis of References and Contributions. Reference Focus on Financial Metrics (RiskReturn) Includes Strategic Ecosystem Factors Dynamic Decision Models Incorporates AI Usability Empirical Validation Integration of ESG Factors Broader Strategic Alignment Yang and Yan [1]Yes Yes No No No Yes Yes Autio [2] Yes Yes Yes No No No Yes Buehring and Bishop [3]Yes Yes Yes No No No Yes He et al. [4] Yes No Yes Yes No No No Loke et al. [5] No No Yes Yes No No No Sumar and Karlsson [9]Yes Yes Yes No Yes No Yes Markowitz [10]Yes No No No No No No Adner [20], Adner and Kapoor [23] No Yes Yes No No No Yes Jacobides et al. [26]No Yes No No No No Yes Wei et al. [27] No Yes Yes No No No No Milhomem and Dantas [14] Yes No No No No No No Trunk et al. [35]No Yes Yes Yes No No No Adesina et al. [36]No Yes Yes Yes No No No Danesh and Ryan [40]No Yes Yes Yes No No Yes Talmar et al. [28]No Yes Yes Yes No No No This paper Yes Yes Yes Yes Yes Yes Yes Foundational studies, such as the work of Markowitz [10] , primarily address financial metrics like risk and return without considering broader strategic factors. Adner [20] , Adner and Kapoor [23] , Autio [2] and Jacobides et al. [26] emphasize ecosystem creation but focus on viability and desirability, overlooking feasibility and sustainability as critical dimensions. Similarly, He et al. [4] as well as Milhomem and Dantas [14] explore financial viability but do not integrate qualitative factors or ESG considerations. In contrast, Yang and Yan [1] and Kuhlman and Farrington [34] delve into sustainability, but their work does not incorporate quantitative optimization or ecosystem-specific dynamics. The contributions of Urli and Terrien [8] , West et al. [12] and Adesina et al. [36] emphasize AI-driven optimization and dynamic decision models, yet they lack practical insights into usability for non-technical decision-makers and fail to align these tools with strategic corporate goals. Computers 2024,13, 322 6 of 15 3. Mathematical Model The CEOs problem described above can be modeled as a rich POP, where CEOs are optimizing for conflicting objectives (risk minimization and return maximization), while aiming certain levels of desirability, feasibility, and sustainability. Consider a set of n projects, P={ 1, 2, . . . , n} . For each project i∈P , CEOs must decide how much to invest in it, denoted by xi , which represents the percentage of the total available budget invested in project i . The investment can range from 0% (no investment) to 100% of the budget, i.e., xi∈[ 0, 1 ] for all i∈P . The objective is to minimize the risk of the portfolio, which includes only the projects with a positive investment. A binary variable, zi∈ { 0, 1 } , takes the value 1 if project i has been selected by the CEO to be included in the portfolio, and 0 otherwise. The risk is measured in terms of the covariance matrix of the project investments. Additionally, the expected returns of the selected projects must exceed a given threshold R>0. The problem also considers the following constraints: • Desirability constraint: The desirability level of the portfolio must meet a predefined threshold, D>0. • Sustainability constraint: The sustainability level of the portfolio must meet a specific threshold, S>0. • Technical feasibility constraint: The portfolio must meet a certain technical feasibility level, T>0. • Budget constraint: The total percentage of the budget invested across all projects should not exceed the available budget. • Portfolio size constraint: The number of projects in the portfolio must be within user-defined limits, nmin and nmax. • Minimum/maximum investment constraints: If project i is included in the portfolio, a minimum investment li> 0 must be made, and at most a maximum investment ui≥liis allowed. Let σij represent the covariance between projects i and j for all i , j∈P . Additionally, for each project i∈P , let ri> 0 denote the expected return, di> 0 the desirability value, si> 0 the sustainability value, and ti> 0 the technical feasibility value. The rich version of the POP to minimize risk can be formulated as follows: Minimize: n ∑ i=1 n ∑ j=1 xixjσij (minimize risk) (1) Subject to: n ∑ i=1 xiri≥R(return constraint) (2) n ∑ i=1 dixi≥D(desirability constraint) (3) n ∑ i=1 sixi≥S(sustainability constraint) (4) n ∑ i=1 tixi≥T(technical feasibility constraint) (5) n ∑ i=1 xi≤1 (budget constraint) (6) Computers 2024,13, 322 7 of 15 nmin ≤ n ∑ i=1 zi≤nmax (portfolio size constraint) (7) lizi≤xi≤uizi∀i∈P(minimum/maximum investment constraint) (8) zi∈ {0, 1} ∀i∈P(binary decision variables for project selection) (9) xi≥0∀i∈P(non-negativity constraint) (10) The solution to the problem will be presented using a Pareto frontier, where each curve represents the risk-return combination of projects for a given set of values associated with the desirability, sustainability, and technical feasibility constraints. At each point on the frontier, there is a precise allocation of projects, along with their specific investment percentages. One of the key contributions of our paper is that it allows CEOs to understand the impact of improving one objective at the expense of another. This enables the quantification of trade-offs, showing the cost in terms of risk and return when incorporating a more desirable, sustainable, or feasible project into the portfolio of projects for constructing the ecosystem. 4. Solving Approach The mathematical model proposed in the previous section is a mixed-integer quadratic programming problem. It can be solved with exact optimization algorithms, which are available in different commercial and open source software. In our case, we have modeled the problem using Pyomo, a Python-based and open-source optimization modeling language (https://www.pyomo.org/, accessed on 3 December 2024). Once modeled, we have solved it using the Gurobi optimization engine (https://www.gurobi.com/, accessed on 3 December 2024). The process begins by initializing the problem parameters required for the optimization problem, such as desirability, technical feasibility, and sustainability thresholds (Listing 1). Listing 1. Define problem parameters. 1from pyomo . environ import ConcreteModel , RangeSet , Var , NonNegativeReals , Objective , minimize 2from pyomo . environ import Expression , Constraint , Binary , SolverFactory , value , Param 3import numpy as np 4 5# Define par a met e rs 6n_min = 1 7n_max = num_assets 8min_xi = 0. 0 9max_xi = 1 .0 10 big_M = 1 e20 11 12 des_threshold = 0 .0 # d e s i r a b i l i t y t h r es h ol d 13 fea_threshold = 0.0 # t e c h n i c a l f e a s i b i l i t y t h r e s h o l d 14 sus_threshold = 0.0 # s u s t a i n a b i l i t y t hr e s h o l d 15 np . random . seed ( 42 ) # s e t t h e s e e d to a s p e c i f i c va lue f o r r e p r o d u c i b i l i t y 16 a=7 # each a s s e t l e v e l i s random between a and b 17 b = 10 # b > a 18 des_lev el s = a + ( b − a ) *np . random . rand ( num_assets ) 19 fe a _ l e v e l s = a + ( b − a ) *np . random . rand ( num_assets ) 20 su s_l eve ls = a + ( b − a ) *np . random . rand ( num_assets ) Subsequently, we generate the optimization model using Pyomo v6.8.2., defining the objective function and the constraints (Listing 2). Computers 2024,13, 322 8 of 15 Listing 2. Objective function and constraints. 1# Ge nerate the Pyomo−Gurobi model 2model = ConcreteModel () # d e f i n e th e Pyomo model 3model . I = RangeSet ( 1 , num_assets ) # d e f i n e s e t o f i ndexes 4model . x = Var ( model . I , domain=NonNegativeReals ) # d e f i n e v a r i a b l e s 5 6# Def in e t h e o b j e c t i v e f un ct io n 7 8def portf_risk_rule(model) : 9p_risk = sum(sum( cov_matrix [ i −1][ j −1] *model . x [ i ] *model . x [ j ] for jin model . I ) for iin model . I ) 10 return p_risk 11 12 # E x p ressions to compute p o r t f o l i o o b j e c t i v e s 13 model . po r t f _ r i s k = Objective ( rule= por tf_ ris k_r u le , sense=minimize ) 14 model . portf_ re turn = Expression ( expr=sum( avg_returns [ i −1] *model . x [ i ] for iin model . I ) ) 15 model . portf_des = Expression ( expr=sum( des_ le ve ls [ i −1] *model . x [ i ] for iin model . I ) ) 16 model . port f_f ea = Expression ( expr=sum( fe a _ l e vels [ i −1] *model . x [ i ] for iin model . I ) ) 17 model . portf_sus = Expression ( expr=sum( sus _le vel s [ i −1] *model . x [ i ] for iin model . I ) ) 18 19 # Budget c o n t r a i n t 20 model . sum_weights_cons = Constraint ( expr = sum( model . x [ i ] for iin model . I ) <= 1 . 0 ) 21 22 # Return t h r e s h o l d c o n s t r a i n t 23 model . return_threshold = Param ( i n i t i a l i z e =0.0 , mutable=True ) 24 model . return_cons = Constraint ( expr = model . portf_ re turn >= model . return_threshold ) 25 26 # P o s i t i v e we igh ts c o n s t r a i n t 27 model . low_weight_cons = Constraint ( model . I , rule=lambda model , i : model . x [ i ] >= 0) 28 29 # P o r t f o l i o d e s i r a b i l i t y l e v e l c o n s t r a i n t 30 model . des_cons = Constraint ( expr = model . portf_des >= des_threshold ) 31 32 # P o r t f o l i o t e c h n i c a l f e a s i b i l i t y l e v e l c o n s t r a i n t 33 model . fea_cons = Constraint ( expr = model . po r tf_ fea >= fea_threshold ) 34 35 # P o r t f o l i o s u s t a i n a b i l i t y l e v e l c on st ra in t 36 model . sus_cons = Constraint ( expr = model . portf_sus >= sus_threshold ) 37 38 # C a r d i n a l i t y c o n s t r a i n t s 39 model . i s _ a s s e t _ s e l e c t e d = Var ( model . I , within=Binary ) 40 for iin model . I : # i n i t i a l i z e t h e bin ary v a r i a b l e 41 model . i s _ a s s e t _ s e l e c t e d [ i ] = 0 42 43 def i s_ a s se t _ se l ec t e d_ r u le _ 1 ( model , i ) : # i f x [ i ] > 0 then i s _ a s s e t _ s e l e c t e d [ i ] == 1 44 return model . x [ i ] <= model . i s _ a s s e t _ s e l e c t e d [ i ] 45 model . is _a ss et _s el ec ted _c on s_ 1 = Cons traint ( model . I , r ule= is _ as s et _ se le c te d _r u le _ 1 ) 46 47 def i s_ a s se t _ se l ec t e d_ r u le _ 2 ( model , i ) : # i f x [ i ] == 0 then i s _ a s s e t _ s e l e c t e d [ i ] == 0 48 return model . i s _ a s s e t _ s e l e c t e d [ i ] <= model . x [ i ] *big_M 49 model . is _a ss et _s el ec ted _c on s_ 2 = Cons traint ( model . I , r ule= is _ as s et _ se le c te d _r u le _ 2 ) 50 51 def count_selected_assets_rule(model) : 52 return sum( model . i s _ a s s e t _ s e l e c t e d [ i ] for iin model . I ) 53 model . num_selected_assets = Expression ( rule= co unt _se lec ted _a sse ts_ rul e ) 54 model . p or tf _si ze _lb _c ons = Constraint ( expr = model . num_selected_assets >= n_min ) 55 model . portf_size_ub_cons = Constraint ( expr = model . num_selected_assets <= n_max ) 56 57 # Minimum and maximum i nve stmen t c o n s t r a i n t s 58 model . up_weight_cons = Constraint ( model . I , rule =lambda model , i : model . x [ i ] <= max_xi ) 59 60 def selected_asset_min_invest_rule (model, i ) : 61 return model . x [ i ] >= min_xi *model . i s _ a s s e t _ s e l e c t e d [ i ] 62 model . se lected _asset_ min_inv est_co ns = Constraint ( model . I , rule = selected_asset_min_invest_rule) Finally, we choose Gurobi 12.0 as the optimization solver and use it to find the optimal solution for each return threshold (Listing 3). Once the algorithm finalizes, the results are printed and visualized in a Pareto graph. The code prints a Pareto curve containing 2000 points. Each point contains the optimal portfolio allocation for a given configuration of parameters. Computers 2024,13, 322 9 of 15 Listing 3. Solve the optimization problem. 1# Choose a s o l v e r e ngine 2solver = SolverFactory ( ’ gurobi ’ ) 3 4# Con struct t he Pa r eto f r o n t i e r 5r e s u l t s = [ ] 6 7k = 0 8for threshold in return_thresholds : 9k=k+1 10 11 # Re set s e l e c t e d a s s e t s from th e model 12 for iin model . I : # r e s e t a l l i s _ a s s e t _ s e l e c t e d t o 0 13 model . i s _ a s s e t _ s e l e c t e d [ i ] = 0 14 model . return_threshold = threshold # update t he r et ur n t h r e s h o l d p ara met er 15 16 # Find o pt im al p o r t f o l i o f o r d e f in e d re tu rn t h r e s h o l d 17 sol ver . solve ( model ) 18 19 # E x t r a c t t h e s o l u t i o n i n f o r m a t i o n 20 opt_weights = [ round ( value ( model . x [ i ] ) , 4) for iin model . I ] 21 risk = round( value ( model . p o r t f _ r i s k ( ) ) , 6) 22 exp_return = round( value ( model . portf_re turn ) , 6) 23 des = round( value ( model . portf_des ) , 1) 24 fea = round ( value ( model . po rtf _fe a ) , 1) 25 sus = round( value ( model . portf_sus ) , 1) 26 n_se lected = value ( model . num_selected_assets ) 27 28 # Save s o l u t i o n i nf o rm a ti o n in d i c t i o n a r y 29 r e s u l t s . append ( { ’ Return ’ : exp_return , ’ Risk ’ : risk , ’Desirab ’ : des , ’Feasib ’ : fea , 30 ’ Sustainab ’ : sus , ’N selected ’ : n_selected , ’Weights ’ : opt_weights } ) 5. Computational Experiments and Results The proposed Pyomo-Gurobi algorithm is executed in Python. We use a standard personal computer, manufactured by ASUSTeK COMPUTER INC. in Taipei (Taiwan), with an Intel Core i7 CPU at 2.5 GHz and 12 GB RAM with Windows 10 to run all tests. 5.1. Creation of the Database for the Experiment In order to validate the correctness and effectiveness of our algorithm we have used a realistic database, which is described in Chang et al. [48] for the risk-return variables. These authors constructed five test data sets considering the stocks involved in five different capital market indices drawn from around the world. Specifically, he considered Hang Seng (Hong Kong), DAX 100 (Germany), FTSE 100 (UK) and S&P 100 (USA). The indices have been extensively used in computational experiments by scholars within the related body of research Tasgetiren and Suganthan [49] and Lu and Vasko [50] . The data were sourced from Datastream to obtain weekly price data for the stocks in these indices with a time frame from March 1992 to September 1997. The data cleaning process was run to eliminate stocks with missing values from the analysis. Return is calculated as the average of 52 periods, each of which covers one week, covering a period of 5.5 years. This approach smooths out erratic fluctuations and provides a solid perspective on company returns. Risk is measured through standard deviation, making it possible to identify variability in investment returns. With the data of returns and risk we have calculated the respective covariance matrix. As a result, there were 291 values available for each stock to compute returns and covariances. The datasets varied in size, ranging from 31 to 225 stocks. All the test problems are publicly available from OR-Library (https://people.brunel.ac.uk/~mastjjb/jeb/info.html, accessed on 19 October 2024). Due to the fact that there is no public and objective information on the desirability, feasibility and sustainability factors, we have decided to use a random creation of these variables. The values will be generated with the NumPy 2.1 library of Python. The establishment of corresponding thresholds will be a discretion of the user (CEOs). In this paper we use 3 different thresholds for each variable and for the 5 indices.