A multiple stakeholder-based target-oriented robust optimization approach and its applications

In real-world decision-making, multiple stakeholders often participate, each with diverse and sometimes conflicting interests, which may fall exclusively under the expertise of individual decision-makers. Existing multiple-criteria decision-making (MCDM) methods can accommodate multiple criteria but typically fail to reconcile conflicting stakeholder priorities into a satisficing solution. To address this gap, this paper proposes the multiple stakeholder-based target-oriented robust-optimization (MS-TORO) approach, which explicitly embeds stakeholder interests into an optimization framework that minimizes deviations among priorities. The implementation procedure involves eliciting ordinal stakeholder preferences, parameterizing trade-offs, and solving the optimization model to generate an aggregated solution. Three case studies demonstrate the applicability and viability of MS-TORO, showing that it effectively produces solutions that satisfy the performance targets defined by each decision-maker across all criteria.

1 Introduction

When a decision-maker has a set of desirable alternatives to choose from, a choice is made based on the notion of subjective value or utility of alternatives as in the decision-making theory [1]. Often, when a single criterion cannot sufficiently assess the viability of alternatives, decision-makers refer to multiple criteria decision-making (MCDM) as a field of operational research and a very important branch of decision-making theory to analyze alternatives given a set of multiple and even conflicting criteria [2]. To a relatively greater extent, the theory of MCDM problems is centered on the simultaneous achievement of several objectives subject to diverse decision criteria, which can also be modeled as an optimization problem, among other techniques [3]. While most objectives involved in multi-objective optimization cases are conflicting in nature due to different performance targets of a given set of stakeholders, it is imperative to consider that solutions generated may reach Pareto-optimality or non-domination. By non-dominated solutions, or Pareto optimal solutions, a set of non-inferior solutions in the objective space that defines a boundary beyond which none of the objectives can be further improved without having to sacrifice at least one of the other objectives [4]. Typically, classical methods such as weighting method (parametric method), goal programming method, genetic algorithms, and ε-constraint method are used to generate a Pareto set of solutions. However, several disadvantages in the use of such methods may arise including, but not limited to, computational effort and inability of the method to retain the type of optimization model solved (e.g., pure integer linear programming) [3].

Practically, MCDM problems can be classified according to the solution space of the problem being either continuous or discrete. In continuous problems, multiple objective decision-making methods (MODM) are used. MODM problems involve optimization problems. On the other hand, for discrete cases, multi-attribute decision-making (MADM) methods are employed [5]. MADM methods, or is also commonly termed discrete MCDM, involve a set of feasible alternatives, a set of decision-making criteria, and a corresponding score of alternatives with respect to each criterion. The MCDM tool centers on the achievement of selecting the best (i.e., being most desirable or most important) alternative among the set of feasible alternatives given, ranking the alternatives based on some metric, and portfolio characterization, to name a few. The final selection of an alternative is characterized by its overall value which can be obtained with the use of various methods in the literature. Generally, a simple additive weighted value function is used to signify the overall value of an alternative should weights (i.e., priority vectors) across all criteria are considered [6].

However, the manner of setting weights to criteria has become a major issue in the decision-making domain. Therefore, in the last few decades, several MCDM methods are developed to introduce how the weights of criteria are obtained. In fact, early formulations of MCDM methods sought to extract priority vectors among criteria including analytic hierarchy process (AHP) [7–9], analytic network process (ANP) [10], a technique for order of preference by similarity to ideal solution (TOPSIS) [11–15], Elimination Et Choix Traduisantla REalité (Elimination and Choice Expressing REality) [16–18], VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) [19], and preference ranking organization method for enrichment evaluations (PROMETHEE) [20–23], and best-worst method (BWM) [5], to name a few. It is also worth emphasizing that the fuzzy logic theory is further integrated into the classical MCDM methods to account for the subjective judgment of decision-makers [24]. With fuzzy set numbers, the complexity and the uncertainty of objective criteria in the midst of human thinking become convenient to set up by means of linguistic scales within which hold a corresponding set of fuzzy values [25, 26].

These MCDM methods possess outstanding capabilities in terms of obtaining the priority vectors of criteria especially with the use of pairwise comparison. However, these methods are only deemed appropriate when all concerned decision-makers have the capacity to assess the given set of criteria. That is when decision-makers are influenced or have a role to play in the fulfillment of criteria in the decision process. For instance, given a decision problem consists of three criteria on cost, reputation, and time savings. When decision-makers can perform a pairwise comparison among criteria in terms of its impact with respect to the decision process as a whole and to the decision-makers in specific, then the weighting system provided by these MCDM methods may prove to be necessary and appropriate. In cases when the decision-makers have little to no information about certain criteria, then no reliable and valid judgment can be made on such criteria in light of the expertise of decision-makers. For example, when a decision-maker does not have necessary and direct responsibility (or influence) to a particular criterion, then such a decision-maker cannot make an assessment of the criteria accordingly.

To provide a clearer understanding of the performance characteristics of the proposed MS-TORO framework, we compared it with established MCDM methods including AHP, BWM, TOPSIS, and VIKOR (see Table 1). Unlike these conventional techniques, which primarily rely on preference ranking or quantitative scoring and often require external aggregation to handle multiple stakeholders, MS-TORO is formulated as an optimization model with explicitly defined parameters that aim to minimize deviations among competing stakeholder interests. This feature enables the framework to systematically reconcile conflicts and produce solutions that balance priorities across diverse participants. In terms of computational efficiency, MS-TORO operates with low to moderate complexity, as the ordinal rankings and structured trade-off procedures allow efficient solution of the optimization problem for practical problem sizes. Moreover, while AHP, BWM, TOPSIS, and VIKOR require precise numerical inputs and are less suited for intangible or hard-to-measure resources, MS-TORO effectively accommodates qualitative and ordinal judgments within its parameterized optimization structure. Ranking stability is also enhanced, as the model's explicit objective of minimizing stakeholder deviations mitigates sensitivity to small inconsistencies in judgments. These features highlight MS-TORO's methodological advantages, particularly in multi-stakeholder resource allocation problems where conventional MCDM approaches may fail to systematically reconcile conflicting interests.

Table 1

Table 1. Comparative analysis on the key characteristics of MCDM methods and MS-TORO approach.

In practical cases, there are decision processes involving multiple stakeholders which goals and interests are not mutually related. One interest, otherwise considered as a decision criterion, that plays a significant role in the decision-making process may not necessarily be the same criterion considered by another decision-maker. As such, it is only appropriate for a decision-maker to assess a decision criterion with which one has expertise. Otherwise, the decision-maker cannot effectively perform an accurate pairwise comparison across a set of criteria via MCDM methods. Therefore, there is a need for a decision framework to be developed to address such a limitation in the literature involving independent decision criteria. Such a framework can prove to be beneficial in practice especially that certain situations such as the one previously mentioned actually do exist, thereby, making the previously established methodologies inadequate in such an area. To add, note further that each MCDM method is only applicable to some types of decision problems [27]. The type of MCDM method that can address the choice problem by looking into different decision criteria that can only be assessed by a respective decision-maker has not yet been developed in the extant literature.

At a relatively larger scope, decision-making also has its roots in several social science theories. For instance, the stakeholder theory has provided a monumental avenue for scholars and practitioners alike to objectively examine the dynamics of stakeholders within an organization as well as their interests. Since its inception in the 1980s as introduced by Freeman [28], critics have not only recognized its contribution to the field of business ethics, corporate social responsibility, and strategic management, to name a few, but also noticed flaws inherent to its original model. Such flaws include a non-convergent definition of a “stakeholder” and their “interests,” and unavailability of a generally established approach to balancing stakeholder interests, among others [29–32]. The most pressing of these flaws that has become a central focus of further extensions to supplement the stakeholder theory refers to the approach of balancing stakeholder interest [31, 33–37]. It is imperative to consider that in a decision-making process, multiple decision-makers with diverse and often conflicting interests are required to convene at a single, unified solution that does not only satisfy their individual auxiliary goals but also achieve their overall goal. In some instances, the fulfillment of one stakeholder's interest may demean another's. As a result, there exists an imbalance in interests of stakeholders. When such an imbalance occurs, an impending conflict arises, thereby, putting an organization into a chaotic situation. Therefore, an arithmetic-based weighting system including the principles of Kantianism [38], property rights [39], and social contract theory [40] has been early on implemented to formally accommodate all the interests of stakeholders despite the theoretical objections to its very nature of balancing stakeholder interests. In addition, the satisficing approach to decision-making has also become a suitable guiding principle to create a satisfactory decision rather than opting for an approximate best solution. Other than that, uncertainties in the decision-making have yet to be addressed via stochastic modeling approaches with potentially problematic assumptions on probability density functions of parameters which nature is difficult to distinguish due to data unavailability, for instance.

While other scholars have proposed sophisticated analytical techniques to provide a weighting scheme that is representative of the interests of stakeholders [41], it appears to be insufficient due to the tendency of stakeholder interests having an intrinsic value that demands to be satisfied to some extent [39, 42]. The satisficing concept of decision-making, due to Simon [43], has unfortunately not been integrated into the analytical techniques previously proposed to balance stakeholder interest. On the other hand [44], developed TORO approach which allows the achievement of certain system targets in a robust environment; hence, significantly modeling the degree of satisfaction obtained for each target. One caveat of this approach, although successfully demonstrated across various cases, is its singular metric which represents system targets. In the involvement of multiple stakeholders in organizations with conflicting interests, it is imperative to develop a framework that not only balances stakeholder interests but also looks into the achievement of certain targets.

Building on these insights, the motivation of this study is to develop a decision-making framework capable of systematically balancing conflicting stakeholder interests while simultaneously achieving performance targets across multiple criteria. To address this, we propose the MS-TORO approach [45], which extends TORO to multi-stakeholder MCDM problems by allowing simultaneous optimization of multiple objectives in a computationally tractable and scalarized manner. A key feature of the proposed model is the translation of multiple objectives into system constraints, where each bound (upper or lower, depending on the direction of optimization) represents the deviation from the individual optimal solutions. The primary objective function is designed to minimize these deviations, denoted as θ, across all objectives, ensuring that stakeholder interests are balanced while achieving the desired performance targets. This framework contributes to the literature by integrating multi-stakeholder trade-off reconciliation, satisficing principles, and an optimization-based methodology that is applicable to both tangible and intangible decision criteria.

2 Multiple stakeholder-based target-oriented robust-optimization MCDM approach

The proposed MS-TORO model can handle decision-making processes that involve collaboration among stakeholders, also referred to as decision-makers, which corresponding decision criteria are independent. By independence means that one decision criterion can only be best judged and elicited by a particular decision-maker. Other decision-makers to which a decision criterion does not belong are not considered to be suitable experts to frame the decision function. As an example, let A, B, and C be decision criteria involved in a decision process participated by decision-makers D, E, and F. Suppose that the decision criteria A, B, and C corresponds to the interests of decision-makers D, E, and F, respectively. As such, only decision-maker D can most suitably frame decision criterion A with respect to the decision problem. Then, decision-maker E is the only one who can develop decision criterion B. Decision-maker F is concerned about achieving decision criterion C with respect to the decision problem. In other words, decision-maker D is only interested in decision criterion A, and therefore cannot elicit judgment to both B and C. This is the same manner that decision-maker E cannot elicit judgment to decision criterion A and C while decision-maker F can only elicit judgment to C and not A and B.

Other developed MCDM methods in the literature are designed in a way that the decision criteria considered are evaluated by multiple decision-makers with respect to one another as though each decision-maker can suitably assess each criterion. These MCDM methods are only deemed appropriate when such a case is evident. However, when a certain criterion is only accounted for by a specific decision-maker, the use of previously established MCDM methods may be inadequate. The proposed MS-TORO model can very well-handle decision problems involving decision-makers whose concern is only limited to a particular criterion and not necessarily all criteria at hand. Furthermore, the model can also portray how the decision criterion, also referred to as interests of decision-makers, is balanced in an equitable manner such that in the process of generating an agreeable aggregated solution, no criterion is made better or worse off than others. The deviation metric further demonstrates the risk profile of decision-makers as well as the flexibility one can tolerate while accommodating the other's interests.

2.1 The general form of MS-TORO model

In this section, the proposed MS-TORO model is presented in its general form as well as its case-specific form in the context of the green building promotion, supplier selection, and post-departure aircraft rerouting problem which will be later used as a means to demonstrate the model. The rest of the discussions are lifted from the detailed description in the work of Brown et al. [46].

To begin with, the general form of the proposed MS-TORO model is shown in Equation 1. Here, the objective function is set to minimize the deviation of performance targets as represented by the theta symbol, θ. This deviation is a metric from a closed interval of 0 to 1. This performance target denotes the value that a stakeholder may want to achieve. That is, when τ_k as the soft constraint representing the target of stakeholder k, is high, the value of the metric θ will be zero. This implies that stakeholders are more conservative with the target objectives. Otherwise, in more strict cases where τ_k is lowered, then θ will be forced to approach 1. Here, stakeholders are more rigorous to set a higher target.

Under deterministic cases of the decision problem, τ_k is represented by the optimal solution (i.e., targets) generated with respect to each stakeholder. Note that the constraint, c_ijx_n ≤ τ_k(1 + θ) only holds when the function that represents the interest stakeholders are to be minimized (e.g., cost). Otherwise, the constraint c_ijx_n ≥ τ_k/(1 + θ) holds (e.g., profit). The constraint a_ijx_n ≥ b_ij represents all the other system constraints considered in the decision problem. The expression c_ijx_n is defined by the objective function which relates to the interest of stakeholders for every parameter c and variable x. The parameters and n denotes the number of variables x. Additionally, a binary constraint can also be added as may be necessary.

Also note that, τ₁ signifies the lowest possible target performance metric that stakeholders may set while τ₀ represents the highest possible target performance metric. The α values relate to the uncertainty levels that may be realized across the range of 0 to 1.

$\begin{array}{c}{min\theta }_{0\le \theta \le 1}\\ \text{subject to}:& (1)\end{array}$

$\begin{array}{c}\sum c_{ij}{x}_j\le {\tau }_k(1+\theta )for1\le j\le n\\ \sum c_{ij}{x}_j\ge {\tau }_k/(1+\theta )for1\le i\le kand1\le j\le n\\ \sum a_{ij}{x}_j\le b_{i}for1\le i\le kand1\le j\le n\\ x_n\ge 0\\ \tau =\alpha {\tau }_1+(1-\alpha ){\tau }_0\end{array}$

For clarity and ease of reference, Table 2 summarizes all symbols, parameters, and decision variables used throughout the manuscript. This unified notation provides readers with a clear understanding of the model components, performance targets, deviation metrics, and stakeholder-related variables, ensuring consistent interpretation of the MS-TORO framework and the results presented in the case studies.

Table 2

Table 2. Notations used in MS-TORO formulation.

2.2 A step-by-step guide to implement MS-TORO approach

As a straightforward account of comparison between the traditional TORO framework and the proposed MS-TORO framework, the detailed step-by-step guideline of the proposed model is outlined in Figure 1 and further elaborated in the following sections. The step boxes highlighted in the figure represent the novel steps designed specifically for the proposed MS-TORO model.

Figure 1

Figure 1. Implementation roadmap of the proposed MS-TORO approach.

As an overview, note that the proposed framework explicitly defines a step to which the direction of the decision criteria is formulated depending on the orientation of the performance targets. Furthermore, it puts forward the need for allowing decision-makers to set the targets analytically or objectively via the deterministic versions of their initial model in mind. Lastly, the deviation metric and aggregated solutions are generated to reflect how the decision criteria, or interests of stakeholders in a practical lens, are balanced and what solutions are considered to be agreeable among them. The specific steps of the proposed MS-TORO model are presented as follows.

Suppose there are k stakeholders, also termed as decision-makers, each having their own individual decision criteria and performance targets represented by τ. Suppose further that the decision problem can be modeled as in a choice problem represented by a pure integer linear programming model. Such a decision problem is initiated under several objective functions subject to system constraints. The proposed MS-TORO model facilitates the selection of alternatives based on an established set of criteria.

2.2.1 Step 1: establish relevant decision alternatives

Suppose that there are n alternatives involved in the decision problem where {x₁, x₂, x₃, …, x_n}. Decision-makers aim to establish an aggregated solution from among the given alternatives, x_n. For the case of this sample problem, the optimization model takes the form of an integer linear programming model to ensure a choice type of decision model.

2.2.2 Step 2: build the decision criteria system involved in the problem

The decision criteria system reflects the interests of decision-makers which is typically established as a target function being oriented toward either minimizing or maximizing directions. The coefficients of the decision criteria, c_n, further reflect the parameters involved in achieving a particular decision-maker interest. Suppose there are n decision criteria where {x₁, x₂, x₃, …, x_n}. Embedding the parameters into the decision criteria, the equation takes the form of c_nx_n ≤ τ_k(1 + θ) and/or c_nx_n ≥ τ_k/(1 + θ) for a minimize and maximize objective, respectively.

2.2.3 Step 3: set up the direction of decision criteria

The direction of the decision criteria specified in the earlier step takes on a right-hand side value of ≤ τ_k(1 + θ) for a target function that involves a minimize direction or ≥ τ_k/(1 + θ) for a target function which involves a maximize direction.

2.2.4 Step 4: set up relevant system constraints

Suppose further that there are n set of constraints where {a₁, a₂, a₃, …, a_n} corresponds to each parameter with a right-hand side value of b_n. Also, a constraint enforcing binary variables is set as in x_n ∈ 0, 1. The system constraints take on a_nx_n ≤ b_n or a_nx_n ≥ b_n, whichever is applicable.

2.2.5 Step 5: set target performance for each decision criteria (optional)

The target performance, τ, may be set analytically by decision-makers as may be deemed necessary. When such a course is opted, this step may be skipped. Otherwise, the target can be set in a more objective manner as follows:

2.2.5.1 Set target performance for each decision criterion

Suppose that decision-makers aim to minimize (or maximize) a decision criterion, such can be translated into an objective function being min(ormax)c_nx_n subject to system constraints. The complete linear programming model to generate the optimal value, or the target, for each decision criteria may be formulated as in Equation 2

$\begin{array}{c}min(ormax)c_n{x}_n\\ \text{subject to}:& (2)\end{array}$

$\begin{array}{l}\sum a_{ij}{x}_n\le b_{i}for1\le i\le kand1\le j\le n\\ \sum a_{ij}{x}_n\ge b_{i}for1\le i\le kand1\le j\le n\\ x_n\in 0,1\end{array}$

2.2.5.2 Run the model and generate results

A linear programming model solver can be used to run the model and generate the optimal value equivalent to the target performance of each decision-maker. Other than the target metric, a solution set can also be obtained which represents the supposed preference of decision-makers should their own decision criteria is considered as the sole criterion in the decision problem. The model in Section 2.2.5.1. may be rerun for a number of decision criteria as may be necessary. That is, every decision criterion that requires a target may be run using such a model. In summary, a target reflecting the decision criteria of decision-makers along with the solution sets specific to suit their individual criterion is generated at this step. These model outputs may be used for further comparative analyses with respect to the aggregated preference which will be generated from the proposed MS-TORO model.

2.2.6 Step 6: set objective function subject to the decision criteria and system constraints to form the MS-TORO model

In this step, the objective function takes on to minimize θ representing the deviation from the target set by decision-makers with respect to the decision criteria. This deviation metric is a member of a closed interval [0, 1]. For a decision-maker who seeks to minimize a target performance, a higher value of θ implies a risk profile being more rigid and strict to the achievement of the target. On the other hand, a lower value of θ signifies a more conservative decision-maker with respect to the achievement of the target. For a decision-maker who seeks to maximize a target performance, the value of θ is interpreted in the opposite manner. The complete MS-TORO model can be presented as in Equation 3. Note that in particular, the MS-TORO model takes on its robust counterpart as in Equation 3 where alpha, α, denotes the degree of uncertainty. The MS-TORO model is run under various realizations of uncertainty from 0 to 1. It is important to emphasize that the MS-TORO model is capable of providing a solution set based on various uncertainty levels. This contribution is deemed to be significant for stakeholders to arrive at a solution given that capacity of routes cannot be readily determined based on its probability distribution function. Therefore, the solution set provided by the MS-TORO model becomes necessary as a guideline for stakeholders to follow in the event of uncertain capacity.

$\begin{array}{c}min\text{θ }\\ \text{subject to}:& (3)\end{array}$

$\begin{array}{c}{c}_{ij}{x}_j\le {\tau }_k(1+\theta )for1\le i\le kand1\le j\le n\\ c_{ij}{x}_j\ge {\tau }_k/(1+\theta )for1\le i\le kand1\le j\le n\\ \sum a_{ij}{x}_n\le b_{i}for1\le i\le kand1\le j\le n\\ \sum a_{ij}{x}_n\ge b_{i}for1\le i\le kand1\le j\le n\\ \tau =\alpha {\tau }_1+(1-\alpha ){\tau }_0\\ x_n\in 0,1\\ \theta \in 0,1\end{array}$

2.2.7 Step 7: run the model and generate results

A linear programming model solver can be used to run the model. As outputs, the value of the deviation metric θ can be obtained together with the solution set based on the choice problem. The deviation metric θ represents the degree to which the target set by each decision-maker for each decision criterion can be stretched to reach a feasible solution that is also aggregable to other decision-maker's decision criteria. A θ equal, or approaching, to 0 entails a completely favorable scenario as this signifies that a feasible aggregated solution is obtained while completely satisfying the targets set by decision-makers. On the other hand, a θ equal, or approaching, to 1 implies the maximum deviation that the model can tolerate while still meeting the system constraints and producing feasible solutions.

3 Case studies

This section presents three practical case studies to illustrate the applicability of the proposed MS-TORO model. The proposed MS-TORO model is implemented in an ASUS laptop computer under the Windows 10 Home Single Language operating system with Intel Core i7 8th Gen processor and a 16.0 GB RAM. A Demo version of LINGO Win64 version 19.0.32 is used.

3.1 Case study 1: green building promotion analysis

Consider a green building promotion problem consisting of five alternatives. The decision-making process is composed of three stakeholders each assigned to oversee a particular area of the organization. Decision-maker A's major concern is on the financial aspect of selecting a green building alternative. Decision-maker B, on the other hand, seeks to maximize the marketability of the green building. Then, decision-maker C aims to minimize the travel time from their key points to the selected green building alternative. These interests are further summarized in Table 3. Furthermore, the organization also has to consider the following areas in the decision-making process (see Table 4).

Table 3

Table 3. Characteristics of decision criteria (case study 1).

Table 4

Table 4. System constraints for case study 1.

Following step 1, the relevant decision alternatives considered in this case are the candidate buildings. Then, step 2 requires building the decision criteria involved in the decision problem. These decision criteria are composed of the “acquisition cost,” “customer reachability,” and “travel time from suppliers” which each correspond to the interests of the decision-makers, respectively. In step 3, the direction of the decision criteria is established. Note that the decision criteria on both “acquisition cost” and “travel time from suppliers” are sought to be minimized by the decision-makers. Therefore, these decision criteria will take on the general form of as in Equation 4. On the other hand, the remaining decision criterion on “customer reachability” is aimed to be maximized, thus, the right-hand side of this criterion will take on Equation 5. Plugging in the parameters for the decision criteria, Equations 6–8 show the complete formulation with the right-hand side of the equation.

$\begin{array}{cc}{c}_{ij}{x}_j\le {\tau }_k(1+\theta )& (4)\end{array}$

$\begin{array}{cc}{c}_{ij}{x}_j\ge {\tau }_k/(1+\theta )& (5)\end{array}$

$\begin{array}{cc}1000x_1+1500x_2+3500x_3+2000x_4+4000x_5\le {\tau }_1(1+\theta )& (6)\end{array}$

$\begin{array}{cc}200x_1+100x_2+150x_3+250x_4+290x_5\le {\tau }_2(1+\theta )& (7)\end{array}$

$\begin{array}{cc}120x_1+200x_2+70x_3+90x_4+90x_5\ge {\tau }_3/(1+\theta )& (8)\end{array}$

For step 4, the relevant system constraints are set up. To formulate the constraints, data from Table 3 are utilized and translated as in Equations 9 and 10 as follows. Also note that the uncertain parameters considered in this work is the capacity of the buildings and travel time of customers. Supposing that this uncertain parameter has a lowest possible value, τ₀, equal to one-third of its initially set target, τ₁, so does Equation 9 present. For the uncertain parameter being capacity, this follows the formulation ατ₁ + (1 − α)τ₀. The highest value of these parameters is given in Tables 3, 4 while the lowest possible values are taken from the lowest possible capacity due to uncertainty indicated in the second row of Table 4. Thus, Equations 9–11 represent the constraints in capacity under uncertainty, extra space, and labor availability, respectively.

$\begin{array}{c}(300\alpha +(1-\alpha )(100))x_1+(200\alpha +(1-\alpha )(66))x_2\\ +(190\alpha +(1-\alpha )(63))x_3+(250\alpha +(1-\alpha )(83))x_4\\ +(250\alpha +(1-\alpha )(83))x_5\le 200& (9)\end{array}$

$\begin{array}{cc}100x_1+100x_2+50x_3+100x_4+90x_5\le 100& (10)\end{array}$

$\begin{array}{cc}75x_1+80x_2+50x_3+90x_4+150x_5\le 85& (11)\end{array}$

Suppose for this case a performance target is intuitively set by decision-makers as shown in Table 3. Therefore, step 5 may no longer be necessary to obtain the targets. As such, Equations 6–8 are now translated into Equations 12–14 with account for the performance targets of decision-makers for each decision criteria.

$\begin{array}{cc}1000x_1+1500x_2+3500x_3+2000x_4+4000x_5\le 2500(1+\theta )& (12)\end{array}$

$\begin{array}{cc}200x_1+100x_2+150x_3+250x_4+290x_5\le 250(1+\theta )& (13)\end{array}$

$\begin{array}{c}(240\alpha +(1-\alpha )(150))x_1+(400\alpha +(1-\alpha )(300))x_2\\ +(140\alpha +(1-\alpha )(100))x_3+(180\alpha +(1-\alpha )(150))x_4\\ +(180\alpha +(1-\alpha )(150))x_5\ge 350/(1+\theta )& (14)\end{array}$

To complete the MS-TORO model, step 6 is carried out as in Equation 15. Note that the last three constraints represent that the model (a) only aims to select one buildings alternative, (b) sets the value of the deviation metric from 0 to 1, and (c) has binary variables.

$\begin{array}{c}min\theta \\ \text{subject to}:& (15)\end{array}$

$\begin{array}{c}1000x_1+1500x_2+3500x_3+2000x_4+4000x_5\le 2500(1+\theta )\\ 200x_1+100x_2+150x_3+250x_4+290x_5\le 250(1+\theta )\\ (240\alpha +(1-\alpha )(150))x_1+(400\alpha +(1-\alpha )(300))x_2\\ +(140\alpha +(1-\alpha )(100))x_3+(180\alpha +(1-\alpha )(150))x_4\\ +(180\alpha +(1-\alpha )(150))x_5\ge 350/(1+\theta )\\ (300\alpha +(1-\alpha )(100))x_1+(200\alpha +(1-\alpha )(66))x_2\\ +(190\alpha +(1-\alpha )(63))x_3+(250\alpha +(1-\alpha )(83))x_4\\ +(250\alpha +(1-\alpha )(83))x_5\le 200\\ 100x_1+100x_2+50x_3+100x_4+90x_5\le 100\\ 75x_1+80x_2+50x_3+90x_4+150x_5\le 85\\ x_1+x_2+x_3+x_4+x_5=1\\ 0\le \theta \le 1\\ x_1,x_2,x_3,x_4,x_5\in \left\{0,1\right\}\end{array}$

By solving Equation 15, as in step 7, the following results shown in Table 5 is generated. The aggregated solution indicates that, considering the decision criteria of “acquisition cost,” “customer reachability,” and “travel time from suppliers,” Building 2 is selected as the optimal alternative. Across uncertainty realizations from 0.0 to 1.0 (with increments of 0.2, 0.5, and 0.7), the deviation metric θ quantifies the extent to which stakeholders must adjust their individual performance targets to achieve a compromise solution. Specifically, θ increases as uncertainty rises, reflecting that stakeholders are willing to relax their targets proportionally to maintain feasibility and reach agreement. However, when uncertainty reaches its maximum (1.0), θ does not increase further, indicating that stakeholders cannot adjust their targets beyond a certain point without violating feasibility. This demonstrates the decision-making significance of θ: it captures the balance between stakeholder flexibility and constraints, providing insight into how much compromise is required to satisfy all interests. In practical terms, decision-makers can interpret changes in θ as guidance for adjusting performance targets strategically, ensuring that solutions remain balanced, achievable, and aligned with the collective goals of all stakeholders.

Table 5

Table 5. Results of the MS-TORO model (case study 1).

3.2 Case study 2: supplier selection

Suppose that there are three decision-makers convening to select the most favorable supplier from among four alternatives. This decision problem follows the main interest of each decision-maker in terms of maximizing “certification” of suppliers, maximizing capability for “product development,” and minimizing “defect rate.” These interests, also termed decision criteria, have various attributes with respect to each alternative supplier (see Table 6). Also, note that the decision-makers have their respective targets set for each decision criterion due to some limitations in resources. There are also several constraints to which the decision process is subject to such as “cost,” “lead time,” and “customer base” as presented in Table 8.

Table 6

Table 6. Characteristics of decision criteria (case study 2).

For step 1 of the MS-TORO framework, the decision alternatives are identified being the candidate suppliers with various characteristics in terms of “certifications,” “product development,” “defect rate,” “cost,” “lead time,” and “customer base.” After establishing the alternatives, the decision criteria are then defined in step 2. For this case, the decision criteria involved are “certifications,” “product development,” and “defect rate.” These decision criteria each correspond to the interests of the decision-makers. It is also imperative to emphasize that these decision criteria are assumed to belong to separate decision-makers (i.e., stakeholders) which can only assess the decision criterion belonging under their responsibility. As an illustration, decision-maker A is mainly interested to fulfill the criterion of maximizing the “certification” portfolio of suppliers as his expertise dictates. Decision-makers B and C cannot satisfactorily evaluate the decision criterion on “certification” due to a lack of expertise in this respect. Then, in step 3, the direction of the decision criteria is defined. For decision criteria that are sought to be maximized, that is, “certification” and “product development,” the right-hand side of these criteria will take on the form as in Equations 16 and 17 with the parameters plugged in. On the other hand, for “defect rate” which is to be minimized, the right-hand side will take on Equation 18

$\begin{array}{cc}29x_1+18x_2+20x_3+12x_4\ge {\tau }_1/(1+\theta )& (16)\end{array}$

$\begin{array}{cc}0.50x_1+0.25x_2+0.30x_3+0.19x_4\ge {\tau }_2/(1+\theta )& (17)\end{array}$

$\begin{array}{cc}0.05x_1+0.05x_2+0.02x_3+0.10x_4\le {\tau }_3(1+\theta )& (18)\end{array}$

To continue with the MS-TORO framework, step 4 involves setting up the relevant system constraints. Using the data found in Table 7, the following constraints are formulated as in Equations 19–21. Note that Equation 20 represents the uncertain parameter for this case study.

$\begin{array}{cc}85x_1+100x_2+50x_3+75x_4\le 90& (19)\end{array}$

$\begin{array}{c}(10\alpha +(1-\alpha )(8))x_1+(10\alpha +(1-\alpha )(8))x_2\\ +(12\alpha +(1-\alpha )(9))x_3+(7\alpha +(1-\alpha )(4))x_4\le 10& (20)\end{array}$

$\begin{array}{cc}97x_1+70x_2+80x_3+90x_4\ge 60& (21)\end{array}$

Table 7

Table 7. System constraints for case study 2.

Similar to the first case, the performance targets of the supplier selection problem in this part are also predetermined. As can be seen in Table 7, the targets for each decision criterion are identified. Therefore, step 5 plugs in the targets to the model. See Equations 22–24.

$\begin{array}{cc}29x_1+18x_2+20x_3+12x_4\ge 23/(1+\theta )& (22)\end{array}$

$\begin{array}{cc}0.50x_1+0.25x_2+0.30x_3+0.19x_4\ge 0.30/(1+\theta )& (23)\end{array}$

$\begin{array}{cc}0.05x_1+0.05x_2+0.02x_3+0.10x_4\le 0.03(1+\theta )& (24)\end{array}$

To proceed with the MS-TORO framework, the decision criteria and system constraints are embedded into the model in Equation 25. Notice that there are three constraints in the model that signify the selection of only one supplier, restrict the value of the deviation metric from 0 to 1, and ensure that variables take on a binary form.

$\begin{array}{c}min\theta \\ \text{subject to}:& (25)\end{array}$

$\begin{array}{l}29x_1+18x_2+20x_3+12x_4\ge 23/(1+\theta )\\ 0.50x_1+0.25x_2+0.30x_3+0.19x_4\ge 0.30/(1+\theta )\\ 0.05x_1+0.05x_2+0.02x_3+0.10x_4\le 0.03(1+\theta )\\ 85x_1+100x_2+50x_3+75x_4\le 90\\ (10\alpha +(1-\alpha )(8))x_1+(10\alpha +(1-\alpha )(8))x_2\\ +(12\alpha +(1-\alpha )(9))x_3+(7\alpha +(1-\alpha )(4))x_4\\ \le 10\\ 97x_1+70x_2+80x_3+90x_4\ge 60\\ x_1+x_2+x_3+x_4=1\\ 0\le \theta \le 1\\ x_1,x_2,x_3,x_4\in \left\{0,1\right\}\end{array}$

Solving Equation 25, as explained in step 7, generates the following results in Table 8. The MS-TORO model identifies Supplier 3 as the optimal choice while simultaneously satisficing the diverse interests of decision-makers, quantified by a deviation metric θ of 0.15 at uncertainty levels between 0.0 and 0.2. Supplier 3 meets the combined priorities of stakeholders in terms of “certifications,” “product development,” and “defect rates.” The deviation metric θ indicates the extent to which individual performance targets must be adjusted to reach a compromise solution. For example, decision-maker A initially set 23 as the target number of “certifications.” Applying the deviation metric, the adjusted target becomes 20 [23/(1 + 0.15)] to accommodate other stakeholders. Similarly, the target for “product development” is adjusted from 0.30 to 0.26 [0.30/(1 + 0.15)], and the target for “defect rates” is adjusted from 0.03 to 0.0345 [0.03 × (1 + 0.15)]. When uncertainty increases from 0.5 to 1.0, the model selects Supplier 1, requiring stakeholders to adjust their targets by approximately 67% to maintain a feasible and balanced solution. These results highlight the decision-making significance of θ, demonstrating how the model quantitatively reconciles conflicting stakeholder priorities and indicates the degree of compromise required. Furthermore, the results illustrate the robustness of MS-TORO under varying levels of uncertainty, providing actionable insight into how decision-makers can implement the approach in real-world scenarios while preserving alignment among multiple interests.

Table 8

Table 8. Results of the MS-TORO model (case study 2).

3.3 Case study 3: aircraft rerouting process

In a post-departure aircraft rerouting problem, suppose that there are three stakeholders involved in the decision-making process. During air traffic congestion which calls for rerouting of flights, the decision-makers has to choose from four alternate routes with various characteristics in length and cost. Each decision-maker is concerned with the fulfillment of their individual goals such as minimizing operating costs, maximizing utilization of routes, and minimizing delays. Also see Table 9 for more details about the case. The selection of an alternate route is further subject to constraints in capacity and the workload of air traffic controllers (see Table 10).

Table 9

Table 9. Characteristics of decision criteria (case study 3).

Table 10

Table 10. System constraints for case study 3.

Following the step-by-step guideline in implementing the MS-TORO model, we begin with identifying the relevant decision alternatives as in step 1. For this case, the alternatives are represented by the alternate routes that can be taken by en-route flights in the case of air traffic congestion. The decision of selecting an alternate route is subject to decision criteria which are sought to be fulfilled by the decision-makers. These decision criteria are composed of “operating cost,” “utilization,” and “delays.” The third step involves setting the direction of the decision criteria with respect to the interests of decision-makers. Note that the “operating cost” and “delays” are ought to be minimized, therefore, the right-hand side of these criteria will take on the form of Equations 26 and 28. As for the criterion on “utilization” which is expected to be maximized, the right-hand side will take on Equation 27. Plugging in the parameters of the alternatives with respect to the decision criteria, the complete formulation for the criteria is shown as follows. Take note, further, that the utilization is considered as an uncertain parameter.

$\begin{array}{cc}460x_1+256x_2+300x_3+400x_4\le {\tau }_1(1+\theta )& (26)\end{array}$

$\begin{array}{c}(78\alpha +(1-\alpha )(8))x_1+(75\alpha +(1-\alpha )(8))x_2\\ +(59\alpha +(1-\alpha )(9))x_3+(56\alpha +(1-\alpha )(4))x_4\\ \ge {\tau }_2/(1+\theta )& (27)\end{array}$

$\begin{array}{cc}14x_1+16x_2+10x_3+10x_4\le {\tau }_3(1+\theta )& (28)\end{array}$

In step 4, the system constraints are established. Table 10 contains the data about these constraints and are translated as in Equations 29–33. In here, note that the capacity of alternate routes is set to be lesser or equal to the number of flights under consideration (i.e., 50). Further, the workload of air traffic controllers should also not exceed their mental capacity (i.e., 10).

$\begin{array}{cc}42x_1\le 50& (29)\end{array}$

$\begin{array}{cc}44x_2\le 50& (30)\end{array}$

$\begin{array}{cc}31x_3\le 50& (31)\end{array}$

$\begin{array}{cc}27x_4\le 50& (32)\end{array}$

$\begin{array}{cc}6x_1+7x_2+5x_3+4x_4\le 10& (33)\end{array}$

Unlike the first case in which targets are already intuitively set by decision-makers, this case does not readily present the targets of each decision-maker with respect to their individual decision criterion. Therefore, step 5.a. is carried out. Note that there are three decision-makers in this case, thus, three separate optimization models have to be developed which corresponds to the individual interests of these decision-makers subject to the exact, same system constraints. The first model in Equation 34 shows the interest of the decision-maker A to minimize the operating cost of the decision problem. On the other hand, Equation 35 corresponds to the interest of decision-maker B to maximize the utilization of routes. Lastly, Equation 36 represents decision-maker C's interest to minimize delays. These models are solved separately which will provide an optimal value for each interest then representing the performance targets of each decision criterion.

$\begin{array}{cc}min460x_1+256x_2+300x_3+400x_4& (34)\end{array}$

$\begin{array}{l}42x_1\le 50\\ 44x_2\le 50\\ 31x_3\le 50\\ 27x_4\le 50\\ 6x_1+7x_2+5x_3+4x_4\le 10\\ x_1+x_2+x_3+x_4=1\\ x_1,x_2,x_3,x_4\in \left\{0,1\right\}\end{array}$

$\begin{array}{cc}max78x_1+75x_2+59x_3+56x_4& (35)\end{array}$

$\begin{array}{l}42x_1\le 50\\ 44x_2\le 50\\ 31x_3\le 50\\ 27x_4\le 50\\ 6x_1+7x_2+5x_3+4x_4\le 10\\ x_1+x_2+x_3+x_4=1\\ x_1,x_2,x_3,x_4\in \left\{0,1\right\}\end{array}$

$\begin{array}{cc}min14x_1+16x_2+10x_3+10x_4& (36)\end{array}$

$\begin{array}{l}42x_1\le 50\\ 44x_2\le 50\\ 31x_3\le 50\\ 27x_4\le 50\\ 6x_1+7x_2+5x_3+4x_4\le 10\\ x_1+x_2+x_3+x_4=1\\ x_1,x_2,x_3,x_4\in \left\{0,1\right\}\end{array}$

By solving Equations 34–36 separately as in Section 2.2.5.2, the following results are obtained (see Table 11). For each decision criterion, the optimal alternate route is generated and its corresponding optimal value. Such optimal values are then used to represent the objective performance targets of each decision criterion. Notice that setting the individual decision criterion as objective functions that are subject to the same exact system constraints produces different preferences for alternate routes. Specifically, in terms of operating cost, the best route to be selected in order to generate optimal solution is route 2. Contrastingly, in terms of utilization rates, route 1 should be selected. On the other hand, taking route 3 provides the least delays. These results apparently present the inherent conflict in solutions among diverse decision criteria.

Table 11

Table 11. Performance targets for each decision criterion.

To proceed with the MS-TORO framework, step 6 is performed. The objective function, as shown in Equation 37 is to minimize the deviation metric subject to the decision criteria as a constraint and other system constraints.

$\begin{array}{c}min\theta \\ \text{subject to}:& (37)\end{array}$

$\begin{array}{l}460x_1+256x_2+300x_3+400x_4\le 256(1+\theta )\\ 78x_1+75x_2+59x_3+56x_4\ge 78/(1+\theta )\\ 14x_1+16x_2+10x_3+10x_4\le 10(1+\theta )\\ 42x_1\le 50\\ 44x_2\le 50\\ 31x_3\le 50\\ 27x_4\le 50\\ 6x_1+7x_2+5x_3+4x_4\le 10\\ x_1+x_2+x_3+x_4=1\\ 0\le \theta \le 1\\ x_1,x_2,x_3,x_4\in \left\{0,1\right\}\end{array}$

Solving Equation 37, as shown in step 7, the results of the model are summarized in Table 12. The proposed MS-TORO model produced an aggregated solution selecting Route 3 with a deviation metric θ of 0.32. As observed when analyzing each decision criterion individually (see Table 11), different routes may appear optimal in isolation. To reconcile these conflicting criteria and arrive at an aggregated solution that satisfices all criteria to a practical extent, the deviation metric is introduced. For instance, the “operating costs” criterion with an initial target of $256 is adjusted upward to $338 [256 × (1 + 0.32))]. The “utilization” criterion, initially set at 78%, is adjusted downward to 59% [78/(1 + 0.32)]. For the “delays” criterion, the initial target of 10 min is adjusted to 13 min [10 × (1 + 0.32))]. These adjustments illustrate the decision-making significance of θ, as they quantify the extent to which stakeholders must compromise their individual targets to achieve a balanced, feasible solution. The model ensures that the selected alternative, Route 3, simultaneously respects system constraints and approximates each criterion as closely as possible, demonstrating the ability of MS-TORO to reconcile multiple, often conflicting stakeholder objectives while providing actionable guidance for decision-makers.

Table 12

Table 12. Results of the MS-TORO model (case study 3).

4 Synthesis of the key features of MS-TORO model

The MS-TORO framework enables finding balance among the diverse, often conflicting, interests of stakeholders by introducing multiple system targets. Through the MS-TORO framework, key research gaps, solution approaches that focus on single stakeholder perspective, in the literature specifically to decision problems involving multiple stakeholders can be well-addressed. The behavior of MS-TORO, particularly its robustness to uncertainty and ability to adjust performance targets to achieve a compromise solution, aligns with recent advances in robust optimization, including regularization methods for load reconstruction with hybrid uncertainties [47], load-dependent structural state reconstruction-oriented and reliability-based sensor placement optimization [48], and natural frequency- and surface accuracy-targeted uncertain on-orbit assembly planning [49]. These studies similarly emphasize the incorporation of uncertainty into optimization models, ensuring solution reliability and robustness, which mirrors how MS-TORO handles stakeholder risk profiles and varying uncertainty levels.

As an illustration of the applicability and viability of the MS-TORO model, three hypothetical scenarios are performed. These scenarios involve green building promotion analysis, supplier selection, and aircraft rerouting process and altogether consider multiple stakeholders each with their own individual interests which they seek to fulfill. Also, note that these stakeholders have specifically defined performance targets corresponding to each decision criterion. For an agreeable, unified solution to be generated, stakeholders, also referred to as decision-makers, must be keen on being able to adjust their performance targets to accommodate the interests of other decision-makers. Such a compromised solution paves the way for arriving at a satisficing and balanced solution. This satisficing metric is represented by a deviation metric, θ. The risk profile of stakeholders is known from being lenient and conservative toward the target to being more rigid and strict toward the achievement of the target. Such information on the risk profile of stakeholders can be used to further understand the behavior of stakeholders during the decision-making process.

By a satisficing solution, the risk profile of decision-makers can also be known. This implies that due to the incorporation of other interests, which may be conflicting, stakeholders assert the achievement of their individual interests to a degree that is favorable among other stakeholders. It is interesting that the model serves no bias in the preference of decision-makers. In other words, the model generates solutions that are not dependent on individual preference of decision-makers but strictly follow an objective of minimizing the target deviation.

It is important to note that the model has a tendency to generate local optima when used with other solvers (e.g., Minos and Knitro solver from AMPL) due to its characteristic being NP-complete in nature. Specifically, the proposed model follows a linear quadratic integer programming. In such cases that it is not possible to force the model to return binary variables for the decision variables, the magnitude of the decision variable dictates its selection (i.e., a higher magnitude corresponds to higher preference).

Furthermore, the model declares the uncertainty level from a closed interval [0,1] in order to understand how the model behaves across different realizations of uncertainty. For case study 1, the uncertainty level may not have an apparent influence on the activated decision variable given that no change is observed across all uncertainty levels. In contrast, the results of case study 2 and 3 effectively show that the activated decision variable shifts when the uncertainty level also does. It is also worthy to emphasize that this setup is restrictive with several variables which may result to local optimal solutions, In order to provide an additional perspective of the proposed model's capability, the models of the respective case studies are re-run with the uncertainty level declared as a decision variable. The logic behind such conversion from a parameter to a decision variable is to allow the model generate results that are also sensitive to the maximum uncertainty level that can be tolerated by the model as achieves the least possible deviation metric. Table 13 summarizes the results of the MS-TORO re-run with uncertainty level set as a decision variable. The uncertainty level indicated in this table pertains to the highest possible system perturbation that the model can handle given that the objective is retained to minimize the deviation metric. It can be seen that for case study 1, the deviation metric is 0.00000 while the uncertainty level is 0.93273 leading to the selection of Building 2. Such results imply that the case may experience 0.93253 uncertainty level and does not warrant any deviation from the targets set by the decision-makers. In fact, comparing these results to Table 5, it confirms that even up to an uncertainty level of 1.00000, the model can still generate optimal solutions selecting Building 2 throughout any realization of uncertainty. For case study 2, the model can take as far as 0.30076 uncertainty level with a deviation metric of 0.15000 before it shifts its choice from Supplier 3 to Supplier 1 (i.e., with reference to the results in Table 8). For case study 3, the uncertainty level of 1.0000 is generated as the model reaches the deviation metric of 0.32203, thereby, selecting Route 3 as its optimal solution. Again, the results are comparable to that of Table 12 which points out the least deviation metric of 0.32203 under an uncertainty level of 1.0000.

Table 13

Table 13. Results of the MS-TORO model with uncertainty level set as a decision variable.

Overall, the MS-TORO framework offers decision-makers a structured and transparent approach for balancing conflicting stakeholder interests while achieving performance targets. The deviation metric θ provides a clear measure of how well each stakeholder's objectives are satisfied and allows assessment of solution feasibility under varying conditions. By incorporating stakeholder risk profiles and uncertainty levels, the model provides insight into the flexibility and robustness of decisions, enabling stakeholders to anticipate potential trade-offs and adjustments. Re-running the case studies with uncertainty treated as a decision variable further demonstrates the framework's practical applicability, revealing the maximum perturbation each system can tolerate while maintaining minimal deviation. Collectively, these features offer actionable guidance for real-world implementation, supporting decision-makers in achieving balanced, satisficing solutions that accommodate multiple interests. Furthermore, the proposed MS-TORO method also demonstrates computational efficiency and decision-making accuracy. Its scalarized optimization formulation allows simultaneous consideration of multiple objectives, eliminating repeated single-objective analyses and reducing computational effort. Accuracy is reflected in θ, which quantifies the extent to which stakeholder performance targets must be adjusted to reach a compromise solution. Across the case studies, MS-TORO consistently generates feasible and balanced solutions, even under varying levels of uncertainty. These results indicate that the model efficiently handles complex multi-stakeholder problems while reliably producing solutions that satisfy the diverse interests of all decision-makers.

5 Concluding remarks and future works

As organizations strive to flourish in terms of resources and operational aspects, the participation of multiple stakeholders has become of primary importance to carry out complex and urgent decision-making processes. These stakeholders have diverse, often conflicting interests along with their overall goals set by the organization in general. Such diversity and conflict rooted in maximizing each stakeholder's utility and individual gains have resulted in internal issues in the organization including difficulty to survive, for one. Several theories in the literature attempted to examine the behavior of individuals in an organization such as the stakeholder theory, helix theory, stewardship theory, and agency theory, among others. Therefore, scholars have made efforts to develop a scheme that aims to balance stakeholder interests despite initial contentions of the concept being thin, unrealistic, and aesthetic in sense.

However, even with strong efforts to balance stakeholder interests in terms of assessing, weighing, and addressing competing claims of stakeholders, none of the prior strategies including sophisticated analytical tools, best practices, and holy books seemed to not be adequate as expected. In later years, specific mathematical representations of balancing interests have been introduced including maximizing expected utility, maximizing the probability of uncertain consequences, and incorporating multiple targets. Further extensions on such models have also emerged which considered a new set of decision criteria—satisficing and aspiration measures—from concepts of Brown and Sim [50] and Brown et al. [46] to Bongo and Sy [45]'s novel target-oriented robust optimization (TORO) framework. Still, none of these outstanding contributions have satisfactorily captured the interests of stakeholders more so balance these diverse, conflicting interests.

It is, therefore, the goal of this paper to develop a framework termed as multiple stakeholder-based TORO (MS-TORO) approach in order to mainly provide a methodology for balancing stakeholder interests. The proposed MS-TORO model seeks to intersect the perspectives of stakeholders and ultimately provide overall solutions that are adherent to these individual interests. Such adherence is represented by a target metric that is designed to denote the minimum deviation from the individual optimal solutions generated for each stakeholder. In order to illustrate the proposed framework, a hypothetical case study is conducted based on the decision-making process of analyzing probable green buildings, assessing suppliers, and selecting an alternate route for the aircraft rerouting procedure. The key results of these case studies indicate that the proposed MS-TORO model can simultaneously handle various interests of stakeholders and provide an aggregated and satisficing solution.

To highlight the main contributions of the proposed model to the body of knowledge, the following points are accounted for. One, the proposed MS-TORO model leverages on its capability to incorporate multiple stakeholders into the decision-making process while taking into consideration the major mathematical prowess of the classical robust optimization approach and its recent developments involving target-setting (i.e., TORO approach). By formulating the model to include the diverse goals of multiple stakeholders, the results are ensured to be all-encompassing, thus, overall satisficing the interests of these stakeholders to some extent that is acceptable to them. Prior works in the literature involving multiple goals and choice problems are often modeled as multi-objective optimization models which do not necessarily provide an insight on the extent of adjustment that each stakeholder has to make in order to arrive at an aggregable solution. Second, realistic life applications are often more concerned about achieving the goals of decision-makers to an extent rather than plainly achieving these as a whole or not at all. The deviation metric introduced in this model provides the decision-makers an extent of goal adjustment that they need to implement in order to accommodate all other goals considered.

While the proposed MS-TORO model lays claim to balance stakeholder interests and address decision problems involving multiple decision-makers each with a corresponding decision criterion, there are also certain computational limitations that the model has. For one, among the set of alternatives involved, the model can only solve a choice problem and therefore cannot handle ranking or outranking of alternatives unlike TOPSIS or PROMETHEE, for instance. For another, the decision criteria have to be accurately translated into a system constraint (or an objective function) to ensure the reliability and validity of model outputs. Also, the model assumes that the decision-makers have an equal priority to the fulfillment of the decision problem. Given such limitations, the proposed MS-TORO model can be further extended and integrated into existing MCDM methods as an auxiliary phase to provide a more comprehensive view of the decision problem. For example, the MS-TORO model can objectively extract and trim down viable alternatives prior to its further evaluation in TOPSIS, ELECTRE, or PROMETHEE methodologies.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

JM: Conceptualization, Investigation, Writing – original draft. CH: Writing – original draft, Funding acquisition, Project administration, Supervision. MS: Formal analysis, Software, Investigation, Conceptualization, Writing – original draft. DA: Writing – original draft, Visualization. MH: Resources, Funding acquisition, Project administration, Writing – original draft, Supervision. MM: Conceptualization, Visualization, Resources, Writing – original draft. PS: Software, Investigation, Conceptualization, Writing – original draft. MB: Writing – review & editing, Methodology.

Funding

The author(s) declared that financial support was received for this work and/or its publication. The authors wish to thank the following entities for the support provided in the completion of this research work. To the Philippine Department of Science and Technology (DOST) Engineering Research and Development for Technology (ERDT) for the graduate scholarship grant provided to MB. To De La Salle University-Manila for resource use. Also, warm thanks to Cebu Technological University for funding the publication fees of this article.

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

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