Full length articleTwo-stage stochastic minimum cost consensus models with asymmetric adjustment costs
Introduction
Consensus reaching process (CRP) exerts more and more important efforts on group decision making (GDM), which is just an interesting question for study. The selected decision-makers (DMs) give their opinions or scores on an issue through their subjective judgment, and then they agree through negotiation. As representatives of different groups, the views of DMs vary due to the differences in the focus of the group’s concerns. In the process of reaching a consensus, the DMs first give their initial opinions based on their knowledge, ability, and the actual situation. The DMs participate in the discussion, under the coordination of the moderator, revise their original viewpoints, and eventually come to an agreement. Achieving the ultimate consensus requires multiple rounds of discussions and negotiations. A consensus through negotiation and communication of each DM is determined.
Several researchers have pointed out that it is important to consider the costs and deviations of modifying experts’ opinions to reach an agreement in solving CRP. Dong et al. [1] proposed a minimum adjustment consensus model (MACM), in which the aim is to preserve as much as possible of the DMs’ original opinions. And Ben-Arieh et al. [2] put forward a minimum cost consensus model (MCCM). However, a linear cost function is utilized to solve the solution of the MCCM, and it does not use mathematical optimization models. No matter in MACM or MCCM, it is easy to see that the total adjustments or total cost of CRP is better as low as possible. Further, MACM and MCCM are combined into a new model in [3]. As mentioned previously, the unit adjustment cost in the up and down direction is identical or symmetrical in the MCCMs. Nevertheless, the unit adjustment cost often varies in different directions. Based on this research idea, to reflect the difference in the upward and downward adjustment direction of unit cost, [4] studies the MCCMs in asymmetric adjustment costs. This means that the unit adjustment cost is directional and asymmetric. Regarding the consensus costs, based on the Stackelberg game, the MRMCCM proposed by Zhang et al. [5] is from the perspective of both the moderator and experts. And this bi-level optimization model takes into account the behaviors of the moderator and experts simultaneously. In conclusion, we point out that it is important to get the solution of the optimization-based consensus model. In recent work, Zhang et al. [6] made a detailed overview of minimum adjustment and cost consensus models in GDM. Besides, research on consensus cost has attracted more and more scholars’ attention (see [7], [8], [9]).
Nowadays, studies have gradually focused on the uncertain consensus model under the group decision-making environment. Under the environment of uncertain opinions, Tan, et al. [10] studied the consensus cost optimization models with budget constraints. Uncertainty factors of these models are set as follows: set the expert opinion or consensus opinion as an interval value, and the consensus opinion or expert opinion obeys the normal distribution. Gong, et al. [11] gave a new interval preference relation based on the uncertainty theory, and then the linear uncertainty distribution is used to describe interval decisions. Therefore, two consensus models with linear uncertain preference relations on the minimum deviation among belief degrees were investigated. Owing to previous studies rarely considered expert opinion as a random distribution, Gong, et al. [12] developed the minimum cost consensus models with cost chance constraints in five different scenarios. In particular, Liu’s [13], [14] uncertainty theory is combined with the minimum cost consensus optimization models, and the uncertainty distribution is applied to the estimation of expert uncertainty preferences. The proposed models are of great significance in dealing with consensus decisions. Han, et al. [15] proposed robust consensus models that rely on a robust optimization method to overcome the uncertainty of data of the consensus problem. Since the unit adjustment cost is usually indeterminate, Li, et al. [16] put forward an interactive consensus process.
As we stated earlier, the existing researches on uncertain consensus optimization models have some limitations. First of all, most studies in the field of uncertain consensus have only focused on symmetric adjustment cost. However, previous studies of the uncertain minimum cost consensus model have not dealt with the constraints of asymmetric cost. Therefore, the research would have been more useful if a wider range of the uncertain minimum cost consensus model with directional constraints had been explored. Secondly, the research on the optimization of uncertain consensus is limited to the uncertainty of expert opinion, while ignoring the uncertainty of other parameters. However, in the actual decision-making environment, the adjustment cost of the expert opinion, the degree of tolerance of deviation adjusted by expert opinion, and a range of thresholds within which experts can make cost-free adjustments may all be uncertain. Therefore, these uncertain parameters should be taken into account in the actual consensus-building process. Finally, another problem with this uncertain consensus modeling is that it fails to take two-stage stochastic programming into account. However, two-stage stochastic programming is an important group part of uncertainty theory, and it is widely used in a variety of practical decision-making problems. This shows that it is necessary to propose these models, which do not use the interval values or random distributions to fit the indeterminate parameters. Instead, the decision is divided into multiple scenarios. When decision-makers make decisions, they take multiple scenarios into consideration to ensure the robustness of the decision, which can help them better provide the optimal solution to a problem that needs to be addressed.
However, whether the unit cost is symmetric or asymmetric, only the distance between expert opinions and consensus is taken into consideration in the above models. Moreover, the above research on uncertainty in MCCMs is mainly supported by fuzzy and preference relations in the decision-making process. Various uncertain scenarios with some probabilities are ignored. They did not consider building two-stage stochastic programming with consensus constraints are considered to get the optimal solution, where the experts may have a variety of uncertain opinions or uncertain unit adjustment cost or any other uncertain parameters under different situations. To highlight the motivation for our research, we make the process for a project evaluation for a company as an example. In a project evaluation, the company selects 5 DMs to evaluate and rate a project and to get a consensus opinion. To achieve an acceptable consensus on the final solution, the moderator estimates the unit upward and downward adjustment cost and persuades DMs to modify their opinions. Nevertheless, scores of project evaluation may change with the change of specific time and economic development, which leads to the changes in experts’ scores based on the scores given in advance. At the same time, the unit adjustment cost estimated by the moderator will also change. Additionally, acting in accordance with the predicted changes in the external environment, the possibility of several situations can be given. For the reason that in the face of the future environment and economic situation that cannot be accurately forecast, they will take a variety of scenarios with some probabilities into account when making decisions. Project evaluation score obtained based on this will be an ideal consequence, since it includes a variety of different scenarios at the same time, which makes the final consensus score more convincing. This case demonstrates the need for better strategies for solving the uncertain consensus problem with various different scenarios. Before we did not get information about the external factors that influencing score change, the best way we can do is to take all different scenarios into account. When different scenarios with probability are considered, the two-stage stochastic minimum cost consensus models with asymmetric costs are put in place to solve the problem for the most part.
In fact, experts cannot modify preferences in a single direction. In most cases, due to the difference of expert opinions in the direction of adjustment, the cost coefficient of expert adjustment is asymmetric. Several research results about MCCMs with asymmetric cost obtained from the [4]. In addition, Cheng, et al. also analyzed the impact of experts’ compromise limits and tolerance behaviors on these models. In fact, even if there is a dominant leader in the decision-making process, he cannot take into account all the influencing factors in an activity, which is involving multiple decision-makers. Because of the knowledge reserve of decision-makers, the historical information amount of reference and the specific decision-making environment is diversified. Thus, the above two points lead to a large number of uncertain factors in the group decision-making process, which brings a challenge to the decision-makers and moderator. Besides, it has been recognized that consensus optimization in an uncertain environment requires explicit consideration of the uncertainty present in the system. This makes the study of uncertainty in the case of consensus is particularly important. However, there is still a lack of research on MCCMs with asymmetric cost in an uncertain environment. Because the decision-makers cannot accurately predict the likely future outcomes, stochastic programming has obvious advantages in dealing with indeterminate decision problems. As we know, the two-stage stochastic programming is a very important mathematical model that models planning under uncertainty, what changes uncertainty will bring, what challenges uncertainty may bring, and what problems can be addressed. There is a growing body of literature that recognizes the importance of constructing two-stage stochastic programming. [17], [18] attached great importance to the rationality of establishing two-stage stochastic programming. In recent decades, this problem has been extensively studied (see [19], [20], [21], [22]). With gradually deepening the study, it has been applied in more fields, such as financial planning (see [23], [24]), supply chain management (see [25], [26], [27], [28]).
However, few researchers have been able to draw on any systematic research into the uncertain two-stage stochastic minimum cost consensus model (TSCM). Therefore, the specific objective of this study was to further investigate this kind of uncertain consensus model. A key issue about the two-stage stochastic MCCM is to divide the set of decisions into two groups. The consensus opinion has to be taken by a moderator before the experiment, which is called the first-stage decision. The second-stage decisions are the deviations of the experts to adjust their opinions after consultation. Finally, all experts involved in decision-making reach an agreement. So, the purpose of this paper is tantamount to investigate two-stage stochastic programming that was applied to the MCCM with asymmetric adjustment costs in different scenarios.
There are several important areas where this study makes an original contribution to consensus optimal optimization problem, which is include:
(1) Taking into account the uncertainty of the minimum cost consensus model with directional constraints, the two-stage stochastic minimum cost consensus optimization models are presented from three different perspectives. For example, uncertainty about asymmetric costs, uncertainty about initial preferences, uncertainty about compromise limits, uncertainty about tolerance behaviors.
(2) The two-stage stochastic model can be notoriously difficult to resolve. Considering the degree of difficulty in handling the two-stage stochastic programming models, an L-shaped algorithm is given to solve the resulting models. Also, the comparisons between the L-shaped algorithm and CPLEX reveal that the given method does a better job, and timeless. This solution solved by an L-shaped algorithm can help DMs to obtain the optimal opinion in CRP.
(3) The relationship between the given two-stage stochastic consensus models the traditional MCCM and the MCCMs with asymmetric adjustment costs is provided. Moreover, under certain conditions, these three consensus models can be transformed into each other.
(4) The feasibility of the proposed two-stage stochastic consensus models with asymmetric costs is verified by negotiations on water pollution in cities around the Taihu Basin in China. Additionally, local and global sensitivity and comparison analysis are also presented.
The paper is structured as follows. In Section 2, we provide a basic introduction to the two-stage stochastic programming and the consensus optimization models. In Section 3, the three types of two-stage stochastic minimum cost consensus models with asymmetric adjustment costs are formulated. In Section 4, we present an L-shaped algorithm to solve the proposed two-stage stochastic consensus models. In Section 5, the computational results of practical application examples are presented. In the context of asymmetric costs, not only the comparison results between the given method and CPLEX are bestowed, but also the comparison results between the given two-stage stochastic consensus models with MCCMs in asymmetric costs are given. Moreover, the EVPI and VSS are reflected in the application instances. In Section 6, the local and global sensitivity analysis of the proposed two-stage stochastic minimum cost consensus models with asymmetric adjustment costs is presented. Moreover, in global sensitivity analysis, the comparison between the three proposed models for the previously given models is also analyzed. In Section 7, we explore the connection between the basic MCCM and MCCMs with asymmetric adjustment costs and the proposed two-stage stochastic consensus models. Finally, Section 8 introduces the summary and gives suggestions for future research work.
Section snippets
Preliminaries
In this section, we review some results on the two-stage stochastic programming and give the development of consensus optimization models, such as MACM, the traditional MCCM, and the three kinds of MCCMs with asymmetric adjustment costs.
Two-stage stochastic consensus models
Under the premise of asymmetric adjustment cost, since the above three minimum cost consensus problems do not explore different situations, we establish three two-stage stochastic minimum cost consensus models. The common feature of the three models is the fact that they pick up the effect of the uncertainty of adjustment costs. The difference between them is that TSCM-DC studies the uncertainty of experts’ initial opinion, -TSCM-DC analyzes the uncertainty of experts’ adjustment limits, and
L-shaped algorithm for solving two-stage stochastic consensus models
The most common decomposition method in stochastic programming is the Benders decomposition method and its improvement, which are referred to as L-shaped algorithm. For a detailed explanation of the standard L-shaped algorithm, see [18]. In view of the excellent performance of the L-shaped algorithm in the solution process and the consideration of small-scale group decision problems in this paper, the L-shaped algorithm is invoked as the preferred method to solve the presented two-stage
The applications of the proposed models
The proposed two-stage stochastic consensus models can be applied effectively in the CRP. All numerical computations are carried out on a notebook computer (Intel i5 CPU and 8 GB RAM) using MATLAB 2018a. By comparing the running time of the L-shaped algorithm and CPLEX, it takes less time to utilize the given method for solving the proposed two-stage stochastic consensus models. Comparative experimental results verify that the L-shaped algorithm is effective.
The proposed two-stage stochastic
Sensitivity analysis of the proposed models
To further understand the two-stage stochastic minimum cost consensus models with asymmetric adjustment costs, we proceeded to a sensitivity analysis of key parameters by taking Examples 5.1–5.3. There are two schools of sensitivity analysis, local sensitivity, and global sensitivity. We will present the local and global sensitivity analysis to the proposed TSCMs with asymmetric adjustment costs to investigate the effect of the uncertain upward and downward adjustment cost, and the compromise
Relationship between MCCM, MCCM with asymmetric costs and TSCM with asymmetric costs
We briefly discussed the relationship between the traditionalMCCM, the MCCM with asymmetric costs, and the given two-stage stochastic MCCM with asymmetric costs in this section. The main relationships could be drawn as follows:
(1) The given TSCM-DC is two-stage stochastic programming based on the model (5), which considered various scenarios in CRP. Every one of the scenarios of TSCM-DC is the model (5), and a variety of different scenarios of the model (5) combined into TSCM-DC. The TSCM-DC
Conclusion and future work
In this paper, we explore the three kinds of two-stage stochastic MCCMs with asymmetric adjustment costs. By introducing the theory of two-stage stochastic programming, we investigate the TSCM-DC, -TSCM-DC, and TB-TSCM-DC. The TSCM-DC is based on the model (5) and we assume that each expert has not one opinion, in other words, we consider each expert has several different scenarios with a certain probability. Besides, we assume that each expert has a variety of different scenarios about the
CRediT authorship contribution statement
Huanhuan Li: Conceptualization, Methodology, Writing - original draft. Ying Ji: Methodology, Writing - review & editing, Funding acquisition. Zaiwu Gong: Conceptualization, Methodology, Writing - review & editing. Shaojian Qu: Methodology, Funding acquisition, Writing - review & editing, Funding acquisition.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
This work is supported by a research grant from the National Social Science Foundation of China (No. 17BGL083).
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