Two-stage stochastic integrated adjustment deviations and consensus models in an asymmetric costs context

Abstract

Decision-makers usually have a variety of unsure situations in the environment of group decision-making. In this paper, we resolve this difficulty by constructing two-stage stochastic integrated adjustment deviations and consensus models (iADCMs). By introducing the minimum cost consensus models (MCCMs) with costs direction constraints and stochastic programming, we develop three types of iADCMs with an uncertainty of asymmetric costs and initial opinions. The factors of directional constraints, compromise limits and free adjustment thresholds previously thought to affect consensus separately are considered in the proposed models. Different from the previous consensus models, the resulting iADCMs are solved by designing an appropriate L-shaped algorithm. On the application in the negotiations on Grains to Green Programs (GTGP) in China, the proposed models are demonstrated to be more robust. The proposed iADCMs are compared to the MCCMs in an asymmetric costs context. The contrasting outcomes show that the two-stage stochastic iADCMs with no-cost threshold have the smallest total costs. Moreover, based on the case study, we give a sensitivity analysis of the uncertainty of asymmetric adjustment cost. Finally, conclusion and future research prospects are provided.

Keywords

Two-stage stochastic integrated adjustment deviations and consensus model directional constraints uncertain adjustment costs uncertainty initial opinions L-shaped algorithm

1 Introduction

Group decision-making (GDM) is making effective decisions when multiple decision-makers (DMs) make decisions and participate in decision-making analysis simultaneously [1, 2]. A key point during this process is that each individual in the group gives full play to his/her wisdom. A group decision problem contains two main elements. One is the project which is scored by expert technicians, and the other is the members involved in the decision, also known as the DMs. During the implementation of the decision, representative members are selected to provide opinions on a certain problem to be solved. The resulting decision is based on the opinions of all members. In order to reach an agreement on a certain issue, members need to negotiate, discuss, and modify their opinions and preferences several times, eventually form an agreement on the problem [3]. Thus, group decisions are often superior to individual decisions, because they are a synthesis of multiple viewpoints and opinions. This type of process is named the process of reaching consensus [4]. Furthermore, the normal progress of the consensus process requires the supervision of a moderator, who not only participates in the entire consensus reaching process but also plays a role of supervision and guidance in this process, so as to ultimately promote the development of the consensus opinion towards the expected opinion.

In order to promote consensus, the moderator in group decision-making needs to compensate for the experts to modify their initial opinions. In this process, the moderator hopes to promote consensus with the least cost, while experts hope to get the maximum compensation by modifying their opinions. Based on the facilitator’s perspective, Ben-Arieh et al. [5] first proposed the minimum cost consensus model (MCCM), which is represented by a linear function, and it is not strictly optimization model in the mathematics sense. To explore the minimum cost of achieving a final consensus which satisfies all participants, many researchers put forward many new idea. It is sometimes impossible to ensure that all experts accept the consensus in the process of consensus building. Based on this idea, Ben-Arieh et al. [6] presented a model of the maximum number of experts who accept the consensus on a certain consensus issue. In addition, a number of scholars indicated that studying the consensus costs and deviations on the process of forming consensus is extremely important. The minimum adjustment consensus model (MACM) was originally presented by Dong et al. [7], which keeps as much information as possible from DMs. Subsequently, Zhang et al. [8] proposed a new consensus model, which is combined with MACM and MCCM. Zhang et al. [9] have done the review according to an overview on feedback mechanisms with minimum adjustment or cost in consensus reaching, and they extend some open problems on these. Besides, the mechanisms of consensus cost involved are creating increased interest among the scholars [10, 11]. In practical decision-making problems, the choice of experts to adjust their opinions is two-way rather than one-way. Usually, when experts revise their initial opinions, there are two alternative directions, that is, to increase their opinion values at one direction or decrease their opinion values in the opposite direction. Based on this modeling idea, Cheng et al. [12] proposed the minimum cost consensus modeling under the background of asymmetric consensus costs. The proposed model reflects the direction of consensus cost by dividing it into upward or downward adjustment directions. However, this modeling method is a deterministic form of modeling. In fact, the process of reaching consensus is not only affected by experts’ knowledge reserve, historical information and decision-making ability, but also by the decision-making environment, which leads to the uncertainty of the whole decision-making process. There is little minimum cost consensus modeling of uncertain asymmetric adjustment cost, which makes this research direction particularly necessary.

In the actual decision-making environment, there may be differences in the personal opinions provided by the decision-making experts who are important participants. Therefore, it is of great importance for the experts to adjust and modify their opinions to get a satisfactory solution. The final consensus can be determined by improving the initial opinions, which is a time-consuming and costly process. During this period, adjustments of opinions includes increasing or decreasing the opinion value based on the given initial opinions. However, when opinions are modified, the unit consensus cost of different directions and the initial opinions of experts are usually uncertain, which indicates that it is also unknown. For example, a decoration company needs to make a comprehensive evaluation on the selected wallpaper. Given this, there are a number of considerations, such as environmental protection, beauty, material and quality. Suppose the score given by the expert here is the sum of several index score. Due to the limitations of the selected experts’ professional and knowledge, some experts are only familiar with one or more of the fields. However, the given score may be uncertain for whose experts not familiar with other domains, which lead to uncertainty of the comprehensive score. In order to get an evaluation score that is still satisfactory to everyone, the relevant personnel who organize this evaluation need to provide some compensation to let the experts voluntarily modify their score. On the other hand, when experts face unfamiliar fields, they usually have no particular stubborn preference, so they tend to be persuadable and conforming [13]. Therefore, unit consensus cost that the moderator needs to pay each DM is relatively low. From this point of view, unit adjustment cost of each DM is also variable and uncertain. In the actual decision-making process, the individual’s initial opinions and unit adjustment cost are variable. Therefore, it is of great significance to deal with the consensus optimization problem with these two uncertain parameters at the same time.

In this paper, under multiple scenarios of initial opinions and unit adjustment cost, we study three types of integrated adjustment deviations and consensus models (iADCMs), which may be easy to arrive at a unified consensus on a group decision problem for DMs. Our approach aims to reach consensus by adjusting opinions in an asymmetric adjustment cost context. The theoretical methods to deal with uncertainty include interval analysis (see [14 –16]), fuzzy sets (see [17, 18]), probability theory (see [19 –21]), uncertainty theory (see [22, 23]), and robust optimization method (see [24 –26]) and so on. However, these modeling of uncertain consensus systems have certain limitations. On the one hand, most of the unsure consensus studies done till today are confined in the uncertainty of opinions, while most of research does not focus on the other uncertain parameters. Owing to that complex consensus models representing GDM processes, require multiple parameters that may be stochastic and uncertain. On the other hand, two-stage stochastic programming as a powerful theory to handle uncertainty information, and this kind of problem is widely used in resource allocation, financial economic and other practical fields. With the wide application in the field of decision-making, it is particularly important to carry out in-depth research on integration of two-stage stochastic programming (TSP) and MCCMs with direction cost constraints. However, in the actual group decision-making problems with uncertain factors, the decisions did not occur concurrently, but instead in sequence, especially some more complex group decision-making problems. Usually, some unexpected decisions must be made before the realization of random variables. Many of our day-to-day decisions are broken down into two stages. Two-stage stochastic programming with recourse is an effective method to deal with uncertain factors in GDM. This lead to the establishment of its rationality has also attracted great attention (see [27 –29]). The main idea of this method is to divide the decisions into two stages. Here, the decision-making of the second stage is closely related to the decision-making of the first stage. This theory developed rapidly, and this approach of stochastic programming as deal with uncertain problems has spread into various disciplines from financial planning (see [30, 31]) to supply chain management (see [32, 33]). More recently, a significant volume of research has been published in this area. As an important branch of mathematics, two-stage stochastic programming is mainly used to deal with uncertainty problems, and has been successfully applied to resource allocation, financial economy, and other fields. Compared with other methods dealing with uncertainty, two-stage stochastic programming has unique advantages. This paper draws a lesson from stochastic programming, so we classified decisions in two stages. Taken together, the process of reaching consensus is thought to a two-stage stochastic program. In an asymmetric cost context, the proposed two-stage stochastic iADCMs are a challenging stochastic program with uncertain unit adjustment costs and initial opinions. More specifically, in our iADCMs the here-and-now decision is the consensus given by the moderator of integrating various factors, while the wait-and-see decision is to adjust the opinions for each unit adjustment cost and opinion scenario.

Due to the uncertainty in DMs’ opinions and unit consensus cost, besides, the adjustment costs are either up or down regulated. So how to effectively reach consensus is an important issue to be resolved. Given that nowadays there is still no solution to solve the MCCMs with directional constraints under multiple scenarios, a need for modelling a two-stage stochastic programming is imperative. However, there is a lack of research on this topic. Thus, further research is required. The so-called two-stage stochastic iADCMs can be considered to contain two kinds of decisions. One is relatively long-term, which is the decision to be made in the first stage,also known as tactical level decision-making. The other is very specific, which is the decision to be made in the second stage, also known as operational level decision-making. In this paper, we introduce consensus opinion as the tactical level decision variable and deviations of experts as the operational level decision variables. The consensus is given by the moderator in advance, and adjustment deviations of the experts are given after consultation. This paper investigates three iADCMs in an asymmetric costs context.

The present paper makes the following main contributions:

(1)We put forward three different optimization models of comprehensive adjustment bias and consensus, which are proposed based on two-stage stochastic programming and MCCM. There are some common characteristics of these models. First of all, their adjustment processes are directional, and differences may exist in the adjustment costs. Secondly, there are some uncertainties regarding unit consensus cost and individual initial opinion. The uncertainty here is referring that the mentioned parameters are divided into multiple scenarios. In addition, the goal of solving the model is to obtain a consensus on the basis of minimum adjustment cost. Finally, compared with the traditional consensus cost model, the main different from those of three types of models is that the three models are two-stage stochastic programming applications in consensus models.

(2) Given the difficulty in handling the uncertain consensus problems, a L-shaped algorithm is adopted in our study which is matched with the proposed two-stage stochastic iADCMs. Moreover, the accuracy and effectiveness of the L-shaped algorithm are verified by comparing it with the results of the solver CPLEX. The results suggest that there is no difference in outcomes between the two manners. Not only that, our novel algorithm can effectively reduce the running time.

(3) A study based on the Gains to Green Program in China is performed to evaluate the usefulness of the proposed two-stage stochastic iADCMs. In particular, we consider the issue of determining consensus and individual adjustment bias when adjustment costs and initial opinions are uncertain. The results of the examples indicated that these models are used in computing consensus to get a better consensus. The factor is that our models take different scenarios into account, which is more accordant with practical circumstances.

(4) In contrast to the MCCMs with determined parameters, results show that our two-stage stochastic iADCMs are more robust and more stable. Further changes to the opinions were made depend on the consensus opinion, which solved from the given models.

The outline of this paper is as follows. Section 2, we review some basic concepts about TSP and MCC models. Section 3, we present three new consensus models that consider the uncertainty of asymmetric cost and initial opinions to help develop a broad consensus on recommendations. Section 4, a L-shaped algorithm with the proposed two-stage consensus programming is used to solve the resulting uncertainty iADCMs. Section 5, the specific numerical experimental results are shown along with a discussion. Section 6, we explore the comparison results between our models and the existing deterministic consensus models. Section 7 demonstrates the local and global sensitivity analyses on several key parameters of the proposed two-stage stochastic iADCMs. Finally, we points out the concluding, research challenges and future directions in the last section.

2 Preliminaries

This section gives a brief review of TSP and MCCMs of symmetric and asymmetric costs. The analysis and description of these models have constructive significance for the modeling of two-stage stochastic integrated adjustment deviations and consensus optimization model in the following section.

2.1 Two-stage stochastic programming

Two-stage stochastic programming has become an important problem area. The basic idea is that once the uncertain events have unfolded, further design or operational adjustments can be made through values of the second-stage or alternatively called recourse variables at a particular cost. [27] gives a detailed description of stochastic programming.

Definition 2.1. [27] The basic TSP can be described as $min c^{T} x + E_{ξ} [min q (ω)^{T} y (ω)]$ (1a) $s . t . Ax = b,$ (1b) $W (ω) y (ω) = h (ω) - T (ω) x,$ (1c) $x \geq 0, y (ω) \geq 0 .$ (1d) Here $c \in ℝ^{n_{1}}$ , $b \in ℝ^{m_{1}}$ , $A \in ℝ^{m_{1} \times n_{1}}$ are the constants, $q (ω) \in ℝ^{n_{2}}, W (ω) \in ℝ^{m_{2} \times n_{2}}, T (ω) \in ℝ^{m_{2} \times n_{1}}, h (ω) \in ℝ^{m_{2}}$ are the vectors and matrices which are set with randomness, and ω denotes a scenario or possible outcome.

The decision of stochastic programming is divided into two stages. The variables x are called first-stage variables, and are known as the “here and now" decision, which must be made before the outcome of stochastic variables ω is observed. The variables y are called second-stage variables, which can be calculated after the outcome of ω is known.

2.2 Cost consensus models

Optimization-based consensus models have its origin in the work of Dong et al. [7], who initiated a MACM. The consensus reaching criteria of MACM is based on minimum total modifications. The MACM to reach consensus given by the following formula $min φ = \sum_{i = 1}^{n} | o_{i} - o_{i}^{'} |$ $s . t . o^{'} = F_{π} (o_{1}^{'}, \dots, o_{n}^{'}),$ $| o_{i}^{'} - o^{'} | \leq ε, i \in N,$ $\sum_{i = 1}^{n} π_{i} = 1,$ $π_{i} \in [0, 1],$ where the F_π is a aggregation function, and π = (π₁, ⋯ , π_n) ^T is a weighting vector.

When DMs are making decisions, in order to reach an agreement, it is usually necessary to modify the initial opinions. Therefore, in this process, on the one hand, each DM hopes to make the minimum adjustment. On the other hand, the moderator hopes to minimize the total compensation for all experts. Based on these two concerns, Ben et al [5] introduced a linear cost function to obtain a better consensus, which can be expressed as follows $f = \sum_{i = 1}^{n} c_{i} | o_{i} - o^{'} |,$ where c_i is the unit adjustment cost of expert e_i, o_i is the initial opinion of e_i, o′ is the consensus opinion. The above linear function, |o_i - o′| represents the deviation that e_i need to adjust, c_i|o_i - o′| represents the cost that moderator needs to pay compensation to the e_i, and the line function f represents the total cost that the moderator pays for e_i.

It’s not hard to summarize that the level of unit adjustment cost of c_i each expert e_i is closely associated with the divergence degree of expert opinions, and the degree of obstinacy more pronounced in high-cost than in low-cost. When o_i > o′, the compensation obtained by expert e_i is c_i (o_i - o′); when o_i < o′, the compensation obtained by expert e_i is c_i (o′ - o_i). However, the linear function has no directional constraint of expert opinion, and the unit adjustment cost of each expert is the same in two different directions. However, in most decision-making problems, this consensus model of symmetric adjustment costs can not meet the need of actual requirements. Because consensus cost differs for different setting of adjustment orientation. Based on the above motivation, Cheng’s minimum cost consensus models under asymmetric cost context [12] makes up the shortcomings mentioned above.

When a single expert adjusts his opinions, he does not modify his opinion in a single direction, but increasing or decreasing opinion in two different directions. Let $δ_{i}^{+} = (o_{i} - o^{'})^{+}$ and $δ_{i}^{-} = (o^{'} - o_{i})^{+}$ represent the amount of change in upward adjustment opinion, where a⁺ = max {0, a}, we have $δ_{i}^{+} \cdot δ_{i}^{-} = 0$ . And $c_{i}^{U}$ indicates the unit cost in the direction of increasing opinion value, while $c_{i}^{D}$ indicates the unit cost in the opposite direction. Thus, in [12], MCCM-DC (directional constraint) is described as follows $min_{o^{'}, δ_{i}^{+} δ_{i}^{-}} φ = \sum_{i = 1}^{n} c_{i}^{D} δ_{i}^{+} + c_{i}^{U} δ_{i}^{-}$ (2a) $s . t . o^{'} + δ_{i}^{+} - δ_{i}^{-} = o_{i}, i \in N,$ (2b) $o_{1} \leq o^{'} \leq o_{n},$ (2c) $δ_{i}^{+} \geq 0, δ_{i}^{-} \geq 0, i \in N,$ (2d) The optimal solution of model (2) is $o^{*} = (o^{'}, δ_{1}^{+}, δ_{1}^{-}, \dots, δ_{i}^{+}, δ_{i}^{-}, \dots, δ_{n}^{+}, δ_{n}^{-})^{T}$ .

The innovation of this model is to divide the unit adjustment cost into up and down according to the direction. However, this model can only be applied to certain decision-making environment. For uncertain decision-making problems, this model can not be solved, which is one of its disadvantages. We will propose a consensus optimization model in an uncertain environment in Section 3.1, which will make up for the shortcomings of MCCM-DC.

On the basis of model (2), the author further increases the relevant constraints to reach an optimal consensus. From the experience of decision-making, in order to reach the goal of consensus in the decision-making environment, the selected experts who express their opinions will not modify their opinions indefinitely. When experts adjust their opinions, they are only allowed to modify their own opinions within a certain range. It is assumed that ε_i is the maximum range of acceptable distance of expert e_i, which refers to the range of expert opinion and consensus opinion. This means that the bigger the ε_i is, the bigger of the acceptable adjustment distance. On the basis of the above analysis, the ε-MCCM-DC [12] is the following one $min_{o^{'}, δ_{i}^{+} δ_{i}^{-}} φ = \sum_{i = 1}^{n} c_{i}^{D} δ_{i}^{+} + c_{i}^{U} δ_{i}^{-}$ (3a) $s . t . o^{'} + δ_{i}^{+} - δ_{i}^{-} = o_{i}, i \in N,$ (3b) $o_{1} \leq o^{'} \leq o_{n}$ (3c) $0 \leq δ_{i}^{+}, δ_{i}^{-} \leq ε_{i}, i \in N$ (3d) The above modeling method produces an optimal solution $o^{*} = (o^{'}, δ_{1}^{+}, δ_{1}^{-}, \dots,$ $δ_{i}^{+}, δ_{i}^{-}, \dots, δ_{n}^{+}, δ_{n}^{-})^{T}$ .

The biggest difference between the model (2) and model (3) is that model (3) involves accounts for the deviation between DM’s initial opinion and consensus opinion. And the tolerance limit has the following characteristics: the smaller the value of ε_i, the more extreme the expert is in the attitude of revising opinions. On the contrary, the higher the value of ε_i, the more open the expert is to be a revision suggestion. This model adds the adjustment limit allowed by each expert on the consensus optimization model, which makes the model (3) more in line with the psychology of actual DMs. However, it does not take into account the uncertainty characteristics of personal opinions and directional costs. Therefore, a top priority for this is to build a two-stage stochastic iADCMs with maximum acceptable distance to fill in the gaps according to the determined uncertain scenarios. Specific features of the proposed model are described in the Section 3.2.

In addition to two MCCMs for asymmetric costs described above, a new consensus optimization model is proposed in [12]. In this model, experts are allowed to modify their opinions for free within a certain range. In practice, it is assumed that, within self-defined limits θ_i, each expert has a limit within which he or she is allowed to adjust his or her opinion without the need for total compensation from the moderator. Thus, the TB-MCCM-DC with a no-cost threshold is constructed as follows $\begin{matrix} min_{o^{'}} & φ = \sum_{o_{i} < o^{'} - θ_{i}} c_{i}^{U} (o^{'} - θ_{i} - o_{i}) \\ + \sum_{o_{i} > o^{'} + θ_{i}} c_{i}^{D} (o_{i} - θ_{i} - o^{'}) \end{matrix}$ (4a) $s . t . o^{'} \in O$ (4b) The meaning of this model is that if the deviation of experts’ opinions is within the free adjustment threshold, then the moderator does not need to pay additional compensation. Otherwise, the moderator needs to pay a certain cost to the experts, and the experts will agree to adjust the initial opinions towards the consensus.

Let’s $u_{i}^{-} = (o^{'} - θ_{i} - o_{i})^{+}, u_{i}^{+} = (o_{i} - o^{'} + θ_{i})^{+}, v_{i}^{-} = (o^{'} + θ_{i} - o_{i})^{+}, v_{i}^{+} = (o_{i} - o^{'} - θ_{i})^{+}$ , where $u_{i}^{-} \in [0, o^{'} - θ_{i}], u_{i}^{+} \in [o^{'} - θ_{i}, o^{'}], v_{i}^{-} \in [o^{'}, o^{'} + θ_{i}], v_{i}^{+} \in [o^{'} + θ_{i}, \infty], u_{i}^{+} \cdot u_{i}^{-} = 0$ and $v_{i}^{+} \cdot v_{i}^{-} = 0$ . Thus, the model (4) also can be illustrated as follows $min_{o^{'}, u_{i}^{+} u_{i}^{-}, v_{i}^{+} v_{i}^{-}} φ = \sum_{i = 1}^{n} c_{i}^{U} u_{i}^{-} + c_{i}^{D} v_{i}^{+}$ (5a) $s . t . u_{i}^{+} - u_{i}^{-} = o_{i} - o^{'} + θ_{i}, i \in N,$ (5b) $v_{i}^{+} - v_{i}^{-} = o_{i} - o^{'} - θ_{i}, i \in N,$ (5c) $o^{'}, u_{i}^{+}, u_{i}^{-}, v_{i}^{+}, v_{i}^{-} \geq 0, i \in N .$ (5d) o^* is the optimal solution obtained from model (5), where $\begin{matrix} o^{*} = & (o^{'}, u_{1}^{-}, u_{1}^{+}, v_{1}^{-}, v_{1}^{+}, \dots, u_{i}^{-}, u_{i}^{+}, v_{i}^{-}, v_{i}^{+}, \dots, \\ u_{n}^{-}, u_{n}^{+}, v_{n}^{-}, v_{n}^{+})^{T} . \end{matrix}$

The consensus model (5) not only describes the asymmetric cost but also takes into account the threshold of free modification opinions. However, this model does not consider a variety of uncertainties. To improve on existing model, there is a need to add a variety of scenarios into the model (5) involved in uncertainty opinions and costs.

3 Two-stage stochastic integrated adjustment deviations and consensus models

Since none of the previous consensus optimization models take into account the uncertainty of some parameters, three types of two-stage stochastic integrated adjustment deviation and consensus model in this section will be given, which is based on stochastic programming and MCCMs in an asymmetric costs context. For the sake of convenience, the three proposed uncertain consensus models labeled as TSS (two-stage stochastic)-MCCM-I, TSS-MCCM-II, and TSS-MCCM-III. The common feature of the proposed models is that the adjustment costs of different directions and the initial opinions of experts are set to multiple and varied scenarios. Decision time emerges as an important aspect in consensus reaching process. In the following study, according to a temporal order, decisions are classified into two stages. In the following models, o′ and $δ_{i}^{+}, δ_{i}^{-}$ (or $v_{i}^{+}, u_{i}^{-}$ ) are the decision variables in first and second stage, respectively. The parameters and variables used in this section are given in Table 1 below.

Table 1
Additional Notation and Description of the iADCMs

Notation Description

Uncertain Parameter

$c_{i}^{D} (ω)$ uncertain unit downward adjustment cost of e_i

$c_{i}^{U} (ω)$ uncertain unit upward adjustment cost of e_i

o_i (ω) uncertain initial opinion of e_i

Decision Variable

o′ the expected consensus opinion of the moderator

$δ_{i}^{+}$ the downward adjustment deviation

$δ_{i}^{-}$ the upward adjustment deviation

$u_{i}^{-}$ the upward adjustment deviation

$u_{i}^{+}$ no need to adjust

$v_{i}^{-}$ no need to adjust

$v_{i}^{+}$ the downward adjustment deviation

Notation	Description
$c_{i}^{D} (ω)$	uncertain unit downward adjustment cost of e_i
$c_{i}^{U} (ω)$	uncertain unit upward adjustment cost of e_i
o_i (ω)	uncertain initial opinion of e_i
Decision Variable
o′	the expected consensus opinion of the moderator
$δ_{i}^{+}$	the downward adjustment deviation
$δ_{i}^{-}$	the upward adjustment deviation
$u_{i}^{-}$	the upward adjustment deviation
$u_{i}^{+}$	no need to adjust
$v_{i}^{-}$	no need to adjust
$v_{i}^{+}$	the downward adjustment deviation

3.1 Two-stage stochastic MCCM-I

The MCCM-DC ignores the uncertainty of $c_{i}^{D}, c_{i}^{U}, o_{i}$ . In order to make up this shortcoming, let $ξ (ω)^{T} = {c_{i}^{D} (ω)^{T}, c_{i}^{U} (ω)^{T}, o_{i} (ω)^{T}},$ and with the analysis of TSP (1), we give the following TSS-MCCM-I:

$min_{o^{'}, δ_{i}^{+}, δ_{i}^{-}} φ = E_{ξ} [\sum_{i = 1}^{n} (c_{i}^{D} (ω) δ_{i}^{+} + c_{i}^{U} (ω) δ_{i}^{-})]$ (6a) $s . t . o^{'} + δ_{i}^{+} - δ_{i}^{-} = o_{i} (ω), i \in N,$ (6b) $o_{1} \leq o^{'} \leq o_{n},$ (6c) $δ_{i}^{+} \geq 0, δ_{i}^{-} \geq 0, i \in N,$ (6d) where o₁ = min {o₁ (ω)}, o_n = max {o_n (ω)}.

The meaning of objective function formula (6a) of TSS-MCCM-I is as follows: It represents the total adjustment cost under various situations, which are provided by the moderator. c_i represents the directional cost coefficient of the objective function and (6a) denotes the expected total adjustment cost of the second stage problem. Constraint (6b) means that expert e_i’s amendments o_i (ω) in the upward or downward direction, and finally get a consensus o′. Constraint (6c) means that the final consensus o′ is within the scope of the initial opinion o_i (ω) , i = 1, 2, ⋯ , n, which is given considering all scenarios. The implication of constrained (6d) is the case that all the experts’ deviations in both directions are non-negative numbers.

Theorem 3.1. The translation model for model (6) is set up as following optimization model $min_{o^{'}} Q_{1} (o^{'})$ $s . t . o_{1} \leq o^{'} \leq o_{n},$ where $Q_{1} (o^{'}) = E_{ξ} Q_{1} (o^{'}, ξ (ω)),$ where ξ (ω) = {c (ω) , o (ω)}, and $Q_{1} (o^{'}) = min_{δ} c (ω)^{T} δ$ $s . t . W δ = o (ω) - o^{'},$ $δ \geq 0,$ where $c (ω) = (c_{1}^{D} (ω), c_{1}^{U} (ω), \dots, c_{n}^{D} (ω), c_{n}^{U} (ω))^{T}$ , o (ω) = (o₁ (ω) , ⋯ , o_n (ω)) ^T, o′ = {o′, o′, ⋯ , o′}, $δ = (δ_{1}^{+}, δ_{1}^{-}, \dots,$ $δ_{n}^{+}, δ_{n}^{-})^{T}$ , and $W = {[\begin{matrix} 1 & - 1 & 0 & 0 & \dots & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & - 1 & \dots & 0 & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & \dots & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & 1 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & \dots & 0 & 0 & 1 & - 1 \end{matrix}]}_{n \times 2 n .}$

Proof. A model is given below, which is equivalent to model (6). $min_{o^{'}, δ_{1}^{+}, δ_{i}^{-}} φ = E_{ξ} (c_{1}^{D} (ω) δ_{1}^{+} + c_{1}^{U} (ω) δ_{1}^{-} + \dots$ $+ c_{n}^{D} (ω) δ_{n}^{+} + c_{n}^{U} (ω) δ_{n}^{-})$ $s . t . δ_{1}^{+} - δ_{1}^{-} = o_{1} (ω) - o^{'},$ $δ_{2}^{+} - δ_{2}^{-} = o_{2} (ω) - o^{'},$ $\dots$ $δ_{n}^{+} - δ_{n}^{-} = o_{n} (ω) - o^{'},$ $o_{1} \leq o^{'} \leq o_{n},$ $δ_{i}^{+}, δ_{i}^{-} \geq 0, i \in N .$ According to the definition of TSP, let $c (ω) = (c_{1}^{D} (ω), c_{1}^{U} (ω), \dots, c_{n}^{D} (ω), c_{n}^{U} (ω))^{T}, o (ω) = (o_{1} (ω), \dots, o_{n} (ω))^{T}$ , and $W = {[\begin{matrix} 1 & - 1 & 0 & 0 & \dots & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & - 1 & \dots & 0 & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & \dots & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & 1 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & \dots & 0 & 0 & 1 & - 1 \end{matrix}]}_{n \times 2 n .}$ Besides, let o′ = {o′, o′, ⋯ , o′}, $δ = (δ_{1}^{+}, δ_{1}^{-}, \dots,$ $δ_{n}^{+}, δ_{n}^{-})^{T}$ . Thus, we have $min_{o^{'}} E_{ξ} Q (x, ξ)$ $s . t . o_{1} \leq o^{'} \leq o_{n},$ where Q (x, ξ) = min {c (ω) ^Tδ|Wδ = o (ω) - o′, δ ≥ 0} . ■

3.2 Two-stage stochastic MCCM-II

The MCCM-DC with maximum adjustment distance is mentioned before in Section 2, which denotes as ε_i. Nevertheless, model (3) does not take into account the uncertainty of $c_{i}^{U}, c_{i}^{D}, o_{i}$ . Therefore, similar to the previous section, let the uncertainty parameters $ξ^{T} (ω) = {c_{i}^{D} (ω), c_{i}^{U} (ω), o_{i} (ω)}$ . Thus, we develop the following TSS-MCCM-II: $min_{o^{'}, δ_{i}^{+}, δ_{i}^{-}} φ = E_{ξ} [\sum_{i = 1}^{n} (c_{i}^{D} (ω) δ_{i}^{+} + c_{i}^{U} (ω) δ_{i}^{-})]$ (7a) $s . t . o^{'} + δ_{i}^{+} - δ_{i}^{-} = o_{i} (ω), i \in N,$ (7b) $o_{1} \leq o^{'} \leq o_{n},$ (7c) $0 \leq δ_{i}^{+}, δ_{i}^{-} \leq ε_{i}, i \in N .$ (7d)

The obvious difference between TSS-MCCM-I and TSS-MCCM-II is the addition of constraint (7d), which represents the deviation allowed by each expert within a certain range.

Theorem 3.2. Model (7) can be regraded as an equivalent form to the following model $min_{o^{'}} Q_{2} (o^{'})$ $s . t . o_{1} \leq o^{'} \leq o_{n},$ where $Q_{2} (o^{'}) = E_{ξ} Q_{2} (o^{'}, ξ (ω)),$ and $Q_{2} (o^{'}) = min_{δ} c (ω)^{T} δ$ $s . t . W δ = o (ω) - o^{'},$ $0 \leq δ \leq ε,$ where $c (ω) = (c_{1}^{D} (ω), c_{1}^{U} (ω), \dots, c_{n}^{D} (ω), c_{n}^{U} (ω))^{T}$ , ε = (ε₁, ⋯ , ε_n) ^T, o (ω) = (o₁ (ω) , ⋯ , o_n (ω)) ^T, o′ = (o′, ⋯ , o′) ^T, $δ = (δ_{1}^{+}, δ_{1}^{-}, \dots,$ $δ_{n}^{+}, δ_{n}^{-})^{T}$ , and $W = {[\begin{matrix} 1 & - 1 & 0 & 0 & \dots & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & - 1 & \dots & 0 & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & \dots & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & 1 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & \dots & 0 & 0 & 1 & - 1 \end{matrix}]}_{n \times 2 n .}$ Thus the optimal solution is o^* = (o′, δ) ^T.

Proof. By using the similar proof method applied in Theorem 3.1, and we obtain this result.■

3.3 Two-stage stochastic MCCM-III

When making practical decisions, experts usually have a threshold of no-cost for adjustment, which the moderator does not need to pay a certain amount of compensation; on the contrary, beyond this range, the moderator needs to pay compensation to persuade experts to modify their initial opinions. The model (5) mentioned above takes this situation into account, but it does not consider the uncertainty of $c_{i}^{U}, c_{i}^{D}, o_{i}$ . Now consider the case that the factor $ξ^{T} (ω) = {c_{i}^{D} (ω), c_{i}^{U} (ω), o_{i} (ω)}$ is stochastic.

Here, the threshold adjusted by expert e_i for free is denoted as θ_i. Let’s $u_{i}^{+} = (o_{i} (ω) - o^{'} + θ_{i})^{+}, u_{i}^{-} = (o^{'} - θ_{i} - o_{i} (ω))^{+}, v_{i}^{+} = (o_{i} (ω) - o^{'} - θ_{i})^{+}, v_{i}^{-} = (o^{'} + θ_{i} - o_{i} (ω))^{+}$ , where $u_{i}^{-} \in [0, o^{'} - θ_{i}], u_{i}^{+} \in [o^{'} - θ_{i}, o^{'}], v_{i}^{-} \in [o^{'}, o^{'} + θ_{i}], v_{i}^{+} \in [o^{'} + θ_{i}, \infty], u_{i}^{+} \cdot u_{i}^{-} = 0$ and $v_{i}^{+} \cdot v_{i}^{-} = 0$ . Then the TSS-MCCM-III is presented as follows:

$min_{o^{'}, u_{i}^{+}, u_{i}^{-}, v_{i}^{+}, v_{i}^{-}} φ = E_{ξ} [\sum_{i = 1}^{n} (c_{i}^{U} (ω) u_{i}^{-} + c_{i}^{D} (ω) v_{i}^{+})]$ (8a) $s . t . u_{i}^{+} - u_{i}^{-} = o_{i} (ω) - o^{'} + θ_{i}, i \in N,$ (8b) $v_{i}^{+} - v_{i}^{-} = o_{i} (ω) - o^{'} - θ_{i}, i \in N,$ (8c) $o_{1} \leq o^{'} \leq o_{n},$ (8d) $o^{'}, u_{i}^{+}, u_{i}^{-}, v_{i}^{+}, v_{i}^{-} \geq 0, i \in N .$ (8e)

When o_i (ω) ∈ [o′ - θ_i, o′ + θ_i], participation in the modification of opinion is voluntary and free of charge; when o_i (ω) ∈ [0, o′ - θ_i], expert e_i is compensated $c_{i}^{U} (ω) (o^{'} - θ_{i} - o_{i} (ω))$ ; otherwise, when o_i (ω) ∈ [o′ + θ_i, + ∞], expert e_i compensated $c_{i}^{D} (ω) (o^{'} - θ_{i} - o_{i} (ω))$ .

Theorem 3.3. Model (8) is further transformed into the following programming $min_{o^{'}} Q_{3} (o^{'})$ $s . t . o_{1} \leq o^{'} \leq o_{n}$ where $Q_{3} (o^{'}) = E_{ξ} Q_{3} (o^{'}, ξ (ω)),$ and $Q_{3} (o^{'}) = min_{δ} c (ω)^{T} δ$ $s . t . W δ = o (ω) + θ - o^{'},$ $δ \geq 0,$ where $c (ω) = (0, c_{1}^{D} (ω), c_{1}^{U} (ω), 0, \dots, 0, c_{i}^{D} (ω), c_{i}^{U} (ω), 0, \dots, 0, c_{n}^{D} (ω), c_{n}^{U} (ω), 0)^{T}$ , θ = (θ₁, θ₁, ⋯ , θ_i, θ_i, ⋯ , θ_n, θ_n) ^T, o′ = (o′, o′, ⋯ , o′, o′, ⋯ , o′, o′) ^T, o (ω) = (o₁ (ω) , o₁ (ω) , ⋯ , o_i (ω) , o_i (ω) , ⋯ , o_n (ω) , o_n (ω)) ^T, $δ = (u_{1}^{+}, u_{1}^{-}, v_{1}^{+}, v_{1}^{-}, \dots, u_{i}^{+}, u_{i}^{-}, v_{i}^{+}, v_{i}^{-}, \dots, u_{n}^{+}, u_{n}^{-}, v_{n}^{+}, v_{n}^{-})^{T}$ , and $W = {[\begin{matrix} 1 & - 1 & 0 & 0 & \dots & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & - 1 & \dots & 0 & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & \dots & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & 1 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & \dots & 0 & 0 & 1 & - 1 \end{matrix}]}_{2 n \times 4 n .}$ o^* = (o′, δ) ^T is the optimal solution to the model presented above.

Proof. Similarly proofs are omitted for brevity.■

4 L-shaped algorithm to solve two-stage stochastic integrated adjustment deviations and consensus models

In view of the complexity and difficulty of solving the model (6)-(8), we use the L-shaped algorithm (see [29]) to address such problems. The biggest difference between our next algorithm and the original L-shaped algorithm is that we implement it into our two-stage stochastic consensus model, and we give the specific implementation steps. We can write the proposed models in the following general form, which is a special structure of the model (1). $min c (ω)^{T} δ$ $s . t . o_{1} \leq o^{'} \leq o_{n},$ $W δ = h (ω) - o^{'}$ $δ \geq 0,$ where h (ω) = o (ω) in the TSS-MCCM-I, and h (ω) = (o (ω) , ε) in the TSS-MCCM-II, and h (ω) = o (ω) + θ in the TSS-MCCM-III.

In order to give the detailed steps of the algorithm, we will first give a set of valid feasibility cuts and optimality cuts, which are based on duality theory of line programming. Consider the following programming $min_{δ} c (ω)^{T} δ$ $s . t . W δ = h (ω) - o^{'},$ $δ \geq 0 .$ The “optimality cuts" can be generated by $θ \geq E_{ξ} {π^{T} (h (ω) - o^{'}},$ where π is the dual solutions. If $o^{'} \in K_{2}, Q (o^{'}) \geq + \infty$ , solve the following linear programming to obtain the simple multipliers σ^v $min w^{'} = e^{T} ν^{+} + e^{T} ν^{-}$ (9a) $s . t . W δ + I ν^{+} - I ν^{-} = h_{k} - (o^{'})^{v},$ (9b) $δ \geq 0, ν^{+} \geq 0, ν^{-} \geq 0,$ (9c) where e^T = (1, ⋯ , 1). Then a “feasibility cut" is generated (see [2]) $(σ^{v})^{T} o^{'} \geq (σ^{v})^{T} h_{k} .$

Based on the above analysis, the master problem can be described as $min θ$ (10a) $s . t . o_{1} \leq o^{'} \leq o_{n},$ (10b) $(σ^{v})^{T} o^{'} \geq (σ^{v})^{T} h_{k},$ (10c) $θ \geq E_{ξ} {π^{T} (h (ω) - o^{'}} .$ (10d) $θ \in ℝ .$ (10e)

algorithm 1 L-shaped algorithm

1: Initialization: v← 0, r ← 0, s ← 0, θ ← - ∞

2: Solve the master problem (10) to get a solution ((o′) ⁰, θ)

3: while solve the problem (9) and compute the optimal value w′ do

4: if w′ = 0 then

5: add optimality cuts (10d)

6: s ← s + 1

7: else

8: add a feasibility cut (10c)

9: r ← r + 1

10: end if

11: Solve the master problem (10)

12: if (10) is infeasible then

13: Stop. The original optimization problem is infeasible

14: else

15: Let (o′) ^v+1 be its solution, θ^v+1 denotes the optimal objective value

16: end if

17: v ← v + 1

18: end while

19: return optimial solution (o′) ^v, and optimal value θ^v

It was theoretically proven that the L-shaped algorithm is a convergent algorithm, against the convergence criteria given in [29]. Thus, the algorithm can be proven to converge after a finite number of iterations. For brevity, we omit the proof of the convergences. See detailed proof at [29].

5 Numerical analysis of the two-stage stochastic iADCMs

In this paper, two-stage stochastic iADCMs are proposed to solve uncertain decision-making problems in practice, which often exists in the real world. In both theory and practice, three proposed deductive models are applied to analyze the policy of Grains to Green Program (GTGP). The results are demonstrated that the two-stage stochastic iADCMs proposed in this paper have better practicality. A notebook computer (Intel i5 CPU and 8GB RAM) is selected to conduct the numerical experiment. In order to further verify the accuracy of our algorithm, we further compare the algorithm of this paper with the optimization solver CPLEX. The results show, a L-shaped algorithm and CPLEX yielded exactly the same results. However, in terms of time, the given method is faster than the solver CPLEX. This just shows the effectiveness and accuracy of the L-shaped algorithm.

The emergence of a series of ecological problems, such as soil erosion, desertification, and reduction of biodiversity, poses a serious threat to ecological security [34]. As far as we know, the state has paid more and more attention to environmental protection. So, in 1999, the state officially opened the prelude to returning farmland to forest, promoting the area transformation between the farmland ecosystem and forest and grassland ecosystems [35]. This project can improve vegetation coverage and strengthen the protection of agricultural land, especially the rehabilitation of soil and restoration of degraded land. In recent years, the GTGP which is a major ecological construction project invested in China has achieved remarkable results mainly because it relies on national policy subsides with huge economic input. These cropland quotas are allocated by the government to each province who in turn allocates them between their prefecture (city), country (district), township (subdistrict), and village (neighbourhood). The government provides cropland subsides to these farmers who stop cultivating these lands to compensate for their loss.

However, not all the farmers who participated consented to the compensation policy. In case of discrepancies and disagreement, it will be resolved by consensus. With this condition, these farmers provide their expected subsidies per mu of land according to the subsidy policies of other regions. Due to this background, their expected subsidies and the compensatory cost for adjustment are uncertainty, and negotiation costs for the government are necessary to reach consensus. Therefore, on the basis of the GTGP, the subsidies of farmers are modified, and the local government and farmers from an uncertain iADCM system. Here, the local government and farmers can be viewed as the moderator and DMs [20]. Finally, the farmers of the selected fields need to reach a consensus, which can satisfy all farmers and the government. Through this process of negotiation and adjustment, a consensus was reached and the GTGP was completed. In the next part, we will describe the GTGP and solve it by using the two-stage stochastic iADCMs. In this paper, we will refer to the relevant values in [20]. Furthermore, let C = {c₁, ⋯ , c_n} be the set of unit adjustment costs in the proposed iADCMs, where c_i (i ∈ N) is randomly generated based on $N (μ_{c}, σ_{c}^{2})$ , and {μ_c, σ_c} represent the expected value and variance of unit adjustment cost, respectively. This idea was inspired from the [36]. Here we assume that the value of μ_c is the adjustment cost in the first scenario, and the value of σ_c is 0.2.

Example 5.1. (Example of the TSS-MCCM-I)

We hypothesized that four farmers {e₁, e₂, e₃, e₄} and local government e′ participate in negotiation. The local government will play the role of moderator in this process and provide a certain amount of compensation to persuade farmers to modify their opinions. We assume that the uncertain parameters divide into three scenarios. The uncertainty opinion of each farmer is o₁ = {17, 18.5, 20} , o₂ = {16, 18, 21} , o₃ = {22, 23, 24} , o₄ = {18, 20.5, 23} (1000RMB/mu). The first scenario assumes that each farmer unit up-ward adjustment cost is $w^{U} = {w_{1}^{U}, w_{2}^{U}, w_{3}^{U}, w_{4}^{U}} = {0.1, 0, 0.5, 0.9} (1000 RMB)$ , and their unit down-ward adjustment cost is $w^{D} = {w_{1}^{D}, w_{2}^{D}, w_{3}^{D}, w_{4}^{D}} = {0.5, 0.2, 0.9, 1.5} (1000 RMB)$ , respectively. The unit costs data of another two scenarios are randomly generated from normal distribution. Besides, assume that the generative probabilities of different situations are p = {0.2, 0.4, 0.4}, respectively. This issue will be addressed later, using model-based analyses of TSS-MCCM-I. $min_{o^{'}, δ_{i}^{+}, δ_{i}^{-}} φ = E_{ξ} [\sum_{i = 1}^{n} (c_{i}^{D} (ω) δ_{i}^{+} + c_{i}^{U} (ω) δ_{i}^{-})]$ $s . t . δ_{1}^{+} - δ_{1}^{-} = o_{1} (ω) - o^{'},$ $δ_{2}^{+} - δ_{2}^{-} = o_{2} (ω) - o^{'},$ $δ_{3}^{+} - δ_{3}^{-} = o_{3} (ω) - o^{'},$ $δ_{4}^{+} - δ_{4}^{-} = o_{4} (ω) - o^{'},$ $16 \leq o^{'} \leq 24,$ $δ_{i}^{+} \geq 0, δ_{i}^{-} \geq 0, i = 1, 2, 3, 4 .$

We obtain the optimal solution o′ = 23 and φ = 3159.9 through the L-shaped method. This means that all farmers adjust their initial opinions and finally agree with the consensus of 23 (1000 RMB/mu), and the total cost of local government is 3159.9 RMB. In order to validate the accuracy of the proposed algorithm, we use the solver CPLEX to solve this problem. The numerical results are shown in Table 2. We find that the optimal solution of the two methods is the same, but the solution time of the algorithm is shorter, which shows that our algorithm works more efficiently.

Table 2
Comparison results associated with the modeling approach

Model Method o′ ϕ time (sec.)

TSS-MCCM-I L-shaped algorithm 23.0000 3.1599 1.9679

CPLEX 23.0000 3.1599 2.4705

TSS-MCCM-II L-shaped algorithm 20.0000 5.3758 0.4542

CPLEX 20.0000 5.3758 2.5422

TSS-MCCM-III L-shaped algorithm 22.0000 0.9431 2.0282

CPLEX 22.0000 0.9431 2.3864

Model	Method	o′	ϕ	time (sec.)
TSS-MCCM-I	L-shaped algorithm	23.0000	3.1599	1.9679
	CPLEX	23.0000	3.1599	2.4705
TSS-MCCM-II	L-shaped algorithm	20.0000	5.3758	0.4542
	CPLEX	20.0000	5.3758	2.5422
TSS-MCCM-III	L-shaped algorithm	22.0000	0.9431	2.0282
	CPLEX	22.0000	0.9431	2.3864

Example 5.2. (Example of the TSS-MCCM-II)

The setting of uncertain parameters ( $c_{i}^{U} (ω), c_{i}^{D} (ω), o_{i} (ω)$ ) and probabilities are the same as in Example 5.1. In addition, based on TSS-MCCM-II, we assume that the corresponding limited compromises of four farmers are ε = {ε₁, ε₂, ε₃, ε₄} = {3, 4, 4, 3} (1000RMB/mu). We initiate our analysis establishing the iADCM based on TSS-MCCM-II, which can be written as follows: $min_{o^{'}, δ_{i}^{+}, δ_{i}^{-}} φ = E_{ξ} [\sum_{i = 1}^{n} (c_{i}^{D} (ω) δ_{i}^{+} + c_{i}^{U} (ω) δ_{i}^{-})]$ $s . t . δ_{1}^{+} - δ_{1}^{-} = o_{1} (ω) - o^{'},$ $δ_{2}^{+} - δ_{2}^{-} = o_{2} (ω) - o^{'},$ $δ_{3}^{+} - δ_{3}^{-} = o_{3} (ω) - o^{'},$ $δ_{4}^{+} - δ_{4}^{-} = o_{4} (ω) - o^{'},$ $0 \leq δ_{1}^{+}, δ_{1}^{-} \leq 3,$ $0 \leq δ_{2}^{+}, δ_{2}^{-} \leq 4,$ $0 \leq δ_{3}^{+}, δ_{3}^{-} \leq 4,$ $0 \leq δ_{4}^{+}, δ_{4}^{-} \leq 4,$ $16 \leq o^{'} \leq 24 .$

By calculation, the optimal solution and the optimal value are obtained. Thus, four farmers reached a consensus o′ = 20 (1000RMB/mu) and the compensation paid by the local government is 5375.8 RMB. We compared our algorithm with the solver CPLEX, and the results of the comparison are available in Table 2. It further verify the accuracy of our algorithm.

Example 5.3. (Example of the TSS-MCCM-III)

Assuming that each river expert has a certain limiting threshold, which is denoted as θ = {θ₁, θ₂, θ₃, θ₄} = {2, 2, 2, 1} (1000RMB/mu). And the setting of uncertain initial opinion value and the unit uncertain adjustment costs are consistent with Example 5.1. Besides, the probability of three situations is p = {0.2, 0.4, 0.4}. According to the given constraints, to address this problem, we employed a TSS-MCCM-III. $min_{o^{'}, u_{i}^{+}, u_{i}^{-}, v_{i}^{+}, v_{i}^{-}} φ = E_{ξ} [\sum_{i = 1}^{n} (c_{i}^{U} (ω) u_{i}^{-} + c_{i}^{D} (ω) v_{i}^{+})]$ $s . t . u_{1}^{+} - u_{1}^{-} = o_{1} (ω) - o^{'} + 2,$ $v_{1}^{+} - v_{1}^{-} = o_{1} (ω) - o^{'} - 2,$ $u_{2}^{+} - u_{2}^{-} = o_{2} (ω) - o^{'} + 2,$ $v_{2}^{+} - v_{2}^{-} = o_{2} (ω) - o^{'} - 2,$ $u_{3}^{+} - u_{3}^{-} = o_{3} (ω) - o^{'} + 2,$ $v_{3}^{+} - v_{3}^{-} = o_{3} (ω) - o^{'} - 2,$ $u_{4}^{+} - u_{4}^{-} = o_{4} (ω) - o^{'} + 1,$ $v_{4}^{+} - v_{4}^{-} = o_{4} (ω) - o^{'} - 1,$ $16 \leq o^{'} \leq 24,$ $o^{'}, u_{i}^{+}, u_{i}^{-}, v_{i}^{+}, v_{i}^{-} \geq 0, i = 1, 2, 3, 4 .$

Using the presented numerical algorithm, we solved this problem, and numerical results of the TSS-MCCM-III are solved as o′ = 22 (1000RMB/mu) and φ = 943.1 RMB. In the solution methods’ comparison, we compare our algorithm with CPLEX. Table 2 shows the results for the outcomes. However, in terms of time, the L-shaped algorithm is faster than the CPLEX.

In Table 2, by comparing the total consensus costs calculated by the three two-stage stochastic iADCMs, it can be clearly seen that the total cost of TSS-MCCM-II is the largest, while the total cost of TSS-MCCM-III is the smallest. This is because TSS-MCCM-II allows experts to have a certain compromise limit, which may lead to more costs to reach a consensus. On the contrary, TSS-MCCM-III allows experts to have a no-cost threshold to adjust opinions, which makes the total consensus cost lower. In a word, the tolerances of experts and no-cost adjustment thresholds have an important impact on the iADCMs. It is found that the tolerance of experts in GDM has a significant impact on the total cost, and the total costs of persuading experts to modify their opinions is higher.

6 Comparison results

In this section, under the background of asymmetric consensus cost, we compare the deterministic minimum cost consensus models (MCCM-DC, ε-MCCM-DC, TB-MCCM-DC) with uncertainty minimum cost consensus models (TSS-MCCM-I, TSS-MCCM-II, TSS-MCCM-III).

Under the context of asymmetric costs, in each case, the proposed iADCMs corresponds with one of the Cheng’s MCCMs. In order to compare the models, three uncertain scenarios in Example 5.1-5.3 are applied to the deterministic minimum cost consensus model. And the results are presented in Table 3. According to the numerical result, an obvious feature can be observed, that is, the minimum total cost of two-stage stochastic iADCMs may be lower or higher than the deterministic MCCMs when considering three kinds of uncertain scenarios in the consensus process. In addition, there are also differences in consensus between deterministic and stochastic cases. This is mainly because of the deterministic consensus models only consider a single situation. However, the two-stage stochastic consensus model takes multiple scenarios into account, which makes the model more complex and the numerical results more robust. Under the joint action of various situations, the total cost of consensus is obviously higher. However, the uncertainty of concrete decision-making problems in real life makes the two-stage stochastic consensus model more practical.

Table 3
Comparison results of different cost consensus models

Model (2) TSS-MCCM-I Model (3) TSS-MCCM-II Model (5) TSS-MCCM-III

o′ 20.5 23 20.5 20 21 22

φ 3.6000 3.1599 3.6000 5.3738 0.2500 0.9431

	Model (2)	TSS-MCCM-I	Model (3)	TSS-MCCM-II	Model (5)	TSS-MCCM-III
o′	20.5	23	20.5	20	21	22
φ	3.6000	3.1599	3.6000	5.3738	0.2500	0.9431

7 Sensitivity analysis

In this section, we analyze the sensitivity of the three proposed two-stage stochastic consensus optimization models for the uncertain adjustment costs in the upward or downward direction, respectively. Data of sensitivity analysis are all from Example 5.1-5.3. For local sensitivity analysis, only the test upward/downward adjustment costs of individual experts are allowed to change, and other test parameters are held constant. We perform global sensitivity analysis on the models. To achieve this, all the experts’ test upward/downward adjustment costs are changed simultaneously.

7.1 Sensitivity analysis on downward adjustment cost

For the convenience of marking, $▵ c_{i}^{D}$ is used to indicate the change of unit cost of an individual expert when adjusting opinions in a downward direction. When $▵ c_{i}^{D}$ is greater than 0, it means increasing unit cost; when $▵ c_{i}^{D}$ is less than 0, it means reducing unit cost. Accordingly, ▵c^D is used to denote the value of the change in unit cost of all experts in the upward direction, and at the same time. Here we divide the sensitivity analysis into two parts, including local and global sensitivity analysis.

First of all, we give a local sensitivity analysis of $c_{i}^{D}$ , which includes the effect of increasing or decreasing the cost of downward adjustment opinion on consensus and total compensation cost. In terms of the impact of $c_{i}^{D}$ on consensus opinion, when the unit adjustment cost of a single expert e_i is changed, the consensus opinions of TSS-MCCM-I, TSS-MCCM-II, and TSS-MCCM-III are almost unchanged, which are 23, 20, and 22, respectively. This conclusion can be summarized in Table 4 –7. However, the effect of changing the same value of $c_{i}^{D}$ on the total consensus cost of three proposed two-stage stochastic iADCMs is different. When the value $c_{i}^{D}$ of a single expert changes to the same value, it has a significant impact on TSS-MCCM-II. Since under three situations, the $c_{1}^{U} (ω)$ are all less than o′ = 20, the change of $o_{1}^{D} (ω)$ has no effect on the total cost. In the three uncertain situations, the initial opinions of expert e₂, e₃ and e₄ are greater than the consensus in a certain situation. Compared with TSS-MCCM-I, each expert in TSS-MCCM-II has a set range for opinion adjustment, which only allows adjustment within a certain range. Therefore, they are more stubborn in this process of TSS-MCCM-II, and the total consensus cost is more sensitive to the change of $c_{i}^{D}$ . In addition, we find that the change value of $c_{i}^{D}$ of a single expert has no effect on total consensus cost of TSS-MCCM-III. This is because in TSS-MCCM-III, each expert has a free adjustment threshold, and there is no need for the moderator to compensate the experts when adjusting within this threshold range. Therefore, compared with TSS-MCCM-I, TSS-MCCM-III is more tolerant. The above conclusion can be clearly seen in Fig. 7.1.

Table 4
Effects of $c_{1}^{D}$ on consensus opinion

$c_{1}^{D}$ -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 ⋯

o′(TSS-MCCM-I) 23 23 23 23 23 23 23 23 23 23 23 ⋯

o′(TSS-MCCM-II) 20 20 20 20 20 20 20 20 20 20 20 ⋯

o′(TSS-MCCM-III) 22 22 22 22 22 22 22 22 22 22 22 ⋯

$c_{1}^{D}$	-0.5	-0.4	-0.3	-0.2	-0.1	0	0.1	0.2	0.3	0.4	0.5	⋯
o′(TSS-MCCM-I)	23	23	23	23	23	23	23	23	23	23	23	⋯
o′(TSS-MCCM-II)	20	20	20	20	20	20	20	20	20	20	20	⋯
o′(TSS-MCCM-III)	22	22	22	22	22	22	22	22	22	22	22	⋯

Table 5

Effects of $c_{2}^{D}$ on consensus opinion

$▵ c_{2}^{D}$	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	⋯
o′(TSS-MCCM-I)	23	23	23	23	23	23	23	23	23	23	⋯
o′(TSS-MCCM-II)	20	20	20	20	20	20	20	20	20	20	⋯
o′(TSS-MCCM-III)	22	22	22	22	22	22	22	22	22	22	⋯

Table 6

Effects of $c_{3}^{D}$ on consensus opinion

$▵ c_{3}^{D}$	-0.6	-0.5	-0.4	-0.3	-0.2	-0.1	0	0.1	0.2	0.3	0.4	0.5	⋯
o′(TSS-MCCM-I)	22	22	23	23	23	23	23	23	23	23	23	23	⋯
o′(TSS-MCCM-II)	20	20	20	20	20	20	20	20	20	20	20	20	⋯
o′(TSS-MCCM-III)	22	22	22	22	22	22	22	22	22	22	22	22	⋯

Table 7

Effects of $c_{4}^{D}$ on consensus opinion

$▵ c_{4}^{D}$	-1.4	-1.2	-1	-0.8	-0.6	-0.4	-0.2	0	0.2	0.4	0.6	⋯
o′(TSS-MCCM-I)	22	22	22	23	23	23	23	23	23	23	23	⋯
o′(TSS-MCCM-II)	20	20	20	20	20	20	20	20	20	20	20	⋯
o′(TSS-MCCM-III)	21.5	21.5	21.5	22	22	22	22	22	22	22	22	⋯

Fig. 7.1

Effects of individual downward adjustment cost on the total consensus cost.

Secondly, global sensitivity analysis of $c_{i}^{D}$ will be presented, which is also described in terms of consensus opinion and total consensus cost. According to Table 8, it can be found that the optimal consensus values of the three uncertain consensus cost models are invariant within the range of the given unit cost changes, and the consensus results remain relatively high among all the initial opinions, which are 23, 20 and 22, respectively. Furthermore, it can be seen from Fig. 7.2 that the unit costs have an obvious impact on TSS-MCCM-II within a certain range of changes, but there is no special change for the other models. The main reason for this phenomenon is that TSS-MCCM-II has stronger constraints and higher requirements for experts to adjust their opinions. However, experts in TSS-MCCM-I and TSS-MCCM-III are more moderate in modifying their opinions. In addition, in terms of total consensus cost, TSS-MCCM-III is lower than TSS-MCCM-I, which is mainly due to the fact that each expert in the previous model has a threshold for free adjustment, so the total cost required is relatively lower.

Table 8

Effects of c^D on consensus opinion

▵c^D	-0.2	-0.1	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	⋯
o′(TSS-MCCM-I)	23	23	23	23	23	23	23	23	23	23	23	⋯
o′(TSS-MCCM-II)	20	20	20	20	20	20	20	20	20	20	20	⋯
o′(TSS-MCCM-III)	22	22	22	22	22	22	22	22	22	22	22	⋯

Fig. 7.2

Effects of global downward adjustment costs on the total consensus cost

Fig. 7.3

Effects of individual upward adjustment cost on the total consensus cost

Fig. 7.4

Effects of global upward adjustment costs on the total consensus cost

7.2 Sensitivity analysis on upward adjustment cost

In real situation, we assume that the increase and decrease of each expert’s unit upward adjustment cost are decided by the moderator according to the decision problems. Next, we show the impact of the change in the cost of the expert’s $c_{i}^{U}$ on the consensus opinion and the total consensus cost. Subsequently, a sensitivity analysis of unit upward cost is performed. We will give the sensitivity analysis of the unit upward adjustment cost locally and globally. Through the sensitivity analysis, we can clearly and intuitively understand the result of the test parameter to the optimal solution, which is also of great significance to the implementer.

On the one hand, we first give the sensitivity analysis of the change of individual adjustment cost $c_{i}^{U}$ . When the $c_{i}^{U}$ is increased or reduced respectively, the consensus opinion of an expert is changing towards the direction of low initial opinion, and the three uncertain consensus cost models all have this characteristic. This consequence can be seen clearly in Table 9 –12. When the adjustment cost is increased, the total consensus cost of expert e₁, e₂, e₄ increases. The reason why the three two-stage stochastic iADCMs have this result is that the initial opinions given by these experts are lower than the consensus. However, from the perspective of expert e₂, the total consensus cost does not change with the change of $c_{2}^{U}$ . This is because the three different initial opinions provided by expert e₂ are larger than or equal to the consensus under three situations. As a result, the $c_{2}^{U}$ has no effect on its total cost. In addition to the above characteristics, we also find that the most obvious trend is TSS-MCCM-II, followed by TSS-MCCM-I. TSS-MCCM-II limits the scope of the expert’s opinion adjustment, which makes this model have stronger constraints, and experts stick to their initial opinions. Compared with TSS-MCCM-I, TSS-MCCM-III is more relaxed. In the latter model, the experts not only have no restrictions on opinion modification but also do not charge any fee for modifying opinions within a certain threshold, which leads to the above results. The above observations from Fig. 7.3.

Table 9
Effects of $c_{1}^{U}$ on consensus opinion

$▵ c_{1}^{U}$ -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 ⋯

o′(TSS-MCCM-I) 23 23 23 23 23 22 22 22 21 20.5 20.5 20.5 20.5 ⋯

o′(TSS-MCCM-II) 20 20 20 20 20 20 20 20 20 20 20 20 20 ⋯

o′(TSS-MCCM-III) 22 22 22 22 22 22 22 21.5 21.5 21.5 21.5 21.5 21.5 ⋯

$▵ c_{1}^{U}$	-0.1	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0	1.1	⋯
o′(TSS-MCCM-I)	23	23	23	23	23	22	22	22	21	20.5	20.5	20.5	20.5	⋯
o′(TSS-MCCM-II)	20	20	20	20	20	20	20	20	20	20	20	20	20	⋯
o′(TSS-MCCM-III)	22	22	22	22	22	22	22	21.5	21.5	21.5	21.5	21.5	21.5	⋯

Table 10

Effects of $c_{2}^{U}$ on consensus opinion

$▵ c_{2}^{U}$	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0	1.1	1.2	⋯
o′(TSS-MCCM-I)	23	23	23	23	22	22	22	21	21	21	21	21	21	⋯
o′(TSS-MCCM-II)	23	20	20	20	20	20	20	20	20	20	20	20	20	⋯
o′(TSS-MCCM-III)	22	22	22	22	22	22	21.5	21.5	21.5	21.5	21.5	21.5	21	⋯

Table 11

Effects of $c_{3}^{U}$ on consensus opinion

$▵ c_{3}^{U}$	-0.3	-0.2	-0.1	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	⋯
o′(TSS-MCCM-I)	23	23	23	23	23	23	23	23	23	23	23	23	⋯
o′(TSS-MCCM-II)	20	20	20	20	20	20	20	20	20	20	20	20	⋯
o′(TSS-MCCM-III)	22	22	22	22	22	22	22	22	22	22	22	22	⋯

Table 12

Effects of $c_{4}^{U}$ on consensus opinion

$▵ c_{4}^{U}$	-0.8	-0.6	-0.4	-0.2	0	0.2	0.4	0.6	0.8	1	1.2	1.4	⋯
o′(TSS-MCCM-I)	23	23	23	23	23	23	23	22	22	22	21	20.5	⋯
o′(TSS-MCCM-II)	20	20	20	20	20	20	20	20	20	20	20	20	⋯
o′(TSS-MCCM-III)	22	22	22	22	22	22	22	21.5	21.5	21.5	21.5	21.5	⋯

On the other hand, we focus on global sensitivity analysis of c^U to total consensus cost. According to Table 13, it can be found that when the c^U change, the consensus of TSS-MCCM-I and TSS-MCCM-III changes in the direction of decreasing. However, the consensus of TSS-MCCM-II will not change, which is mainly because the model is more stubborn to experts, more insistent on their initial opinions, and will not easily change their views. As a result, the consensus remains unchanged within the range of cost changes given. When the focus is on the total consensus cost, (Fig. 7.4) we find that the total cost of the three uncertain iADCMs increases with the change of c^U. However, the changing trend of the second model is more obvious. This is because the experts are more stubborn, so the moderator needs to spend more money persuading them to change their initial opinions until a mutual consensus is reached.

Table 13

Effects of c^U on consensus opinion

▵c^U	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0	⋯
o′(TSS-MCCM-I)	23	23	22	21	20.5	20.5	20.5	20.5	20.5	20.5	20.5	⋯
o′(TSS-MCCM-II)	20	20	20	20	20	20	20	20	20	20	20	⋯
o′(TSS-MCCM-III)	22	22	21.5	21.5	21.5	21	21	20.5	20.5	20.5	20.5	⋯

8 Conclusion

This research investigates the use of cost consensus problem with a directional costs context in two-stage stochastic iADCMs that combines uncertainty of adjustment costs and original opinions. By introducing TSP and asymmetric MCCMs, we construct two-stage stochastic iADCMs that divides the model into three classes bases on these uncertainty parameters (TSS-MCCM-I, TSS-MCCM-II, and TSS-MCCM-III). Three uncertain consensus optimization models are all based on 3 variables: increase and decrease the unit cost of opinion values, and the initial opinions. There are multiple possible scenarios for adjustment costs, compromise limits, thresholds and opinions change in changing decision environments, and arbitrary situation with a certain probability. Besides, the TSS-MCCM-II provides each expert a maximum acceptable adjustment distance. What’s more, TSS-MCCM-III assumes that there is an interval for no-cost modification of personal opinion, and there is no compensation charge for adjustment within the interval.

The distinctive feature of the proposed models is that consider the costs of direction and initial opinions with uncertain properties. The proposed two-stage stochastic iADCMs all divide the decision-making process into two stages. The first-stage decisions include the expected consensus, while the latter stage determines how to adjust their opinion. Besides, we solve the stochastic consensus models using a presented L-shaped algorithm. We take GTGP as the background application and apply it to the presented two-stage stochastic iADCMs. The results of numerical experiments show that our model can provide a better reference for DMs in an uncertain environment. By comparison with the deterministic consensus model, we find that the proposed uncertain consensus optimization models have more robust numerical results. Therefore, in the complex and changeable decision environment, the proposed models are more suitable for consensus reach in an uncertain environment. In practice, the above methodologies are applied in group decision work.

There are several directions for future research based on the current study.

1. When we deal with uncertain decision-making problems, it is very important to find the worst-case solution [37]. Therefore, it is a promising research direction to study the robust consensus optimization problem under the uncertain environment of directional adjustment and initial opinions.

2. In the uncertain environment, it is worth paying attention to the risk problem caused by the uncertain parameters. In order to deal with this difficulty, many scholars have studied the two-stage stochastic programming of risk aversion (see [38, 39]). Therefore, it is also very meaningful direction to explore the risk-neutral two-stage stochastic iADCMs problem.

3. This paper studies the group decision-making problem based on a small-scale. However, in the process of application, decision-makers are usually faced with large-scale decision-making problems. Therefore, it is more important to study the uncertain consensus optimization model based on large-scale groups.

4. The linguistic preference relation is an effective method to deal with subjective judgements, where DMs can express their opinions by using linguistic variables [40, 41]. Consensus modeling based on linguistic preference relation as an important approach to treat consensus-building has been used widely in recent years. Therefore, the binding of stochastic programming and consensus with linguistic preference relations will be a hot reach direction in the field of consensus reaching process when we face uncertain environments.

Footnotes

Acknowledgment

The work is supported by a research grant from the National Social Science Foundation of China (No.17BGL083).

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Notation	Description
Uncertain Parameter
$c_{i}^{D} (ω)$	uncertain unit downward adjustment cost of e_i
$c_{i}^{U} (ω)$	uncertain unit upward adjustment cost of e_i
o_i (ω)	uncertain initial opinion of e_i
Decision Variable
o′	the expected consensus opinion of the moderator
$δ_{i}^{+}$	the downward adjustment deviation
$δ_{i}^{-}$	the upward adjustment deviation
$u_{i}^{-}$	the upward adjustment deviation
$u_{i}^{+}$	no need to adjust
$v_{i}^{-}$	no need to adjust
$v_{i}^{+}$	the downward adjustment deviation

Two-stage stochastic integrated adjustment deviations and consensus models in an asymmetric costs context

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Two-stage stochastic programming

Table 3 Comparison results of different cost consensus models Model (2) TSS-MCCM-I Model (3) TSS-MCCM-II Model (5) TSS-MCCM-III o′ 20.5 23 20.5 20 21 22 φ 3.6000 3.1599 3.6000 5.3738 0.2500 0.9431

7.1 Sensitivity analysis on downward adjustment cost

Table 4 Effects of c 1 D on consensus opinion c 1 D -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 ⋯ o′(TSS-MCCM-I) 23 23 23 23 23 23 23 23 23 23 23 ⋯ o′(TSS-MCCM-II) 20 20 20 20 20 20 20 20 20 20 20 ⋯ o′(TSS-MCCM-III) 22 22 22 22 22 22 22 22 22 22 22 ⋯

Footnotes

Acknowledgment

References

Table 3
Comparison results of different cost consensus models

Model (2) TSS-MCCM-I Model (3) TSS-MCCM-II Model (5) TSS-MCCM-III

o′ 20.5 23 20.5 20 21 22

φ 3.6000 3.1599 3.6000 5.3738 0.2500 0.9431

Table 4
Effects of $c_{1}^{D}$ on consensus opinion

$c_{1}^{D}$ -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 ⋯

o′(TSS-MCCM-I) 23 23 23 23 23 23 23 23 23 23 23 ⋯

o′(TSS-MCCM-II) 20 20 20 20 20 20 20 20 20 20 20 ⋯

o′(TSS-MCCM-III) 22 22 22 22 22 22 22 22 22 22 22 ⋯