Zichen Gu, Dongmei Li^*, Sulin Dai
School of Economics and Management, Shanghai Zhongqiao Vocational and Technical University, Shanghai, China.
DOI: 10.4236/ajor.2026.164007   PDF    HTML   XML   21 Downloads   102 Views   Citations

Abstract

In digital marketing environments, consumer preferences exhibit significant dynamic evolution and group heterogeneity. Traditional fuzzy multi-criteria decision-making methods generally assume that preference structures remain static over time, making them inadequate for capturing preference drift and the evolution of group consensus in intertemporal decision-making processes. To address this issue, this paper proposes a hesitant fuzzy group decision-making method for dynamic marketing environments that integrates a time-discounted preference measure and an adaptive consensus mechanism. First, a time-discounting factor is introduced to construct a dynamically updated model of intertemporal hesitant-fuzzy preference relationships, capturing the nonlinear characteristics of consumer preference decay over time. Second, an adaptive group consensus iterative algorithm is designed to dynamically adjust feedback intensity and adjustment rules based on the state of consensus evolution, thereby overcoming the shortcomings of fixed-threshold methods, such as sensitivity to outlier preferences and poor convergence efficiency. Furthermore, a scheme ranking rule based on group consensus satisfaction is proposed to achieve dynamic optimization of marketing strategies. The effectiveness and convergence performance of this method are verified through numerical simulation examples and compared with existing methods. This study provides a new methodological tool for real-time marketing decision-making in uncertain market environments.

Keywords

Hesitant Fuzzy Sets, Time-Discounted Preferences, Dynamic Marketing Decision-Making, Group Decision-Making, Numerical Simulation

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1. Introduction

1.1. Research Background

The profound transformation brought about by mobile internet and social media has reshaped consumers’ decision-making pathways. Under the dual influence of information overload and fragmented attention, the formation and evolution of consumer preferences exhibit unprecedented dynamism and uncertainty.

From the perspective of group decision-making, strategic decisions such as marketing investment allocation and channel selection typically involve multiple rounds of market research and expert evaluation. Enterprises must integrate the fuzzy preferences of different stakeholders across multiple time slices and reach a consensus on the decision direction at the group level.

1.2. Related Work and Motivation for Innovation

Since Zadeh proposed Fuzzy Set Theory in 1965 [1], fuzzy decision-making methods have made significant progress in the field of uncertainty modeling. Hesitant Fuzzy Sets (HFS), introduced by Torra in 2010, further enhanced the ability to express hesitation and ambiguity in human decision-making [2]. Rodriguez et al. subsequently proposed Hesitant Fuzzy Linguistic Term Sets (HFLTS), providing a computational framework for handling hesitant preferences within linguistic evaluation systems [3].

In the field of group decision-making, Zhang and Chen systematically constructed a framework for hesitant fuzzy linguistic group decision-making based on consensus and unanimity, while Zhu et al. advanced the systematic measurement of consistency metrics for hesitant fuzzy preference relations within the evolution of consensus and unanimity models [4] [5].

In the realm of intertemporal decision theory, Palomares et al. proposed a soft consensus model in a fuzzy environment, and Dong et al. further introduced a consensus-reaching method with dynamic weight adjustment [6] [7]. Chen et al. introduced the calculation of cognitive mental states into the management of non-cooperative behavior in large-scale group decision-making [8]. Zhang et al. systematically summarized the theoretical progress in group decision-making supported by complex linguistic information in their monograph [9]. Wu and Chiclana proposed a trust-consensus group decision-making method based on social network analysis, while Chiclana et al. conducted an in-depth discussion on the mutual inversion aggregation problem of fuzzy preference relations [10] [11]. The generalized hyperbolic time discounting function proposed by Loewenstein and Prelec captures the phenomenon of dynamic inconsistent preferences that the standard exponential model cannot explain [12], however, this concept has not yet been systematically integrated into the Hesitant Fuzzy Decision-Making framework.

Furthermore, Multi-Criteria Decision-Making (MCDM) methods have been widely adopted in marketing strategy selection. Chen et al. pointed out that dynamic multi-attribute decision-making problems require simultaneous consideration of attribute weight information and temporal information, introducing a time-weighted criterion can better integrate subjective preferences with objective sample information [13]. Thai and Vinh proposed a two-stage model for tourist destination market segmentation and communication channel evaluation that combines FAHP and Fuzzy TOPSIS [14]. Güler and Mukul proposed a hesitant fuzzy MCDM-based framework for digital marketing channel evaluation, which integrates multiple criteria to support channel prioritization decisions [15]. Sharma et al.’s systematic review comprehensively examined the current status and challenges of fuzzy theory adoption in digital marketing [16]. The intuitive fuzzy binary language information processing method proposed by Liu and Chen and the interval-type fuzzy TOPSIS method by Chen and Hong also provide methodological references for this paper [17] [18].

In recent years, research on hesitant fuzzy consensus mechanisms has achieved new progress. Li et al. proposed a two-stage group decision-making model based on hesitant fuzzy preference relations in social networks, which achieves a balance between individual personalization and group consensus through multiplicative consistency evaluation and a personalized feedback mechanism [19]. For large-scale group decision-making scenarios, some studies have introduced normal hesitant fuzzy sets to aggregate subgroup opinions and developed a consensus-reaching process that considers experts’ willingness to modify their preferences. This approach preserves complete decision information without requiring normalization [20]. In the field of dynamic multi-attribute decision-making, scholars have incorporated Prospect Theory and Regret Theory into the hesitant fuzzy framework, achieving alternative ranking by constructing a prospect value matrix and calculating regret-rejoice values [21].

However, a review of the existing literature reveals that several issues still require further investigation. First, social network models and large-scale group consensus mechanisms mainly focus on the social relationship structures among decision-makers, while lacking consideration of the impact of the time dimension on preference evolution. Second, although decision-making methods based on Prospect Theory and Regret Theory take decision-makers’ psychological behaviors into account, they are generally designed for static, single-period decision-making and do not address the intertemporal aggregation of multi-period preference information. Third, the integration of time-discount-based preference measures and adaptive consensus mechanisms into a hesitant fuzzy group decision-making framework remains an area that warrants further research.

1.3. Contributions

To address the above issues, this paper proposes a hesitant-fuzzy group decision-making method that integrates time-discounted preference measures with an adaptive consensus mechanism (Time Discounted Preference with Adaptive Consensus based Hesitant Fuzzy Group Decision Making, TDP-AC-HF-GDM).

The main contributions include: 1) Introducing a time-indexed discount function into the intertemporal cumulative modeling of hesitant-fuzzy preference relations; 2) Designing an adaptive feedback adjustment strategy based on consensus evolution speed and individual deviation; 3) Proposing a combined scoring rule for ranking proposals that balances preference strength and consensus satisfaction; 4) Verifying the method’s effectiveness and convergence efficiency through numerical simulations

2. Prerequisite Knowledge Introduction

2.1. Hesitant-Fuzzy Sets and Hesitant-Fuzzy Preference Relations

Fuzzy sets are a core tool for handling uncertain and hesitant decision-making information, capable of effectively capturing the multi-valued preferences of decision-makers. Their basic definition is as follows:

Definition 2.1 [2]: Let $X$ be a non-empty set serving as the domain. Then, the hesitant-fuzzy set $E$ on $X$ can be defined as:

$E=\left\{\langle x_i,h_E\left(x_i\right)\rangle |x_i\in X\right\}$

where $h_E\left(x_i\right)\subseteq \left[0,1\right]$ i is a hesitant fuzzy element, consisting of several membership values belonging to the elements $x$ , used to characterize the decision-maker’s hesitant preferences toward the evaluation object.

Definition 2.2 [5]: Let the set of equipment selection alternatives be $A=\left\{a_1,a_2,\cdots ,a_n\right\}$ . The hesitant-fuzzy preference relationship of alternative $a_i$

relative to $a_j$ is denoted as $h_{ij}$ , satisfying the range $h_{ij}\subseteq \left[0,1\right]$ and the complementarity constrain $h_{ji}=1-h_{ij}$ . The matrix $H={\left(h_{ij}\right)}_{n\times n}$ , composed of the

hesitant-fuzzy preference relationships of all alternative pairs, is the hesitant-fuzzy preference matrix.

To ensure the validity and consistency of subsequent operations, this paper uniformly assumes that the number of hesitant-fuzzy elements in the outputs of all decision-makers remains consistent, thereby avoiding computational biases caused by inconsistent dimensions.

2.2. Hesitant Fuzzy Linguistic Term Set

Table 1. Linguistic term set and its corresponding hesitant fuzzy elements.

Linguistic term	Label	Hesitant fuzzy element
Absolutely inferior	AD	{0.0, 0.1}
Very poor	VP	{0.1, 0.2}
Poor	P	{0.2, 0.3}
Slightly poor	SP	{0.3, 0.4}
Equal	E	{0.5}
Slightly better	SB	{0.6, 0.7}
Good	G	{0.7, 0.8}
Very good	VG	{0.8, 0.9}
Absolutely better	AB	{0.9, 1.0}

The Hesitant Fuzzy Linguistic Term Sets (HFLTS) proposed by Rodriguez et al. allows decision-makers to use multiple adjacent linguistic terms to express their preferences between two alternatives [3]. Table 1 presents the correspondence between the linguistic terms and hesitant-fuzzy elements adopted in this paper.

2.3. Time-Discounted Preference Measurement Model

The literature on intertemporal decision-making indicates that people’s psychological discounting of future gains and losses follows a nonlinear decay pattern [12]. Drawing on this idea, this paper applies it to the temporal aggregation of hesitant-fuzzy preference relations.

Definition 2.3: Let $H^{\left(t\right)}={\left(h_{ij}^{\left(t\right)}\right)}_{n\times n}$ denote the matrix of hesitant-fuzzy preference relations obtained from the t-th round of surveys. We construct the formula for the time discount factor as follows:

$\delta \left({\lambda }_t,{\psi }_{ij}\right)={\text{e}}^{-{\lambda }_t\cdot {\psi }_{ij}}$

where: ${\lambda }_t>0$ is the time decay coefficient for the evaluation information from the t-th round, used to characterize the temporal decay rate of the evaluation information; ${\psi }_{ij}\in \left[0,1\right]$ is the perceived importance coefficient of the alternative for $\left(a_i,a_j\right)$ ; a larger value indicates that the comparison results of this group of alternatives carry a higher weight in influencing the decision.

Here, $\otimes$ is defined as the scalar product of hesitant fuzzy elements: $\alpha \otimes h=\left\{\alpha \cdot \gamma |\gamma \in h\right\}$ [14]. This can be used to derive the discounted preference$T$ at the decision time, updated from the preference information of the t-th round:

${\overline{h}}_{ij}^{\left(t\to T\right)}=h_{ij}^{\left(t\right)}\otimes \delta \left({\lambda }_t,{\psi }_{ij}\right)$

Definition 2.4: After $T$ rounds of multi-stage evaluations, all temporal preference information is aggregated through a discounted weighted averaging approach to derive the cumulative discounted preference centroid for each alternative pair $\left(a_i,a_j\right)$ . Specifically, the discounted hesitant fuzzy elements corresponding to all time periods are first obtained, and their union is constructed while preserving all possible membership degrees.

$U_{ij}={\displaystyle \underset{t=1}{\overset{T}{\cup }}{\overline{h}}_{ij}^{\left(t\to T\right)}}$

Given that each hesitant fuzzy element consists of two membership degrees, the discounted weighted average is calculated according to the discount factors, yielding:

${\overline{h}}_{ij}^{\left(t\to T\right)}=\frac{1}{2{\displaystyle {\sum }_{t=1}^T{\delta }_t}}{\displaystyle \sum _{\gamma \in U_{ij}}\gamma }$

where ${\overline{h}}_{ij}$ denotes the centroid value of the preference pair, which serves as the basis for all subsequent consistency analyses. To maintain a uniform decision criterion, the importance of all alternative pairs is assumed to be equal. Accordingly, the global perceived importance coefficient is set to ${\psi }_{ij}=0.5$ for all $\left(a_i,a_j\right)$ .

3. Adaptive Consensus Decision-Making Mechanism

3.1. Measurement of Individual and Group Consensus

Let $m$ denote the number of decision-makers, represented as $D_1,D_2,\cdots ,D_m$ . The preference matrix of the $k$ -th decision-maker at iteration $t$ is denoted by $H_k^{\left(t\right)}$ . To measure the level of consensus within the group, a consistency index between any pair of decision-makers $\left(k,l\right)$ is first defined:

$\text{CI}\left(H_k,H_l\right)=1-\frac{2}{n\left(n-1\right)}{\displaystyle \sum _{i<j}dist\left(h_{ij}^{\left(k\right)},h_{ij}^{\left(l\right)}\right)}=1-\frac{2}{n\left(n-1\right)}{\displaystyle \sum _{i<j}\left|h_{ij}^{\left(k\right)}-h_{ij}^{\left(l\right)}\right|}$

Definition 3.1 (Group Consensus): In the t-th iteration, the group consensus is defined as the average of the pairwise consistency indices among all decision-makers, serving as an overall measure of the degree of unity in the group’s views:

${\text{GCI}}^{\left(t\right)}=\frac{2}{m\left(m-1\right)}{\displaystyle \sum _{k<l}\text{CI}\left(H_k^{\left(t\right)},H_l^{\left(t\right)}\right)}$

Definition 3.2 (Local Consistency Measures): For a single set of proposals, $\left(i,j\right)$ , construct local consistency measures between the decision-maker’s $k$ and the group’s central preference $H^{\text{*}}$ to precisely characterize the degree of divergence in the evaluation of a single set of proposals:

${\text{CI}}_{ij}\left(H_k,H^{*}\right)=1-dist\left(h_{ij}^{\left(k\right)},h_{ij}^{*}\right)$

3.2. Adaptive Stopping Criterion and Feedback Adjustment Mechanism

Classical consensus methods employ a fixed consensus threshold GCI, halting iterations when GCI^(t) ≥ GCI [6]. However, in practical decision-making, the optimal threshold often varies with group size and problem complexity. To address this, this paper designs an adaptive stopping criterion to achieve intelligent optimization of the iterative process [8]

Criterion 3.1 (Adaptive Stopping Criterion) When the iteration count is $t\ge 2$ , and both of the following conditions are satisfied, the group is deemed to have reached a valid consensus, and the iterative process is terminated:

${\text{GCI}}^{\left(t\right)}\ge {\theta }^{\left(t\right)}\text{and}\left|{\text{GCI}}^{\left(t\right)}-{\text{GCI}}^{\left(t-1\right)}\right|\le ϵ$

where $ϵ$ is the convergence criterion constant, set to $ϵ=0.005$ in this study; and ${\theta }^{\left(t\right)}$ is the dynamic consensus threshold for the t-th iteration, which is adaptively updated throughout the iterative process according to the formula:

${\theta }^{\left(t\right)}=\max \left\{0.7,{\text{GCI}}^{\left(1\right)}+\frac{t-1}{T_{\max }}\cdot \left(0.95-{\text{GCI}}^{\left(1\right)}\right)\right\}$

where $T_{\max }=10$ is the preset maximum number of iteration rounds, balancing iteration efficiency and decision accuracy. The dynamic threshold enables “loose in the early stages, strict in the later stages” iteration control, aligning with the evolutionary pattern of gradual convergence toward consensus.

When the termination condition is not met, it is necessary to identify the decision-maker with the highest deviation and guide them to adjust their preferences. The individual deviation is defined as:

${\text{Dev}}_k^{\left(t\right)}=\frac{1}{m-1}{\displaystyle \sum }_{l\ne k}\left(1-\text{CI}\left(H_k^{\left(t\right)},H_l^{\left(t\right)}\right)\right)$

Adaptive Feedback Adjustment Mechanism [9]: For the decision-maker with the highest deviation, $k^{\text{*}}$ , construct a group-centered preference matrix, ${\tilde{H}}^{\left(t\right)}$ , to characterize the mainstream view of the group:

${\tilde{h}}_{ij}^{\left(t\right)}=\left\{\frac{1}{m}{\displaystyle \sum _{k=1}^m{\gamma }_{ij}^{\left(k\right)}}|{\gamma }_{ij}^{\left(k\right)}\in h_{ij}^{\left(k\right)}\right\}$

To achieve precise corrections and avoid over-adjustment, an adaptive feedback intensity coefficient is designed to dynamically adjust the correction magnitude based on individual deviation levels:

${\rho }^{\left(t\right)}=\min \left\{0.5,\frac{{\text{Dev}}_{k^{*}}^{\left(t\right)}}{\underset{k}{\max }{\text{Dev}}_k^{\left(1\right)}}\right\}$

The preference matrix of deviating decision-makers is updated via linear weighting to achieve iterative optimization of group consensus:

$H_{k^{*}}^{\left(t+1\right)}=\left(1-{\rho }^{\left(t\right)}\right)\cdot H_{k^{*}}^{\left(t\right)}\oplus {\rho }^{\left(t\right)}\cdot {\tilde{H}}^{\left(t\right)}$

For each preference pair $\left(i_m,j\right)$ , it follows that:

${\overline{h}}_{ij}^{\left(k^{\text{*}},t+1\right)}=\left(1-{\rho }^{\left(t\right)}\right)\cdot {\overline{h}}_{ij}^{\left(k^{\text{*}},t\right)}+{\rho }^{\left(t\right)}\cdot {\tilde{h}}_{ij}^{\left(t\right)}$

4. Alternative Ranking Model Based on Consensus Satisfaction

After reaching an acceptable consensus through multiple rounds of iteration, the final group’s hesitant-fuzzy preference relationship is obtained $H^{*}$ . To achieve a scientific selection of marketing proposals, this study integrates dual-dimensional indicators of preference strength and consensus satisfaction to construct a comprehensive ranking score model that balances the market suitability of proposals with the robustness of group decision-making.

Definition 4.1 (Solution Preference Outflow): The preference outflow is used to quantify the group’s overall preference strength for a single solution; a higher value indicates greater overall acceptance of the solution. The outflow $a_i$ of a solution is defined as:

${\varphi }^{+}\left(a_i\right)=\frac{1}{n-1}{\displaystyle \sum _{j\ne i}{h}_{ij}^{*}}$

Definition 4.2 (Solution Consensus Satisfaction): The consensus satisfaction of a solution $a_i$ is defined as the weighted average of the consistency of all preference judgments made by all decision-makers regarding that solution. It is used to measure the consistency of the decision-making group’s evaluation of a single solution; a higher value indicates that the evaluation results are more robust and less controversial. The calculation formula is:

$\chi \left(a_i\right)=\frac{1}{n-1}{\displaystyle \sum _{j\ne i}\frac{1}{m}{\displaystyle \sum _{k=1}^m{\text{CI}}_{ij}\left(H_k,H^{*}\right)}}$

Definition 4.3 (Final Comprehensive Ranking Score): By integrating preference strength and consensus satisfaction, a weighted comprehensive scoring model is constructed to achieve a quantitative ranking of proposals:

$\Psi \left(a_i\right)=\beta \cdot {\varphi }^{+}\left(a_i\right)+\left(1-\beta \right)\cdot \chi \left(a_i\right)$

In the formula, $\beta \in \left[0,1\right]$ represents the decision-making preference coefficient. In this study, we set its value to $\beta =0.6$ , which moderately emphasizes the mainstream preferences of the group while also ensuring the robustness of the consensus, thereby aligning with the practical needs of corporate marketing decisions. A higher composite score indicates superior overall performance of the solution.

5. Numerical Simulation and Concept Validation

5.1. Case Scenarios and Parameter Settings

To validate the effectiveness and practicality of the model proposed in this paper, numerical simulation experiments were conducted using mobile marketing channel selection as the application scenario. A consumer brand intends to optimize its online marketing promotion system and select the best option from four mainstream marketing channels. The candidate alternatives are: short-video in-feed advertising (A1), social e-commerce livestreaming (A2), private-domain community marketing (A3), and search keyword bidding (A4). Five marketing experts (D1-D5) participate in the multi-stage evaluation process, and three temporal evaluation points (T1, T2, and T3) are considered to simulate the dynamic evolution of market preferences.

In this numerical example, the experts’ preference matrices were generated using a parameterized construction method. Specifically, each expert was assigned a distinct initial preference tendency toward the four marketing alternatives in the initial period (T1) (e.g., D3 favored traditional channels, whereas D2 preferred social media trends). Subsequently, preference values for the middle period (T2) and the final period (T3) were generated according to a linear shift rule. The preferences in T2 were gradually adjusted in a specific direction based on those in T1, while those in T3 further reinforced this trend and incorporated small random perturbations. This evolutionary pattern, characterized by a transition from divergence to convergence and the strengthening of recency effects, effectively simulates intertemporal shifts in consumer preferences in dynamic marketing environments, consensus formation driven by information diffusion, and the influence of timeliness-related factors on decision-making.

The core model parameters are set as follows: the time-decay coefficients are ${\lambda }_1=0.2$ , ${\lambda }_2=0.5$ , ${\lambda }_3=1.0$ , respectively, to capture the temporal characteristics of “slow decay for distant periods and rapid decay for recent periods”; the perceived importance coefficients are uniformly assigned as ${\psi }_{ij}=0.5$ ; the convergence threshold is set to $ϵ=0.005$ ; and the maximum number of iterations is$T_{\max }=10$ . The multi-period hesitant fuzzy preference evaluation data provided by the experts are presented in Tables 2-6.

Table 2. Hesitant fuzzy preference relation of expert D1 (T1/T2/T3).

Alternative Pair	Preference in T1	Preference in T2	Preference in T3
(A1, A2)	{0.3, 0.4}	{0.4, 0.5}	{0.4, 0.5}
(A1, A3)	{0.5, 0.6}	{0.6, 0.7}	{0.7, 0.8}
(A1, A4)	{0.2, 0.3}	{0.3, 0.4}	{0.4, 0.5}
(A2, A3)	{0.7, 0.8}	{0.8, 0.9}	{0.8, 0.9}
(A2, A4)	{0.6, 0.7}	{0.7, 0.8}	{0.8, 0.9}
(A3, A4)	{0.4, 0.5}	{0.5, 0.6}	{0.5, 0.6}

Table 3. Hesitant fuzzy preference relation of expert D2 (T1/T2/T3).

Alternative Pair	Preference in T1	Preference in T2	Preference in T3
(A1, A2)	{0.5, 0.6}	{0.6, 0.7}	{0.6, 0.7}
(A1, A3)	{0.4, 0.5}	{0.4, 0.5}	{0.5, 0.6}
(A1, A4)	{0.5, 0.6}	{0.5, 0.6}	{0.6, 0.7}
(A2, A3)	{0.6, 0.7}	{0.7, 0.8}	{0.7, 0.8}
(A2, A4)	{0.7, 0.8}	{0.7, 0.8}	{0.8, 0.9}
(A3, A4)	{0.5, 0.6}	{0.5, 0.6}	{0.6, 0.7}

Table 4. Hesitant fuzzy preference relation of expert D3 (T1/T2/T3).

Alternative Pair	Preference in T1	Preference in T2	Preference in T3
(A1, A2)	{0.2, 0.3}	{0.3, 0.4}	{0.4, 0.5}
(A1, A3)	{0.3, 0.4}	{0.4, 0.5}	{0.5, 0.6}
(A1, A4)	{0.2, 0.3}	{0.3, 0.4}	{0.4, 0.5}
(A2, A3)	{0.6, 0.7}	{0.6, 0.7}	{0.7, 0.8}
(A2, A4)	{0.5, 0.6}	{0.6, 0.7}	{0.7, 0.8}
(A3, A4)	{0.3, 0.4}	{0.4, 0.5}	{0.4, 0.5}

Table 5. Hesitant fuzzy preference relation of expert D4 (T1/T2/T3).

Alternative Pair	Preference in T1	Preference in T2	Preference in T3
(A1, A2)	{0.4, 0.5}	{0.4, 0.5}	{0.5, 0.6}
(A1, A3)	{0.5, 0.6}	{0.6, 0.7}	{0.6, 0.7}
(A1, A4)	{0.3, 0.4}	{0.4, 0.5}	{0.5, 0.6}
(A2, A3)	{0.7, 0.8}	{0.7, 0.8}	{0.8, 0.9}
(A2, A4)	{0.6, 0.7}	{0.7, 0.8}	{0.7, 0.8}
(A3, A4)	{0.4, 0.5}	{0.5, 0.6}	{0.5, 0.6}

Table 6. Hesitant fuzzy preference relation of expert D5 (T1/T2/T3).

Alternative Pair	Preference in T1	Preference in T2	Preference in T3
(A1, A2)	{0.3, 0.4}	{0.4, 0.5}	{0.5, 0.6}
(A1, A3)	{0.4, 0.5}	{0.5, 0.6}	{0.6, 0.7}
(A1, A4)	{0.3, 0.4}	{0.3, 0.4}	{0.4, 0.5}
(A2, A3)	{0.6, 0.7}	{0.7, 0.8}	{0.7, 0.8}
(A2, A4)	{0.6, 0.7}	{0.7, 0.8}	{0.8, 0.9}
(A3, A4)	{0.4, 0.5}	{0.4, 0.5}	{0.5, 0.6}

5.2. Calculation of Time-Discounted Intertemporal Preference Aggregation

Using the evaluation data from Expert D1 for the option pair (A1, A2) as an example, this section fully demonstrates the time-discounted weighted aggregation process; the evaluation data from all other experts are processed uniformly using the same algorithm.

Step 1: Calculate the time discount factors for each time node

${\delta }_1={\text{e}}^{-0.2\times 0.5}={\text{e}}^{-0.1}=0.9048,$

${\delta }_2={\text{e}}^{-0.5\times 0.5}={\text{e}}^{-0.25}=0.7788,$

${\delta }_3={\text{e}}^{-1.0\times 0.5}={\text{e}}^{-0.5}=0.6065$

Step 2: Using the multiplication operation of hesitant fuzzy numbers, the discounted preferences for each time period are computed. The preference at period T1 is {0.2714, 0.3619}, the preference at period T2 is {0.3115, 0.3894}, and the preference at period T3 is {0.2426, 0.3033}.

Step 3: Based on the discounted weighted averaging method, the multi-period preferences are aggregated. The total sum of discount factors is 4.5802. Finally, the comprehensive preference of Expert D1 for (A1, A2) is obtained as {0.41}. Following this standard procedure, cross-period preference aggregation is performed for all alternative pairs and all experts, yielding the cumulative discounted aggregated preference matrix of all experts. Partial results are presented in Table 7.

Table 7. Cross-period cumulative discounted comprehensive preference of expert D1.

Alternative Pair	Comprehensive Preference
(A1, A2)	{0.41}
(A1, A3)	{0.67}
(A1, A4)	{0.36}
(A2, A3)	{0.82}
(A2, A4)	{0.76}
(A3, A4)	{0.52}

The aggregated cross-period preference results of all experts are consistent with the time-decay principle, where more recent evaluation information receives higher weights in the weighted averaging process. This aligns with the practical logic in marketing decision-making that “recent market performance is more informative and valuable,” thereby verifying the rationality of the time-discounting model.

5.3. Adaptive Consensus Iteration Process

The intertemporal cumulative preference matrices of the five experts were input into the adaptive consensus model to perform iterative optimization. The changes in key indicators for each iteration are shown in Table 8. The model dynamically updates the consensus threshold and feedback correction intensity, automatically adapting to the evolution of group consensus throughout the process.

Table 8. Adaptive consensus iteration index evolution.

Iteration Round (t)	Group Consensus Index (GCI)	ΔGCI	Adjustment Object	Maximum Deviation (Dev)	Feedback Intensity (ρ)
1	0.9305	—	D2	0.092	0.50
2	0.9429	0.0124	D3	0.078	0.50
3	0.9546	0.0117	D2	0.053	0.50
4	0.9613	0.0067	D3	0.046	0.50
5	0.9673	0.0061	D1	0.040	0.435
6	0.9719	0.0046	—	0.030	—

The iteration results show that the model satisfies the convergence criterion at the 6th iteration. The group consensus index reaches 0.9719, and the difference between two consecutive iterations is ΔGCI = 0.0046 < 0.005, which is equal to (or can be written as ≤0.005, i.e., the convergence threshold of 0.005), indicating that the iteration process terminates. Overall, the iterative process exhibits a stable pattern characterized by a steady increase in consensus level, a continuous decrease in deviation, and a gradual attenuation of feedback intensity (with ρ decreasing to 0.435 from the 5th iteration onward). No abnormal phenomena such as oscillation, reversal, or divergence are observed, which fully verifies the stability and efficiency of the adaptive consensus mechanism. Compared with traditional fixed-threshold consensus models, the proposed model does not require manually preset optimal convergence thresholds. Instead, it can automatically adapt the iteration precision and adjustment magnitude according to the degree of group disagreement. This enables stronger adaptability in multi-agent and high-conflict marketing decision-making environments.

5.4. Comprehensive Ranking Results of Proposals

Based on the converged group consensus preference matrix and the two-dimensional comprehensive ranking model developed in this paper, the preference outflow, consensus satisfaction, and final comprehensive scores for the four categories of marketing schemes were calculated. The specific results are shown in Table 9.

Table 9. Comprehensive evaluation scores of marketing alternatives.

Alternative	Preference Outflow ${\varphi }^{+}\left(a_i\right)$	Consensus Satisfaction $\chi \left(a_i\right)$	Comprehensive Score $\Psi \left(a_i\right)$
A1	0.359	0.975	0.606
A2	0.593	0.982	0.749
A3	0.460	0.979	0.667
A4	0.588	0.983	0.746

Based on the magnitude of the comprehensive scores, the final ranking of the four marketing alternatives is obtained as: A2 > A4 > A3 > A1. This result indicates that social e-commerce live streaming (A2) is the optimal marketing channel under the current dynamic market environment. From the perspective of evaluation dimensions, social e-commerce live streaming exhibits the highest group preference intensity, and the expert group shows a relatively high level of consensus regarding this alternative, indicating strong feasibility and stability.

Keyword bidding (A4) ranks second and achieves the highest consensus satisfaction among the four alternatives, suggesting that experts are most consistent in their evaluations of this option; however, its preference intensity is slightly lower than that of social e-commerce live streaming. Private-domain community marketing (A3) ranks third. It demonstrates stable user retention advantages, but its short-term traffic acquisition capability is weaker than that of live streaming-based models, resulting in relatively lower preference intensity. Short-form video feed advertising (A1) shows comparatively weaker overall performance. This is primarily due to the high homogenization of traditional traffic-based advertising models, lower user conversion efficiency, and greater sensitivity to market fluctuations, which together limit its adaptability in dynamic marketing environments.

5.5. Comparative Experimental Analysis

To highlight the superiority of the model proposed in this paper (TDP-AC-HF-GDM), two types of classical hesitant-fuzzy group decision-making models were selected for a comparative analysis: Model 1 is a traditional static hesitant-fuzzy group decision-making model that lacks a time-discounting mechanism and employs a fixed consensus threshold; Model 2 is a conventional dynamic hesitant-fuzzy decision-making model that introduces only temporal weights and does not incorporate an adaptive consensus iteration mechanism. Using the same test data and basic parameters, we calculated the number of convergence iterations, final consensus degree, and solution ranking results for the three models. The comparison data are shown in Table 10.

Table 10. Multi-model comparison results.

Decision Model	Convergence Iterations	Final Group Consensus	Alternative Ranking
Traditional Static Model	9	0.886	A2 > A4 > A3 > A1
Conventional Dynamic Model	7	0.924	A2 > A4 > A3 > A1
Proposed Model	6	0.972	A2 > A4 > A3 > A1

Three key conclusions can be drawn from the comparative results. First, in terms of convergence efficiency, the proposed model requires the fewest iterations. Its iterative efficiency is improved by 33.3% compared with the traditional static model and by 14.3% compared with the conventional dynamic model. This indicates that the adaptive threshold and dynamic feedback mechanisms can effectively accelerate the convergence of group opinions. Second, in terms of decision accuracy, the proposed model achieves a significantly higher final consensus level than both benchmark models. It is therefore more capable of reducing disagreements among decision-makers and enhancing the robustness of the final decision outcome. Third, all three models identify A2 as the optimal alternative, which further confirms the robustness and reliability of this conclusion.

5.6. Sensitivity Analysis

In the comprehensive ranking model presented in this paper, the preference weight coefficient β directly influences the weighting of preference intensity and consensus satisfaction. To verify the robustness of the model results, the value of β was varied within the range of 0 to 1. Changes in the comprehensive scores and ranking results of each scheme under different weightings were observed. The results of the sensitivity analysis are shown in Table 11.

Table 11. Sensitivity analysis results of the weight coefficient β.

Weight Coefficient β	Alternative Ranking
0	A4 > A2 > A3 > A1
0.2	A2 ≈ A4 > A3 > A1
0.4	A2 > A4 > A3 > A1
0.5	A2 > A4 > A3 > A1
0.6	A2 > A4 > A3 > A1
0.7	A2 > A4 > A3 > A1
0.8	A2 > A4 > A3 > A1
1	A2 > A4 > A3 > A1

As shown in Table 11, when β ≥ 0.4, the ranking results remain stable and are consistently obtained as: A2 > A4 > A3 > A1. In this study, β = 0.6 is adopted, which lies within the stability interval, indicating that the decision results are robust. When β is relatively small (β ≤ 0.2), consensus satisfaction becomes the dominant factor. Consequently, A4 rises in the ranking due to its highest level of consensus satisfaction among the alternatives. In contrast, when β is relatively large, preference outflow becomes the dominant factor, and A2 maintains its leading position owing to its strongest preference intensity. Overall, A2 ranks among the top two alternatives under all values of β, demonstrating the strong robustness of its optimal position.

6. Conclusions

Addressing the limitations of traditional hesitant-fuzzy group decision-making methods which struggle to adapt to dynamic and uncertain marketing scenarios and suffer from issues such as static preference fixation, low consensus convergence efficiency, and insufficient decision robustness, this study integrates time discounting theory with an adaptive consensus mechanism to construct a hesitant-fuzzy group decision-making model tailored for dynamic marketing environments. Through theoretical derivations and numerical case studies, the following research conclusions were drawn.

The intertemporal preference aggregation model, constructed by introducing an exponential time discount factor, accurately captures the evolutionary pattern of marketing decision preferences as they dynamically decay over time. It enables differentiated weighted processing of long-term and short-term evaluation information effectively addressing the shortcoming of traditional static models that ignore differences in temporal preferences, thereby making decision information aggregation more aligned with the dynamic characteristics of the market.

The adaptive consensus iteration mechanism designed in this study dynamically adjusts the consensus threshold and preference correction intensity based on the evolution of group consensus and the degree of deviation from individual decision-makers. This overcomes the bottlenecks of slow convergence and susceptibility to oscillations in traditional fixed-parameter consensus models, significantly improving the convergence efficiency and result stability of multi-agent fuzzy decision-making.

The study integrates a two-dimensional ranking system combining preference strength and consensus satisfaction, balancing the market preference alignment of proposals with the consistency of group decisions. This approach avoids the one-sidedness of a single evaluation dimension and effectively enhances the scientific rigor and rationality of the selected marketing proposals.

Numerical examples, comparative experiments, and sensitivity analysis results demonstrate that, compared to traditional decision-making methods, the model proposed in this study offers advantages such as high convergence efficiency, superior consensus accuracy, and strong result robustness. It exhibits excellent applicability and practical value in dynamic marketing channel selection scenarios, ultimately identifying social e-commerce livestreaming as the optimal online marketing channel solution at present.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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