Dynamic Hybrid Multi-Attribute Group Decision-Making with Two Reference Points for Electricity Sales Package Recommendation

Abstract

Given that existing electricity sales package recommendation methods mainly consider scenarios where customers are familiar with all the attributes of the packages, they tend to overlook the mixed nature of quantitative and qualitative attributes, psychological factors, and the determination of attribute weights in the decision-making process. To address these limitations, this paper proposes a dynamic hybrid multi-attribute group decision-making with two reference points for electricity sales package recommendation, combining prospect theory. The specific approach is as follows: First, the AAP clustering technique is used to divide the sample customer set and establish a customer load information database, from which customer characteristic profiles are extracted to identify sample customers similar to the target customer. Second, considering the mixed nature of quantitative and qualitative attributes of electricity sales packages and the different characteristics of these attributes, the three-parameter interval gray number and triangular fuzzy number are used to characterize them. A mixed target model is proposed as an external reference point. Next, time reference points are set by using the mean and average development speed to examine the dynamic development of the electricity sales packages. Then, the vector angle method is used to assess the similarity between individual decision-making and group decision-making, establishing an appropriate customer weight adjustment model. The comprehensive prospect value of each package is calculated under the two reference points by comparing the values, which are then used to rank all electricity sales packages. Finally, a case study of customers from a region in China is conducted to verify the accuracy and effectiveness of the proposed recommendation method.

1. Introduction

With the deepening of electricity market reforms and the implementation of high-quality development principles, the power industry is undergoing a gradual transition toward green and low-carbon development. Against this backdrop, the liberalization of the electricity retail market is becoming a core aspect of the industry’s market-oriented reform. Electricity, as a clean energy source, possesses high customer stickiness and significant potential for developing diverse value-added services. These services, which are continuously being explored and innovated, often include electricity packages, integrated energy management, and energy-saving services, all of which hold substantial market value [1]. Electricity sales packages reflect innovative service and operational models driven by internet thinking in a highly competitive market and are a necessary means for power retail companies to enhance economic efficiency and gain market competitiveness. As a vehicle for delivering retail services, the competitiveness of electricity sales packages is a key issue that retail companies must address to establish a foothold in the market. Studies indicate that, since 2007, the German electricity market has offered customers more than 9000 electricity sales packages [2,3,4]. In Texas, USA, over 1820 fixed-rate electricity sales packages and corresponding pricing strategies have been provided through specific platforms since the development of electricity market reforms [5,6,7]. Similarly, following market reforms, the UK introduced various time-of-use electricity pricing packages for customer selection [8]. In China, electricity retail companies have also designed diverse electricity pricing packages targeting both industrial and commercial users [9] as well as residential customers [10]. Reference [11] introduces China’s first personalized customization service platform, which offers tailor-made packages to customers. However, in market-based transactions, the wide variety and complexity of retail electricity packages pose challenges for customers in identifying suitable options, leading to increased information costs associated with their decision-making process. Under these circumstances, to enhance market share, profitability, and competitiveness, electricity retail companies must adopt accurate and scientific methods to recommend customized electricity sales packages that precisely meet customers’ needs.

Currently, existing electricity sales package recommendation methods can be divided into two types: direct recommendation methods and indirect recommendation methods. The direct recommendation rule for electricity sales packages is based on online analysis platforms that recommend packages by matching customer information with existing electricity sales packages according to the similarity of electricity usage, selecting the most economical solution to recommend to customers. Examples of such platforms include iSelect [12], Power to Choose [13], Check24 [14], Energy Made Easy [15], etc. Although the implementation of online recommendation platforms is simple and easy, this method only considers customers’ electricity costs and ignores the diversity of customer evaluation information. For customers with complex electricity usage characteristics, there is still no suitable recommendation solution. Apart from the direct cost-based recommendation method, indirect recommendation methods for electricity sales packages mainly involve algorithm-based recommendations. Key research directions include collaborative filtering-based models [16,17], matrix factorization-based models [18], deep learning-based models [19], etc. For example, reference [20] uses a fuzzy C-means clustering algorithm to classify customers, and based on the similarity between the target customer and historical customers in the same category, as well as historical customers’ ratings of electricity sales packages, it predicts the ratings for the target customer to recommend the appropriate package. Reference [21] represents the electricity usage characteristics of residential customers using the energy consumption characteristics of household appliances and designs a recommendation system for residential electricity sales packages based on a Bayesian mixed collaborative filtering algorithm. In addition, reference [22] analyzes the impact of factors such as pricing methods, time-of-use pricing, average price, and the proportion of green electricity on package selection, and introduces an implicit package scoring and customer profiling recommendation method. The above indirect recommendation algorithms for electricity sales packages provide ideas for recommending these packages. Among them, accurate and efficient customer clustering is a prerequisite for improving the accuracy of electricity sales package recommendations. However, the aforementioned clustering methods all require the number of clusters to be pre-set, which results in lower clustering efficiency and accuracy. Reference [23] proposes a multi-attribute group decision-making method for electricity sales packages based on two-stage density clustering (TSDC) and minimum adjustment distance consensus. This method does not require pre-setting the number of clusters, making it a promising approach. Although these indirect recommendation methods consider customers’ electricity costs, behavior characteristics, and preferences, they only consider the scenario where customers are familiar with all attributes of the packages, which is often unrealistic. Additionally, when customers face decision-making risks in multi-attribute group decision-making under uncertainty, they often exhibit some risk preferences. Therefore, integrating their risk preferences into multi-attribute decision-making is a meaningful task. Furthermore, in the problem of recommending electricity sales packages, the phenomenon of mixed multiple attribute values is common and cannot be ignored. For example, when customers evaluate electricity retail packages, they are often influenced by multiple factors such as electricity price, service, and electricity usage characteristics. Among these, electricity price can be represented by numerical values or estimations, while service is often described using fuzzy language. In such multi-attribute decision-making situations with uncertainty, reference [24] proposed a dynamic multi-attribute decision-making method under the three-parameter gray number information based on prospect theory, and reference [25] proposed a hybrid gray target decision-making method based on the “convergence-divergence” concept. In the problem of recommending electricity sales packages, there is not only uncertainty but also different sources of uncertainty in quantitative and qualitative attributes. The former has clear boundary values, but the specific value within the range is unclear, represented as “uncertain connotation, clear extension”, while the latter can only be expressed using fuzzy language such as “good” or “better”, and the boundaries between “good” and “better” are unclear, represented as “clear connotation, unclear extension.” Therefore, it is important to separate the descriptions of quantitative and qualitative attributes.

Therefore, considering the limitations of both direct and indirect methods, it is highly suitable and effective to use the multi-attribute group decision-making (MADM) method for electricity sales package recommendations. Multi-attribute decision-making has always been a focus in decision science.

In 1979, Kahneman et al. introduced prospect theory and the concept of “reference points” [26]. Prospect theory highlights the psychological bias in human perception when facing risks, where individuals tend to “overestimate low-probability events and underestimate high-probability events.” This theory can explain many phenomena that traditional expected utility theory cannot, and when applied to multi-attribute decision-making, it effectively addresses the preference issues of decision-makers. For example, in reference [27], the cumulative prospect theory and the VIKOR method are incorporated into the Pythagorean hesitant fuzzy risk multi-attribute decision-making model to address bounded rational behavior among customers. In reference [28], the hesitant fuzzy set prospect theory evaluation method is used to compare evaluation index values under various scenarios, mitigating the impact of individual decisions on corporate emissions reduction, and identifying the recommended energy reduction plan. Reference [29] proposes an electricity sales package recommendation method based on the prospect advantage-disadvantage degree and Choquet integral. The reference point, as the basis for decision-makers’ judgment and choices, directly influences the decision outcome. Currently, research on reference point setting in multi-attribute decision-making can be divided into two main categories. One category assumes that reference points exist by default within the decision-making method itself. For example, in the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method, the positive and negative ideal points serve as reference points. Additionally, methods like the Analytic Hierarchy Process (AHP), Data Envelopment Analysis (DEA), Weighted Sum Method, Weighted Product Method, and Elimination and Choice Translating Reality (ELECTRE) all implicitly incorporate the idea of reference points [30]. In these methods, although reference points are not explicitly mentioned, they are assumed to exist objectively and play a role. The other category involves decision-makers subjectively setting reference points. In references [31,32], the authors use decision-makers’ expected values as reference points and propose multi-attribute decision-making methods based on prospect theory. From the above research, it is evident that studies on reference points in multi-attribute decision-making are relatively scarce, and the setting of reference points is usually performed in static decision-making scenarios. However, in dynamic multi-attribute decision-making problems, reference points often change dynamically over time, along with the continuous evolution of the decision environment. Therefore, it is necessary to set a dynamic reference point that is related to time. However, current research on the dynamic reference point setting in multi-attribute decision-making is almost non-existent.

To address this, this paper proposes a new electricity sales package recommendation method—dynamic hybrid multi-attribute group decision-making with two reference points for electricity sales package recommendation (DH-MAGDM). The specific steps are as follows: First, a customer load information database is established by constructing a sample customer profiling system and segmenting the sample customer set using the AAP clustering method. This database facilitates the extraction of characteristic profiles of electricity users and supports the identification of sample customers similar to the target customer. Next, quantitative and qualitative attributes are characterized using three-parameter interval gray numbers and triangular fuzzy numbers, respectively, to capture their distinct features. A hybrid target model (external reference point) is constructed by combining gray numbers and fuzzy numbers, while a temporal reference point is defined using average values and mean growth rates to reflect the dynamic evolution of options. A customer weight adjustment model is then developed by examining the similarity between individual and group decision-making among sample customers. Additionally, time weights are determined using the entropy weight method. This process yields comprehensive evaluation scores for all electricity sales packages from the perspectives of different customer categories, enabling a ranked recommendation of packages. Finally, a case study involving customers in a specific region of China is conducted to validate the feasibility and practicality of the proposed recommendation method. The results demonstrate its effectiveness in accurately and scientifically recommending electricity sales packages tailored to customer needs.

Based on the above, this paper proposes a dynamic hybrid multi-attribute group decision-making with two reference points for electricity sales package recommendation (DH-MAGDM). The main contributions of this paper are as follows:

(1)

A sample customer clustering method, the AAP clustering algorithm, is used to avoid the randomness, subjectivity, and related errors present in existing clustering algorithms, thereby improving the accuracy of identifying customers similar to the target customers.

(2)

Considering the uncertainty in customers’ evaluation information for electricity sales packages, the quantitative attributes and qualitative attributes are characterized using three-parameter interval gray numbers and triangular fuzzy numbers, respectively. A dynamic multi-attribute group decision-making model based on the combination of three-parameter interval gray numbers and triangular fuzzy numbers is proposed.

(3)

A time reference point is set using the mean and average growth rate to reflect the dynamic development of electricity sales packages, effectively improving the accuracy of group decision-making for electricity sales packages.

The remainder of this paper is organized as follows. Section II presents the methodology, specifically proposing a sample customer clustering method based on the customer profile label system and AAP, as well as a dynamic multi-attribute group decision-making model based on two reference points. Section III covers the results and discussion, where a case study is conducted on customers in a certain region of China to test the feasibility of the proposed method, followed by a brief discussion of the results. Finally, Section IV concludes the paper, outlining the limitations of the study and future research directions.

2. Methodology

2.1. Classification of Customer Groups

2.1.1. Building Sample Customer Portrait Label System

First, to help readers better understand the DH-MAGDM proposed in this paper, a complete diagram covering all the applied methodological procedures is shown in Figure 1.

The energy usage and consumption habits of different types of customers vary, as reflected in their differing preferences and selection outcomes for electricity sales packages. To better distinguish the practical differences in electricity demands and systematically describe customers’ electricity usage characteristics from multiple perspectives, this study employs a sample customer set $R=\{R_1,R_2,\dots ,R_i,\dots ,R_I\}$. The load dataset for sample customers on a given day of a specific month is represented as $P=\{P_{R_1},P_{R_2},\dots ,P_{R_i},\dots ,P_{R_I}\}$, where the load data for customer $R_i$ is $P_{R_i}=\{P_{R_i,1},P_{R_i,2},\dots ,P_{R_i,m},\dots ,P_{R_i,M}\}$. Here, M represents the total number of hours in a month, and I represents the total number of customers in the sample set.

This study employs 6 key metrics to characterize sample customers, which are defined as follows:

• Monthly Load Rate: The Monthly Load Rate $b_{R_i}^1$ is used to reflect the fluctuation of electricity load within a month. It is calculated as the ratio of the average monthly load ($P_{R_i(av)}$) to the maximum load ($P_{R_i(\max )}$), as expressed in the following formula:

  $$b_{R_i}^1={\displaystyle \frac{P_{R_i(av)}}{P_{R_i(max)}}}={\displaystyle \frac{{\displaystyle \sum _{m=1}^M{P}_{R_i,m}}/M}{P_{R_i(max)}}}.$$

  (1)

A larger value of $b_{R_i}^1$ indicates that customer $R_i$ has a higher ratio of average power load to maximum power load within the month, signifying greater fluctuations in their power load.

2.

Peak-Time Load Rate: The Peak-Time Load Rate $b_{R_i}^2$ is defined as the ratio of the electricity consumption during peak hours to the total electricity consumption for customer $R_i$. The peak load during the time period 8:00–16:00 in a given month is represented by $P_{R_i,h.m}$. The formula is expressed as follows:

$$b_{R_i}^2={\displaystyle \frac{{\displaystyle \sum _{m=1}^M{P}_{R_i,h.m}}}{{\displaystyle \sum _{m=1}^M{P}_{R_i,m}}}}.$$

(2)

A larger $b_{R_i}^2$ value indicates a higher utilization rate of electricity during peak hours throughout the month. This also indirectly suggests smaller fluctuations in the monthly electricity load, as the customer’s consumption is more concentrated during peak periods.

3.

Off-Peak Load Rate: The Off-Peak Load Rate $b_{R_i}^3$ represents the ratio of electricity consumed during off-peak hours to the total electricity consumption for customer $R_i$. The off-peak load during the time period 22:00–5:00 (the following day) in a given month is denoted as $P_{R_i,l.m}$. The formula is expressed as follows:

$$b_{R_i}^3={\displaystyle \frac{{\displaystyle \sum _{m=1}^M{P}_{R_i,l.m}}}{{\displaystyle \sum _{m=1}^M{P}_{R_i,m}}}}.$$

(3)

A larger $b_{R_i}^3$ value indicates a higher utilization rate of electricity during off-peak hours throughout the month. This also indirectly suggests smaller fluctuations in the monthly electricity load, as the customer’s consumption is more concentrated during off-peak periods.

4.

Flat-Time Load Rate: The Flat-Time Load Rate $b_{R_i}^4$ represents the ratio of electricity consumed during flat (non-peak and non-off-peak) hours to the total electricity consumption for customer $R_i$. The flat-time load for a specific day in a month, excluding peak and off-peak hours, is denoted as $P_{R_i,k.m}$. The formula is expressed as follows:

$$b_{R_i}^4={\displaystyle \frac{{\displaystyle \sum _{m=1}^M{P}_{R_i,k.m}}}{{\displaystyle \sum _{m=1}^M{P}_{R_i,m}}}}.$$

(4)

A larger $b_{R_i}^4$ value suggests that the customer has largely avoided electricity consumption during peak and off-peak periods throughout the month, favoring flat-period usage instead.

5.

Maximum Load Utilization Hours: The Maximum Load Utilization Hours $b_{R_i}^5$ reflects the time utilization efficiency of a customer’s load. It is calculated as the ratio of the total monthly electricity consumption to the maximum monthly load, as expressed in the following formula:

$$b_{R_i}^5={\displaystyle \frac{{\displaystyle \sum _{m=1}^M{P}_{R_i,m}}}{P_{R_i(max)}}}.$$

(5)

A larger $b_{R_i}^5$ indicates that customer $R_i$ has a higher load time utilization efficiency for the month, meaning their electricity usage is more effectively distributed relative to their maximum load.

6.

Monthly Average Electricity Stability: The Monthly Average Electricity Stability $b_{R_i}^6$ reflects the degree of variation in a customer’s daily electricity load over a month. It is calculated as the sum of the squared differences between the daily electricity load and the monthly average electricity load. The formula is expressed as follows:

$$b_{R_i}^6={\displaystyle \frac{{\displaystyle \sum _{n=1}^T{\left(P_n-P_{R_i(av)}\right)}^2}}{T}}.$$

(6)

where $P_n$ is the electricity load on the n-th day, and T is the total number of days in the month.

Based on the six sample customer electricity usage characteristic indicators, the profile of customer $R_i$ is represented by a feature vector $\overrightarrow{G_{R_i}}=[b_{R_i}^1,b_{R_i}^2,\dots ,b_{R_i}^6]$. Using the feature vector $\overrightarrow{G_{R_i}}$, a load information database for sample customers can be constructed, providing a standard for matching customer characteristic information in subsequent analyses.

2.1.2. Sample Customer Clustering Based on AAP Algorithm

Considering the large number of customers, it is necessary to cluster the sample customers based on their constructed profiles to reduce computational complexity. The Adaptive Affinity Propagation (AAP) clustering algorithm [33] overcomes the limitations of the standard Affinity Propagation (AP) algorithm, such as the difficulty in selecting the “preference” parameter and the inability to automatically eliminate oscillations. It demonstrates better performance, particularly in handling complex or large-scale clustering tasks.

Compared to traditional clustering methods, AAP clustering can automatically determine the number of clusters based on the similarity of customer data without the need to predefine the number of categories. It does not rely on randomly selected initial centers but instead uses information propagation to automatically identify cluster centers. This eliminates the sensitivity to initial conditions inherent in traditional clustering algorithms. Additionally, it can more accurately handle complex and nonlinear customer data structures and identify appropriate cluster boundaries in such structures, providing more precise and reliable clustering results.

Due to these advantages, using the AAP clustering method is well-suited for clustering sample customers based on their profiles.

AAP clustering only requires the input of similarity between customer profiles. The similarity $s(R_i,R_j)$ between profiles of customers $R_i$ and $R_j$ represents the degree to which $R_j$ is suitable as a clustering center for $R_i$. The self-similarity $s(R_i,R_i)$, which indicates the reference level of customer $R_i$’s own profile, is usually set to the median of $R_i$’s similarity with other customer profiles, ensuring the minimum number of clusters.

Some customers may tend to give higher ratings to electricity retail packages, while others may be more critical and give lower ratings. This issue of rating scale inflation can affect the accuracy of similarity calculations. However, the Pearson correlation coefficient can address this issue, as it accounts for the overall rating tendencies of customers rather than just the absolute values of the ratings. By calculating the Pearson correlation coefficient between customers, it is possible to more accurately identify customers with similar interests. In recommendation systems, if two customers have similar interests, a retail electricity package adopted by one customer is likely to be adopted by the other.

Therefore, this paper uses the Pearson correlation coefficient to calculate the similarity between the target customer and the sample customers, as follows:

$$s(R_i,R_j)={\displaystyle \frac{{\displaystyle \sum _{x=1}^6(b_{R_i}^x-\overline{b_{R_i}})}(b_{R_j}^x-\overline{b_{R_j}})}{\sqrt{{\displaystyle \sum _{x=1}^6{(b_{R_i}^x-\overline{b_{R_i}})}^2}}\cdot \sqrt{{\displaystyle \sum _{x=1}^6{(b_{R_j}^x-\overline{b_{R_j}})}^2}}}}\hspace{1em}i\ne j,$$

(7)

where $b_{R_i}^x$ and $b_{R_{\mathrm{j}}}^x$ represent the x-th portrait lable of customers $R_i$ and $R_j$, respectively, while $\overline{b_{R_i}}$ and $\overline{b_{R_j}}$ denote the average values of the profile labels for customers $R_i$ and $R_j$, respectively.

The absolute value of $s(R_i,R_j)$ indicates the strength of the linear relationship between two customers. The closer the correlation coefficient $s(R_i,R_j)$ is to 1, the stronger the correlation between the two customers. When $s(R_i,R_j)$ is closer to 1, it suggests a stronger relationship, meaning that the similarity is higher, and it is more likely that both customers will choose similar packages.

The detailed steps of the traditional AP clustering algorithm are as follows:

Step 1: Initialize the responsibility and availability of customer profiles, as shown in Equations (8) and (9). Here, the responsibility $r(R_i,R_j)$ represents the degree to which customer $R_j$ is suitable as the cluster center for customer $R_i$, and the availability $a(R_i,R_j)$ represents the suitability of customer $R_i$ selecting customer $R_j$ as the cluster center.

$$r(R_i,R_j)=s(R_i,R_j)-\max \{s(R_i,R_j)\}.$$

(8)

$$a(R_i,R_j)=0.$$

(9)

Step 2: Update the responsibility and availability of customer profiles for the new iteration. The results of the N-th iteration are shown as follows:

$$r_N(R_i,R_j)=\lambda r_{(N-1)}(R_i,R_j)+(1-\lambda )\left(s(R_i,R_j)-{\max }_{j^{\prime }(j^{\prime }\ne j)}\left\{a_{(N-1)}(R_i,R_{j^{\prime }})\right\}+s(R_i,R_{j^{\prime }})\right),$$

(10)

$$\left\{\begin{array}{l}{a}_N(R_i,R_j)=\lambda a_{(N-1)}(R_i,R_j)+(1-\lambda )\min \left\{0,r_N(R_j,R_j)+{\displaystyle \sum _{i^{\prime }(i^{\prime }\ne i,j)}\max }\left\{0,r_N(R_{i^{\prime }},R_j)\right\}\right\}\hfill \\ a_N(R_j,R_j)=\lambda a_{(N-1)}(R_j,R_j)+(1-\lambda ){\displaystyle \sum _{i^{\prime }(i^{\prime }\ne j)}\max }\left\{0,r_N(R_{i^{\prime }},R_j)\right\}\hfill \end{array}\right.,$$

(11)

where $\lambda$ is the damping factor, used to accelerate the convergence of the AP clustering and reduce the impact of numerical oscillations. The value of $\lambda$ ranges from [0.5, 1), and N represents the number of iterations during the AP clustering process for the customers.

Step 3: Check if the iteration of the responsibility and availability for the user profiles has stabilized. The cluster center set $C_{center,N}$ after the N-th iteration is obtained from Equation (12). If $C_{center,N}=C_{center,N-1}$, then the stability count $\delta =\delta +1$; otherwise, $\delta =0$. If $\delta$ reaches the maximum predefined stability count ${\delta }_{\max }$, proceed to Step 5; otherwise, proceed to Step 4.

$$r_N(R_j,R_j)+a_N(R_j,R_j)>0.$$

(12)

Step 4: If N reaches the predefined maximum number of iterations $D_{\max }$, proceed to Step 5. Otherwise, set N = N + 1 and return to Step 2.

Step 5: Obtain the cluster center set $C_{center}$, and use Equation (12) to determine whether customer $R_j$ is a cluster center.

AAP clustering algorithm is based on the traditional AP clustering algorithm, where in each iteration, the self-similarity of the users is updated, and the step size is dynamically adjusted according to the number of clusters, as follows:

$$s(A_j,A_j)=s(A_j,A_j)-s_{\rho }=s(A_j,A_j)-{\displaystyle \frac{{\mu }_m}{10\sqrt{Q+50}}},$$

(13)

where $s_{\rho }$ is the update step size, ${\mu }_m$ is the median similarity between $R_j$ and other user profiles, and Q is the number of clusters.

Based on the clustering results obtained from each iteration, the clustering quality metric $Z_{Q(av)}$ is calculated as follows:

$$Z_{Q(av)}={\displaystyle \frac{1}{I}}{\displaystyle \sum _{i=1}^I{Z}_Q}(R_i)={\displaystyle \frac{1}{I}}{\displaystyle \sum _{i=1}^I\frac{d_{out}(R_i)-d_{in}(R_i)}{\max (d_{out}(R_i),d_{in}(R_i))}},$$

(14)

where $Z_Q(R_i)$ is the tightness of user $R_i$ with its corresponding cluster center when the number of clusters is Q, $d_{out}(R_i)$ is the average inter-cluster distance, representing the average distance between user $R_i$ and user profiles from other clusters, and $d_{in}(R_i)$ is the average intra-cluster distance, representing the average distance between user $R_i$ and other user profiles within the same cluster.

Compare the clustering quality metric $Z_{Q(av)}$ for different numbers of clusters. The number of clusters that corresponds to the maximum value of $Z_{Q(av)}$ is considered the optimal number of clusters.

The detailed flowchart of the AAP-based customer clustering method is shown in Figure 2.

2.2. Setting of Reference Points

First, let t represent the phase in which the sample customers give comprehensive ratings for electricity sales packages, where $t=1,2,\dots ,h$, and the corresponding weights are ${\lambda }_t$, with $0\le {\lambda }_t\le 1$ and ${\displaystyle \sum _{t=1}^h{\lambda }_t}=1$. The set of electricity sales packages is $A=\{A_1,A_2,\dots ,A_k,\dots ,A_K\}$, and the sample customer set is $R=\{R_1,R_2,\dots ,R_i,\dots ,R_I\}$, with corresponding customer weights ${\mu }_i$, where $i=1,2,\dots ,I$ and $0\le {\mu }_i\le 1$. For each sample customer within a given category, the sum of the customer weights within that category equals 1. The set of attributes used to comprehensively evaluate the electricity sales packages is $C_R=\{C_{R,1},C_{R,2},\dots ,C_{R,j},\dots ,C_{R,J}\}$. Without loss of generality, consider the first y attributes as quantitative attributes and the remaining $J-y$ attributes as qualitative attributes. Let the sets of quantitative and qualitative attributes be $\{C_1,C_2,\dots ,C_y\}$ (where $1\le y\le J$) and $\{C_{y+1},C_{y+2},\dots ,C_J\}$, respectively. The weights of each attribute are ${\omega }_j$, where $j=1,2,\dots ,J,0\le {\omega }_j\le 1$, and ${\displaystyle \sum _{j=1}^J{\omega }_j}=1$.

Next, in this paper, the quantitative attributes and qualitative attributes are described using three-parameter interval gray numbers and triangular fuzzy numbers, respectively. The specific calculation formulas are given below [34].

Let $a(\otimes )=[a_1,a_2,a_3]$ and $b(\otimes )=[b_1,b_2,b_3]$ be three-parameter interval gray numbers (triangular fuzzy numbers), then

$$d(a(\otimes ),b(\otimes ))=k_1|a_2-b_2|+k_2|a_1-b_1|+(1-k_1-k_2)|a_3-b_3|,$$

(15)

where $d(a(\otimes ),b(\otimes ))$ represents the distance between the three-parameter interval gray number (triangular fuzzy number) $a(\otimes )$ and $b(\otimes )$, with $0\le k_1\le 0.5$ and $0.5\le k_2\le 1$. This formula still applies when either $a(\otimes )$ or $b(\otimes )$ degenerates into a real number.

Below is the method for comparing the size of three-parameter interval gray numbers (triangular fuzzy numbers).

Let $a(\otimes )=[a_1,a_2,a_3]$ and $b(\otimes )=[b_1,b_2,b_3]$ be three-parameter interval gray numbers (triangular fuzzy numbers), where $d\prime (a_3-a_2)$ represents the distance between the real numbers $a_3$ and $a_2$. The comparison rules are as follows:

• When $a_2>b_2$, then $a(\otimes )>b(\otimes )$.

2.

When $a_2=b_2$, the comparison is based on the following:

• If $d\prime (a_3-a_2)-d\prime (a_2-a_1)>d\prime (b_3-b_2)-d\prime (b_2-b_1)$, then $a(\otimes )>b(\otimes )$.
• If $d\prime (a_2-a_1)=d\prime (b_2-b_1)$ and $d\prime (a_3-a_2)=d\prime (b_3-b_2)$, then $a(\otimes )=b(\otimes )$.
• If $d\prime (a_3-a_2)-d\prime (a_2-a_1)>d\prime (b_3-b_2)-d\prime (b_2-b_1)$ and $d\prime (a_3-a_2)<d\prime (b_3-b_2)$, then $a(\otimes )>b(\otimes )$.

For the qualitative attributes of the electricity sales packages, the linguistic variables for the comprehensive ratings and their corresponding triangular fuzzy numbers are shown in

Table 1. Attribute linguistic variables and corresponding triangular fuzzy numbers.

Language Variables	Very Poor	Poor	Slightly Poor	Fair	Slightly Good	Good	Very Good
Corresponding triangular fuzzy numbers	(0, 1, 2)	(1, 2, 3)	(2, 3, 4)	(3, 4, 5)	(4, 5, 6)	(5, 6, 7)	(6, 7, 8)

.

2.2.1. External Reference Points

In the case of a mixture of gray numbers and fuzzy numbers, in order to minimize the loss caused by data processing, this paper proposes a hybrid target model, where both positive and negative target values contain gray numbers and fuzzy numbers. Here, $u_{kj}$ represents the attribute value of the k-th electricity sales package for the j-th attribute.

Let $u_{kj}^{+}=\max \{u_{kj}(\otimes )|1\le k\le K,j=1,2,\dots ,J\}$, where the corresponding attribute value is denoted as $u_{kj}^{+}$.

$$u^{+}(\otimes )=\{u_1^{+},u_2^{+},\dots ,u_J^{+}\}=\{u_{k1}^{+}(\otimes ),u_{k2}^{+}(\otimes ),\dots ,u_{kJ}^{+}(\otimes )\},$$

(16)

where $u^{+}(\otimes )$ is the optimal effect vector of the decision, referred to as the positive target.

Let $u_{kj}^{-}=\min \{u_{kj}(\otimes )|1\le k\le K,j=1,2,\dots ,J\}$, where the corresponding attribute value is denoted as $u_{kj}^{-}$.

$$u^{-}(\otimes )=\{u_1^{-},u_2^{-},\dots ,u_J^{-}\}=\{u_{k1}^{-}(\otimes ),u_{k2}^{-}(\otimes ),\dots ,u_{kJ}^{-}(\otimes )\},$$

(17)

where $u^{-}(\otimes )$ is the worst-case effect vector of the decision, referred to as the negative target.

2.2.2. Time Reference Points

The determination of time weights in this paper mainly draws on the method presented in [35]. For the setting of the temporal reference point, the approach mainly considers combining the average value with the average growth rate. This is because the average value of each stage can effectively reflect the average level of that stage, while the growth rate reflects the impact of dynamic changes on the results.

Let $a(\otimes )=[a_1,a_2,a_3]$ be a three-parameter interval gray number, defined as follows:

$$a^{\prime }(\otimes )=E(a(\otimes ))={\displaystyle \frac{1}{4}}{a}_1+{\displaystyle \frac{1}{2}}{a}_2+{\displaystyle \frac{1}{4}}{a}_3=a(\otimes ).$$

(18)

This definition transforms the three-parameter interval gray number into a real number, and it still applies when it degenerates into a real number.

In the t-th stage, the decision evaluation mean is $\overline{u_{jt}^{\prime }(\otimes )}={\displaystyle \frac{1}{K}}{\displaystyle \sum _{k=1}^K{u}_{kjt}^{\prime }}(\otimes ),j=1,2,\dots ,p$, then the equation is

$$1+\overline{f}+{\overline{f}}^2+\dots +{\overline{f}}^{(h-1)}={\displaystyle \frac{{\displaystyle \sum _{t=1}^h\overline{u_{jt}^{\prime }(\otimes )}}}{\overline{u_{j1}^{\prime }(\otimes )}}}.$$

(19)

The positive root of Equation (19) is referred to as the average growth rate.

Let $s_{jt}(\otimes )=\overline{u_{j1}^{\prime }(\otimes )}\times \overline{f_j^{t-1}}$, where $j=1,2,\dots ,J$, then the time reference point is

$$s_t(\otimes )=\{s_{1t}(\otimes ),s_{2t}(\otimes ),\dots ,s_{Jt}(\otimes )\},t=1,2,\dots ,h.$$

(20)

2.2.3. Fusion of Two Reference Points

The “value function” and “decision weights” together determine the magnitude of the prospect value [26]. Therefore, the prospect value function can be expressed as follows:

$$V={\displaystyle \sum _{k=1}^K\pi }(p_k)v(x_k).$$

(21)

The weight function is expressed as follows:

$$\pi (p)={\displaystyle \frac{p^{\gamma }}{{(p^{\gamma }+{(1-p)}^{\gamma })}^{\frac{1}{\gamma }}}}..$$

(22)

The value function is typically expressed as follows:

$$v(x)=\left\{\begin{array}{ll}{x}^{\alpha },\hfill & x\ge 0;\hfill \\ -\theta {(-x)}^{\beta },\hfill & x<0.\hfill \end{array}\right..$$

(23)

For the parameters in the above equation, the literature [36] has determined through extensive experiments that $\alpha =\beta =0.88$ and $\theta =2.25$. For Equation (22), when facing gains, $\gamma =0.61$ and when facing losses, $\gamma =0.69$.

The value function has the following three main characteristics [26]:

1.

In practical decision-making, under normal circumstances, decision-makers tend to choose the current situation as the reference point, and the value function is relative to the reference point. When the outcome is perceived as a loss, people generally prefer risk; when the outcome is perceived as a gain, people generally prefer to avoid risk. Therefore, a decision-maker’s attitude toward risk is often determined by the specific situation at the time of making the decision and is not fixed or unchanging.

2.

The value function curve generally exhibits an “S” shape. As shown in Figure 3, it can be observed that when the decision outcome is a gain, decision-makers tend to avoid risk; when the decision outcome is a loss, decision-makers generally seek risk. Additionally, as the gains or losses increase, the value curve gradually levels off, indicating that decision-makers’ attitude toward gains or losses follows a principle of diminishing marginal utility.

3.

Decision-makers’ perception of losses is greater than that of gains. In other words, when people face equal amounts of gain and loss, they tend to feel the loss more intensely.

Figure 3. Value function.

Figure 3. Value function.

For the above external reference points, the prospect value of each customer’s evaluation of each alternative relative to the external reference points is derived. In other words, the prospect values of the alternative solutions relative to the positive and negative reference points are obtained as follows:

1.

When the attribute value is a “three-parameter interval grey number”, the prospect value is given by

$$\left({(V_{kji}^1(\otimes ))}_1\right)=v_{kj}^{(+)}\left(u_{kji}^{(\otimes )}\right){\pi }_{kj}^{(+)}(p(\otimes ))+v_{kj}^{(-)}\left(u_{kji}^{(\otimes )}\right){\pi }_{kj}^{(-)}(p(\otimes )),1\le j\le y.$$

(24)

2.

When the attribute value is a “triangular fuzzy number”, the prospect value is given by

$$\left({(V_{kji}^2(\otimes ))}_1\right)=v_{kj}^{(+)}\left(u_{kji}^{(\otimes )}\right){\pi }_{kj}^{(+)}(p(\otimes ))+v_{kj}^{(-)}\left(u_{kji}^{(\otimes )}\right){\pi }_{kj}^{(-)}(p(\otimes )),y\le j\le J.$$

(25)

The prospect value of a single electricity sales package for each attribute is

$${(V_{kji}(\otimes ))}_1={(V_{kji}^1(\otimes ))}_1+{(V_{kji}^2(\otimes ))}_1.$$

(26)

Similarly, based on the time reference points derived above, the prospect value of each customer’s evaluation of each electricity sales package relative to the time reference points is calculated. The specific formula is as follows:

1.

When the attribute value is a “three-parameter interval grey number”, the prospect value is given by

$$\left({(V_{kji}^1(\otimes ))}_2\right)=v_{kj}^{(+)}\left(s_{kji}^{(\otimes )}\right){\pi }_{kj}^{(+)}(p(\otimes ))+v_{kj}^{(-)}\left(s_{kji}^{(\otimes )}\right){\pi }_{kj}^{(-)}(p(\otimes )),1\le j\le y.$$

(27)

2.

When the attribute value is a “triangular fuzzy number”, the prospect value is given by

$$\left({(V_{kji}^2(\otimes ))}_2\right)=v_{kj}^{(+)}\left(s_{kji}^{(\otimes )}\right){\pi }_{kj}^{(+)}(p(\otimes ))+v_{kj}^{(-)}\left(s_{kji}^{(\otimes )}\right){\pi }_{kj}^{(-)}(p(\otimes )),y\le j\le J.$$

(28)

The formula for calculating the prospect value of each electricity sales package relative to the time reference points is as follows:

$${(V_{kji}(\otimes ))}_2={(V_{kji}^1(\otimes ))}_2+{(V_{kji}^2(\otimes ))}_2.$$

(29)

Currently, in existing research on decision-making problems with multiple reference points, the author believes that handling each reference point independently allows for a more accurate assessment of the results [37]. Therefore, let the prospect value based on the external reference point be $V_1$, and the prospect value based on the temporal reference point be $V_2$. The overall prospect value is then

$$V=l\cdot V_1+(1-l)\cdot V_2,$$

(30)

where l represents the decision-maker’s preference level for the external reference point.

2.3. Determination of Weights

2.3.1. Customer Weights

In the context of group decision-making, the method for adjusting the weights of each customer is provided in [38], which proposes an algorithm based on gray relational analysis theory for adjusting customer weights. The literature [39] uses entropy-based weighting coefficients for weight adjustment. This paper attempts to adopt a new approach, offering a reasonable and practical weight adjustment algorithm.

Basic Idea of Customer Weight Adjustment Using the Projection Method: The individual decision results of customers and the group decision results are treated as vector sequences. The individual decision results are considered a comparison sequence, while the group decision results serve as a reference sequence. By analyzing the angle between the comparison sequence and the reference sequence, the similarity between the two vectors is determined. The smaller the angle between them, the higher the similarity, indicating that the deviation between the individual decision result of the customer and the group decision result is smaller. As a result, the weight of the customer is considered larger. The steps for this method are as follows:

Step 1: Based on the normalized customer weights and attribute weights obtained from initialization, calculate the individual decision results of the customers and the group decision results as follows:

1.

In the t-th stage, the individual decision score for each plan based on the two reference points is given by

$$x_{it}(i)=l\cdot V_1(k)+(1-l)\cdot V_2(k).$$

(31)

2.

In the t-th stage, the group decision score for each plan based on the two reference points is given by

$$x_{0t}(k)={\displaystyle \sum _{i=1}^I{x}_{it}}(k){\mu }_i.$$

(32)

Step 2: Treat the group decision result as the reference sequence $x_{0t}$, and the individual decision result of the customer as the comparison sequence $x_{it}$, where $i=1,2,\dots ,I$.

$$\begin{array}{l}{x}_{0t}=(x_{0t}(1),x_{0t}(2)\dots ,x_{0t}(K))\\ x_{it}=(x_{it}(1),x_{it}(2),\dots ,x_{it}(K))\end{array}.$$

(33)

Step 3: Calculate the angle between the comparison sequence and the reference sequence. The specific calculation formula is as follows:

$$cos\phi =\cos \langle x_{it},x_{0t}\rangle ={\displaystyle \frac{{\displaystyle \sum _{k=1}^K{x}_{it}}(k)x_{0t}(k)}{\sqrt{{\displaystyle \sum _{k=1}^K(x_{it}(}k){)}^2}\sqrt{{\displaystyle \sum _{k=1}^K(x_{0t}(}k){)}^2}}}.$$

(34)

Step 4: Adjust customer weights. In order to reflect both the wisdom of the group decision and take into account the opinions of each customer, the customer weights are adjusted. The specific adjustment direction is as follows:

$${\mu }_i^{\prime }={\displaystyle \frac{\cos \langle x_i,x_0\rangle }{{\displaystyle \sum _{i=1}^I\cos }\langle x_i,x_0\rangle }}.$$

(35)

Step 5: Calculate the distance between $x_{0t}$ and $x_{0t}^{\prime }$. After adjusting the customer weights using the steps mentioned above, and incorporating Equation (26) along with the updated ${\mu }_{0t}^{\prime }$ value, the new group decision result is calculated as follows:

$$x_{0t}^{\prime }=(x_{0t}^{\prime }(1),x_{0t}^{\prime }(2)\dots ,x_{0t}^{\prime }(K)).$$

(36)

Let $L(x_{0t},x_{0t}^{\prime })$ represent the distance between $x_{0t}$ and $x_{0t}^{\prime }$. It is defined as follows:

$$L(x_{0t},x_{0t}^{\prime })=\sqrt{{\displaystyle \sum _{k=1}^K(x_{0t}(}k)-x_{0t}^{\prime }(k){)}^2}.$$

(37)

Step 6: Set a threshold r, and compare $L(x_{0t},x_{0t}^{\prime })$ with r as follows:

• If $L(x_{0t},x_{0t}^{\prime })\le r$, the group decision result is considered stable, and the process can stop.
• If $L(x_{0t},x_{0t}^{\prime })>r$, further adjustments to the customer weights are needed, and the process returns to Step 4 for further iterations.

2.3.2. Time Weights

The entropy weight method is an excellent objective weighting method among all objective weighting approaches. It calculates the entropy value of each indicator based on the variation in the values of the indicators and their overall impact and then determines the weights. This method can objectively reflect the differences and importance of different indicators. Therefore, it is very reasonable to use the entropy weight method to determine the time weights.

First, the decision results for K electricity sales packages are arranged in sequence according to h time stages, forming a $K\times h$ matrix E, i.e.,

$$E=\left[\begin{array}{cccc}{x}_{01}(1)& x_{02}(1)& \cdots & x_{0h}(1)\\ x_{01}(2)& x_{02}(2)& \cdots & x_{0h}(2)\\ ⋮& ⋮& \ddots & ⋮\\ x_{01}(K)& x_{02}(K)& \cdots & x_{0h}(K)\end{array}\right].$$

(38)

Next, calculate the entropy value for each stage.

$$e_t=-{\displaystyle \frac{1}{\ln K}}\cdot {\displaystyle \sum _{k=1}^K\left[\frac{x_{0t}(k)}{{\displaystyle \sum _{k=1}^K{x}_{0t}}(k)}\cdot \ln \frac{x_{0t}(k)}{{\displaystyle \sum _{k=1}^K{x}_{0t}}(k)}\right]}.$$

(39)

Finally, the weight for each stage can be calculated using the entropy values.

$${\lambda }_t={\displaystyle \frac{1-e_t}{h-{\displaystyle \sum _{t=1}^h{e}_t}}}.$$

(40)

2.4. Determination of the Optimal Electricity Sales Package for Sample Customer

After categorizing the sample customers, we can use the dynamic hybrid multi-attribute group decision-making algorithm based on two reference points to select the best electricity sales package that suits each category of sample customers. The steps are as follows:

Step 1: Normalize the attribute values of the electricity sales packages based on the type of attribute value.

Step 2: Use the AHP (Analytic Hierarchy Process) method to obtain customer weights ${\mu }_s$ and attribute weights ${\omega }_j$ for the electricity sales packages.

Step 3: For each customer category, calculate the positive and negative target values (i.e., the external reference points) for the evaluation matrix of customer $R_i$ on electricity package $A_k$ regarding attribute $C_{R,j}$.

Step 4: For each customer category, calculate the temporal reference points for customer $R_i$ on electricity package $A_k$ at each time stage.

Step 5: For each customer category, calculate the prospect value $V_1$ for customer $R_i$ regarding each electricity package at the external reference point. For each customer category, calculate the prospect value $V_2$ for customer $R_i$ regarding each electricity package at the temporal reference point. S For each customer category, calculate the combined prospect value V for customer $R_i$ regarding each electricity package at both the external and time reference points.

Step 6: Adjust the customer weights within each category using Equations (31)–(37).

Step 7: Determine the time weights for each stage using Equations (38) and (40).

Step 8: Combine the results using the formula $Scor{e}_k={\displaystyle \sum _{t=1}^h{\lambda }_t}{x}_{0t}(k)$ to calculate the comprehensive score for each electricity sales package. Compare the scores to determine the most suitable electricity sales package for each customer category.

In summary, the proposed electricity sales package decision-making method can be divided into two stages: constructing a load information database for the sample customer set and making decisions for the target customer’s electricity sales package.

3. Results and Discussion

3.1. Case Studies

This section uses users from a certain region in China as an example to validate the electricity sales package recommendation method based on dynamic hybrid multi-attribute group decision-making with two reference points. The load data of 500 typical customers $R=\{R_1,R_2,\dots ,R_{500}\}$ collected by smart meters from 1–31 January 2020, is used as the basis. A leave-one-out cross-validation method is applied for validation of the proposed method. In each iteration, one user is selected as the new user, and the remaining 499 users are used as the sample set for the decision-making process.

Based on the data from the provincial trading center and the actual conditions of the users, the electricity sales company provides the following set of packages: $A=\{A_1,A_2,A_3,A_4\}$. The attributes of these electricity sales packages are detailed in the appendix. Considering that the main goal of this paper is to study the recommendation method for electricity sales packages, the load data mentioned above is directly used as the new user’s monthly load for analysis. The electricity sales package attributes considered in this paper include electricity price, additional premium, value-added services, and reward policies. The meanings of these attributes are provided in

Table 2. Electricity sales package set provided by electricity retail company.

Electricity Sales Package	Unit Price/(Yuan·(kW·h)−1)	Additional Premium/%	Value-Added Service	Incentive Policy
$A_1$	0.53 (<500 kW·h)	10	Energy-saving management	On-time settlement of electricity discount 7%
	0.67 (500–1000 kW·h)
	0.67 (500–1000 kW·h)
$A_2$	0.56 (<500 kW·h)	15	Power quality-improvement services	Reward power ratio of 10%
	0.63 (500–1000 kW·h)
	0.87 (>1000 kW·h)
$A_3$	0.85 (peak: 10:00–12:00, 13:00–19:00)	10	Energy-saving management	On-time settlement of electricity discount 7%
	0.60 (flat: 06:00–10:00, 12:00–13:00, 19:00–22:00)
	0.30 (valley: 22:00–morrow 06:00)
$A_4$	0.80 (peak: 10:00–12:00, 13:00–19:00)	15	Power quality-improvement services	Reward power ratio of 10%
	0.65 (flat: 06:00–10:00, 12:00–13:00, 19:00–22:00)
	0.35 (valley: 22:00–morrow 06:00)

.

3.2. Electricity Sales Package Recommendation Analysis

3.2.1. Sample Customer Cluster Analysis

Based on the AAP clustering algorithm proposed earlier, with a damping factor $\lambda =0.8$, the sample customers are clustered, resulting in three main categories: commercial electricity customers, industrial customers, and residential customers. Based on the clustering results of the sample customers, the distances between the target customer $R_4$’s profile and the cluster centers of the three categories (commercial electricity customers, industrial customers, and residential customers) are calculated as 0.3563, 0.2097, and 0.5466, respectively. Thus, it can be concluded that the target customer’s similar customers are those in the residential customer category.

The clustering results of the three categories of customers—commercial electricity customers, industrial electricity customers, and residential electricity customers—are shown in Figure 4. In addition, the results are as follows:

1.

The characteristic of commercial electricity customers is a relatively low electricity load, as the service industry generally does not require large energy-intensive equipment, resulting in a lower overall electricity load. The electricity demand in the service industry is usually stable, without significant fluctuations. The workforce typically works during the day, so electricity consumption is concentrated in daytime hours, with higher usage during the day. Therefore, the peak demand is mainly concentrated around noon, with a smaller peak occurring at 18:00 in the evening.

2.

The characteristic of industrial electricity customers is that the industrial sector typically requires the extensive use of various machines, equipment, and heavy machinery, resulting in a high electricity load. The electricity demand in the industrial sector tends to experience smaller fluctuations, with heavy industry showing a relatively stable production load. Industrial electricity consumption is generally high, with electricity primarily used for production facilities and operations. Whether during the day or at night, energy consumption remains relatively high.

3.

The characteristic of residential electricity customers is that their electricity load is relatively low, as residential electricity usage typically meets basic household needs such as lighting, appliances, heating, and air conditioning. Therefore, the electricity load for residential customers is relatively low. Residential electricity demand is concentrated in specific time periods, typically during morning and evening peak hours, as people leave for work in the morning and return home to rest in the evening, leading to higher electricity consumption. The power demand in residential households is mainly focused on lighting, home appliances, and entertainment devices, all of which generally have lower power ratings. Since residents typically return home around 16:00, the electricity load tends to be higher during this time.

Figure 4. Sample customer clustering results.

Figure 4. Sample customer clustering results.

3.2.2. Electricity Sales Package Recommendation and Effects Evaluation

According to the dynamic hybrid multi-attribute group decision-making method based on two reference points proposed earlier, with l = 0.5 and r = 0.0001, the electricity sales packages are ranked in terms of their desirability. By considering the similarity between the target customer and the similar customers, the normalized comprehensive score for each electricity sales package can be obtained for the target customer, as shown in

Table 3. Normalized results of target customers’ overall score for electricity sales packages.

Electricity Sales Package	${\mathit{A}}_1$	${\mathit{A}}_2$	${\mathit{A}}_3$	${\mathit{A}}_4$
Target customers’ overall score	0.8752	0.3553	1.0000	0.7215

.

As shown in Table 3, target customer $R_4$ has the highest satisfaction with $A_3$. In fact, target customer $R_4$ belongs to the residential electricity customer type. Since residential customers are typically away during the day for work and at home in the evening, their electricity usage is concentrated between 16:00 and 22:00. Residential customers usually focus on electricity costs and bill discounts. Therefore, compared to high energy quality value-added services, residential users are more inclined toward energy-saving management services. This result aligns with the theoretical analysis and further demonstrates the accuracy and applicability of the proposed electricity sales package recommendation method for different types of electricity consumers.

Based on the above results, the electricity sales packages provided to the target customer and the ranking results are shown in

Table 4. Recommended ranking results and simulated actual ranking results.

Electricity Sales Package	Recommendation Ranking of This Paper	Actual Ordering of Simulations
$A_1$	2	3
$A_2$	4	4
$A_3$	1	1
$A_4$	3	2

.

As seen in Table 4, for customer $R_4$, the package ranking results obtained by the method proposed in this paper are mostly consistent with the simulated actual ranking results, with a root mean square error (RMSE) of 0.2371. The difference in the ranking of $A_1$ and $A_4$ arises because, while the costs of $A_1$ and $A_4$ are similar and the types of value-added services and reward policies are the same, the additional premium of package $A_1$ is lower. As a result, the comprehensive score for package $A_1$ is higher, indicating that the ranking results obtained by the method align with the actual situation.

Taking the residential electricity customer $R_4$ as an example, the customer’s overall scores for each electricity sales package are shown in Figure 5.

As seen in Figure 5, the comprehensive score for electricity sales package $A_3$ is the highest for customer $R_4$, with a score of 6.6467, indicating that the customer’s satisfaction with the package falls between “Good” and “Very Good.” In fact, customer $R_4$ is a residential electricity customer who is particularly concerned about electricity costs. Therefore, they are more satisfied with the electricity sales package $A_3$, which offers lower costs, certain electricity bill discounts, and low additional premiums.

Additionally, compared to high-quality energy services, this type of customer has a greater demand for energy-saving management services. Since $A_3$ fully meets their electricity consumption needs, the analysis and calculation results align with these findings.

Similarly, the overall scores for other residential electricity customers regarding the electricity sales packages are shown in Figure 6.

3.3. Comparison of Different Methods

3.3.1. Analysis of Effects of Different Clustering Algorithms

To verify the effectiveness of the AAP clustering algorithm in terms of clustering performance, the clustering quality metrics of the AAP and K-means clustering algorithms are compared across different customer scales, as shown in Figure 7.

3.3.2. Comparison of Different Electricity Sales Package Recommendation Methods

To further verify the rationality and feasibility of the proposed electricity sales package recommendation method, a comparison is made between the proposed method and the following two recommendation methods. The recommendation results for each type of target customer under each method are calculated and shown in Figure 8.

• Method 1: The evaluation attributes of the electricity sales packages only consider cost, with the corresponding attribute weight set to 1, while the weights for other attributes are set to 0. This method is based on AAP and the dynamic hybrid multi-attribute group decision-making method with two reference points for recommending electricity sales packages.
• Method 2: This method does not consider the differences in the evaluation attributes. The attribute weights for all electricity sales packages are set to be equal. It is based on the AAP and dynamic hybrid multi-attribute group decision-making method with two reference points for recommending electricity sales packages.

As seen in Figure 8, the comprehensive score ranking results of the electricity sales packages obtained by the proposed recommendation method have the smallest root mean square error (RMSE), with all RMSE values being less than 1. The results of Method 1, which only considers cost, show the largest deviation compared to the other two methods, with the maximum deviation being 1.5293. This indicates that considering the diversity of electricity customers’ evaluation information significantly reduces the deviation in the recommendation results.

Additionally, Figure 8 shows that if the differences between the attributes of the electricity sales packages are not considered, the recommendation performance of Method 2 is clearly inferior to that of the proposed method, which accounts for both the differences in attributes and the incomplete weight information.

Figure 8. Root mean square error of recommended ranking.

Figure 8. Root mean square error of recommended ranking.

4. Conclusions

With the continuous development of the electricity market, electricity sales package companies are facing increasingly fierce market competition and must possess higher business capabilities. The selection of electricity sales packages is influenced by many subjective and objective factors, including the correlation between the decision-making information of electricity sales package functions and customers’ risk preferences. Therefore, this paper proposes a dynamic hybrid multi-attribute group decision-making method with two reference points for electricity sales package recommendation (DH-MAGDM) to address these issues, providing a basis for accurate electricity sales package recommendations. The method has the following characteristics:

• The proposed method for identifying similar users based on user profiling and AAP clustering has the advantages of high accuracy, high clustering efficiency, strong adaptability, and no need to predefine the number of clusters. These features ensure that the identification results can more effectively reflect the profile characteristics of the target users.
• The proposed electricity sales package recommendation method is based on dynamic hybrid multi-attribute group decision-making with two reference points, which comprehensively considers multiple stages of multi-attribute decision-making. It effectively avoids decision errors caused by the abnormal value of any attribute in static decision-making. At the same time, it takes a more comprehensive approach to the decision-making process and the development trends of alternative options, leading to more scientific decisions. This method effectively enables the electricity sales company to accurately recommend the most suitable electricity sales package for each category of target customers, which helps the company improve user satisfaction, increase user retention, and enhance market competitiveness.

In future research, we can further explore and improve the electricity sales package recommendation method proposed in this paper. Regarding parameter selection, sensitivity analysis can be conducted on the decision-maker’s preference level l to enhance the model’s applicability. In terms of validation strategy, the leave-one-out cross-validation (LOOCV) method used in this paper is typically suitable for small datasets, so it can be replaced with a more suitable k-fold cross-validation method for large-scale customer profiling studies, thereby improving the model’s accuracy and generalization ability in practical applications. In the comparison of electricity sales package methods, comparisons with deep learning-based recommendation models or collaborative filtering techniques can be incorporated to strengthen the effectiveness of the method. Additionally, significance tests such as ANOVA or t-tests can be conducted on the results to ensure that the performance improvements are not due to randomness. Furthermore, more factors affecting the choice of electricity sales packages, such as customers’ personal preferences, family structure, and electricity usage habits, can be considered in the model to further enhance the personalization and accuracy of the recommendations.