A novel performance measurement approach based on trust context using fuzzy T-norm and S-norm operators: The case study of energy consumption

Abstract

In today’s economic environment, performance and efficiency assessment is essential for organizations in order to survive and raise their market share. Energy efficient consumption is a major issue in the energy planning of each country which is a big concern of managers, hence, exploitation of a strong approach for efficiency evaluation and assessment seems necessary in the energy section. In this study, a novel performance assessment model is proposed based on the concept of trust, using two popular fuzzy operators called T-norm and S-norm. The developed model is applied for a real case study of energy consumption efficiency assessment for 36 countries. An adaptive network based fuzzy inference system (ANFIS) is used to measure the efficiencies. Also, to predict efficiency rates of the future time periods, a regression model is applied as a time series model. The obtained results indicate the superiority and applicability of the proposed methodology. To the best of our knowledge, this is the first study that proposes a novel performance measurement approach based on trust context by using fuzzy T-norm and S-norm operators.

Keywords

Performance assessment measures adaptive network based fuzzy inference system (ANFIS)trust fuzzy t-norm and s-norm operators time series modeling linear regression

Introduction

In today’s changing world, severe competitiveness among economic entities (decision making units (DMUs)) persuades them to constantly improve their performance. In such a complex situation, managers require some measures to evaluate the current conditions of organizations in order to monitor the performance and make right decisions. Besides, in order to have a better prediction of the future trend, the available historical data of the organization’s performance should be taken into consideration. It makes the performance and efficiency assessment of any organization to be important remarkably (Azadeh et al., 2007a).

Energy is one of the key economic elements of a nation that should be considered to provide economic growth and consequently community development. Energy efficiency is the goal to reduce the amount of energy required to provide products and services which concerns energy planners. Hence, efficiency assessment of energy use is an essential issue to improve the performance of energy section, especially in developing countries.

In the field of efficiency assessment, there are a vast number of methodologies that can be applied to determine the boundaries of efficiencies (Velicer and Fava, 2003). The approach of each methodology has some positive and some negative characteristics. The main shortcoming of these approaches is providing purely numerical values in which a wide evaluation is needed to make the results meaningful and applicable (León et al., 2003). In this situation, using the concept of trust in the field of performance assessment would be appropriate to cover some of the weaknesses. It provides explicit qualitative scales rather than just providing purely numerical data which would be more tangible and apprehensible for end users. This attitude makes trust a key component in the efficiency assessment process that enhances the efficiency and effectiveness of the managers in making decisions. Before explaining the definition of trust, some concepts should be defined as follows:

Trust agent is an entity that believes in another entity for a specific context and in a specific timeslot.

Trusted agent is an entity that gives a specific context in a specific timeslot in which other entities have belief.

Timeslot is a time duration between two time pints in which the trust value for a specific context and between two entities is determined.

In the above definitions, context relates to the essence of the service given by an entity.

A comprehensive definition of trust is provided by Chang et al. (2006) as follows: “The belief that the trusting agent has in the trusted agent’s willingness and capability to deliver a mutually agreed service in a given context and in a given timeslot”.

Unlike general beliefs about the similarity of security and trust, they have different meaning; security makes a safe environment without unconfident agents, while trust aids the trusting agent to select a trustee agent. In contrast to objective measurable scales, trust is the subjective idea of trusting agent about the trusted agent which shows the possibility of continuing their interaction (Chang et al., 2006; Luo et al., 2009a).

Two major attitudes can be considered in the trust modeling: trust prediction and trust determination. Trust prediction is the process of determining trust value of an entity, either qualitatively or quantitatively, at a point of future time and trust determination is the process of establishing trust value of an entity, either qualitatively or quantitatively, either during the current time or in the past (Jamasb and Pollitt, 2001).

Prediction and forecasting have two categories: (1) prediction of whether interactions between two parts have been formed and therefore the trust between two agents will exist. (2) Predicting trust values and prediction of the level of trusting agent satisfaction (Azadeh et al., 2015b; Raza et al., 2011). Prediction and forecasting is a necessary research challenge and miscellaneous approaches have been proposed to forecast future values in different areas such as health system (Wu et al., 2010), earthquake magnitude prediction (Panakkat and Adeli, 2007), software reliability (Tian and Noore, 2004), failure time (Tahir et al., 2014), climate prediction (Pangburn, 2010), and energy (Aras, 2008; Azadeh et al., 2015a; Balat, 2004; Das et al., 2011; Fan et al., 2014; Zhao et al., 2015). A most valuable case to select a suitable partner is by predicting trustworthiness of an agent.

There are different numerical scales to quantify the amount of trust that the trust agent has in the trusted agent. By this quantification, trust value can be easily incorporated in the calculations of efficiency assessment. The numerical scale applied in the current study is demonstrated in Table 1. To deal with the human-centric information that trust values represent, fuzzy context as a frontier approach would be helpful. Hence, in the current study, two fuzzy operators, i.e. T-norm and S-norm are exploited to model trust value. One of the main aims of this study is demonstrating efficiency score as the trust values in the field of energy use efficiency and performance assessment. The proposed approach is applied to assess the energy use efficiency of the countries under study.

Table 1.

Numerical scales to quantify the amount of trust.

Scale number	Semantics	Trust value
1	Very untrustworthy	1
2	Untrustworthy	2
3	Partially trustworthy	3
4	Trustworthy	4
5	Very trustworthy	5

To the best knowledge of the authors, it is the first time that these two fuzzy operators are used in the area of trust for the performance assessment of real entities.

The remainder of this study is organized as follows: in the next section, the previous works related to this study are reviewed. In “Results” section, the proposed methodology for efficiency assessment is described. In “Discussion” section, a real case study is presented to show the application of the proposed methodology. Finally, in “Conclusions and policy implications” section, the conclusions of the study are discussed.

Literature review

The related literature can be categorized into three sections: performance assessment, energy consumption, and trust. Performance measurement and efficiency assessment is of great importance for optimal management and control activities of complex systems. There are a vast number of methodologies with different approaches in the literature which are used to determine the efficiency boundaries (Jamasb and Pollitt, 2001). These approaches are classified into parametric and non-parametric approaches. The parametric methodologies include the estimation of both deterministic and stochastic boundary functions (SFF) and they are based on the econometric regression theory which has been widely accepted in the econometrics field while the non-parametric ones comprise data envelopment analysis (DEA) and free disposal hull (FDH) which are based on a mathematical programming approach (Azadeh et al., 2011).

All the proposed methodologies have some strengths and some weaknesses. Although the efficiency frontier assessment methods are known as reliable analysis tools and they have been used widely in previous studies (Cook and Green, 2005; Golany et al., 1994; Goto and Tsutsui, 1997; Knittel, 2002; Lam and Shiu, 2001; Olatfubi and Dismukes, 2000; Park and Lesourd, 2000; Sanhueza et al., 2004; Sueyoshi and Goto, 2001), the required assumptions for some of these methods are restrictive and conflicting conclusions of efficiency are often resulted by using different methods due to the unsuitability of the assumptions (Azadeh et al., 2011). For example, the DEA frontier is so sensitive to the presence of the outliers and statistical noise which indicates that the frontier derived from DEA analysis may be warped if the data are contaminated by statistical noise (Bauer, 1990).

A review of the energy consumption literature shows that there are a vast number of studies on the energy consumption (Jorgenson et al., 2014; Wang, 2014). Some of them about energy prediction are reported here. Gürbüz et al. (2013) used classical and neural network approaches to predict energy consumption of Turkey based on artificial bee colony algorithm. Kavaklioglu (2011) used support vector regression methodology to predict and model Turkey’s electricity consumption. Azadeh et al. (2008) focused on prediction of the annual electricity consumption in energy intensive manufacturing industries. In their study, they showed that artificial neural network can be used for long-term prediction. Ekonomou (2010) described an artificial neural network method for the long-term prediction of Greek energy consumption.

Trust is a concept that has been taken into consideration in a number of areas especially electronic commercial fields. In this context, making decisions depends on the expectations created due to the information of former behaviors that is called reputation (Carbo et al., 2003). Abdul-Rahman and Hailes (2000) prepared a trust model based on a reputation mechanism, or word-of-mouth that would allow agents to decide which other agents’ opinions they trust more and would allow agents to progressively adapt their understanding of another agent’s subjective recommendations. Azadeh et al. (2015c) presented an efficient model for the analysis of outputs of performance measurement methodologies, using trust that provides explicit qualitative scales rather than pure numerical data. They introduced implementing the concept of trust in terms of performance assessment and measurement. Singh and Sinha (2010) presented a statistical predictive Trust model based on the Local Learning technique and famous methodologies of the Markov model. Carbo et al. (2003) developed a trust management mechanism tackling with uncertain information to apply it for a multi-agent system of merchants, recommenders, and buyers, using fuzzy set theory. They represented the benefits of using fuzzy sets to manage reputation in multi-agent systems.

Luo et al. (2009b) developed a model of trust based on fuzzy recommendation similarity, to evaluate and to quantify the trustworthiness of nodes, which consists of five kinds of fuzzy trust recommendation relationships. Siegrist et al. (2003) proposed a dual-mode model of social trust and confidence which was examined in the applied context of electromagnetic field risks. Structural equation modeling procedures and data from a random sample of 1,313 Swiss citizens between 18 and 74 years old were exploited. The obtained results delineated that the dual-mode model of social trust and confidence could be used as a common framework in the field of trust and risk management. Guo et al. (2011) presented a novel extensible trust evaluation model. The contribution of their paper is a formal general Extensible Trust Evaluation Model for Cloud computing environments (ETEC Model) with a method of comprehensive time-variant evaluation for calculating direct trust and a method of space-variant evaluation for expressing recommended trust. Teacy et al. (2005) developed a model of trust and reputation to ensure good interactions among software agents in large scale open systems. To compute trust, probability theory was applied with respect to the past interactions between agents. In the absence of sufficient personal experiences, the model used reputation information gathered from third parties. Kim et al. (2010) developed a model of trust for efficient allocation and reconfiguration of computing resources satisfying various user requests. They analyzed reliability based on historical information of servers in a Cloud data center.

Wang et al. (2012) presented a general model of trust, called RLM; they applied the Robust Linear Markov model to trust aggregation and representation. Huynh et al. (2006) proposed a trust and reputation model called FIRE which integrated a number of information sources to produce a comprehensive assessment of an agent’s likely performance in open systems. The model incorporated interaction trust, role-based trust, witness reputation, and certified reputation to provide trust metrics in most circumstances.

An overview of the literature shows that there is no study that uses trust in the field of performance assessment. Forasmuch as most determinant agents are probable, absence of a trust approach was felt in the field of determination of assessment agents and assessment approaches of energy consumption. For this reason one of the main aims of this paper is presentation of a trust value approach in energy consumption and efficiency prediction. Hence, in the current study, a trust based model is proposed for the efficiency assessment problem using fuzzy T-norm and S-norm operators. The proposed approach covers some of the shortcomings of the previous mentioned assessment techniques through preparing explicit qualitative scales rather than just providing purely numerical data. To show the applicability and superiority of the methodology, it is applied for a real case study of efficiency assessment for energy consumption in 36 countries. The features of this study are compared with previous studies and depicted in Table 2.

Table 2.

Comparison between studies of energy consumption.

Study	Methodology	Predictive approach	Qualitative consideration	Practicability	Trust value	Confidence level
This study	Fuzzy, T-norm and S-norm, ANFIS	✓	✓	✓	✓	✓
Li et al. (2012)	grey theory, Adaptive grey forecasting	✓		✓
Gürbüz et al. (2013)	Artificial neural network, Linear and nonlinear regression, artificial bee colony	✓		✓
Azadeh et al. (2007b)	Integration of artificial neural networks and genetic algorithm	✓		✓
Ekonomou (2010)	Artificial neural networks	✓		✓
Azadeh et al. (2008)	Artificial neural networks, the mean absolute percentage error (MAPE), ANOVA and Duncan’s multiple range test	✓		✓

Methodology

For the efficiency assessment, a methodology is developed based on fuzzy T-norm and S-norm operators as illustrated in Figure 1. The main steps and sub-steps of the methodology are as follows:

Figure 1.

T-norm and S-norm calculation.

Step 1: Data collection

In order to characterize the structure of the problem, the scenarios should be determined with regard to the number of DMUs and the duration of each timeslot. On the basis of the number of DMUs, the following classes of data sets are defined:

Class 1: Very small data sets which contain 10 DMUs

This type of data sets is defined for a very small organization or an organization with very little importance for the analyzer.

Class 2: Small data sets which contain 20 DMUs

This type of data sets is defined for a small organization or an organization with low importance for the analyzer.

Class 3: Medium data sets which contain 30 DMUs

This type of data sets is defined for a medium size organization or an organization with mediocre importance for the analyzer.

Class 4: Large data sets which contain 40 DMUs

This type of data sets is defined for a large size organization or an organization with high importance for the analyzer.

Class 5: Very large data sets which contain 50 DMUs

This type of data sets is defined for a very large size organization or an organization with very high importance for the analyzer. On the basis of the duration of the timeslots, the following classes of data sets are defined:

Class 1: Short-term data set

This type of data sets contains the efficiency rate of 10 years (T = 10).

Class 2: Middle-term data set

This type of data sets contains the efficiency rate of 20 years (T = 20).

Class 3: Long-term data set

This type of data sets contains the efficiency rate of 40 years (T = 40).

According to the mentioned data sets for timeslots and DMUs, 15 scenarios are defined as shown in Table 3.

Table 3.

Fifteen different scenarios.

Number of DMUs	Timeslot
	10 years	20 years	40 years
	(Short-term)	(Middle-term)	(Long-term)
10 DMUs	Short-term analysis with very small size set	Middle-term analysis with very small size set	Long-term analysis with very small size set
20 DMUs	Short-term analysis with small size set	Middle-term analysis with small size set	Long-term analysis with small size set
30 DMUs	Short-term analysis with medium size set	Middle-term analysis with medium size set	Long-term analysis with medium size set
40 DMUs	Short-term analysis with large size set	Middle-term analysis with large size set	Long-term analysis with large size set
50 DMUs	Short-term analysis with very large size set	Middle-term analysis with very large size set	Long-term analysis with very large size set

Step 2: Efficiency calculations

To measure each DMU’s efficiency rate, an input oriented non-parametric efficiency frontier analysis method based on adaptive network-based fuzzy inference system (ANFIS) is applied. To implement this method, following steps are necessary:

Determine input(s) and output(s) for each DMU.

Use ANFIS method to estimate relation between input(s) and output(s).

Calculate weigh of DMUi (W_i) as follows:

Calculate the error between the real output ( $P real (i)$ ) and ANFIS model output ( $P ANFIS * (i)$ ) in the selected period as follows:

E i = P real (i) - P ANFIS (i) i = 1, \dots, n

(3)

Shift frontier function from ANFIS for obtaining the effect of the largest positive error which is one of the unique features of this algorithm:

The largest $E' i$ indicate the DMU with the best performance. Suppose that DMU _k have the largest $E' i$ and we have:

E' k = max (E' i)

(5)

Therefore, the value of the shift for each of the DMUs is different and is calculated by:

The efficiency scores take values between 0 and 1 which are calculated as follows:

Step 3: Prediction

Efficiency rates of the current, previous, and following years (predicted), along with the average efficiency and standard deviation, are the five main inputs in trust modeling, however, in different situations their relation with final trust value varies. To predict the efficiency of the following years will be calculated using linear regression as a time series modeling.

The trust value of a specific year depends on a number of factors, including the efficiency rates of the current and previous years, the average efficiency rate of the timeslot, and the standard deviation of the timeslot. The required efficiency rates are calculated in the prior step using ANFIS method. After calculating the average efficiency rate and the standard deviation of the timeslots, the efficiency rate of the following year is predicted by an appropriate regression model using Wolfram Mathematica 9.0.

Time series analysis is a statistical method of analyzing data from repeated observations of a single unit or individual at regular intervals over a large number of observations; and time series forecasting is the use of a model to predict future values based on previously observed values (Mirtalaei et al., 2012).

In statistics, the definition of linear regression is a method to model the relationship between a numeral dependent variable y and one or more expositive variables X. One expositive variable is named simple linear regression. For more than one expositive variable, it is named multiple linear regression (This case differs from multivariate linear regression where multiple dependent variables are predicted rather than one scalar variable).

In linear regression, datum is modeled using linear predicting functions, and unclear model parameters are guessed and estimated from the datum. Such models are named linear models. Most widely, linear regression points to a model in which the contingent mean of y given the quantity of X is an affine function of X. Less widely, linear regression could point to a model in which the median, or some other quintile of the contingent distribution of y given X is declared as a linear function of X. Similar to all types of regression analysis, linear regression concentrates on the conditional probability distribution of y given X, rather than on the joint probability distribution of y and X, which is the amplitude of multivariate analysis.

Step 4: T-norm and S-norm

In this step, two fuzzy operators, i.e. S-norm and T-norm are applied to model trust in the assessment of efficiency rates. These two operators have some characteristics as follows:

The operator ^ is an S-norm, if it satisfies four conditions:

Commutativity: $x \circ y = y \circ x$

Monotony: $if x \leq y \Rightarrow x \circ z \leq y \circ z$

Associativity: $x \circ (y \circ z) = (x \circ y) \circ z$

Neutrality of 0: $0 \circ x = x for x \in [0, 1]$

And the operator ^ is a T-norm, if it satisfies four conditions:

Commutativity: $x \circ y = y \circ x$

Monotony: $if x \leq y \Rightarrow x \circ z \leq y \circ z$

Associativity: $x \circ (y \circ z) = (x \circ y) \circ z$

Neutrality of 1: $1 \circ x = x for x \in [0, 1]$

Suppose that the efficiency rate of the current year is E_t, the previous year

E t - 1

, the following year

E t + 1

, the average efficiency rate

E ave

and standard deviation SD. The T-norms and S-norms operators are shown in Table 4.

Table 4.

T-norms and S-norms.

T-norm	Formula	S-norm	Formula
Minimum	$Min {a, b}$	Maximum	$Max {a, b}$
Product	$a . b$	Probabilistic sum	$a + b - a . b$
Lukasiewicz	$Max {0, a + b - 1}$	Bounded sum	$Min {1, a + b}$
Nilpotent minimum	${Min (a, b) a + b > 1 0 otherwise$	Nilpotent maximum	${Max (a, b) a + b < 1 1 otherwise$
Hamacher product	$\frac{a . b}{a + b - a . b}$	Einstein sum	$\frac{a + b}{1 + a . b}$

Then, the modified efficiency rate (M.E) is calculated as shown in Figure 1. The conceptual framework of this step is illustrated in Figure 2.

Figure 2.

Conceptual framework of step 4.

Step 5: Trust value calculation

This step consists of two main steps as follows:

Step 5.1: Initial trust value calculation

The M.E which was computed in step 4 is used to calculate the trust value. The value can be proposed in the scale, as shown in Table 1. Then, the procedure depicted in Figure 3 is applied to calculate the trust value.

Figure 3.

Initial trust value calculation.

Step 5.2: Confidence level computation and determining modified trust value

Maturity, tendency, and density are here metrics for the confidence level that will be used in order to calculate the modified trust value. Each of them can have a value of 1, 2 or 3, where 1 is the lowest value, 2 is the normal value, and 3 is the highest one.

Maturity (M): Maturity refers to the total life span of an entity for which efficiency rates are provided (Raza et al., 2011).

Tendency (Te): Tendency is a metric projecting the time to which the largest data belongs. It means that for each data set, we determine the k-th largest data (k = 1,…,j) and then examine the year to which the data belongs (Raza et al., 2011).

Density (D): Density is the metric that compares the mean and median of the data set. It determines whether or not our data set contains any low rates. Imagine the median is larger than mean value; this means that there a number of data whose values are very low that have decreased the value of mean (Raza et al., 2011). Then, the confidence level is calculated as follows:

ConfidenceLevel = \frac{M + Te + D}{Max (M) + Max (Te) + Max (D)}

(8)

ConfidenceLevel = \frac{M + Te + D}{3 + 3 + 3}

(9)

This value to determine the modified trust value as: Modified Trust Value = Confidence Level × Trust Value. Figure 4 projects the mentioned procedure.

Figure 4.

Modified trust value determination.

Results

To show the applicability of the proposed methodology, assessment of energy consumption efficiency is implemented as a real case study.

Step 1

Table 5 includes energy consumptions data for 36 different countries.

Table 5.

Energy consumptions.

Country	Index	2008	2009	2010	2011
Argentina	Energy use	0.996822532	0.966334636	0.984222769	1
	Population	0.97415449	0.982687973	0.991295728	1
	GDP	0.732176029	0.688620514	0.826680779	1
Australia	Energy use	5729.344288	5606.73274	5552.251454	5504.757368
	Population	21384400	21778800	22065300	22323900
	GDP	1.05226E + 12	9.23499E + 11	1.13826E + 12	1.38415E + 12
Brazil	Energy use	1296.343071	1242.767653	1362.056423	1371.149594
	Population	191765567	193490922	195210154	196935134
	GDP	1.65354E + 12	1.62017E + 12	2.14304E + 12	2.47665E + 12
Canada	Energy use	7945.464211	7451.792937	7354.746497	7303.252859
	Population	33317662	33726915	34126547	34483975
	GDP	1.50268E + 12	1.33758E + 12	1.57704E + 12	1.77779E + 12
Chile	Energy use	1800.816389	1735.225415	1802.835093	1939.720133
	Population	16831184	16991729	17150760	17308449
	GDP	1.79627E + 11	1.71727E + 11	2.17312E + 11	2.50994E + 11
China	Energy use	1601.031137	1717.272837	1881.379825	2029.362967
	Population	1324655000	1331260000	1337705000	1344130000
	GDP	4.52183E + 12	4.99126E + 12	5.94979E + 12	7.31443E + 12
Colombia	Energy use	646.8650381	672.7863754	694.0508817	671.4911462
	Population	45153037	45802561	46444798	47078792
	GDP	2.43982E + 11	2.33822E + 11	2.87018E + 11	3.36346E + 11
Germany	Energy use	4075.429664	3824.665928	4032.548446	3811.472082
	Population	82110097	81902307	81776930	81797673
	GDP	3.62369E + 12	3.29864E + 12	3.28447E + 12	3.60083E + 12
Denmark	Energy use	3494.903271	3323.882714	3480.25491	3230.788149
	Population	5493621	5523095	5547683	5570572
	GDP	3.43881E + 11	3.10545E + 11	3.13366E + 11	3.33616E + 11
Egypt, Arab Rep.	Energy use	954.4086452	930.8428211	942.3598673	978.0366565
	Population	75491922	76775023	78075705	79392466
	GDP	1.62818E + 11	1.88984E + 11	2.18888E + 11	2.35984E + 11
Spain	Energy use	3051.501441	2782.319842	2772.864891	2719.471144
	Population	45555716	45908594	46070971	46174601
	GDP	1.59342E + 12	1.45596E + 12	1.38011E + 12	1.47688E + 12
France	Energy use	4113.695325	3917.427607	4015.872526	3867.530238
	Population	64371099	64702921	65031235	65371613
	GDP	2.83179E + 12	2.61969E + 12	2.54832E + 12	2.77972E + 12
Ghana	Energy use	387.6158426	393.396662	412.609399	425.0354925
	Population	23110139	23691533	24262901	24820706
	GDP	28528046011	25977853492	32174210793	39564970070
Greece	Energy use	2706.980292	2608.921487	2442.21332	2364.846793
	Population	11237094	11282760	11307502	11299976
	GDP	3.41594E + 11	3.21016E + 11	2.92305E + 11	2.89627E + 11
Indonesia	Energy use	796.627581	841.2317566	877.9259553	857.2892162
	Population	234243489	237486894	240676485	243801639
	GDP	5.10245E + 11	5.3958E + 11	7.09266E + 11	8.46483E + 11
India	Energy use	538.8408308	586.7887972	600.3055621	613.7188502
	Population	1174662334	1190138069	1205624648	1221156319
	GDP	1.2241E + 12	1.36537E + 12	1.71092E + 12	1.87285E + 12
Ireland	Energy use	3367.370189	3221.403642	3177.940915	2887.699301
	Population	4425683	4458942	4474356	4576748
	GDP	2.6203E + 11	2.24042E + 11	2.05904E + 11	2.20823E + 11
Iran, Islamic Rep.	Energy use	2811.863073	2902.015617	2829.319822	2812.692225
	Population	72660887	73542954	74462314	75424285
	GDP	3.55988E + 11	3.62661E + 11	4.22568E + 11	5.1406E + 11
Italy	Energy use	2941.635988	2738.843472	2814.63883	2757.011417
	Population	59832179	60192698	60483385	60723569
	GDP	2.30731E + 12	2.11115E + 12	2.04195E + 12	2.19236E + 12
Japan	Energy use	3878.906008	3701.641139	3915.965787	3610.371413
	Population	127704040	127557958	127450459	127817277
	GDP	4.84921E + 12	5.03514E + 12	5.49538E + 12	5.89679E + 12
Korea, Rep.	Energy use	4636.383276	4659.794396	5058.965918	5231.916973
	Population	48949000	49182000	49410000	49779000
	GDP	9.31402E + 11	8.3406E + 11	1.01489E + 12	1.11447E + 12
Kuwait	Energy use	10673.77465	10811.72639	10892.60959	10408.2795
	Population	2702221	2850102	2991580	3124705
	GDP	1.47402E + 11	1.05911E + 11	1.19935E + 11	1.60913E + 11
Malaysia	Energy use	2673.993643	2513.747591	2569.169717	2639.431881
	Population	27302348	27790324	28275835	28758968
	GDP	2.30988E + 11	2.02251E + 11	2.46823E + 11	2.87934E + 11
Netherlands	Energy use	4837.166893	4729.159957	5020.998479	4637.767016
	Population	16445593	16530388	16615394	16693074
	GDP	8.70811E + 11	7.96333E + 11	7.74658E + 11	8.36074E + 11
Norway	Energy use	6250.801558	6166.249441	6614.177792	5680.652555
	Population	4768212	4828726	4889252	4953088
	GDP	4.53885E + 11	3.78849E + 11	4.21236E + 11	4.91065E + 11
Oman	Energy use	7201.99551	6863.480128	8262.49372	8356.291743
	Population	2593523	2663224	2802768	3024774
	GDP	60566970579	46866060196	58813004375	69971912138
Peru	Energy use	525.5568192	594.4341566	656.348822	694.9951556
	Population	28625628	28934303	29262830	29614887
	GDP	1.26823E + 11	1.26923E + 11	1.53545E + 11	1.76812E + 11
Portugal	Energy use	2299.879415	2271.501706	2213.028701	2186.622069
	Population	10622413	10632482	10637346	10556999
	GDP	2.51925E + 11	2.34119E + 11	2.27447E + 11	2.37574E + 11
Paraguay	Energy use	697.2287546	705.1328083	741.2928515	739.0061336
	Population	6236005	6347383	6459721	6573097
	GDP	18504760911	15954961410	20028375554	26007966687
Saudi Arabia	Energy use	5843.643707	6555.936876	7043.845991	6738.415779
	Population	26366358	26796375	27258387	27761728
	GDP	5.19797E + 11	4.29098E + 11	5.26811E + 11	6.69507E + 11
Singapore	Energy use	5199.457164	5666.540821	6752.33636	6452.326331
	Population	4839400	4987600	5076700	5183700
	GDP	1.78924E + 11	1.94131E + 11	2.172E + 11	2.45024E + 11
Sweden	Energy use	5379.959103	4883.20049	5471.798737	5190.339026
	Population	9219637	9298515	9378126	9449213
	GDP	4.86159E + 11	4.05783E + 11	4.62903E + 11	5.39278E + 11
Thailand	Energy use	1626.579013	1618.958019	1768.443378	1789.636774
	Population	66185340	66277335	66402316	66576332
	GDP	2.72578E + 11	2.63711E + 11	3.18908E + 11	3.45672E + 11
Turkey	Energy use	1399.897839	1370.847901	1457.397913	1539.293382
	Population	70363511	71241080	72137546	73058638
	GDP	7.30337E + 11	6.14554E + 11	7.31144E + 11	7.74775E + 11
United States	Energy use	7487.930303	7055.603934	7162.354598	7032.346172
	Population	304093966	306771529	309326225	311587816
	GDP	1.42193E + 13	1.38983E + 13	1.44194E + 13	1.49913E + 13
Venezuela, RB	Energy use	2474.984666	2434.89293	2599.640819	2379.528027
	Population	28120312	28583040	29043283	29500625
	GDP	3.156E + 11	3.29419E + 11	3.93807E + 11	3.16482E + 11

Step 2

The efficiency rates which are calculated based on the algorithm explained in step 2 are reported in Table 6.

Table 6.

Efficiency rates by ANFIS.

DMU	Country	2008	2009	2010	2011
1	Argentina	0.69979718	0.75530931	0.85171943	0.94831806
2	Australia	0.73803089	0.81080753	0.86814833	0.9180371
3	Brazil	0.66732878	0.66550883	0.84227248	0.90525207
4	Canada	0.7853405	0.69555026	0.73654803	0.77146633
5	Chile	0.7164953	0.6863725	0.83713816	1
6	China	0.52546652	0.64936243	0.83315628	1
7	Colombia	0.68387579	0.78135711	0.87876047	0.97261283
8	Germany	0.69868406	0.91764763	0.71216768	0.88610382
9	Denmark	0.94127579	0.91181344	0.94439495	0.82105959
10	Egypt, Arab Rep.	0.64288764	0.74455104	0.84055108	0.9329499
11	Spain	0.91163125	0.8887266	0.88893286	0.88454948
12	France	0.92432381	0.9178494	0.92719652	0.91680967
13	Ghana	0.64774423	0.78365103	0.96970789	1
14	Greece	0.9560605	0.96058059	0.86496533	0.79149938
15	Indonesia	0.64057922	0.91220368	1	0.98727088
16	India	0.67191388	0.69173094	0.69950142	0.70724455
17	Ireland	0.96701008	0.88303528	0.86331294	0.73107439
18	Iran, Islamic Rep.	0.66790359	0.77372157	0.83927207	0.92374589
19	Italy	0.93029298	0.88800383	0.9075052	0.89626426
20	Japan	0.63232092	0.91080816	0.64997982	1
21	Korea, Rep.	0.93943499	0.9272075	0.98287511	0.98775818
22	Kuwait	0.73662278	0.75108585	0.76886641	1
23	Malaysia	0.78354597	0.77404491	0.86076567	0.96988218
24	Netherlands	1	0.88186596	0.81128549	0.84327312
25	Norway	0.85116119	0.85148771	0.8736283	0.83879004
26	Oman	0.62751217	0.65045148	0.80804909	0.95754171
27	Peru	0.6897782	0.80284139	0.91025888	1
28	Portugal	0.90061133	0.91736168	0.92850915	0.80926076
29	Paraguay	0.61645725	0.71542944	0.79130855	0.90114206
30	Saudi Arabia	0.67389943	0.72519796	0.781097	0.8432118
31	Singapore	0.79188588	0.86806052	0.89508631	0.95779284
32	Sweden	0.88047075	0.86034438	0.82180942	0.81259972
33	Thailand	0.8058424	0.80226472	1	1
34	Turkey	0.8267169	0.83768529	0.97879349	1
35	United States	0.95700689	0.95880648	0.96607936	0.96169198
36	Venezuela, RB	0.55118976	0.60118432	0.62960794	0.6969294

Step 3

Predicted efficiency rates of the next year are presented in Table 7.

Table 7.

Predicted efficiency rates.

DMU	Country	Predicted for 2012
1	Argentina	1
2	Australia	0.983096
3	Brazil	0.992724
4	Canada	0.74707
5	Chile	1
6	China	1
7	Colombia	1
8	Germany	0.892846
9	Denmark	0.822619
10	Egypt, Arab Rep.	1
11	Spain	0.8732
12	France	0.918246
13	Ghana	1
14	Greece	0.745952
15	Indonesia	1
16	India	0.721038
17	Ireland	0.679226
18	Iran, Islamic Rep.	1
19	Italy	0.88487
20	Japan	1
21	Korea, Rep.	1
22	Kuwait	1
23	Malaysia	1
24	Netherlands	0.748916
25	Norway	0.850024
26	Oman	1
27	Peru	1
28	Portugal	0.82321
29	Paraguay	0.988568
30	Saudi Arabia	0.896811
31	Singapore	1
32	Sweden	0.783269
33	Thailand	1
34	Turkey	1
35	United States	0.966228
36	Venezuela, RB	0.736138

For example, the fitted regression model for Argentina is developed using the efficiency rates of the previous four years from Table 6. The related results obtained by Wolfram Mathematica 9.0 are illustrated in Figure 5.

Figure 5.

The fitted regression model for Argentina.

As Table 8 depicts, the analysis of variance (ANOVA) of the parameter estimates for this DMU indicates that the fitted model is acceptable at 99% level of confidence level:

Table 8.

ANOVA results.

	Estimate	Standard error	p value
b	0.603292	0.019459	0.001038
a	0.084197	0.007105	0.007046

Step 4

The average and the standard deviation for DMUs are presented in Table 9.

Table 9.

Average and standard deviation for DMUs.

DMU	Country	Average	Standard deviation
1	Argentina	0.8138	0.1095
2	Australia	0.8338	0.0774
3	Brazil	0.7701	0.1224
4	Canada	0.7472	0.0401
5	Chile	0.8100	0.1424
6	China	0.7520	0.2081
7	Colombia	0.8292	0.1244
8	Germany	0.8037	0.1143
9	Denmark	0.9046	0.0576
10	Egypt, Arab Rep.	0.7902	0.1248
11	Spain	0.8935	0.0123
12	France	0.9215	0.0050
13	Ghana	0.8503	0.1655
14	Greece	0.8933	0.0809
15	Indonesia	0.8850	0.1675
16	India	0.6926	0.0152
17	Ireland	0.8611	0.0977
18	Iran, Islamic Rep.	0.8012	0.1080
19	Italy	0.9055	0.0183
20	Japan	0.7983	0.1852
21	Korea, Rep.	0.9593	0.0305
22	Kuwait	0.8141	0.1246
23	Malaysia	0.8471	0.0906
24	Netherlands	0.8841	0.0825
25	Norway	0.8538	0.0145
26	Oman	0.7609	0.1537
27	Peru	0.8507	0.1342
28	Portugal	0.8889	0.0543
29	Paraguay	0.7561	0.1203
30	Saudi Arabia	0.7559	0.0729
31	Singapore	0.8782	0.0687
32	Sweden	0.8438	0.0320
33	Thailand	0.9020	0.1131
34	Turkey	0.9108	0.0913
35	United States	0.9609	0.0040
36	Venezuela, RB	0.6197	0.0608

Step 5

Modified efficiency and initial trust value of T-norms/S-norms for DMUs are shown in Table 10. Confidence levels for each pair of T-norm and S-norm are provided in Table 11. Finally, the modified trust values are calculated in Table 12.

Table 10.

Modified efficiency and initial trust value of T-norms/S-norms.

DMU	Country	Min Max		Product probabilistic		Lukasiewicz bounded		Nilpotent		Hamacher Einstein
DMU	Country	M.E.	Trust value	M.E.	Trust value	M.E.	Trust value	M.E.	Trust value	M.E.	Trust value
1	Argentina	0.8989	4	0.8989	4	0.8989	4	0.9312	4	0.9057	4
2	Australia	0.8890	4	0.8890	4	0.8890	4	0.9423	4	0.8993	4
3	Brazil	0.8588	4	0.8588	4	0.8588	4	0.9189	4	0.8732	4
4	Canada	0.7549	3	0.7549	3	0.7549	3	0.8857	4	0.8016	3
5	Chile	0.9237	4	0.9237	4	0.9237	4	0.9237	4	0.9237	4
6	China	0.8989	4	0.8989	4	0.8989	4	0.8989	4	0.8989	4
7	Colombia	0.9144	4	0.9144	4	0.9144	4	0.9315	4	0.9180	4
8	Germany	0.8434	3	0.8434	3	0.8434	3	0.9280	4	0.8650	4
9	Denmark	0.8291	3	0.8291	3	0.8291	3	0.9105	4	0.8546	4
10	Egypt, Arab Rep.	0.8810	4	0.8810	4	0.8810	4	0.9229	4	0.8906	4
11	Spain	0.8826	4	0.8826	4	0.8826	4	0.9423	4	0.8966	4
12	France	0.9170	4	0.9170	4	0.9170	4	0.9584	5	0.9244	4
13	Ghana	0.9274	4	0.9274	4	0.9274	4	0.9274	4	0.9274	4
14	Greece	0.7932	3	0.7932	3	0.7932	3	0.8957	4	0.8289	3
15	Indonesia	0.9262	4	0.9262	4	0.9262	4	0.9342	4	0.9278	4
16	India	0.7027	2	0.7027	2	0.7027	2	0.8536	4	0.7630	3
17	Ireland	0.7361	2	0.7361	2	0.7361	2	0.8655	4	0.7874	3
18	Iran, Islamic Rep.	0.8810	4	0.8810	4	0.8810	4	0.9287	4	0.8913	4
19	Italy	0.8930	4	0.8930	4	0.8930	4	0.9481	4	0.9049	4
20	Japan	0.9126	4	0.9126	4	0.9126	4	0.9126	4	0.9126	4
21	Korea, Rep.	0.9749	5	0.9749	5	0.9749	5	0.9825	5	0.9753	5
22	Kuwait	0.9282	4	0.9282	4	0.9282	4	0.9282	4	0.9282	4
23	Malaysia	0.9238	4	0.9238	4	0.9238	4	0.9426	4	0.9272	4
24	Netherlands	0.8235	3	0.8235	3	0.8235	3	0.9216	4	0.8528	4
25	Norway	0.8408	3	0.8408	3	0.8408	3	0.9194	4	0.8636	4
26	Oman	0.8844	4	0.8844	4	0.8844	4	0.9110	4	0.8912	4
27	Peru	0.9341	4	0.9341	4	0.9341	4	0.9341	4	0.9341	4
28	Portugal	0.8188	3	0.8188	3	0.8188	3	0.9046	4	0.8466	3
29	Paraguay	0.8531	4	0.8531	4	0.8531	4	0.9163	4	0.8687	4
30	Saudi Arabia	0.8143	3	0.8143	3	0.8143	3	0.9216	4	0.8425	3
31	Singapore	0.9281	4	0.9281	4	0.9281	4	0.9545	5	0.9321	4
32	Sweden	0.8100	3	0.8100	3	0.8100	3	0.9063	4	0.8413	3
33	Thailand	0.9500	4	0.9500	4	0.9500	4	0.9500	4	0.9500	4
34	Turkey	0.9569	5	0.9569	5	0.9569	5	0.9569	5	0.9569	5
35	United States	0.9611	5	0.9611	5	0.9611	5	0.9808	5	0.9629	5
36	Venezuela, RB	0.6731	2	0.6731	2	0.6731	2	0.8485	3	0.7411	2

Table 11.

Confidence levels for each pair of T-norm and S-norm.

DMU	Country	Maturity	Tendency	Density	Confidence level
1	Argentina	1	3	3	0.7778
2	Australia	1	3	1	0.5556
3	Brazil	1	3	3	0.7778
4	Canada	1	2	1	0.4444
5	Chile	1	3	3	0.7778
6	China	1	3	3	0.7778
7	Colombia	1	3	1	0.5556
8	Germany	1	3	3	0.7778
9	Denmark	1	2	1	0.4444
10	Egypt, Arab Rep.	1	3	1	0.5556
11	Spain	1	2	3	0.6667
12	France	1	2	3	0.6667
13	Ghana	1	3	1	0.5556
14	Greece	1	2	1	0.4444
15	Indonesia	1	3	1	0.5556
16	India	1	3	1	0.5556
17	Ireland	1	2	1	0.4444
18	Iran, Islamic Rep.	1	3	1	0.5556
19	Italy	1	2	3	0.6667
20	Japan	1	3	3	0.7778
21	Korea, Rep.	1	3	1	0.5556
22	Kuwait	1	3	3	0.7778
23	Malaysia	1	3	3	0.7778
24	Netherlands	1	2	3	0.6667
25	Norway	1	3	3	0.7778
26	Oman	1	3	3	0.7778
27	Peru	1	3	1	0.5556
28	Portugal	1	3	1	0.5556
29	Paraguay	1	3	3	0.7778
30	Saudi Arabia	1	3	3	0.7778
31	Singapore	1	3	1	0.5556
32	Sweden	1	2	3	0.6667
33	Thailand	1	3	1	0.5556
34	Turkey	1	3	3	0.7778
35	United States	1	3	3	0.7778
36	Venezuela, RB	1	3	3	0.7778

Table 12.

Modified trust values.

Country (DMU)	Min Max		Product probabilistic		Lukasiewicz bounded		Nilpotent		Hamacher Einstein		Output
Argentina	3.11	T	3.11	T	3.11	T	3.11	T	3.11	T	Trustworthy
Australia	2.22	PT	2.22	PT	2.22	PT	2.22	PT	2.22	PT	Partially trustworthy
Brazil	3.11	T	3.11	T	3.11	T	3.11	T	3.11	T	Trustworthy
Canada	1.33	U	1.33	U	1.33	U	1.78	U	1.33	U	Untrustworthy
Chile	3.11	T	3.11	T	3.11	T	3.11	T	3.11	T	Trustworthy
China	3.11	T	3.11	T	3.11	T	3.11	T	3.11	T	Trustworthy
Colombia	2.22	PT	2.22	PT	2.22	PT	2.22	PT	2.22	PT	Partially trustworthy
Germany	2.33	PT	2.33	PT	2.33	PT	3.11	T	3.11	T	Partially trustworthy
Denmark	1.33	U	1.33	U	1.33	U	1.78	U	1.78	U	Untrustworthy
Egypt	2.22	PT	2.22	PT	2.22	PT	2.22	PT	2.22	PT	Partially trustworthy
Spain	2.67	PT	2.67	PT	2.67	PT	2.67	PT	2.67	PT	Partially trustworthy
France	2.67	PT	2.67	PT	2.67	PT	3.33	T	2.67	PT	Partially trustworthy
Ghana	2.22	PT	2.22	PT	2.22	PT	2.22	PT	2.22	PT	Partially trustworthy
Greece	1.33	U	1.33	U	1.33	U	1.78	U	1.33	U	Untrustworthy
Indonesia	2.22	PT	2.22	PT	2.22	PT	2.22	PT	2.22	PT	Partially trustworthy
India	1.11	U	1.11	U	1.11	U	2.22	PT	1.67	U	Untrustworthy
Ireland	0.89	VU	0.89	VU	0.89	VU	1.78	U	1.33	U	Very untrustworthy
Iran	2.22	PT	2.22	PT	2.22	PT	2.22	PT	2.22	PT	Partially trustworthy
Italy	2.67	PT	2.67	PT	2.67	PT	2.67	PT	2.67	PT	Partially trustworthy
Japan	3.11	T	3.11	T	3.11	T	3.11	T	3.11	T	Trustworthy
Korea	2.78	PT	2.78	PT	2.78	PT	2.78	PT	2.78	PT	Partially trustworthy
Kuwait	3.11	T	3.11	T	3.11	T	3.11	T	3.11	T	Trustworthy
Malaysia	3.11	T	3.11	T	3.11	T	3.11	T	3.11	T	Trustworthy
Netherlands	2.00	PT	2.00	PT	2.00	PT	2.67	PT	2.67	PT	Partially trustworthy
Norway	2.33	PT	2.33	PT	2.33	PT	3.11	T	3.11	T	Partially trustworthy
Oman	3.11	T	3.11	T	3.11	T	3.11	T	3.11	T	Trustworthy
Peru	2.22	PT	2.22	PT	2.22	PT	2.22	PT	2.22	PT	Partially trustworthy
Portugal	1.67	U	1.67	U	1.67	U	2.22	PT	1.67	U	Untrustworthy
Paraguay	3.11	T	3.11	T	3.11	T	3.11	T	3.11	T	Trustworthy
Saudi Arabia	2.33	PT	2.33	PT	2.33	PT	3.11	T	2.33	PT	Partially trustworthy
Singapore	2.22	PT	2.22	PT	2.22	PT	2.78	PT	2.22	PT	Partially trustworthy
Sweden	2.00	PT	2.00	PT	2.00	PT	2.67	PT	2.00	PT	Partially trustworthy
Thailand	2.22	PT	2.22	PT	2.22	PT	2.22	PT	2.22	PT	Partially trustworthy
Turkey	3.89	T	3.89	T	3.89	T	3.89	T	3.89	T	Trustworthy
United States	3.89	T	3.89	T	3.89	T	3.89	T	3.89	T	Trustworthy
Venezuela	1.56	U	1.56	U	1.56	U	2.33	PT	1.56	U	Untrustworthy

Discussion

Some significant facts can be inferred from the obtained results. In this study, five different classes of T-norm and S-norm operators (i.e. Min Max, product probabilistic, Lukasiewicz bounded, nilpotent, and Hamacher Einstein) were applied for the trust value computations. These five sets were taken into consideration due to the role of personality in trust value calculation. As can be seen in Table 12, the results obtained by the first three sets are exactly the same, while Nilpotent and Hamacher Einstein sets provide different values which are greater in some cases. Also, the results of Nilpotent are the largest ones. Figure 6 represents the mentioned outcomes as well. It should be noted that the first three sets are different in calculating F₁, F₂, S(F₂, E_t), and T(F₂, E_t) but the M.E. rates are the same.

Figure 6.

Modified trust values of different T-norms/S-norms.

As shown in Table 13, the highest range for trust value is related to partially trustworthy, which is followed by trustworthy, untrustworthy, very untrustworthy, and very trustworthy. It is considerable that no value for very trustworthy range is generated.

Table 13.

The number of trust value ranges for different T-norms /S-norms.

Trust value range	Min Max	Product probabilistic	Lukasiewicz bounded	Nilpotent	Hamacher Einstein
Very untrustworthy	1	1	1	0	0
Untrustworthy	6	6	6	4	7
Partially trustworthy	18	18	18	17	16
Trustworthy	11	11	11	15	13
Very trustworthy	0	0	0	0	0
Sum	36	36	36	36	36

Conclusions and policy implications

In this paper, a trust-based model for performance assessment which provides explicit qualitative scales instead of representing purely numerical data was proposed to determine the trust value in energy consumption efficiency. The developed model was used for real data of 36 countries as DMUs. To the best knowledge of the authors, it is the first study that proposes an integrated trust model for assessing the performance of real DMUs, based on their size and degree of significance. The use of historical data, current values, prediction, and central and dispersion factors made the trust value determination model to be a comprehensive approach in which more reliable results are obtained by the utilization of the concept of confidence level. The proposed model can be practically used in different fields, such as sociology, business, etc.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was partially supported by a grant from University of Tehran [No. 8106013/1/20]. The authors are grateful for the support provided by the College of Engineering, University of Tehran, Tehran, Iran.

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