Load Forecasting Using Interval Type-2 Fuzzy Logic ...

IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. 10, NO. 2, MAY 2014

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Load Forecasting Using Interval Type-2 Fuzzy Logic Systems: Optimal Type Reduction Abbas Khosravi, Member, IEEE, and Saeid Nahavandi, Senior Member, IEEE

Abstract—This paper aims at using interval type-2 fuzzy logic systems (IT2FLSs) for one-day ahead load forecasting task. It introduces an optimal type reduction (TR) algorithm for IT2FLSs to improve their approximation capability. Flexibility and adaptiveness are the key features of the proposed nonparametric optimal TR algorithm. Lower and upper firing strengths of rules as well as their consequent coefficients are fed into a neural network (NN). NN output is a crisp value that corresponds to the optimal defuzzified output of IT2FLSs. The NN type reducer is trained through minimization of an error-based cost function with the purpose of improving forecasting performance of IT2FLS models. Once the optimal NN-based type reducer is trained, IT2FLS models can be straightforwardly forecast the next-day load demand. Numerical testing using real load datasets indicate IT2FLS models equipped with the new optimal TR algorithm outperform IT2FLS models using traditional TR algorithms in terms of forecast accuracies. This benefit is achieved in no cost, as the computational requirement of the proposed optimal TR algorithm is the same as for traditional TR algorithms. Index Terms—Interval type-2 fuzzy logic system (IT2FLS), neural network (NN), type reduction (TR).

I. INTRODUCTION

O

NE-DAY-AHEAD load forecasting is essential for efficient management and operation of energy systems in deregulated markets. System operators and decision-makers require access to accurate forecasts for completion of tasks such as unit commitment, economic load dispatch, power system security, and generation reserve determination [1]. Precise load forecasting is also vitally important for both utilities and consumers, as electricity prices skyrocket in peak hours. While utilities need load forecasts to modify their bidding strategies, consumers use forecasts to shift their power consumption to avoid high prices. Accurate short- and medium-load forecasting is still a challenging problem. This is due to nonlinear and random behavior of load demands. They are affected by many factors including but not limited to weather conditions, social and economic environment, electricity price amendments, and calendar information (regular workdays and anomalous days). A variety of methods have been proposed in literature to address forecast

Manuscript received May 30, 2013; revised August 18, 2013 and September 17, 2013; accepted September 27, 2013. Date of publication October 24, 2013; date of current version May 02, 2014. Paper no. TII-13-0341. The authors are with the Centre for Intelligent Systems Research (CISR), Deakin University, Geelong VIC 3220, Australia (e-mail: {abbas.khosravi, saeid.nahavandi}@deakin.edu.au). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TII.2013.2285650

accuracy issues in this traditional research topic. Traditional time-series forecasting methods have been widely used for load forecasting [2]–[5]. The key advantage of these models is their simplicity and low computational cost. However, the linearity of these models significantly restricts their applicability in describing the nonlinear and seasonal patterns of loads. Advanced nonlinear methods from the field of artificial intelligence have also been widely employed for electric load forecasting. These models include neural networks (NNs) and fuzzy logic systems (FLSs) [6], [7]. Metaxiotis et al. [8] and Hippert et al. [9] provide a good review of NN applications for short-term load forecasting. Construction of prediction intervals for quantification of uncertainties associated with load forecasts has also been investigated in literature [10]. Support vector machines have also gained popularity in recent years due to their promising nonlinear approximation capabilities [11], [12]. Some studies have also considered the problem of quantifying uncertainties associated with forecasts generated by artificial intelligence methods [13]–[16]. A comprehensive overview of the various models and methods used for load forecasting is provided in [17] and [18]. Broadly speaking, there is no answer to the question of which method is the best forecasting tool. Results demonstrated in studies such as [19]–[23] are contradictory. To the best of our knowledge, there is no general consensus on superiority and dominance of one forecasting method over others. However, it is reasonable to assume universal approximators such as NNs and FLSs are a more appropriate tool for estimating the nonlinear relationships between load demands and independent exogenous variables. This conclusion complies with recent trends observed in the field of load forecasting [17]. In recent years, much attention has been devoted to interval type-2 fuzzy logic system (IT2FLSs) to complete their theory [24] and implement them for real-world applications [25]. Recent theoretical and practical studies confirm that IT2FLSs more appropriately handle uncertainties than their type-1 (T1) counterparts [26], [27]. Wu in [26] discusses adaptiveness and novelty as two fundamental differences between T1 and T2 FLSs. It is argued and shown that a type-1 FLS cannot implement the complex surface of an IT2FLS given the same rule base. As per these, it is reasonable to expect to see more and more applications of IT2FLSs in different fields of science and engineering. Type-2 fuzzy neural systems have also been developed and applied to several data mining problems [28]–[31]. Recent studies on load forecasting report that IT2FLSs possess an excellent approximation capability even better than traditional nonparametric methods such as NNs [32].

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The output of the IT2FLS model is obtained through combining the outcomes of rules [as defined in (1)]

Fig. 1. IT2FLS.

Observing these recent findings, this study focuses on oneday-ahead load forecasting problem using IT2FLS models. It aims to comprehensively investigate effects of TR algorithms on the quality of load forecasts. It also proposes a new nonparametric nonlinear TR algorithm that optimally generates the defuzzified output directly from the firing strengths and consequent lower and upper values of each rule. These values are fed into a NN as inputs to approximate the crisp output. The NN model is trained through minimization of an error-based cost function. Real load data samples from an Australian energy market are used to examine performance of the proposed TR algorithm. The remainder of this paper is laid out as follows. The basics of IT2FLSs are shortly introduced in Section II. We also provide some discussion about TR algorithms in this section. Section III introduces the new TR algorithm. Section IV represents simulations results. Finally, conclusions are provided in Section V. II. IT2FLS MODELS The structure of an IT2FLS is similar to that in T1 FLS. Fig. 1 displays blocks of a typical IT2FLS. It includes a fuzzifier, an if–then rule base, inference engine, and the output processor. The latter has also two components: type reduction and defuzzification. These are similar to components of a T1 FLS with the exception of the type reduction (TR) block. An IT2FLS is characterized by if–then rules. Assume that there are rules in the rulebase of an IT2FLS. The th rule is denoted by [24] as

(1) where . There are input variables in the antecedent part of the system. Also, and are lower and upper coefficients of the consequent part. is an interval T1 set corresponding to the centroid of the IT2 consequent set [24]. is the th IT2 FS which has a lower and upper bound membership function (MF) (2) Assume that the set of inputs is firing strength of a rule is an interval T1 set, where

. The , (3) (4)

where is a t-norm (minimum or product). The singleton fuzzifier is applied to obtain (3) and (4).

(5)

and in (5) are calculated through TR algorithms. More than a dozen TR algorithms have been proposed in literature in recent years [33]. Further information about TR algorithms used in this study can be found in Appendix A. Finally, the crisp output of IT2FLS is generated as the average of and as follows: (6) It is important to highlight that the TR block is the key component of an IT2FLS. It converts an IT2 fuzzy set (FS) into a T1 FS. While studies on different blocks of an IT2FLS have matured in the last decade [24], there is no general consensus on how TR should be performed. This study aims to fill in some gaps towards development of custom-made TR algorithms. The first TR algorithm was proposed by Karnik and Mendel (KM) [34]. It recursively computes the left and right end points (center of sets) generating a T1 interval set. The center of this interval (mean) is used for defuzzification. The KM algorithm is computationally intensive and may not be suitable for fast real-time applications. This is correct in particular when there are many MFs and the rule base is large. To remedy this problem, modified and enhanced KM algorithms have been proposed [35]–[39]. These algorithms reduce the number of iterations required to calculate left and right end points as outputs of an IT2FLS. Despite these enhancements, the computational requirements of a modified KM algorithm is still considerable. Alternative TR algorithms have been proposed in literature to relieve the computational burden of the TR block. Examples are the uncertainty bound (UB) [40], Nie–Tan (NT) [41], Begian–Melek–Mendel (BMM) [42], Coupland-John (CJ) [43], Gorzalczany [44], Liang–Mendel (LM) [45], Wu-Tan [46], Greenfield–Chiclana–Coupland–John [47], Li–Yi–Zhao (LYZ) [48], Du–Ying [49], and Tao–Taur–Chang–Chang (TTCC) [50]. Some algorithms such as BMM and TTCC try to calculate the output as linear combination outputs of two T1 FLSs. Wu [33] compares these algorithms from a computational burden perspective and concludes that they are faster than the original KM algorithm is, in particular when the number of rules is less than ten. Assuming Gaussian MFs, WT, NT, LM, and BMM are the fastest amongst alternative TR algorithms. However, IT2FLS models using these TR algorithms are still computationally more demanding than T1 FLSs. Another interesting finding of [33] is that alternative TR algorithms lead to completely different outputs. It is found that the UB algorithm generates the closest outputs to the KM TR algorithm. As the LM algorithm is an unnormalized one, its output is

KHOSRAVI AND NAHAVANDI: LOAD FORECASTING USING IT2FLSs: OPTIMAL TYPE REDUCTION

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proposed in [54] and [55] can be applied for calculation of these centroids. Therefore, we only need to calculate elements to form for each observation. These are and as defined in (3) and (4), respectively. Using as the input vector, the th desired input–output pattern pair of the new dataset is formed as follows: (8) Fig. 2. Neural network-based type reduction.

significantly different from other algorithms. Also, outputs obtained from LYZ algorithm are different from other outputs. As indicated in [33], no study has the investigated performance of TR algorithms to check the quality of defuzzified outputs they generate. Previous studies mainly attempt to minimize the computational requirement of TR block compared to original KM algorithm. However, there is no discussion on what cost is paid to achieve this goal. As defuzzified outputs are different, it is not clear which TR algorithm is superior in terms of generating more accurate results. III. OPTIMAL TYPE REDUCTION ALGORITHM Here, we discuss the new optimal TR algorithm for IT2FLS models. Two objectives are considered in the design and implementation of the new optimal TR algorithm: 1) improving the prediction and approximation power of IT2FLS models and 2) keeping the computational burden as low as possible. The new TR algorithm directly computes the crisp output of the model using a feedforward NN. Fig. 2 displays the sketch of proposed NN-based TR algorithm. Assume that is a -dimensional explanatory vector with as its elements. Given a random sample, , of observations, and the corresponding values of the response variable T, an IT2FLS model is first fit to the data. Steps described in [51] are followed to build the model equipped with traditional TR algorithms. Training is performed through minimization of mean squared error or mean absolute percentage error cost functions. Metaheuristic optimization algorithms, such as genetic algorithm [52] or simulated annealing [53], are applied for optimal tuning of MF parameters and consequent coefficients (centroids). Once the development of the IT2FLS model is complete, we put aside its traditional TR block and replace it with a feedforward NN-based type reducer. A new dataset is required to optimally develop this block. The feeding vector to NN type reducer is formed for all training samples and is indicated as , . is mathematically represented as (7) and indicate the lower and upper firing strengths where of the th rule as defined in (3) and (4), both calculated for the th sample . and are also the centroid of IT2 FS in the consequent part of rules as defined in (1). The dimension of is , where is the number of rules. Note that and are calculated and recorded in advance. Methods

Thus, the whole idea is to determine a mapping from to , i.e., . The process of developing the NN type reduce is a supervised learning problem, as each datum in the new dataset is labeled. This allows us to train the NN type reducer using traditional backpropagation techniques. Powerful training algorithms, such as Levenberg–Marquardt [56], can be applied to optimally adjust NN parameters. Once this is done, optimal NN-based TR can be used in real time to approximate the crisp output of IT2FLS models. The proposed optimal TR algorithm has some interesting feature. First, it is certainly computationally faster than original KM algorithm and its enhancements. The computational burden required to form vector is the same for all TR algorithms. Once is calculated, it is fed into the trained NN and the crisp output is very quickly generated. No iteration is required and this is all done in one step. Second, it uses the excellent learning and approximation capability of NNs to optimally generate the crisp output. This adds extra value to IT2FLS models, as their TR block is a nonlinear nonparametric model with universal approximation capability. None of previously proposed TR algorithms possesses this feature. The majority of these algorithms calculate a weighted average, , which means they have limited capability to find nonlinear relationship between the firing strengths and centroids of consequent parts of rules and targets. The proposed optimal type reducer here uses the centers of fuzzy sets and firing strengths of rules to calculate the crisp output of an IT2FLS model. Although it functionally operates similar to center-of-set type reducer [45], it is more adaptive and flexible due to using advanced models for optimally estimating the nonlinear mapping between inputs (centers and firing strengths) and the crisp output. IV. SIMULATION RESULTS A. Data This investigation employs historical electric load data collected from Victoria region of the national energy market in Australia. It includes May, August, and December load demands during 2010. These three months represent three different seasons of a year in this region of Australia. They correspond to autumn, winter, and summer, respectively. This selection is made to make the test results less subjective. Load demands with an hour interval (24 data points per day) are displayed in Fig. 3. It is obvious that the load patterns are completely different in these three months. There are more demand spikes in May and August compared to December. The weekend electricity demands are greater in December compared to May and August. The minimum demand occurs in December 25th (Boxing Day).

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TABLE I LAGGED LOADS FOR EACH MODEL

Fig. 4. Experimental procedure for developing models and examining their performance.

Fig. 3. Load demands in Australian electricity market (Victorian region). (a) May 2010. (b) August 2010. (c) December 2010.

The demand then significantly increases towards the end of the year. The datasets used in this study are publicly available at www.aemo.com.au. B. Experimental Setup The purpose of modelling is to develop a one-day-ahead load forecaster. An individual IT2FLS model is developed for each month. It is assumed that no extra information, such as temperature forecasts, is available. Thus, load forecasting is purely done using historical load values. Similar to other data-driven techniques, the performance of the proposed forecasting method depends on the appropriate selection of its set of inputs and its structure. Autocorrelation and partial correlation are applied here for a selection of lagged load values as IT2FLS inputs. As per this, the whole process of modelling can be described as (9)

where and are the 24-h-ahead load forecast and the actual load at time , respectively. is the number of lags determined through correlation analysis of historical data. Lagged values used in experiments conducted in this study are shown in Table I. Performance of the proposed optimal TR algorithm is compared with performance of five traditional TR algorithms. These are KM, NT, UB, BMM, and CJ. A short description of these TR algorithms is provided in Appendix A. The experimental procedure for developing IT2FLS models using traditional and new TR algorithm and examining their performance is shown in Fig. 4. Gaussian MFs with uncertain standard deviations are considered for inputs. Rules and initial values for model parameters are determined using fuzzy C-means clustering [57], [58]. MF parameters are then optimally adjusted using genetic algorithm. The population size is 30 and the cross over factor is set to 0.7. Training process uses the mean-squared error index for adjusting model parameters. Experiments are repeated five times to avoid driving misleading conclusions and to make results statistically meaningful. Mean absolute percentage error (MAPE) is used as the performance index for forecast evaluation. In all examples, 80% of samples are used for training. The remaining 20% are used to examine the forecasting performance of models. We first train IT2FLS models using traditional TR algorithms. Then, the proposed algorithm is applied to develop an optimal NN-based TR model. In each replicate, we compare performance of IT2FLS equipped with traditional TR algorithms and NN-based TR algorithm. Assuming that there are four rules in IT2FLS model, there will be 16 inputs fed to NN-based type reducer. Therefore, the number of neurons for the feedforward single-layer NN models is set to 10 to allow for sufficient learning capacity. Simulations are performed using a Lenovo Thinkpad T420s laptop computer with Intel Core i7-2640M CPU @2.8 G Hz and 8 GB memory, running Windows 7 Professional.


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TABLE II MAPE VALUES FOR IT2FLS MODELS WITH SIX TR ALGORITHMS DEVELOPED FOR LOAD DEMANDS OF MAY 2010

TABLE IV MAPE VALUES FOR IT2FLS MODELS WITH SIX TR ALGORITHMS DEVELOPED FOR LOAD DEMANDS OF DECEMBER 2010

TABLE III MAPE VALUES FOR IT2FLS MODELS WITH SIX TR ALGORITHMS DEVELOPED FOR LOAD DEMANDS OF AUGUST 2010

• In the first replicate of BMM TR algorithm, we have . Such a rare event is due to improper initialization of the NN model used in the proposed TR algorithm. It is important to note that this event does not occur in other replicates of this experiment, as . • and are the best performing models. • According to median values of MAPEs, the maximum improvements are achieved for UB (38%) and KM/CJ (33%) TR algorithms. For the case of December 2010, observe the following. • As before, IT2FLS models equipped with the proposed TR algorithm perform better than IT2FLS models hiring traditional TR algorithm. • For this month, we have and . As before, this is due to improper initialization of NN parameters leading to a NN with superior performance compared to NNs obtained in other replicates of the same experiment. • and are the best performing models. • According to median values of MAPEs, the maximum improvements are achieved for KM (64%) and UB (43%) TR algorithms. As per results in Tables II–IV, there are only 4 out of 75 instances where is greater than of other TR algorithms. It is also important to note that the median values of are always smaller than the median values of MAPEs obtained using other TR algorithms. We also calculate the amount of forecast accuracy improvement using MAPE median values (the last column in Tables II–IV) of all TR algorithms:

C. Results and Discussion Tables II–IV show MAPE values obtained in five replicates of experiments for IT2FLS models equipped with six TR algorithms. The last column of tables is the median value of five MAPEs obtained using each TR algorithm. We evaluate and discuss results month by month for six TR algorithms. For the case of May 2010, observe the following. • Prediction performance of IT2FLS equipped with the optimal TR algorithm is better than those using traditional TR algorithms. Mathematically, . • There is only one replicate out of twenty five replicates where forecasting accuracies are not improved using the proposed method. This is related to replicate four of CJ TR algorithm, . However, is less than 9% greater than . • Amongst traditional TR algorithm, application of CJ and NT TR algorithms leads to the best performing IT2FLS models, as the MAPE median values of these models are less than others. • According to median values of MAPEs, the maximum improvements are achieved for BMM (37%) and UB (34%) TR algorithms. For the case of August 2010, observe the following. • Similar to the previous case, is smaller than MAPEs of other models.

(10) stands for traditional KM, NT, BMM, UB, and CJ where TR algorithms. Fig. 5 displays the improvement values for three months and five TR algorithms. The amount of improvement ranges between 3% (NT TR algorithm for December) to 64% (KM TR algorithm for December). These are strong evidences indicating that the proposed optimal TR algorithm is a better

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TABLE V COMPUTATIONAL TIME OF TRADITIONAL AND PROPOSED TR ALGORITHMS

Fig. 5. Amount of improvement in forecast accuracies.

Fig. 6. Targets and forecasts for replicate two of UB- and NN-based TR algorithms for December 2010.

option for developing reliable IT2FLS models compared to traditional TR algorithms. It is also good to have a graphical view about the performance of the proposed TR algorithm. Fig. 6 displays actual and forecast loads obtained using IT2FLS models equipped with UB and proposed optimal TR algorithms. It is observed that load forecasts generated using the proposed TR algorithm accurately follow the actual load values. Minor mismatched cases are rare, and forecasting error is small in those cases. In contrast, forecasts obtained using the traditional UB TR algorithm are far from being perfect. Forecasting errors are large in particular when there is a sharp peak or fall in demand. Similar patterns are observed for forecasts obtained using other TR algorithms. As per these observations, it is reasonable to conclude that the traditional TR algorithms curb the excellent approximation capabilities of IT2FLS models. This is mainly due to their inflexibility and limited approximation capability. An important aspect of TR algorithms is their computational cost, as this is one of the key motivations to develop faster alternatives. Computational burden of traditional and proposed TR algorithms are also compared here. Comparison is made based on the time required to forecast targets for three month datasets using IT2FLS models. For a fair comparison, we also include the time spent to calculate the intervals of fired rules. Table V represents the required time for calculating the crisp outputs of IT2FLS models. It is obvious that the proposed TR algorithm is computationally very similar to other traditional TR algorithms. It is important to note that, when the number of inputs is increased (May and August), the computational burden of the

proposed TR algorithm remains almost the same as other algorithms. This is due to the fact that calculation of the crisp output is as simple as feeding inputs to the NN model and then computing the NN output. There is no hidden computation in the proposed optimal TR algorithm. Another interesting feature of the proposed optimal TR algorithm is its flexibility and adaptiveness. In contrast to NT and BMM algorithms, the NN-based TR algorithm does not calculate the defuzzified output of IT2FLS using a traditional T1 FS or a linear combination of several traditional T1 FS. More complex NNs (extra layers and neurons) can be used for the cases that relationships between the crisp outputs and firing strengths and centroids are highly nonlinear. Other algorithms do not have this feature, as their structure is fixed. From a learning capacity perspective, the more the number of NN inputs, the bigger the network size. If there are several rules in the IT2FLS rule base which correspond to having several inputs for NN type reducer, a bigger size network should be considered. This can be easily done by adding more neurons or layers to NN type reducer. Low computational cost and excellent approximation capability make the proposed optimal TR algorithm one of the best alternative ones. IT2FLS models using this TR algorithm are more amenable to real time embedded applications. V. CONCLUSION In this paper, the performance of IT2FLS models is improved for the problem of one-day-ahead load forecasting. Better forecast accuracies are achieved through introduction and application of a new NN type reducer. The firing strengths of rules and their interval centroids are fed into an NN to directly and optimally generate the deffuzzied output of IT2FLS models. NN training is performed through minimization of an error-based cost function. Performance of the proposed algorithm is compared with performance of five traditional TR algorithms. Comparative studies are performed using real datasets from Victorian region of Australian energy market. It is found that the proposed algorithm is computationally similar to the traditional TR algorithms. Furthermore, IT2FLS models developed using the proposed algorithm have a much better forecasting performance than IT2FLS models using traditional TR algorithms. In summary, the proposed optimal TR algorithm has several notable advantages. First, it is not computationally intensive. Thus, it can be easily implemented on existing platforms used for load forecasting. Second, it is adaptive, which means it can cope with the changing behavior of loads. Finally, it is flexible.


This feature allows the modelers to modify and apply the proposed type reducer for different energy systems and markets. The proposed type reducer in this paper can be easily adopted and applied as an optimal defuzzifier for type-1 FLSs. It is also possible to simultaneously adjust parameters of both IT2FLS model and NN-based type reducer. This will further improve the forecasting and generalization power of IT2FLS models. It is important to note that the type reduction and defuzzification are performed simultaneously using the prosed NN-based type reducer. This complies with the purpose of this study which is generating accurate point forecasts. Further studies are required to optimally estimate the left and right end points of the type reduced set through an unsupervised learning mechanism. It is highly likely that the proposed type reducer more optimally locates the forecast in the type reduced set. This is in contrast with existing TR methods where they consider the center of the set as the defuzzified output.

where

,

,

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, and

are calculated as follows: (17) (18)

(19)

(20) (21)

APPENDIX TR ALGORITHMS

(22)

A. Karnik–Mendel The KM type reducer [54] iteratively calculates the upper and and are first lower bounds of the center of gravity. sorted in an ascending order. Then, and are calculated as

(23) (24)

(11) (12)

where and are integer switch points in the following condition:

and satisfy

C. Nie–Tan Algorithm Type reduction and defuzzification are performed together in the NT algorithm [41]. The final output of an IT2FLS is computed as follows: (25)

(13) (14) Detailed discussion about calculating switch points ( and ) can be found [35], [54]. The enhanced versions of the KM algorithm [35]–[39] are not investigated in this study, as they generate the same left and right points similar to the original KM algorithm. Therefore, the forecasting performance is the same as original algorithm. B. Wu–Mendel Uncertainty Bound (UB) In contrast to KM algorithm, the UB algorithm [40] provides and a closed-form type reduction. Again, it requires to be rank-ordered. It computes the left and right end points using the following equations:

(15) (16)

is not required by the NT algorithm. As Again, sorting the crisp output is just obtained in one step (no iteration is required), the computational requirement is much less than for the KM algorithm [41]. However, the method does not provide any indication of uncertainties encapsulated by the width of the type-reduced set. D. Begian–Melek–Mendel Algorithm The closed-form TR and defuzzification algorithm proposed by Begian, Melek, and Mendel (BMM) is as follows [42]:

(26)

where and are two coefficients. BMM algorithm computes the crisp output value of IT2FLSs as a linear combination of two T1 fuzzy sets (upper and lower MFs). BMM algorithm does not

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require sorted . An extension of the BMM algorithm when is proposed in [59] (27)

E. Coupland–John Algorithm The CJ algorithm [43] borrows ideas from the field of computational geometry. Rather than using the extension principle, Coupland and John apply and extend the geometric interpretation of the centroid to drive the defuzzified output. The algorithm constructs a polygon using firing strengths and sorted . Assuming there are fired rules, the polygon has points on its boundary, indicated by , . The defuzzified output is the center of the polygon (28)

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Abbas Khosravi (M’07) received the B.Sc. degree in electrical engineering from Sharif University of Technology, Tehran, Iran, in 2002, the M.Sc. degree in electrical engineering from Amirkabir University of Technology, Tehran, in 2005, and the Ph.D. degree from Deakin University, Geelong, VIC, Australia, in 2010. He is currently a Research Fellow with the Centre for Intelligent Systems Research (CISR), Deakin University, Geelong, VIC, Australia. His primary research interests include development and application of artificial intelligence techniques for (meta)modeling, analysis, control, and optimization of operations within complex systems. Mr Khosravi was the recipient of the Alfred Deakin Postdoctoral Research Fellowship in 2011.

Saeid Nahavandi (SM’07) received the B.Sc. (hons.), M.Sc., and Ph.D. degrees in automation and control from Durham University, Durham, U.K. He is the Alfred Deakin Professor, Chair of Engineering, and the Director for the Center for Intelligent Systems Research (CISR), Deakin University, Geelong, VIC, Australia. He has authored and coauthored over 350 peer-reviewed papers in various international journals and conferences. He is an Editorial Consultant Board member for the International Journal of Advanced Robotic Systems and an editor (South Pacific Region) of the International Journal of Intelligent Automation and Soft Computing. He designed the world’s first 3-D interactive surface/motion controller. His research interests include modeling of complex systems, simulation-based optimization, robotics, haptics, and augmented reality. Dr. Nahavandi is a Fellow of Engineers Australia (FIEAust) and IET (FIET). He is an associate editor for the IEEE SYSTEMS JOURNAL.