J Intell Manuf (2011) 22:229–245 DOI 10.1007/s10845-009-0284-8
An integrated artificial neural network-genetic algorithm clustering ensemble for performance assessment of decision making units A. Azadeh · M. Saberi · M. Anvari · M. Mohamadi
Received: 27 February 2009 / Accepted: 17 June 2009 / Published online: 1 August 2009 © Springer Science+Business Media, LLC 2009
Abstract This study proposes a non-parametric efficiency frontier analysis method based on artificial neural network (ANN) and genetic algorithm clustering ensemble (GACE) for measuring efficiency as a complementary tool for the common techniques of the efficiency studies in the previous studies. The proposed ANN GA algorithm is able to find a stochastic frontier based on a set of input–output observational data and do not require explicit assumptions about the functional structure of the stochastic frontier. Furthermore, it uses a similar approach to econometric methods for calculating the efficiency scores. Moreover, the effect of the return to scale of decision making unit (DMU) on its efficiency is included and the unit used for the correction is selected based on its scale (under constant return to scale assumption). Also, in this algorithm, GA is used to cluster DMUs to increase
DMUs’ homogeneousness. It should be noted that data envelopment analysis (DEA) is sensitive to the presence of the outliers and statistical noise. It is also not capable of performing prediction and forecasting. This is shown by two examples related to outlier situations. However, the proposed algorithm is capable of handling outliers and noise and DEA is used as a benchmark to show advantages of the proposed algorithm. Also, the proposed algorithm and conventional algorithm are compared in viewpoint of DEA through statistical t-test. The proposed approach is applied to a set of actual conventional power plants to show its applicability and superiority.
A. Azadeh (B) Department of Industrial Engineering and Center of Excellence for Intelligent-Based Experimental Mechanics, College of Engineering, University of Tehran, P.O. Box 11365-4563, Tehran, Iran e-mail:
[email protected];
[email protected]
Introduction
M. Saberi Institute for Digital Ecosystems & Business Intelligence, Curtin University of Technology, Perth, Australia e-mail:
[email protected] M. Saberi Department of Industrial Engineering, Faculty of Engineering, University of Tafresh, Tafresh, Iran M. Anvari Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran M. Mohamadi Department of Computer Engineering, Iran University of Science and Technology, Tehran, Iran
Keywords Performance assessment · Artificial neural network · Genetic algorithm · Decision making units
The appropriate use of limited resources with the available technology is referred to as technical efficiency. Efficiency frontier analysis has been an important approach of evaluating firms’ performance in private and public sectors. As alternatives to determine the efficiency boundaries, the international experience reports a significant number of methodologies with different approaches and methods to characterize such efficiency (Jamasb and Pollitt 2001). Basically, these methodologies can be classified according to how the frontier is estimated. There are two competing paradigms on efficiency analysis. Parametric and non-parametric approaches are widely-used in the efficiency measurement. The first include the estimation of both deterministic and stochastic frontier function (SFF) which is based on the econometric regression theory and has been widely accepted in the econometrics field. The latter include DEA and Free Disposal Hull (FDH) which are based on a mathematical programming
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230 Table 1 ANN and DEA features
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DEA
ANN
Nonparametric approach
Nonparametric approach
Optimal weights to maximize the efficiency
Optimal weights to derive the best possible fit
No assumptions about the functional form
No assumptions about the functional form
Medium assumptions about functional form and data Medium flexibility
Low assumptions about functional form and data High flexibility
Many theoretical studies/applications on efficiency Very sensitive to the presence of the outliers and statistical noise Forecasting incapability
Few theoretical studies No sensitive to the presence of the outliers and statistical noise Forecasting capability
Global optimization in linear status
Local optimization in nonlinear status
Use continuous and integer inputs and outputs Medium computational expense
Use continuous and classification inputs and outputs
Black box
Black box
approach. Each of these two methodologies has its strength as well as major limitations. In all of these methodologies, the frontier is defined by the most efficient DMU of the sample. Mathematically, the frontier methods are introduced as a high-reliability analysis tool and have been largely used for studies in the electrical and industrial field (Pollit 1995; Sanhueza et al. 2004). There have been many efficiency frontier analysis methods reported in the literature (for instance only about power plants 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). But, the assumptions made for each of these methods are restrictive. Conflicting conclusions of efficiency are often resulted by using the different methods due to the unsuitability of the assumptions. Their frontier is deterministic and sensitive to outliers. The non-parametric approaches make no assumption about the functional form of the frontier. Instead, it specifies certain assumptions about the underlying technology that in combination with the data set allows the construction of the production set. For instance, the DEA frontier is very 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). On the other hand, DEA can hardly be used to predict the performance of other decision-making units. Although, benchmarking in DEA allows for the identification of targets for improvements, it has certain limitations. A difficulty addressed in the literature regarding this process is that an inefficient decision making unit (DMU) and its benchmarks may not be inherently similar to their operating practices. This is primarily due to the fact that the composite DMU that dominates the inefficient DMU does not
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High computational expense
exist in reality. To overcome these problems researchers have utilized performance-based clustering methods for identifying more appropriate benchmarks (Doyle and Green 1994; Talluri and Sarkis 1997). These methods cluster inherently similar DMUs into groups, and the best performer in a particular cluster is utilized as a benchmark by other DMUs in the same cluster. Moreover, Wang indicates that the research to remove drawbacks and limitations in this area has been insufficient (Delgado 2005). However, his proposed algorithm is time consuming (using uncertain λ value and trial and error process and the need to run stochastic frontier analysis (SFA) and DEA jointly). The proposed algorithm uses new and powerful tools and removes the limitations of previous methods (Cook and Green 2005; Costa and Markellos 1997). It further improves SFF and DEA approaches. Table 1 shows the important features of ANN and DEA. Examination of this table shows that each of the two methods has its shortcoming and superiority. In present study, the conventional algorithm (that uses ANN as approximation tool) is improved with proposed algorithm1 (Athanassopoulos and Curram 1996). Also, the weakness of parametric method with respect to outliers (and statistical noise) and forecasting is removed by the proposed algorithm. Previous studies by Azadeh et al. show the applicability of ANN and ANN Fuzzy C-Means in selected power plants (Athanassopoulos and Curram 1996; Azadeh et al. 2006a,b, 2007a). However, this study presents an algorithm which is not only based on ANN but also uses GA as a clustering tool. The most important feature of the GA clustering is ability to extract optimum number of clusters. In addition, ANN Fuzzy C-Means which is proposed in previous studies 1 DEA is used as benchmark to show the advantage of the proposed algorithm.
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(Athanassopoulos and Curram 1996; Azadeh et al. 2006a,b, 2007a) is expanded for output oriented models. The proposed algorithm improves the conventional algorithm (that uses ANN as approximation tool). It should be noted that data envelopment analysis (DEA) is sensitive to the presence of the outliers and statistical noise. It is also not capable of performing prediction and forecasting. However, the proposed algorithm is capable of handling outliers and noise. It is also capable of performing forecasting for a given set of data. In present Study, we are not going to compare the result of DEA and the proposed algorithm and the result of DEA is used only as benchmark. The conventional algorithm requires explicit assumptions about the functional structure of the stochastic frontier. Moreover, the effect of the return to scale of decision making unit (DMU) on its efficiency isn’t included. However, the proposed algorithm doesn’t require explicit assumptions about the functional structure of the stochastic frontier because it uses ANN. Also, several studies show the superiority of ANN versus conventional econometric methods. Moreover, our emphasis is on philosophic drawback of conventional algorithm nor on the higher or lower results of each performance assessment method because this doesn’t show its superiority or weakness. As mentioned in the introduction, DEA has two problems: 1. Not capable of performing prediction and forecasting; 2. Sensitive to the presence of the outliers and statistical noise. Problem number one is removed in proposed algorithm (shown in step 8). Also, problem number two is removed in the proposed algorithm (step 9). The paper is organized as follows. Section “Introduction” provides an introduction to ANNs in efficiency analysis, where neural networks form a promising analysis tool together with known econometric models and non-parametric methods. This section concludes with a review of several published papers about ANNs and efficiency. DEA is introduced in “Data envelopment analysis”. Section “Genetic algorithm clustering ensemble” provides an introduction to GA method for clustering. Section “Clustering ensemble” is dedicated to the proposed ANN GA Clustering Ensemble for efficiency analysis and assessment of DMUs. An empirical illustration for measuring performance of thermal power plants in Iran is carried out in “The case study”. The capability of proposed algorithm in handling outliers and corrupted data is presented in “Examination of proposed algorithm in outliers and corrupted data situation”. The final section of the paper offers conclusions and suggests areas for the future research. Artificial Neural Networks (ANNs) are a promising alternative to econometric models and they are information
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processing paradigms that are inspired by the way biological nervous systems, such as the brain, process information. ANNs, like people, learn by example. ANNs are configured for specific applications, such as pattern recognition, function approximation through learning process. Learning in biological systems involves adjustments to the synaptic connections that exist between the neurons. This is true for ANNs as well. They are made up of simple processing units which are linked by weighted connections to form structures that are able to learn relationships between sets of variables. Nowadays, many researches have done in field of ANN. For example, Wang et al. discuss the computational complexity the convergence rate and the stability of the single parameter dynamic searching algorithm for Multi-layer Neural Networks (Wang et al. 2008). Zhang et al. proposed learning algorithm that maintains some trends of quick descent to either global minimum or a local minimum, and at the same time has some chance of escaping from the local minima by permitting temporary error increases during learning (Zhang et al. 2008). Tamura and Tann propose a midpointvalidation method, which improves the generalization of neural networks (Tamura and Tann 2008). In this study application of ANNs in efficiency analysis is discussed. Within the efficiency literature, few applications have been developed in this field. Commonly, neural network technique is used as a complementary tool for parametric and non-parametric methods such as data envelopment analysis (DEA), to fit production functions andmeasure efficiency under non-linear contexts. In fact, applying ANNs can reduce the restrictive assumptions of each of these methods. This heuristic method can be useful for nonlinear process that has an unknown functional form. There has been a vast literature about ANNs, basically in the empirical field, indicating ANN comparability or superiority to conventional methods for estimating various functions (Azadeh et al. 2006c, 2007b,c; Chiang et al. 1996; Hill et al. 1996; Indro et al. 1999; Jhee and Lee 1993; Kohzadi et al. 1996; Stern 1996; Tang et al. 1991; Tang and Fishwick 1993). This study, uses ANN for estimating production function and consequently for performance evaluation of DMUs (Tang and Fishwick 1993). The efficiency prediction power of ANNs is unique and its flexibility to solve complex problems where the main information lies implicitly in the data is very applicable (Wu et al. 2006). The idea of combination of neural networks and DEA for classification and/or prediction was first introduced by Athanassopoulos and Curram (1996). They treated DEA as a preprocessing methodology to screen training cases in a bank with four inputs and three outputs. After selecting samples, ANN is trained to learn a nonlinear forecasting model. They assume that inefficiency distributions are semi-normal and exponential and conclude that DEA is superior to ANN for measurement purpose. Their study indicates that ANN
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results are more similar with the constant returns to scale and less with the variable returns to scale results. Costa and Markellos (1997) analyzed the London underground efficiency with time series data for 1970–1994 where there are two inputs (fleet and workers) and one output (distance). They explain how the ANNs results are similar to corrected ordinary least squares (COLS) and DEA. They proposed two procedures: (a) similar method to COLS after neural training; (b) by an oversized network until some signal to noise ratio is reached. Then, inefficiency is determined as observation-frontier distance. However, ANN offers advantages in the decision making, the impact of constant versus variable returns to scale or congestion areas (Costa and Markellos 1997). Santin et al. (2004) infer that ANN is superior to econometric approach for frontier estimation. Pendharkar and Rodger (2003) used DEA as a data screening approach to create a sub sample training data set that is ‘approximately’ monotonic, which a key property is assumed in certain forecasting problems. Their results indicate that the predictive power of a trained ANN on the ‘efficient’ training data subset is stronger than the predictive performance of a trained ANN on the ‘inefficient’ training data subset. Santin et al. used a neural network approach for a simulated nonlinear production function and compared its performance with conventional alternatives such as stochastic frontier and DEA in different observations and noise scenarios. The results suggested that ANNs are a promising alternative to conventional approaches to fit production functions and measure efficiency under non-linear contexts. Wu et al. presented a DEA-ANN2 study for performance assessment of branches of a large Canadian bank. Their results are operable to the normal DEA results on the whole (Wu et al. 2006). They concluded that the DEA-ANN approach produces a more robust frontier and identifies more efficient units because better performance patterns are explored. Furthermore, for worse performers, it provides the guidance on how to improve their performance to different efficiency ratings. Ultimately, they concluded that neural network approach requires no assumptions about the production function (the major drawback of the parametric approach) and it is highly flexible. It is concluded that ANN has been viewed as an ideal tool to approximate numerous non-parametric and non-linear problems.
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Data envelopment analysis Data envelopment analysis is a non-parametric method that uses linear programming to calculate the efficiency in a given set of decision-making units (DMUs). The DMUs that make up a frontier envelop, the less efficient firms and the relative efficiency of the firms is calculated in terms of scores on a scale of 0 to 1, with the frontier firms receiving a score of 1. DEA models can be input- or output-oriented and can be specified as constant returns to scale (CRS) or variable returns to scale (VRS). Basic models of DEA The original fractional Charnes Cooper Rhodes (CCR) model (1) evaluates the relative efficiencies of n DMUs ( j = 1, . . ., n), each with m inputs and s outputs denoted by x1 j , x2 j , . . ., xm j and y1 j , y2 j , . . ., ys j , respectively (Charnes et al. 1978). This is achieved by maximizing the ratio of weighted sum of outputs to the weighted sum of inputs as shown in model (1): s u y rm=1 r r o i=1 vi xio s u y rm=1 r r j ≤ 1, i=1 vi xi j
Max θ = s.t.
j = 1, . . . , n, r = 1, . . . , s
(1)
u r , vi ≥ 0, i = 1, . . . , m, r = 1, . . . , s
In model (1), the efficiency of DMUo is θo and u r and vi are the factor weights. However, for computational convenience the fractional programming model (1) is re-expressed in linear program (LP) form as follows in model (2): Max θ = s.t.
s
s r =1
u r yr o
u r yr j −
r =1 m
m
vi xi j ≤ 0,
j = 1, . . . , n,
i=1
(2)
vi xio = 1
i=1
u r , vi ≥ ε, i = 1, . . . , m, r = 1, . . . , s
where ε is a non-Archimedean infinitesimal introduced to ensure that all the factor weights will have positive values in the solution. Model (3) evaluates the relative efficiencies of n DMUs ( j = 1, . . ., n), respectively, by minimizing inputs when outputs are constant. The dual of linear program (LP) model for input oriented CCR is as follows:
2
They defined an algorithm to implement the DEA-ANN model. Moreover, after collecting a data set, CCR method is used to calculate efficiency score of DMUs (Charnes et al. 1978). The preprocessed data set is obtained and is grouped into four categories based on the efficiency scores and the neural network is trained with some groups of data subset until the pre-specified epochs or accuracy is satisfied. Then the trained neural network model is applied to calculate efficiency scores of all DMUs and post process the calculated efficiency scores by regress analysis between DEA-NN results and CCR DEA results.
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Min θ s.t. θ xio ≥ yr o ≤
n
λ j xi j , i = 1, . . . , m,
j=1 n j=1
λj ≥ 0
(3) λ j yr j , r = 1, . . . , s
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The output oriented CCR model is as follows: Max θ
n
s.t. xio ≥
λ j xi j , i = 1, . . . , m,
j=1
θ yr o ≤
(4)
n
λ j yr j , r = 1, . . . , s
j=1
1. Using different clustering algorithms to produce partitions for combination; 2. Changing initialization or other parameters of a clustering algorithm; 3. Using different features via feature extraction for subsequent clustering; 4. Partitioning different subsets of the original data.
λj ≥ 0 If λ j = 1( j = 1, . . ., n) is added to model (3), the Banker Charnes Cooper (BCC) model is obtained which is input oriented and its return to scale is variable. The calculations provide a maximal performance measure using piecewise linear optimization on each DMU with respect to the closest observation on the frontier. The linear programming system for the BCC input-oriented model is given in expression (5), and the output-oriented model is given in expression (6) (refer to Charnes et al. 1978). Min θ s.t. θ xio ≥ yr o ≤
n
λ j xi j , i = 1, . . . , m,
j=1 n
λ j yr j , r = 1, . . . , s
(5)
j=1
n
λj = 1
j=1
λ j ≥ 0, Max θ s.t. xio ≥
n j=1
θ yr o ≤ n
j = 1, . . . , n
λ j xi j , i = 1, . . . , m, n
λ j yr j , r = 1, . . . , s
(6)
j=1
λj = 1
j=1
λ j ≥ 0,
j = 1, . . . , n
Based on the diversity of components, a matrix is computed which contains combinational information. This matrix is co-association matrix. Co-association matrix can be computed by the following steps. Let D be a dataset of N data points in d-dimensional space. Suppose that, k-means algorithm has m running time. The value of co_assocition(i, j) is equal to:
i = j 1 i= j k m
(7)
Out of m times repeating the single clustering algorithm on the samples, the ith and jth samples are placed in the same cluster for the k times. It should be noted that the co-association matrix is symmetry matrix (Strehl and Ghosh 2002a). Numerous hierarchical algorithms can be applied to the co-association matrix to obtain the final partition, including Single Link (SL), Average Link (AL), Complete Link (CL) and voting k-means (Minaei et al. 2004; Strehl and Ghosh 2002b; Topchy et al. 2004). In this paper, we used genetic algorithm (GA) on co-association matrix to obtain the final partitions. The appropriateness of GA versus other hierarchical algorithms is shown in two recent studies (Azimi et al. 2006; Mohammadi et al. 2007). Also, the most important feature of our algorithm is ability to extract the number of optimum clusters. In this paper, ten different k-means algorithms are used to create the ensemble matrix. Then, GA is applied to the ensemble matrix. The superiority of GACE method is:
Genetic algorithm clustering ensemble Clustering ensemble Clustering ensemble method is a common and efficient method for clustering task which is used in several applications and algorithms (Minaei et al. 2004; Strehl and Ghosh 2002a,b; Topchy et al. 2004). The proposed algorithm uses single clustering algorithms on a same dataset and then combines the results and consequently improved solution for clustering task is identified. In order to optimally integrate clustering ensembles in a robust and stable manner, one needs a diversity of component partitions for combination. Generally, this diversity can be obtained from several sources:
1. Combination of the results of single clustering algorithms; 2. Ability to extract the number of optimum clusters; 3. Improved outputs of GACE versus hierarchical algorithms. GA applied on ensemble matrix The searching capability of GAs has been used to appropriately determine the number of cluster and final clusters. Genetic algorithm encompasses three main operators: (1) selection, (2) crossover, and (3) mutation (Davis 1991; Goldberg 1989).
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Representation (definition of individuals)
Selection operation
The first step in defining a GA is to link the “real world “to the “GA world”, that is to setup a bridge between the original problems context and the problem solving space. In this method, the individuals have two parts which are sample part and index part. Sample part contains all samples of dataset and each gene is a representative of a member of database. The length of this part is equal to the number of dataset members. The order of each sample in each individual is random (initialization step). Figure 1 shows an instance of sample part. This instance shows the dataset which has 15 samples with the following order. The boundary of each cluster is determined in index part. The length of this part is equal to number of clusters + 1. Figure 2 shows an instance of the index part. This instance contains 3 clusters. Members from index 1 to 5 belong to cluster 1, members from index 6 to 11 belong to cluster 2 and members from index 12 to 15 belong to cluster 3. Table 2 shows the member of each cluster. The value of genes in the index part is selected at random (initialization step), except first and last genes. The value of the first and last genes always are fixed (first = 0 and last = the number of samples).
The selection process selects individuals from the mating pool directed by the survival of the fitness concepts of natural genetic systems. In this paper, the tournament selection (2) is used as selection method (Davis 1991). In tournament selection (2), two individuals are chosen at random and the individual with better fitness is selected for next population.
5
6
7
12
13
11
2
15
9
1
4
3
8
10
Crossover operation The role of the crossover operation is to create new individuals from old ones. Crossover is a probabilistic process that exchanges information between two parent individuals for generating two child individuals. We use “cut-and-crossfill” as our crossover operation with a fixed crossover probability of μ (Davis 1991). In cut-and-crossfill crossover, a random integer (crossover point) is generated in the range of [1, l − 1] (l is the length of dataset). The portions of the individuals lying to the right of the crossover point are exchange to produce two offspring. This crossover has been used only on sample part. This crossover guarantees that no repeated gene is created and changes the value of repeated genes to the correct values. Mutation operation
14
Fig. 1 An example of sample part
Fig. 2 An example of index part
0
5
Mutation is applied to one individual and delivers a modified mutant (the child). Three Mutation methods are implemented to increase the performance of GA. Intelligence mutation is needed to approximate the results. Examination of Fig. 3 shows that in iteration 1,200 the diversity is decreased and consequently the approximate result is yielded. At each step (iteration) of GA, mutation rate is determined by the following procedure:
11 15
Table 2 Cluster members Cluster 1
Cluster 2
If (iteration < 1,200) then
Cluster 3
Swap rate = 10%; Creep rate = 20%;
Cluster members {5, 6, 7, 12, 13} {11, 2, 15, 9,1,4} {3, 8, 10, 14}
Fig. 3 Diversity rate
Intelligence rate = 0%
90
Diversity Rate
80 70 60 50 40 30 20 10
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00
00
18
00
17
00
16
00
15
00
14
50
12
00
Iteration
11
11
90 0 10 00
80 0
0
70 0
0
60
0
50
40
30 0
20 0
1
10 0
0
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Else, Swap rate = 5%; Creep rate = 10%; Intelligence rate = 20% Mutation operation is selected from swap mutation, creep mutation and Intelligence mutation by the randomize function. This function generates the integer numbers from 1 to 100 to swap mutation rate, creep mutation rate and Intelligence mutation rate. Diversity is essential to the genetic algorithm because it enables the algorithm to search a larger region of the space. Three mutations are applied to create sufficient diversity. Each mutation creates one type of diversity (Davis 1991). Swap mutation This mutation simply changes the position of two samples at random (Aarts et al. 1989). This mutation is used only on sample part.
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which is composed of two parts (Aarts et al. 1989). The first part gets more importance to the individual clusters which are very similar (based on co-association matrix). The second part considers clusters which have low association with the other clusters. Equations 8–10 presents final fitness, penalty cluster and fitness cluster, respectively. I Fitness_cluster(Ck ) Final fitness = k=1 I I I −1 Penalty_Cluster(i, j) i=1 j=i+1
−
Penalty_cluster(Ci , C j ) NC j NCi
i= j co_assocition(i, j) N C ∗ N C − N C ∩ N C 2 i j i j Fitness_cluster(Ck ) (i, j)∈(NCk ×NCk ) i≥ j co_assocition(i, j) 2 =
=
(i, j)∈(NCk ×NCk )
NC − NC k k
Intelligence mutation This mutation is used only on sample part. It is used after some iterations (in final iterations) when the algorithm obtains the approximate results. We introduce this mutation to increase the accuracy of the proposed algorithm. This mutation is named as intelligence mutation, because it selects the best candidate samples for mutation. The steps of special mutation are as follows: • A cluster is chosen at random (C). • A member of C with minimum similarity with other members of C is chosen (X ) (dissimilar sample in the same cluster). • The similarity of X with other cluster is calculated. • X is transformed to the most similar cluster (H ). The boundary of H is updated in the index part. The procedure for calculating similarity is the same as the following section (fitness function). Evaluation function (fitness function) The role of the fitness function is to adapt to the requirements. The proposed algorithm presents a fitness function
(9)
(10)
2
Creep mutation This mutation is used only on index part and it works by adding a small (positive or negative) value to selected gene with probability p. It means that this mutation can change the boundary of the selected cluster (increase or decrease) (Aarts et al. 1989).
(8)
I (I − 1)/2
In Eqs. 8–10 we have: Co_association(i, j): The value of entry (i, j) in the co-association matrix Ck : The kth cluster Nk: The set of Ck members index |Nk |: Number of samples in cluster k i: Number of clusters for current individual Fitness_cluster (Ck ): The fitness of cluster kth Penalty_cluster (Ci , C j ): The average similarity between Ci &C j Penalty: The average similarity between all clusters and V is balance parameter.
Method As mentioned, conventional and existing methods have several limitations for in assessment of DMUs. The sensitivity of those methods about outliers and corrupted data is the main reason to introduce the proposed algorithm. To select a method for performance assessment of a set of DMUs, it is suggested to pay great attention about the data, type of problem and DMUs. An ANN GACE algorithm is proposed to measure DMUs’ efficiency in current period. This algorithm can estimate efficiency by considering input (output) orientation through identifying production (cost) function ANN (same as econometric methods). Also, for simplicity we consider one output (input), but it is easy to extend to it various outputs (inputs). Figure 4 presents the integrated algorithm for performance
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1
Determination of input(s) and output(s) variables
2
Collect relative data set S in all available previous and current periods, normalize all data
3
Divide S in to two subsets: training (S1) and valid (S2) data
4
Use different ANN architecture to estimate relation between input(s) and output(s)
5
Run ANN* for Sc.
2.
3.
6 Cluster DMUs by GACE method into x clusters
Set j = 1
Calculate weigh of DMUi (Wi) i = 1… n for DMUs in jth cluster
7
Calculate the error between the real output and ANN model output for DMUs in jth cluster
8
4. Shift frontier function from ANN for DMUs in jth cluster
9
j=j+1
Yes
j ≤ x No
10
• Train the model using the training data (S). In this study, MLP Levenberg–Marquardt (MLP–LM) training algorithm is used. LM algorithm is selected by noting its capability to reach desired error in an appropriate time frame. • Evaluate the model using the test data (Sc ) and obtain MAPE error.
Calculate efficiency scores for all DMUs
Fig. 4 Integrated ANN-GACE algorithm for performance assessment of DMUs
assessment of DMUs. The proposed algorithm is composed of the following steps:
1.
Determination of input (or inputs) and output (P) variables under input oriented assumption (input (C) and output (or outputs) variables under output oriented assumption) of the model.
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Collect data set S in all available previous periods which describes the input-output relationship for DMUs. Assume that there are n DMUs to be evaluated. Note that the current period data (Sc : test data3 ) does not belong to S. Then obtain the preprocessed data set (S and Sc ) after the data are scaled between 0 and 1. Divide S in to two subsets: training (S1 ) and valid (S2 ) data. The first subset is the training set, which is used for computing the gradient and updating the network weights and biases. The second subset is the validation set. The error on the validation set is monitored during the training process. The validation error will normally decrease during the initial phase of training, as does the training set error. However, when the network begins to over fit the data, the error on the validation set will typically begin to rise. When the validation error increases for a specified number of iterations, the training is stopped, and the weights and biases at the minimum of the validation error are returned. The validation is separated from training data and it should be a good representation of data across the range of outcomes. It is necessary to strike a balance between the training and validation data set sizes. When large amounts of data are available the selection of validation data can be done using a simple random choice. The extrapolation ability of ANN should be used and therefore the data for validation is chosen from the period which is closer to Sc . Use ANN method to estimate relation between input(s) and output(s). For this reason select architecture and training parameters. All networks used in this study have a single hidden layer because the single hidden layer network is found to be sufficient to model any function (Cybenko 1989; Patuwo et al. 1993). To find the appropriate number of hidden nodes, the following steps are performed for networks with one to q nodes in their hidden layer. The value of q is optional and should be changed if after the following next steps, the goal error has not been met.
Then the model which has the lowest error (MAPE) is selected to estimate the production function. The MAPE is calculated with following formula: 3
The test set error is not used during the training, but it is used to compare different ANN models.
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MAPE =
237
N 1 Actualvaluei − Setpointvaluei , N Setpointvaluei
on its efficiency is considered and the unit used for the correction is selected by noting its scale (CRS).
i=1
(N : the number of rows) 5. 6.
7.
Run ANN∗ for Sc . Cluster DMUs by GACE (x cluster are obtained after running GA)and for each of the clusters do the fowling steps. Calculate weight of DMUi (Wi )
Vi = Ci /Average (C1 , . . . , Ci−1 , Ci+1 , . . . , Ccj ) i = 1, . . . , n for input oriented model Wi = Vi / (Vi ) (11) where c j (1 ≤ j ≤ x) is the number of DMUs which belong to jth cluster of x clusters. Calculate the error between the real output (Preal(i) ) and ANN model output (PANN∗(i) ) in the period which you want to assess the efficiency of DMUs (Sc ): E i = Preal(i) − PANN∗ (i) i = 1, . . . , n for input oriented model E i = CANN∗ (i) − Creal(i) i = 1, . . . , n for output oriented model (12) Shift frontier function from neural network to obtain the effect of the largest positive error which is one of the unique features of this algorithm: E k = E i /Wi
i = 1, . . . , n
(13)
This option consists of not considering the largest error, but it calculates the frontier function by noting the DMU scale (Constant Returns to Scale (CRS)). Therefore: The largest E i indicates the DMU with best performance. Suppose that DMUk has the Largest E i and we have: E k i = max(E i )
(14)
Thus, the value of the shift for each of the DMUs is different and is calculated by: Shi = E k∗ Wi /Wk i = 1, . . . , n
Fi = Pi /(P(ANN∗ )i + Shi ) i = 1, . . . , n for input oriented model i = 1, . . . , n for input oriented model (16)
i = 1, . . . , n for input oriented model
9.
Calculate the efficiency scores The efficiency scores take values between 0 and 1. The maximum score is assigned to the unit used for the correction in each cluster:
Fi = (C(ANN∗ )i − Shi )/Ci
Vi = Pi /Average (P1 , . . . , Pi−1 , Pi+1 , . . . , Pcj )
8.
10.
(15)
In this approach in spite of the previous studies (Athanassopoulos and Curram (1996) called this measure “standardized efficiency”) the effect of the scale of DMUs
It should be noted that the efficient unit in cluster A is not more efficient than the efficient unit in cluster B although it may be better than it. In some cases, perhaps in a particular cluster, the error E i for all DMUs is negative. In this situation, by the proposed algorithm, frontier function from neural network is shifted to lower level of production. Moreover, the best unit is the DMU that has the lowest loss with respect to its scale. In this study, it has been noted that comparing various units with various scales and capitals is not logical. For example in power plant, increasing 100,000 MWh in the production of a 2,000 and 50 MW plants is more feasible for the 2,000 MW plant. In fact, if a 50 MW plant can produce 100,000 MWh more than predicted value of the ANN algorithm, its efficiency is more than a plant with the same added production of 2,000 MW plant. ANN modelling capability can be used to analyze DMUs’ status. The inputs of each DMU can be changed to foresee their impacts on the output. Moreover, DMUs can assess their own status with probability of changing inputs. Then DMUs can decide to increase or decrease each input with regard to their affect on efficiency. Also, DMUs can establish relationship with other attributes of DMU such as personnel attributes and yield efficiency (Azadeh and Javanmardi 2007). The case study The proposed algorithm is applied to an actual steam power generation plant. The proposed algorithm is applied and explained to the actual data set. The selected power plants (DMUs) essentially perform the same tasks and are homogeneous. This selection ensures that plants in the sample constitute a homogenous technology, thus forming a suitable sample for applying the proposed algorithm. The DMUs are 19 steam power plants in Iran for 1997–2004. It is assumed that the model is input-oriented because power plants have particular orders to fill (e.g., electricity generation) and hence the input quantities appear to be the primary decision variables.
123
238
J Intell Manuf (2011) 22:229–245
Table 3 Real data indicators for evaluating performance of conventional power plants (1997–2000) and (2001–2004) 1997 IC
1999 ICo
FC
GP
IC
ICo
FC
GP
Esfehan
835
329279
1099958
4363907
835
374496
1226427
4673887
Iranshahr
128
31081
132584
347549
128
53869
209113
650897
Besat
248
70690
345723
1197503
1280
439286
1864230
7005830
1280
64464
301947
1006278
466577
1814283
6677500
Bistoon
640
321182
987583
3905080
640
Tabriz
736
338000
1055127
4194640
736
314416
952885
3684730
376163
1219200
4820225
Ramin
1260
380375
1774332
7298784
1890
Zarand
60
11100
45238
121184
60
401456
1766489
7940506
15460
64018
240
99388
359522
1379867
183005
240
98732
359759
1360083
Rajaei
1000
412982
1487164
6280709
1000
385772
1498393
5888736
Salimi
1760
337938
Firoozi
50
14425
2644540
9962950
1760
320923
2713157
10092838
91093
233787
50
7059
44968
Madhaj
290
114861
73939
242386
1033094
290
87872
392738
1570995
Mofatteh
1000
370550
1083089
4633526
1000
385508
1211224
5142265
Montazeri
800
377682
1420517
5399730
1600
646582
2292959
8839074
Toos
600
232562
1050193
3756918
600
231469
1053040
3760082
Bandarabbas
Beheshti
247.5
Mashahd
133
58106
226835
661598
133
66235
247101
742750
Montazerghaem
626
254320
1057716
3920790
625
227990
940948
3529630
1998 IC
2000 ICo
FC
GP 4788579
IC
ICo
FC
GP
835
386886
1315642
128
67058
264044
834132
247.5
70575
326939
1089433
Esfehan
835
351887.5
1209563
Iranshahr
128
42475
246048
707111
Besat
247.5
67577
318269
1061026
452931.5
1808702
6602310
1280
483799
1936360
7076050
963206
3934980
640
334874
1048605
4181695
Bandarabbas Bistoon
1280 640
317799
5091712
Tabriz
736
357081.5
932911
3698715
736
381788
1241956
4853130
Ramin
1575
390915.5
1453842
6301687
1890
528661
2334391
10238875
Zarand
60
13280
62456
175639
60
14887
61964
172142
240
99060
324733
1252004
240
104456
417468
1576566
Rajaei
1000
399377
1337951
5797072
1000
413835
1507394
6483750
Salimi
1760
329430.5
2642707
9563050
1760
266939
2905615
10492131
Firoozi
50
73872
188691
50
3215
22023
56254
Madhaj
290
304735
1177177
290
108852
393838
1575351
1376720
5821394
Beheshti
10742 80905.5
Mofatteh
1000
378029
1142855
4945512
1000
409486
Montazeri
1200
512132
1620133
6117069
1600
733304
2631552
10247273
Toos
600
232015.5
997084
3553653
600
266989
1123230
3991182
Mashahd
133
62170.5
228970
654738
133
61515
536099
682881
Montazerghaem
626
942865
3469570
625
225110
935678
3426910
241155
2001 IC
2003 ICo
FC
GP
IC
ICo
FC 1353534
Esfehan
835
380422
1315319
5045028
835
383123
Iranshahr
128
65602
272739
865502
256
76538
123
322812.8
GP 5159787 1024768
J Intell Manuf (2011) 22:229–245
239
Table 3 continued 2001 IC Besat Bandarabbas
247.5 1280
2003 ICo
FC
GP
IC 247.5
ICo
97371
402697
1256959
500282
1759773
6306860
1280
521124
102313
FC 451699.9 2067974
GP 1354178 7329520
Bistoon
640
284381
882762
3574670
640
341280
1060294
4175810
Tabriz
736
327362
1015303
4053160
736
343049
1066862
4171440
Ramin
1890
635592
2606317
11327188
1890
595521
2484192
10516063
Zarand
60
22060
86974
243189
60
27925
111874
309276
1647855
Shazand
1300
365280
993077
4061178
1300
665394
Beheshti
240
109320
425810
1572894
240
101397
486229.9
7444968 1728820
Rajaei
1000
442179
1576409
6649454
1000
394500
1419424
6017929
Salimi
1760
337494
2845149
10241907
1760
315385
2681187
10003329
Firoozi
50
12301
87159
217896
50
13707
97763.7
241905
Madhaj
290
110374
396984
1587936
290
89828
299103.3
1196408
Mofatteh
1000
358077
1227777
5064299
1000
413842
1423815
5468079
Montazeri
1600
789265
2939317
11115901
1600
823033
3004335
11640505
Toos
600
265032
1046902
3725620
600
258318
1039862
3692738
Mashahd
120
580363
775823
120
72635
262383
771897
Montazerghaem
625.88
924527
3417070
242139
871940
3155480
67615.5 257902
2002 IC
625.88 2004
ICo
FC
GP
IC
ICo
FC
GP
Esfehan
835
413412
1431905
5486515
835
422673
1519828
5621431
Iranshahr
192
57733
237914
751759
256
140940
453860
1492847
Besat
91143
388880
1217997
1280
366441
1415261
5012560
1280
Bistoon
640
276674
858706
3455480
Tabriz
736
272689
833390
3367600
Ramin
1890
622654
2658594
Zarand
60
31085
111268
1300
607092
1558995
Bandarabbas
Shazand Beheshti
247.5
247.5
139505
489664
1500253
588855
2109162
7196540
640
350154
1101614
4210280
736
361080
1233226
4341330
11494462
1890
636643
2686567
10861867
309696
60
29698
124479
341402
6806304
1300
642909
1650253
7438002
240
111710
463475
1738341
240
85307
404894
1435991
Rajaei
1000
429868
1441756
6079938
1000
421015
1572620
6342203
Salimi
1760
362650
2672674
9867577
1760
301276
3085375
11310817
Firoozi
50
12858
95242
240363
50
13039
98275
212403
Madhaj
290
97091
361081
1444325
290
81674
227104
922587
Mofatteh
1000
398613
1333949
5411964
1000
390708
1335352
5134547
Montazeri
1600
802543
3298201
11273792
1600
796262
2949382
11137177
Toos
600
264541
1107494
3889653
600
271901
1078599
3831065
Mashahd
120
73716
272196
816277
120
63050
229087
665887
Montazerghaem
625.88
159212
903907
3318380
241139
891334
3297100
625.8
IC Install capacity (MW); ICo Internal consumption (MWh); EC Fuel consumption (TJ); GP Gross production (MWh)
Step 1: As shown by several authors, the production function for conventional thermal steam-electric production may be described conveniently within an engineering
framework. In this framework, pertinent inputs are the fuel quantity consumed and installed power, which is the maximum nominal power the plants are initially designed
123
240
J Intell Manuf (2011) 22:229–245
Table 4 The MAPE for the 20 Models Model
Number of neurons in hidden layera
Install Capacity
MAPE
1
1
0.066
2
2
0.045
3
3
0.080
4
4
0.016
5
5
0.051
6
6
0.089
7
7
0.045
8
8
0.019
9
9
0.035
10
10
0.036
11
11
0.028
12
12
0.081
13
13
0.097
14
14
0.044
15
15
0.056
16
16
0.058
17
17
0.094
18
18
0.066
19
19
0.079
20
20
0.056
a Hidden
layer activation function which is defined for this single layer is Tan–Sigmoid. This non-linear feature is introduced at the hidden transfer function. From the previous universal approximation studies, these transfer function must have mild regularity conditions: continuous, bounded, differentiable and monotonic increasing. The most popular transfer function is sigmoid or tanh. TanSig (n) calculates its output according to: n = 2/(1 + exp(−2 × n)) − 1. This is mathematically equivalent to tanh (n). This transfer functions bound the output to a finite range [−1, 1]
for. On the other hand, labor inputs contribute to production through control and maintenance services, which also require some capital. The output is, of course, electrical energy production. But by noting studies about efficiency measurement of thermal power generations in Iran which indicate that labor isn’t an effective factor (for instance, Emami Meibodi 1998), in our study, electric power (in megawatt hour) generated from thermal power plants in each DMU (GP) is used as the output variable, while Install Capacity (IC), Fuel Consumption (FC), and Internal Consumption (ICo) are three inputs used for power generation. Capital is measured in terms of installed thermal generating capacity in megawatt (MW (Fare et al. 1983; Hawdon 1997). Various natural elements have been used as fuel in the production of electric power in various steam plants in Iran (natural gas, gas oil and Mazute). The choice of fuel depends on many factors such as availability, cost and environmental concerns and each fuel has its limitations. Our figures measure fuel consumption in terms of Tera Joule (TJ). In other words, our figures have already adjusted for
123
1 Gross Production Internal Consumption
2 . . .
Fuel Consumption
Input layer
4
Hidden layers
Output layer
Fig. 5 The architecture of the selected ANN model (8th model)
the quality of fuel used in different plants. Internal power is the amount of energy consumed (in megawatt hour) within the site (for electrically powered equipment etc). Step 2: 148 rows of data are collected from 1997 to 2004. Table 3 shows the real data for the inputs and output used in the algorithm. Detailed information about power generation of thermal power plants in Iran, like total output, generation capacity and fuel consumption can be obtained from “Electric Power Industry in Iran” (including transmission and distribution) published by the Tavanir (1997– 2004). Current period (2004) is considered as Sc . Step 3: In our problem, each period have all range of outcomes. Therefore, all data for period can be selected for test or can be sorted by the value of the output variable, partitioned and one validation data point is chosen at random from each partition. In this way the stratification tries to ensure that validation data is chosen across the range of outcomes. S1 is data from 1997 to 2002 (110 rows of data) and S2 is 2003 data (19 rows of data). Step 4: In this study the value of the desired minimum error has been defined between 2 and 4% (96–98% confidence) and the value of q has been defined 20. The error is estimated by Mean Absolute Percentage Error (MAPE). In order to get the best ANN for the electricity production in steam power plants, 20 MLP–LM models are tested to find the best architecture. The architecture of the MLP– LM models and their MAPE values are shown in Table 4. It seems the 4th model (it has four neurons in single hidden layer) has the lowest MAPE error and consequently is chosen as the preferred model. In Fig. 5, the ANN architecture for the preferred network (4th model) is shown. Step 5: Therefore, this model is selected for estimating the electricity production in 2004.
J Intell Manuf (2011) 22:229–245
241
Table 5 Estimation of efficiency scores for all DMUs by proposed algorithm in 2004 DMU name
Preal(i)
PANN(i)
Ei
Wi
E i
Shi
Fi
Ranking
Cluster A Montazerghaem
0.280
0.282
−0.002
0.195
−0.008
0.004
0.982
3
Bistoon
0.359
0.354
0.005
0.274
0.019
0.005
1
1
Toos
0.326
0.325
0.001
0.240
0.004
0.005
0.989
2
Tabriz
0.370
0.398
−0.028
0.292
−0.097
0.006
0.916
4
0.125
0.154
−0.03
0.25
−0.13
0.015
0.70
4
−0.45
0.002
0.36
7
0.01
1
1
Cluster B Besat Firoozi
0.013
0.033
−0.02
0.04
Beheshti
0.119
0.109
0.01
0.16
0.059
Madhaj
0.075
0.084
−0.009
0.12
−0.073
0.007
0.82
2
Zarand
0.025
0.043
−0.018
0.05
−0.3
0.003
0.53
6
Mashhad
0.053
0.071
−0.018
0.1
−0.17
0.006
0.68
5
Iranshahr
0.124
0.15
−0.03
0.24
−0.13
0.014
0.75
3
Salimi
0.972
0.942
0.030
0.175
0.170
0.061
0.969
2
Shazand
0.637
0.601
0.036
0.104
0.350
0.036
1
1
Rajaei
0.543
0.538
0.005
0.092
0.051
0.032
0.952
3
Mofatteh
0.438
0.459
−0.021
0.077
−0.269
0.027
0.902
7
Ramin
0.933
0.967
−0.034
0.180
−0.189
0.063
0.906
6
Bandarabbas
0.616
0.619
−0.003
0.107
−0.025
0.038
0.939
5
Cluster C
Esfahan
0.480
0.518
−0.038
0.088
−0.429
0.031
0.875
8
Montazeri
0.957
0.953
0.004
0.177
0.020
0.062
0.942
4
Step 6: The best partitioning of the data is achieved with 3 clusters. To identify the best fitness, the required parameters on GA algorithm are as follows: • Population size: 100 • Iteration: 1,500 • Crossover rate: 80% Steps 7–10: The results of these steps for each of the clusters are shown in Table 5. It is observed that medium, small and large plants belong to clusters A, B, and C respectively. The proposed algorithm estimates the efficiencies by the best estimated neural networks for 2004. The main results of the proposed algorithm, proposed algorithm without clustering and DEA are summarized in the Table 6. The results are compared through t-test. Several differences are clearly appreciated from the statistical tests. Also, DEA which is a well known performance assessment tool is used for validation with proposed algorithm without clustering. It is compared with the results of proposed and proposed without clustering through paired t-test. • The mean technical efficiencies obtained from the proposed algorithm and DEA is relatively small ( p-value:
0.52; higher p-value). Therefore, the proposed algorithm is verified and validated by DEA. • Mean technical efficiencies obtained from the proposed algorithm without clustering and DEA is significantly different ( p-value: 0.16; small p-value). Therefore, clustering benefits the proposed algorithm.
Examination of proposed algorithm in outliers and corrupted data situation The capability of proposed algorithm in handling outliers and corrupted data is presented. We suppose that imaginary outlier data is yielded from human error. Also, it is assumed that outlier or corrupted data is occurred in inputs variable. Let us suppose that one of input variable of Besat DMU (Fuel Consumption) is changed from 0.13 to 0.013 that is yielded from human error. The result of proposed algorithm and DEA after this error is shown in Tables 7 and 8. MAPE is used to compare the results. Table 9 shows MAPE value for proposed algorithm. Examination of MAPE values show that the proposed algorithm is approximately eight times better than DEA. The same procedure is done for Bandarabbas DMU. Second input variable of Bandaabbas DMU (Internal Consumption) is changed from 0.71 to 0.071. The results
123
242 Table 6 Efficiency results for the conventional power plants in 2004
Table 7 The impact of outlier and corrupted data on proposed algorithm result
J Intell Manuf (2011) 22:229–245
DMU
Efficiency scores by the proposed algorithm
Efficiency scores by the proposed algorithm without clustering
Efficiency scores by DEA-CCR
Bandarabbas
0.93
0.91
0.81
Beheshti
1.00
1
0.92
Besat
0.72
0.71
0.83
Bistoon
1.00
0.92
0.98
Esfahan
0.87
0.83
0.97
Firoozi
0.38
0.38
0.64
Iranshahr
0.75
0.74
0.81
Madhaj
0.819
0.83
0.91
Mashhad
0.68
0.67
0.72
Mofatteh
0.90
0.86
0.80
Montazerghaem
0.98
0.91
0.82
Montazeri
0.94
0.91
1
Rajaei
0.95
0.92
1
Ramin
0.90
0.85
0.99
Salimi
0.969
0.91
1
Shazand
1.00
0.93
1
Tabriz
0.91
0.85
0.82
Toos
0.98
0.91
0.91
Zarand
0.52
0.53
0.81
Mean (μTE )
0.852
0.819
0.881
Proposed algorithm
DMU name
Cluster A
Montazerghaem Bistoon
Cluster B
97.99 100
Original efficiency score (%)
97.99 100
98.94
98.94
Tabriz
91.61
91.61
91.61
Besat
92.20
70.94
70.94
Firoozi
36.48
36.46
36.46
100
100
98.94
100
Madhaj
82.11
82.09
82.09
Zarand
53.78
53.74
53.74
Mashhad
68.78
68.75
68.75
Iranshahr
75.15
75.19
75.19
Salimi Shazand
123
97.99 100
Efficiency score after Bandarabbas noise (%)
Toos
Beheshti
Cluster C
Efficiency score after Besat noise (%)
96.93 100
96.95 100
96.93 100
Rajaei
95.29
95.29
95.29
Mofatteh
90.17
90.16
90.17
Ramin
90.62
90.64
90.62
Bandarabbas
93.56
73.87
93.56
Esfahan
87.50
87.50
87.50
Montazeri
94.34
94.35
94.34
J Intell Manuf (2011) 22:229–245
Table 8 The impact of outlier and corrupted data on DEA result
243
DEA
DMU name
Efficiency score after Besat noise
Original efficiency score
Bandarabbas
0.71
1
0.81
Beheshti
0.87
0.69
0.92
Besat
1
0.72
0.83
Bistoon
0.79
0.81
0.98
Esfahan
0.81
0.68
0.97
Firoozi
0.64
1
0.64
Iranshahr
0.71
0.77
0.81
Madhaj
0.64
0.62
0.91
Mashhad
0.57
0.8
0.72
Mofatteh
0.6
0.81
0.8
Montazerghaem
0.69
0.68
0.82
Montazeri
0.82
0.88
1
Rajaei
0.84
0.68
1
Ramin
0.83
0.65
0.99
Salimi
1
1
1
Shazand
0.75
0.58
1
Tabriz
0.68
0.56
0.82
Toos
0.83
0.71
0.91
Zarand
0.81
0.9
0.81
Table 9 The MAPE value for Besat and Bandarabbas in proposed algorithm and DEA
Besat Bandarabbas
Efficiency score after Bandarabbas noise
Proposed algorithm
DEA
0.015 0.011
0.14 0.13
show that the proposed algorithm is approximately 11 times better that DEA in viewpoint of handling outliers and corrupted data. This two sample show that the results of proposed algorithm are more robust than the DEA. Conclusion A highly unique flexible ANN GA Clustering Ensemble algorithm was proposed to measure the DMUs efficiency scores. It is composed of ten steps. The non-linearity of the neural networks and its universal approximations makes the algorithms highly flexible. Furthermore, it can handle outliers and corrupted data much better than conventional econometric approach and DEA. This is shown by two examples related to outlier situations. Examination of the two example show that the proposed algorithm is 9 and 11 times better than DEA in handling outliers and corrupted data. It is also capable of performing optimization analysis and forecast-
ing for a given set of data. It was applied to a set of actual steam power plants to show its applicability and superiority. Furthermore, the results and rankings were compared with the proposed algorithm without clustering. Results indicate that the proposed algorithm estimates the values of efficiency scores closer to the ideal efficiency in viewpoint of DEA through statistical t-test. In summary, the algorithm is able to find a stochastic frontier based on a set of input–output observational data and do not require explicit assumptions about the functional structure of the stochastic frontier. Furthermore, it uses a similar approach to econometric methods for calculating the efficiency scores. Moreover, the effect of the return to scale of decision making unit (DMU) on its efficiency is included and the unit used for the correction is selected by notice of its scale (under constant return to scale assumption). Also, in this algorithm, for increasing DMUs’ homogeneousness, GA is used to cluster DMUs. Although it is believed that ANNs can be a potential alternative for measuring technical efficiency and can outperform other techniques when the production process is unknown, there is still a lack of both theoretical and empirical work in efficiency analysis and consequently optimization analysis. In future, some cases with more outputs or inputs can be studied and also the proposed algorithm can be expanded by encountering the lack of data. Designing proper algorithms
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for cases with data of only one period is a candidate for future research. In summary, it should be mentioned that pervious studies do not consider the all of following features of the proposed algorithm 1. High flexibility; 2. Handling outlier and statistical noise data and o sensitive to the presence of the outliers and statistical noise; 3. Forecasting capability; 4. Low assumptions about functional form and data and do not require explicit assumptions about the functional structure of the stochastic frontier; 5. Optimum clustering ability; 6. No sensitivity to initial points in clustering section.
Acknowledgments The authors are grateful for the valuable comments and suggestion from the respected reviewers. Their valuable comments and suggestions have enhanced the strength and significance of our paper.
References Aarts, E. H. L., Eiben, A. E., & vanHee, K. M. (1989). A general theory of genetic algorithms. Tech.Rep.89/08, Einndhoven University of Thechnology. Athanassopoulos, A. D., & Curram, S. P. (1996). A comparison of data envelopment analysis and artificial neural networks as tool for assessing the efficiency of decision making units. Journal of the Operational Research Society, 47(8), 1000–1016. Azadeh, A., Ghaderi, S. F., Anvari, M., & Saberi, M. (2006a). Measuring performance electric power generations using artificial neural networks and fuzzy clustering. In G. A. Capolino, L. G. Franquelo (Eds.), Proceedings of the 32nd annual conference of the IEEE industrial electronics society. Paris, France: IECON. Azadeh, A., Ghaderi, S. F., Anvari, M., & Saberi, M. (2007a). Performance assessment of electric power generations using an adaptive neural network algorithm. Energy Policy, 35, 3155–3166. Azadeh, A., Ghaderi, S. F., Anvari, M., Saberi, M., & Izadbakhsh, H. (2006b). An integrated artificial neural network and fuzzy clustering algorithm for performance assessment of decision making units. Applied Mathematics and Computation, 187(2), 584–599. Azadeh, A., Ghaderi, S. F., & Sohrabkhani, S. (2007b). Forecasting electrical consumption by integration of neural network, time series and ANOVA. Applied Mathematics and Computation, 186, 1753–1761. Azadeh, A., Ghaderi, S. F., Tarverdian, S., & Saberi, M. (2006c). Integration of artificial neural networks and GA to predict electrical energy consumption. In G. A Capolino & L. G. Franquelo (Eds.), Proceedings of the 32nd annual conference of the IEEE industrial electronics society-IECON’06. Paris, France: Conservatoire National des Arts & Metiers. Azadeh, A., Ghaderi, S. F., Tarverdian, S., & Saberi, M. (2007). Integration of artificial neural networks and genetic algorithm to predict electrical energy consumption. Applied Mathematics and Computation, 186, 1731–1741. Azadeh, A., & Javanmardi, L. (2007). Studying impact of decision making units feature on efficiency by integration of data
123
J Intell Manuf (2011) 22:229–245 envelopment analysis and data mining tools. In International conference on operations research. German Azimi, J., Mohammadi, M., Movaghar, M., Analoui, A., & Clustering, M. (2006). Ensembles using genetic algorithm. In International workshop on computer architecture for machine perception and sensing, 2006 CAMP (pp. 119–123). Bauer, P. W. (1990). Recent developments in the econometric estimation of frontiers. Journal of Econometrics, 46(1–2), 39–56. Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2(6), 429–444. Chiang, W. C., Urban, T. L., & Baldridge, G. W. (1996). A neural network approach to mutual fund net asset value forecasting. Omega, The International Journal of Management Science, 24, 205–215. Cook, W. D., & Green, R. H. (2005). Evaluating power plant efficiency: A hierarchical model. Computers & Operations Research, 32(4), 813–823. Costa, A., & Markellos, R. N. (1997). Evaluating public transport efficiency with neural network models. Transportation Research C, 5(5), 301–312. Cybenko, G. (1989). Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals and Systems, 2, 303– 314. Davis, L. (Ed.). (1991). Handbook of genetic algorithms. New York: Van Nostrand Reinhold. Delgado, F. J. (2005). Measuring efficiency with neural networks, an application to the public sector. Economics Bulletin, 3(15), 1–10. Doyle, J., & Green, R. (1994). Efficiency and cross-efficiency in DEA: Derivations, meanings and uses. Journal of the Operational Research Society, 45(5), 567–578. Emami Meibodi, A. (1998). Efficiency considerations in the electricity supply industry: The case of Iran. A thesis submitted to university of Surrey for the degree of Doctor of Philosophy. Fare, R., Grosskopf, S., & Lovell, C. A. K. (1983). The measurement of efficiency of production. Boston: Kluwer. Golany, B., Yaakov, R., & Rybak, D. (1994). Measuring efficiency of power plants in Israel by data envelopment analysis. IEEE Transactions on Engineering Management, 41(3), 291–301. Goldberg, D. E. (1989). Genetic algorithms in search optimization and machine learning. New York: Addison-Wesley. Goto, M., & Tsutsui, M. (1997). Comparison of productive and cost efficiencies among Japanese and US electric utilities. Tokyo, Japan: Central Research Institute of Electric Power Industry. Hawdon, D. (1997). Improving the performance of electricity industries in developing countries: Is World Bank policy on deregulation the way forward. Economics Departmental Research Seminar, University of Surrey. Hill, T., O’Connor, M., & Remus, W. (1996). Neural network models for time series forecasts. Management Science, 42(7), 1082–1092. Hornik, K., Stinchcombe, M., & White, H. (1989). Multilayer feed forward networks are universal approximators. Neural Networks, 2, 359–366. Indro, D. C., Jiang, C. X., Patuwo, B. E., & Zhang, G. P. (1999). Predicting mutual fund performance using artificial neural networks. Omega, The International Journal of Management Science, 27, 373–380. Jamasb, T., & Pollitt, M. (2001). Benchmarking and regulation of electricity distribution and transmission utilities: Lessons from international experience. DEA Working Paper, Cambridge, UK: Department of Applied Economics, University of Cambridge. Jhee, W. C., & Lee, J. K. (1993). Performance of neural networks in managerial forecasting. Intelligent Systems in Accounting, Finance and Management, 2, 55–71. Knittel, C. R. (2002). Alternative regulatory methods and firm efficiency: Stochastic frontier evidence from the US electricity
J Intell Manuf (2011) 22:229–245 industry. The Review of Economics and Statistics, 84(3), 530– 540. Kohzadi, N., Boyd, M. S., Kermanshahi, B., & Kaastra, I. (1996). A comparison of artificial neural network and time series models for forecasting commodity prices. Neurocomputing, 10(3), 169–181. Lam, P., & Shiu, A. (2001). A data envelopment analysis of the efficiency of China’s thermal power generation. Utilities Policy, 10, 75–83. Minaei, B., Topchy, A., & Punch, W. F. (2004). Ensembles of Partitions via Data Resampling. In Proceedings of the international conference on information technology, ITCC 04. Las Vegas. Mohammadi, M., Davoodi, R., & Rahmani, A. (2007). A genetic based clustering method. In Proceedings of 12th annual international computer society of iran computer conference (CSICC). Olatfubi, W. O., & Dismukes, D. E. (2000). A data envelopment analysis of the level and determinants of coal-fired electric power generation performance. Utilities Policy, 9, 47–59. Park, S., & Lesourd, J. (2000). The efficiency of conventional fuel power plants in South Korea: A comparison of parametric and non-parametric approaches. International Journal of Production Economics, 63, 59–67. Patuwo, E., Hu, M. Y., & Hung, M. S. (1993). Two-group classification using neural networks. Decision Sciences, 24(4), 825–845. Pendharkar, P. C. J. A., & Rodger, J. A. (2003). Technical efficiencybased selection of learning cases to improve forecasting accuracy of neural networks under monotonicity assumption. Decision Support Systems, 36(1), 117–136. Pollit, M. G. (1995). Ownership and performance in electric utilities. Oxford, UK: Oxford University Press. Rumelhart, D. E., & McClelland, J. L. (1986). Parallel distributed processing: Explorations in the microstructure of cognition, Vol. 1, Foundations. Cambridge, MA: MIT Press. Sanhueza, R., Rudnick, H., & Lagunas, H. (2004). DEA efficiency for the determination of the electric power distribution added value. IEEE Transactions on Power Systems, 19(2), 919–925. Santin, D., Delgado, F. J., & Valino, A. (2004). The measurement of technical efficiency: A neural network approach. Applied Economics, 36, 627–635. Stern, H. S. (1996). Neural networks in applied statistics. Technometrics, 38(3), 205–214. Strehl, A., & Ghosh, J. (2002). Cluster ensembles—acknowledge reuse framework for combining multiple partitions. Journal on Machine Learning Research, 3, 583–617.
245 Strehl, A., & Ghosh, J. (2002). Cluster ensembles—a knowledge reuse framework for combining partitionings. In National Conference on Artificial Intelligence (pp. 93–98). AB, Canada. Sueyoshi, T., & Goto, M. (2001). Slack-adjusted DEA for time series analysis: Performance measurement of Japanese electric power generation industry in 1984–1993. European Journal of Operation Research, 133, 232–259. Talluri, S., & Sarkis, J. (1997). Extensions in efficiency measurement of alternate machine component grouping solutions via data envelopment analysis. IEEE Transactions on Engineering Management, 44(3), 299–304. Tamura, H., & Tann, K. (2008). Midpoint-validation method of neural networks for pattern classification problems. International Journal of Innovative Computing, Information and Control, 4(10), 2475– 2482. Tang, Z., Almeida, C., & Fishwick, P. (1991). A Time series forecasting using neural networks vs Box–Jenkins methodology. Simulation, 57(5), 303–310. Tang, Z., & Fishwick, P. A. (1993). Back-propagation neural nets as models for time series forecasting. ORSA Journal on Computing, 5(4), 374–385. Tavanir Management Organization. (1997–2004). Electric power industry in Iran 1997–2004. Tavanir management organization. Topchy, A., Minaei-Bidgoli B., Jain, A. K., & Punch, W. (2004). Adaptive Clustering ensembles. In Proceedings of the international conference on pattern recognition, ICPR’04 (pp. 272–275). Cambridge, UK. Wang, S. (2003). Adaptive non-parametric efficiency frontier analysis: a neural-network-based model. Computers & Operations Research, 2003(30), 279–295. Wang, X., Zhao, X., & Feng, Y. (2008). A single parameter dynamic searching algorithm for multi-layer neural networks (part II). International Journal of Innovative Computing, Information and Control, 4(11), 2941–2954. Wu, D., Yang, Z., & Liang, L. (2006). Using DEA-neural network approach to evaluate branch efficiency of a large Canadian bank. Expert Systems with Applications, 31, 108–115.Z Zhang, Z., Yang, G., Yi, J., Zhu, Y. Y., & Tang, Z. (2008). A new stochastic dynamic adaptive local search algorithm for Elman neural network. International Journal of Innovative Computing, Information and Control, 4(11), 2927–2940.
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