A fuzzy regression approach to a hierarchical ... - Springer Link
Fuzzy Optim Decis Making (2010) 9:105–122 DOI 10.1007/s10700-010-9072-3
A fuzzy regression approach to a hierarchical evaluation model for oil palm fruit grading A. Nureize · J. Watada
Published online: 21 March 2010 © Springer Science+Business Media, LLC 2010
Abstract Measurement of quality is an important task in the evaluation of agricultural products and plays a pivotal role in agricultural production. The inspection process normally involves a visual examination according to the ripeness standards of crops, and this grading is subject to expert knowledge and interpretation. Therefore, the quality inspection process of fruits needs to be conducted properly to ensure that high-quality fruit bunches are selected for production. However, human subjective judgments during the evaluation make the fruit grading inexact. The objectives of this paper are to build a fuzzy hierarchical evaluation model that characterises the criteria of oil palm fruits to decide the fuzzy weights of these criteria based on a fuzzy regression model, and to help inspectors conduct a proper total evaluation. A numerical example is included to illustrate the computational process of the proposed model. Keywords Fuzzy regression analysis · Fuzzy hierarchical model · Multicriterion · Oil palm fruit grading 1 Introduction The palm oil industry has played a remarkable role in Malaysia’s economic and social development. Accordingly, a current priority of Malaysian policy is to ensure that the yearly surplus of exported palm oil satisfies the growing worldwide market demand for oils and fats (Yusuf and Chan 2004). The increasing demand for palm oil products
A. Nureize (B) · J. Watada Graduate School of Information, Production and System, Waseda University, 2-7 Hibikino, Wakamatsu, Kitakyushu 808-0135, Fukuoka, Japan e-mail: [email protected] J. Watada e-mail: [email protected]
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in local and international markets drives interest in raising the yield of fresh oil palm fruit. Since fresh fruit bunches are the starting input for crude palm oil production, it is therefore imperative that only high-quality fruit bunches be selected and processed (Abdullah et al. 2004; MPOB 2003). Moreover, higher quality fresh fruit bunches (FFB) produce a higher quantity and quality of palm oil (Abdullah et al. 2001). Highquality oil palm fruit can improve the quality and quantity of palm oil products. The presence of unripe bunches results in a lower oil extraction rate, and the overripe bunch affects the fatty fruit acid content. Therefore, to sustain the production rate and production efficiency, a higher quality of oil palm fruit bunches should be used. To accomplish this goal, inspection control should be placed at the entrance of processing plants to ensure that the required characteristics of fruit are satisfied. The grading process is carried out besides the loading ramp inside the mill premises in the presence of a supplier representative. Representative persons from the field and mill must be involved in quality control and be responsible for the quality requirements (Eng and Tat 1985). Grading fresh fruit bunches is a process wherein fruits are assessed and classified according to criteria of ripeness and bunch quality (Yusuf and Chan 2004). In practice, oil palm fruits are inspected and graded by expert inspectors at a mill who have capabilities and experiences in grading fresh fruit bunches and who judge quality by looking individually at the product (Abdullah et al. 2004). Basically, the grading practice involves the inspection of bunch quality, and the estimation of basic extraction rates and graded extraction rates. Consignment of a fresh fruit bunch that has poor quality will be allowed, but subject to a penalty if up to 20 to 30% of the fruit fails to meet the allowable quality limit. The penalty is the percentage to be deducted from a basic extraction rate. Meanwhile, the basic extraction rate is the maximum theoretical percentage of crude palm oil and palm kernels that can be produced from fruit bunches. All the information from the grading process is subsequently transferred to the grading form for documentation. The grading process must be handled properly to select quality fruit and to remove defective units that show signs of noncompliance with the standard criteria. Fruit bunches are evaluated and classified based on standard criteria as set by the Malaysian Palm Oil Board (MPOB). The standard criteria are used by the MPOB to promote quality awareness among the mills, the plantations and the small holding sectors (Yusuf and Chan 2004). Fruit bunches are typically evaluated using visual examination based on standard criteria such as colour, number of detached or attached fruitlets, physical appearance and disease. Each criterion carries a different weight of importance in the evaluation. These weights are necessary and can be used to decide the most important criterion during the fruit evaluation. Generally, the ripeness of an oil palm bunch is determined by using the percentage of detached fruits per bunch (Siregar 1976). However, the colour of fruit also provides valuable information in estimating maturity and ripeness. Colour characteristics do not only indicate the original product quality, but also determine the efficiency of the manufacturing (Abdullah et al. 2004). Therefore, colour becomes an important feature for identifying the ripeness and maturity of oil palm fruit. Several studies available in the literature specifically focus on visual analysis in the evaluation of oil palm fruit. Abdullah et al. (2001) conducted a stepwise discriminant analysis for inspecting the quality features of oil palms through colour image analysis. Meanwhile, the correlation between the oil content and the
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colour of oil palm fruit has also been investigated (Rashid et al. 2002). Accordingly, Abdullah et al. (2004) focused on image acquisition technologies, using a machine vision system and computerised radar tomography to assess the physical properties of oil palm fruit. Alfatni et al. (2008) found that the ripeness of a fruit bunch could be classified into different categories of fruit bunches based on Red, Green, and Blue (RGB) colour intensity. In addition, Abbas et al. (2005) also investigated the feasibility of using moisture measurements to assess the quality of oil palm fruits. However, from the literature, it can be concluded that colour is the main characteristic that plays a pivotal role in determining the quality of the fruit. Currently, human graders are involved directly in the evaluation and grading process in the mills. Even though numerous studies (Abbas et al. 2005; Abdullah et al. 2001; Abdullah et al. 2004; Alfatni et al. 2008; Rashid et al. 2002) have been published regarding automating the grading process to accelerate sorting and evaluation, that kind of technology is still not implemented in Malaysian palm oil mills. For that reason, human grading still remains the most suitable method due to the high cost of advanced machine implementation. In practice, grading experts, whose capability and experience are needed to adequately grade fresh fruit bunches, inspect and grade oil palm fruits at a mill. The skill and experience of human graders are important, as the grading process involves expert visual evaluation. Consequently, accumulated knowledge is useful in the grading process, even though the evaluation is based on several quantitative and qualitative criteria that are influenced by the grader’s experiences and knowledge. Thus, the evaluation involves both accurate and inexact information, since the fruit grading evaluation depends upon subjective human judgments. The objective of this paper is to provide an estimation of weights of attributes by means of fuzzy regression. Moreover, this paper introduces a fuzzy hierarchy evaluation model to assist and improve the quality inspection process as well as to support the decision-making process in the palm oil industry. The remainder of this paper is organised as follows. Related research is reviewed briefly in Section 2. Section 3 explains two widely used methods, namely, AHP and TOPSIS, for comparison with our proposed method using real data. Section 4 describes the fuzzy hierarchical evaluation model. The fuzzy weight in the fuzzy hierarchical evaluation model is assessed by fuzzy regression analysis in Section 5. Section 6 discusses fuzzy hierarchical evaluation decision-making based on our model. Section 7 presents a real application of the model in the evaluation of oil palm grading, and Section 8 concludes this paper with some additional remarks.
2 Overview of related works In decision making process, as the problem grows more complex, decomposition of the problem becomes necessary. Decomposition is the process of dividing a problem into several small sub-problems and arranging them into a hierarchical structure, such as an Analytic Hierarchy Process (AHP). This type of structure can reduce the complexity of the problem, allowing better capture, description and understanding of realistic problems. AHP is a multiattribute approach in decision making that has been successfully applied to many real-world decision making problems (Saaty 1990). The
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AHP structures a multiattribute problem hierarchically, investigates the levels of the hierarchy separately and produces a result in rank order (Irfan and Nilsen 2006). The goal of AHP is to enable us to employ a number of pair-wise comparisons obtained using human judgment. However, many practical cases in a human preference model involve imprecise values, making it difficult to assign exact numerical values for comparison judgments (Irfan and Nilsen 2006; Zadeh 1998). This is due to factors such as incomplete and imprecise subjectivity, which tend to be present to some degree. Therefore, inexact elements included in the AHP decision analysis will add uncertainty and vagueness to the decision process. The Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) is another approach in multiattribute decision making. TOPSIS, which was introduced by Hwang and Yoon (1981), evaluates options geometrically. In this case, alternatives are chosen based on the shortest distance from the positive ideal solution and the longest distance from the negative ideal solution. TOPSIS defines an index called similarity, or relative closeness to the positive-ideal solution and remoteness from the negative-ideal solution. The alternative with the maximum similarity to the positiveideal solution will be selected with priority (Yoon and Hwang 1995). Fuzzy elements have also been introduced and examined in TOPSIS research in order to deal with fuzzy environments (Li 2007). Decision-making situations commonly involve complex, uncertain and imprecise information. Fuzzy decision-making has been tackled successfully to deal with vagueness in linguistics and expressing human knowledge and inference mechanisms in a natural way. Multiple criteria are considered in a decision process. A multicriteria analysis with fuzzy pairwise comparisons is presented in Deng (1999). Kreng and Wu (2007) demonstrated a comprehensive hierarchical framework by using a fuzzy AHP approach and a technique for determining weights for evaluating knowledge portal system development tools. Takahagi (2008) introduces an identification method for fuzzy measures using diamond pairwise comparisons. Meanwhile, Yeh and Chang (2008) presented a new fuzzy multicriteria decision-making approach for evaluating decision alternatives involving subjective judgments made by a group of decision makers. In this study, a pairwise comparison process was used to make comparative judgments, and a linguistic rating method was used to make absolute judgments. A hierarchical weighting method was developed to assess the weights of a large number of evaluation criteria by pairwise comparisons. Enea and Piazza (2004) show that better results can be achieved by considering all the information deriving from the constraints within fuzzy AHP in terms of certainty and reliability. Kuo et al. (2006) present an innovative method, namely, green fuzzy design analysis (GFDA). Their study involves simple and efficient procedures to evaluate product design alternatives based on environmental considerations using fuzzy logic. The hierarchical structure of environmentally-conscious design indices was constructed using the analytical hierarchy process (AHP). The fuzzy multi-attribute decision-making (FMADM) technique is then used to select the most desirable design alternative. On the other hand, Wang et al. (2008) claimed that the priority vectors determined by the extent analysis method do not represent the relative importance of decision criteria or alternatives and that the misapplication of the extent analysis method to fuzzy AHP problems may lead to a wrong decision. They further state that some useful
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decision information such as decision criteria and fuzzy comparison matrices are not considered. Therefore, the evaluation and analysis of the decision must be defined carefully to avoid misleading interpretations. Toyoura et al. (2004) and Watada and Pedrycz (2008) presented fuzzy regression analysis for treating the computation with words. This is essential in the assessment process of experts, who transform the linguistic variables of features and characteristics of an objective into the linguistic expression of the total assessment. A series of multivariate analyses have also been extensively examined and various means are presented for analyzing data in a fuzzy data environment (Watada 2005). Mehran et al. (2005) reviewed relevant articles on fuzzy regression and provided a simple approach to determine the coefficients of a fuzzy linear relationship. Meanwhile, Abdalla and Buckley (2007) applied a new fuzzy Monte Carlo method to a certain fuzzy linear regression problem to estimate the best solution. In this case, the best solution is a vector of triangular fuzzy numbers for the fuzzy coefficients in the model, which minimises one of two error measures. Conventional statistical regression and new fuzzy regression approaches can be used to find relationships among productivity, consumer satisfaction and profitability. In He et al. (2007), the traditional fuzzy linear regression model was applied, producing estimates for the impact coefficients that are consistent with the ordinary least squares results. They then proposed a revised fuzzy linear regression model that improves the goodness-of-fit. In addition, Divakaran and Terence (2005) examine the application of fuzzy sets and fuzzy measure theories to obtain subjective descriptions of indication importance for policy capturing. In their work, the subjective estimates of criteria weights were represented with fuzzy sets and fuzzy measures were applied to determine the importance of criteria and relationships. The study showed that the fuzzy approach yields results consistent with those of linear regression. Decision-making with multiple criteria usually supports the decision making process under numerous and conflicting evaluations. Apart from that, the decision-making process also involves knowledge from experts. As stated in the expert system methodology, knowledge acquisition is the process of obtaining and gathering information from human experts in a particular area and presenting it in an appropriate form implemented on a computer. The gathered information is then analysed to reveal key knowledge, concepts and relationships. The knowledge extracted from human intelligence can be utilised to support decision making without direct consultation from human experts (Ishak and Siraj 2002). The concept of knowledge extraction has been exploited in various applications, discovering and learning new patterns or knowledge. These patterns and rules can be used to guide decision making and forecast the effects of these decisions. For example, in farm management information systems, the agricultural knowledge base for farming information is used to help improve farmer knowledge as well as support the decision-making process (McCown 2002). Girard and Hubert (1999) also explain the effect of decision support and knowledge-based systems on enhancing agricultural decisions. In summary, the integration of different learning and adaptation techniques has in recent years contributed to a large number of new intelligent system designs for overcoming individual limitations and achieving synergistic effects through hybridisation or fusion of these techniques (Saaty 1990). Consequently, a hybrid system is much preferred to a single system, as the hybrid system has more capabilities derived from the multiple techniques adopted.
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3 Evaluation and selection A multicriteria decision-making problem usually requires decision makers to provide qualitative assessments of the performance of each alternative considering various attributes and to find the best solution among all feasible options. There are several techniques available to evaluate the alternatives based on numerous available data samples. Among these, AHP (Saaty 1990) is the most frequently used method because of its ability to evaluate complex multi-attribute alternatives and become a practical tool of multicriteria decision analysis. There has been extensive research in this area that has been successfully applied in real situations (Sugihara and Tanaka 2001). TOPSIS is also one of the most popular of the ideal point methods and is one of the best-known MADM methods (Li 2007). While the AHP concentrates on pairwise comparison judgment, the TOPSIS method is based on an aggregating function, which represents the closeness of the evaluation to the ideal solution. However, the evaluation conducted by the traditional AHP and TOPSIS methods does not consider the interval or fuzzy value. Therefore, in this paper, we selected to evaluate the alternatives and compare the results produced by fuzzy hierarchical evaluation method (FHEM) with interval values for evaluation.
3.1 AHP The AHP process is as follows (Saaty 1994): (1) Construct a pairwise comparison matrix with a scale of relative importance. The pairwise comparison matrix is as follows ⎡
1 ⎢ 1/a12 ⎢ A = [aii ] = ⎢ . ⎣ .. 1/a1m
a12 1 .. . 1/a2m
··· ..
. ···
⎤ a1m a2m ⎥ ⎥ .. ⎥ . ⎦ 1
where ai j = 1 and a ji = 1/ai j ; i, j = 1, 2, . . . , m. Find the relative normalized weight (w j ) of each attribute. Find the maximum eigen value, λ max max −m) Calculate the consistency index as C I = (λ (m−1) Obtain the random index (R I ) for the number of attributes used in the decision making. (6) Calculate the consistency ratio C R = CR II .
(2) (3) (4) (5)
The AHP value using a direct rating evaluation is computed as follows:
Rj =
K i=1
123
a ji wi , for j = 1, 2, . . . , m
(1)
A fuzzy regression approach to a hierarchical evaluation model
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where R j is the sample for the jth alternative, mis the number of alternatives, and K is the number of attributes; a ji denotes the score of the jth alternative related to the ith attribute; and wi denotes the weight of the ith attribute. 3.2 TOPSIS The steps in the general TOPSIS process are as follows (Yoon and Hwang 1995): Step 1: Compute a normalised decision matrix for the ranking. Assume A j is the sample for the jth alternative, j = 1, 2, . . . , n; Fi represents the ith attribute, i = 1, 2, . . . , k, and f ji is a value indicating the performance rating of each alternative solution with respect to each criterion Fi . The structure of the matrix can be expressed as the following:
A1 D = A2 .. . An
⎡ F1 F2 · · · Fk ⎤ f 11 f 12 · · · f 1k ⎢ f 21 f 22 · · · f 2k ⎥ ⎢ ⎥ ⎢ .. .. .. ⎥ .. ⎣ . . . . ⎦ f n1 f n2 · · · f nk
The normalised value r ji is calculated as: f ji r ji = n j=1
f ji2
,
(2)
where j = 1, 2, . . . , n; i = 1, 2, . . . , k Step 2: Calculate the weighted normalised decision matrix by multiplying the normalised decision matrix by its weights. Let wi denote the weight of the ith attribute. The weighted normalised value is calculated as follows: v j = wi r ji .
(3)
Step 3: Determine the positive ideal solution V + and the negative ideal solutionV − , respectively:
+ max v j | j ∈ J , min v j | j ∈ J V + = v+ j , . . . , vn =
− V + = v− min v j | j ∈ J , max v j | j ∈ J , . . . , v n = j
(4)
where J concerning with the positive criteria and J is concerning with the negative criteria.
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Step 4: Find the separation measure using the dimensional Euclidean distance. D + denotes the separation from the positive ideal, and D − is the separation from the negative ideal. The separation measures D + and D − of each alternative are given as follows: D+ j
=
D− j =
k i=1
k i=1
(v ji − vi+ )2 ,
j = 1, . . . , n
(5)
(v ji − vi− )2 ,
j = 1, . . . , n
(6)
Step 5: Calculate the relative closeness of the jth alternative to the ideal solution and rank the alternatives in descending order. The relative closeness of the alternative A j is defined as follows (Chen and Tsao 2008; Byun and Lee 2005; Yoon and Hwang 1995):
Cj =
D− j − D+ j + Dj
, 0 ≤ C j ≤ 1, j = 1, . . . , n
(7)
All alternatives are compared with the positive ideal solution and the negative ideal solution. Larger index values indicate better performance of the alternatives.
4 Fuzzy hierarchical evaluation model The fuzzy hierarchical evaluation model (FHEM) uses an importance scale as stated in conventional AHP method. However, straight forward rating is used in the FHEM instead of pairwise comparison of AHP. Ordinary AHP uses a 5 to 9-point scale for the level of importance to compare the criteria with each other. Meanwhile, triangular fuzzy numbers are used instead of crisp numbers to describe the fuzzy importance level. A triangular fuzzy number is denoted by A = (a, h), using central value a and width h. Table 1 shows the intensity of an importance scale for a crisp number (Saaty 1980) and a fuzzy number. A combination of crisp and fuzzy numbers is used based on the appropriateness for the criteria of the problem, and is assigned to the alternatives to measure their performance against each criterion. The mixture of crisp and fuzzy numbers can give flexibility and extension to an evaluation process, where a suitable judgment scale can be made that corresponds to the criteria. Assume we have K attributes and n samples. Use i to indicate an attribute number and j as a sample number. In order to build the hierarchical evaluation model, let us through the extension principle denote a judgment matrix by A = [a ji ]n×K and a fuzzy weight vector of criteria selection by W = [Wi ]1×K .
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Table 1 Intensity of importance scale used in fruit grading Intensity of importance Crisp value
Definition
Fuzzy value Notation
Membership function A = (a, h)
1˜ 2˜
(1,1)
Equal importance
(2,1)
Equal to moderately importance
3˜ 4˜
(3,1)
Moderate importance
(4,1)
Moderate to strong importance
5˜ 6˜
(5,1)
Strong importance
(6,1)
Strong to very strong importance
(7,1)
Very strong importance
8
7˜ 8˜
(8,1)
Very to extremely strong importance
9
9˜
(9,1)
Extreme importance
1 2 3 4 5 6 7
The total score vector R = [r j ]n×1 of alternatives can be calculated with the following expressions: R = [r j ] = A · WT Rj =
K
a ji · wi ,
(8)
i=1
where T is the transpose of matrix or vector. WhenA, B, C and D denotes fuzzy numbers, we have the following relations: µ AB+C D (T ) =
∨
T =u+v
µ AB (u) ∧ µC D (v)
and µ AB (T ) = ∨ µ A (u) ∧ µ B (v). T =uv
(9)
5 Fuzzy regression model A fuzzy regression model is built in terms of fuzzy numbers and all observed values expressing uncertainty in the system. Thus, a fuzzy regression model can also be called a possibilistic regression model (Tanaka and Watada 1988; Yabuuchi and Watada 1996; Watada 1994, 1996; Watada and Toyoura 2002). In other words, the fuzzy regression model aims to build a model that contains all observed data within the estimated fuzzy numbers. The fuzzy regression is written as follows: Y = [Y j ] = [A1 x j1 + A2 x j2 + · · · + An x jn ] = Axtj x j1 = 1; j = 1, 2, . . . n
(10)
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where regression coefficient Ai is a triangular-shaped fuzzy number Ai = (ai , h i ) with centre ai and width h i . In Eq. (10), x j is a value vector of all criteria for the j-th sample. According to the extension principle, we can rewrite Eq. (10) as follows: Y j = Axtj = (axtj , h|x j |t )
(11)
where |x j | = (|x j1 |, |x j2 |, . . . , |x j K |). The output of the fuzzy regression (10), whose coefficients are fuzzy numbers, results in a fuzzy number. The regression model with fuzzy coefficients can be described using the lower boundary axtj − h|x j |t , centre axtj and upper boundary axtj + h|x j |t . A sample (y j , x j )( j = 1, 2, . . . , n) is defined for the total evaluation with centre y j , width d j as a fuzzy number y j = (y j , d j ), and a value vector of all criteria x j , where the template membership function of fuzzy coefficients is set to L(α), and membership grade is α, which extends to a sample included in the regression model. The inclusion relation between the model and the samples should be written as follows: y j + L −1 (α)d j ≤ axtj + L −1 (α)h|x j |t y j − L −1 (α)d j ≥ axtj − L −1 (α)h|x j |t
(12)
In other words, the fuzzy regression model is built to contain all samples in the model. This problem results in a linear program (LP). Using the notations of observed data (y j , x j ), y j =(y j , d j ), x j =[x j1 , x j2 , . . . , x j K ] for j=1, 2, . . . , n and fuzzy coefficients Ai =(ai , hi ) for i=1, 2, . . . , K , the regression model can be mathematically written as the following LP problem: min
n
a,h j=1
h|x j |t
subject to y j + L −1 (α)d j ≤ axtj + L −1 (a)h|x j |t y j − L −1 (α)d j ≥ axtj − L −1 (a)h|x j |t ( j = 1, 2, . . . , n), h ≥ 0.
(13)
Solving the linear programming problem mentioned above, we have a fuzzy regression. This fuzzy regression contains all samples in its width and results in an expression of all possibilities that the samples embody, which the treated system should contain. It is possible in the formulation of the fuzzy regression model to treat non-fuzzy data with no width by setting the width h j to 0 in the above equations. 6 Fuzzy hierarchical decision making In this study, the general decision process of oil palm fruit grading is enhanced using a hierarchical structure and the fuzzy regression method. This decision-making process consists of five stages:
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selection of high-quality oil palm fruit bunches
Color
Attached Fruitlet
Detached Fruitlet
Surface
Condition
Sample #1
Sample #1
Sample #1
Sample #1
Sample #1
Sample #n
Sample #n
Sample #n
Sample #n
Sample #n
Fig. 1 Hierarchy model for oil palm grading
A. B. C. D. E.
Review related reference and information acquisition. Construct the fuzzy hierarchical evaluation structure. Determine weights using fuzzy regression. Evaluate the alternative samples. Execute decision making and analysis.
6.1 Review related reference and information acquisition The initial step in the decision framework is to review related references to accumulate the key pieces of knowledge in the study domain. With the advancement of technology, greater quantities of information and knowledge have been properly documented and published. These documents can be used as references. Furthermore, expert interviews and brainstorming can also be arranged in order to gain additional insight and validate the findings from published references. In addition, the findings from this step are useful for determining and decomposing the problem hierarchically. This kind of information gathering process is rather similar to the knowledge acquisition step in the expert system methodology. The preliminary study of the oil palm grading process was conducted by reviewing and extracting knowledge from published references consisting of books on oil palm fruit grading process guides, research papers, surveys and reports, which provided secondary information for this project. The information gathered was then represented using an appropriate knowledge model. The basic acquisition procedure consisted of locating each criterion for the grading process within the deterministic tables that contain key pieces of knowledge useful for the next process in this study. 6.2 Construction of the hierarchical estimation model The hierarchical evaluation model in this system consists of the total evaluation, criteria, and alternatives to be evaluated. The main objective was to select a stan-
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Table 2 Descriptive criteria used in oil palm fruit grading Criteria
Description
c1 : Color
Color of the fruitlets
c2 : Attached fruitlets
Number or percentage of attached fruitlets from the fruit bunch
c3 : Detached fruitlets
Number or percentage of detached fruitlets from the fruit bunch
c4 : Surface
External surface of the fruit bunch
c5 : Condition
Fruit bunch condition as a whole
dard quality of oil palm fruit bunches. Several criteria were considered during the process of inspection for quality. Figure 1 illustrates the elements in the evaluation process. 6.3 Weight determination using fuzzy regression Fuzzy regression analysis was used to model an expert evaluation structure. A fuzzy weight value for each criterion was used to build the fuzzy hierarchical structure for the total evaluation of oil palm fruits. Table 2 shows the weights and descriptions of each criterion. In this case study, 20 sample alternatives were used for the weight against each criterion. 6.4 Ranking the alternative samples In this analysis, 20 samples were analyzed in order to obtain the rank of alternatives among the samples. The result obtained from Eq. (13) is used for the input weights for evaluation ranking of oil palm fruit samples. The two evaluation methods compared here are the AHP and TOPSIS algorithms. The outcome from these methods is then scrutinised to evaluate the ranking of oil palm fruit samples. 6.5 Decision making and analysis The preference judgment of each criterion is given by the expert in a straightforward manner using the importance scale of AHP as stated in Table 1. The weights of each criterion are then decided by fuzzy regression instead of pairwise comparison matrix as used in the AHP model. 7 Illustrative example and discussion This section gives an example of FHEM. The data sample and total evaluation are presented in Table 3. The values for each criterion were assigned in a straightforward manner based on an intensity of importance scale, as stated in Table 1. For example, criterion c1 is assigned to 9, which represents the fact that colour has an extremely high importance for the selection of sample fruit. This means that in this case, the color
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Table 3 Data samples with total evaluation given by an expert
Sample yj = yj,dj c2 c1
c3
c4
c5
A1
(9,0.2)
9
5
9
5
5
A2
(9,0.1)
9
5
8
6
6
A3
(8,0.2)
8
8
5
4
4
A4
(5,0.1)
3
8
4
4
5
A5
(6,0.1)
5
8
4
6
7
A6
(7,0.2)
6
5
8
3
6
A7
(8,0.2)
7
7
2
3
3
A8
(8,0.1)
7
6
3
3
2
A9
(6,0.1)
5
7
3
5
5
A10
(5,0.1)
5
5
7
6
8
A11
(8,0.2)
7
5
7
5
3
A12
(7,0.1)
6
5
3
3
6
A13
(5,0.1)
4
8
3
6
6
A14
(5,0.1)
4
5
7
8
8
A15
(6,0.1)
5
3
6
4
8
A16
(7,0.2)
6
4
7
5
5
A17
(4,0.1)
3
3
8
6
6
A18
(5,0.1)
4
4
4
3
3
A19
(6,0.1)
5
3
7
5
2
A20
(8,0.1)
7
5
8
2
8
criterion in alternative 1 for example is more preferable or qualified to be selected as good quality fruit from expert opinion rather than other alternatives. The regression model (13) was applied to the dataset and the weight obtained as shown in Table 4, where ai and h i denote a centre value of weight and its width of criteria ci . Each weight is represented as ci = (ai , h i ), for i = 1, 2, . . . , 5. The evaluations c1 to c5 in Table 3 are the criteria obtained from the experts. From Table 4, the result shows that in the expert’s judgment, Color, Attached Fruitlet and Detached Fruitlet attributes are the most important, with weights of (0.925,0.000), (0.000, 0.224) and (0.075, 0.040), respectively. Other criteria of fruit characteristics were not strongly weighted. The Attached Fruitlet data indicate that this attribute is also important and covers values ranging from 0 to 0.224. This result indicates that experts should also stress attached fruitlet judgment. If, instead, the attached fruitlet showed a weak dominance, then the other criteria might represent strong dominance in the total evaluation. y j = (y j , d j ) is the total evaluation given by the expert. Even though the information can be used for ranking all the alternatives, they cannot provide the weight of each criterion towards the total evaluation in the sample set. Therefore, in this study we show how fuzzy regression can provide the estimated weight which can be used in future for predicting the total evaluation. The weight value yielded by the fuzzy regression model is helpful for assisting the grading process with minimal monitoring by human experts.
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Table 4 Weights of criteria
Center value
Width
a1 = 0.925
h 1 = 0.000
a2 = 0.000
h 2 = 0.224
a3 = 0.075
h 3 = 0.040
a4 = 0.000
h 4 = 0.000
a5 = 0.000
h 5 = 0.014
Table 5 Comparison of the expert evaluation and FHEM Rank
Sample
Expert evaluation, y j = (y j , d j )
Total evaluation by FHEM, y j = ( y˜ j , d j )
1
A1
(9,0.2)
(8.999, 1.48)
2
A2
(9,0.1)
(8.924, 1.44)
3
A3
(8,0.2)
(7.774, 1.99)
4
A20
(8,0.1)
(7.074, 1.44)
5
A11
(8,0.2)
(6.999, 1.40)
6
A8
(8,0.1)
(6.699, 1.46)
7
A7
(8,0.2)
(6.624, 1.65)
8
A6
(7,0.2)
(6.149, 1.44)
9
A16
(7,0.2)
(6.074, 1.18)
10
A12
(7,0.1)
(5.774, 1.24)
11
A10
(5,0.1)
(5.150, 1.40)
12
A19
(6,0.1)
(5.150, 0.95)
13
A15
(6,0.1)
(5.075, 0.91)
14
A5
(6,0.1)
(4.925, 1.95)
15
A9
(6,0.1)
(4.850, 1.69)
16
A14
(5,0.1)
(4.225, 1.40)
17
A18
(5,0.1)
(4.000, 1.06)
18
A13
(5,0.1)
(3.925, 1.91)
19
A17
(4,0.1)
(3.375, 0.99)
20
A4
(5,0.1)
(3.075, 1.95)
Furthermore, the FHEM showed the width of the decision, that is, the range of the evaluation. Therefore, we can see that A1 and A2 have similar evaluations and are not so distinguishable via FHEM, with widths of 1.48 and 1.44, compared to the difference between the two evaluations (8.99 vs. 8.924). In the same way, A3 and A20 have similar expert evaluations and are also not so distinguishable by FHEM, with widths of 1.99 and 1.44, compared to the difference between the two evaluations (7.774 vs. 7.074). As such, the width data indicate that FHEM can play a pivotal role in interpretation. The input data was observed from the experts. Therefore the obtained results show the experts judgments. From the results, a model of the total expert evaluation was obtained. After the weight value for each criterion was derived by means of fuzzy regression, the value was used to estimate the total evaluation based on the fuzzy hier-
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Table 6 Evaluation results obtained using three methods Ranking
Sample
AHP Preference (P j )
Sample
TOPSIS Preference (P j )
Sample
FHEM Preference (P j )
1
A1
0.078
A1
1.000
A1
(8.999, 1.48)
2
A2
0.078
A2
0.987
A2
(8.924, 1.44)
3
A3
0.068
A3
0.827
A3
(7.774, 1.99)
4
A20
0.062
A20
0.668
A20
(7.074, 1.44) (6.999, 1.40)
5
A11
0.061
A11
0.667
A11
6
A8
0.058
A8
0.66
A8
(6.699, 1.46)
7
A7
0.058
A7
0.658
A7
(6.624, 1.65)
8
A6
0.054
A6
0.503
A6
(6.149, 1.44)
9
A16
0.053
A16
0.502
A16
(6.074, 1.18)
10
A12
0.05
A12
0.497
A12
(5.774, 1.24)
11
A10
0.045
A10
0.338
A10
(5.150, 1.40)
12
A19
0.045
A19
0.338
A19
(5.150, 0.95)
13
A15
0.044
A15
0.336
A15
(5.075, 0.91)
14
A5
0.043
A5
0.333
A5
(4.925, 1.95)
15
A9
0.042
A9
0.332
A9
(4.850, 1.69)
16
A14
0.037
A14
0.177
A14
(4.225, 1.40)
17
A18
0.035
A18
0.168
A18
(4.000, 1.06)
18
A13
0.034
A13
0.166
A13
(3.925, 1.91)
19
A17
0.029
A17
0.075
A17
(3.375, 0.99)
20
A4
0.027
A4
0.026
A4
(3.075, 1.95)
archical evaluation model described in Section 5. The estimated results show that this model produces values that are highly similar to the expert evaluation values. Table 5 shows the tabulated results for actual and estimated values. Table 6 shows the evaluation result of the FHEM method compared with the AHP and TOPSIS methods. Let Pi (for i = 1, 2, . . . , n) represent the final preference of alternative Ai when all decision criteria are considered. We obtain the top four final ranking scores of alternatives using the FHEM method, as FHEM A1 FHEM A2 FHEM A3 FHEM A20 . Meanwhile, the AHP method produces AHP A1 AHP A2 AHP A3 AHP A20 and the TOPSIS method gives TOPSIS A1 TOPSIS A2 TOPSIS A3 TOPSIS A20 . Since the comparable methods do not involve the fuzzy weights, the center value of estimated weight of attributes produced by Eq. (13) is used. From the comparison, we see that the FHEM method achieves the same ranking of results as the AHP and TOPSIS methods. However, the ranking results obtained by the FHEM method show added flexibility with the introduction of the width to the evaluation. The width in this evaluation is important as it reflects natural human judgment, which tends to evaluate in interval or fuzzy values rather than crisp and precise judgments.
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8 Conclusions Human expertise is usually involved in decision-making. The judgment experience and knowledge of these experts is unique to each person. However, better understanding of this judgment knowledge, which can be represented by weights of criteria during a decision-making process, can be useful for facilitating the decision-making process with minimal evaluation input from human experts. Apart from that, the fuzzy hierarchical structure is also capable of considering uncertain values in the judgment evaluation. This uncertainty element is important, as the judgment evaluation strongly involves individual human preferences. Quality inspection of oil palm fruit bunches is vital for the production of palm oil. The work described in this paper reveals that fuzzy evaluation in a hierarchy can be effectively used to better facilitate the decision making process during the inspection of oil palm fruit bunch quality. Acknowledgments A. Nureize expresses her appreciation to the University Tun Hussein Onn Malaysia (UTHM) and the Ministry of Higher Education (MOHE) for her study leave, and, to the Malaysian Palm Oil Board (MPOB) for providing research data and discussion.
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Yoon, K. P., & Hwang, C. L. (1995). Multiple attribute decision making: An introduction. Thousand Oaks, CA: Sage Publications. Yusuf, B., & Chan, K. W. (2004). The oil palm and its sustainability. Journal of Palm Oil Research, 16(1), 1–10. Zadeh, L. A., et al. (1998). Roles of soft computing and fuzzy logic in the conception, design and deployment of information/intelligent systems. In O. Kaynak (Ed.), Computational intelligence: Soft computing and fuzzy-neuro integration with applications (pp. 1–9). Germany: Springer.
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A fuzzy regression approach to a hierarchical evaluation model for oil palm fruit grading A. Nureize · J. Watada
Published online: 21 March 2010 © Springer Science+Business Media, LLC 2010
Abstract Measurement of quality is an important task in the evaluation of agricultural products and plays a pivotal role in agricultural production. The inspection process normally involves a visual examination according to the ripeness standards of crops, and this grading is subject to expert knowledge and interpretation. Therefore, the quality inspection process of fruits needs to be conducted properly to ensure that high-quality fruit bunches are selected for production. However, human subjective judgments during the evaluation make the fruit grading inexact. The objectives of this paper are to build a fuzzy hierarchical evaluation model that characterises the criteria of oil palm fruits to decide the fuzzy weights of these criteria based on a fuzzy regression model, and to help inspectors conduct a proper total evaluation. A numerical example is included to illustrate the computational process of the proposed model. Keywords Fuzzy regression analysis · Fuzzy hierarchical model · Multicriterion · Oil palm fruit grading 1 Introduction The palm oil industry has played a remarkable role in Malaysia’s economic and social development. Accordingly, a current priority of Malaysian policy is to ensure that the yearly surplus of exported palm oil satisfies the growing worldwide market demand for oils and fats (Yusuf and Chan 2004). The increasing demand for palm oil products
A. Nureize (B) · J. Watada Graduate School of Information, Production and System, Waseda University, 2-7 Hibikino, Wakamatsu, Kitakyushu 808-0135, Fukuoka, Japan e-mail: [email protected] J. Watada e-mail: [email protected]
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in local and international markets drives interest in raising the yield of fresh oil palm fruit. Since fresh fruit bunches are the starting input for crude palm oil production, it is therefore imperative that only high-quality fruit bunches be selected and processed (Abdullah et al. 2004; MPOB 2003). Moreover, higher quality fresh fruit bunches (FFB) produce a higher quantity and quality of palm oil (Abdullah et al. 2001). Highquality oil palm fruit can improve the quality and quantity of palm oil products. The presence of unripe bunches results in a lower oil extraction rate, and the overripe bunch affects the fatty fruit acid content. Therefore, to sustain the production rate and production efficiency, a higher quality of oil palm fruit bunches should be used. To accomplish this goal, inspection control should be placed at the entrance of processing plants to ensure that the required characteristics of fruit are satisfied. The grading process is carried out besides the loading ramp inside the mill premises in the presence of a supplier representative. Representative persons from the field and mill must be involved in quality control and be responsible for the quality requirements (Eng and Tat 1985). Grading fresh fruit bunches is a process wherein fruits are assessed and classified according to criteria of ripeness and bunch quality (Yusuf and Chan 2004). In practice, oil palm fruits are inspected and graded by expert inspectors at a mill who have capabilities and experiences in grading fresh fruit bunches and who judge quality by looking individually at the product (Abdullah et al. 2004). Basically, the grading practice involves the inspection of bunch quality, and the estimation of basic extraction rates and graded extraction rates. Consignment of a fresh fruit bunch that has poor quality will be allowed, but subject to a penalty if up to 20 to 30% of the fruit fails to meet the allowable quality limit. The penalty is the percentage to be deducted from a basic extraction rate. Meanwhile, the basic extraction rate is the maximum theoretical percentage of crude palm oil and palm kernels that can be produced from fruit bunches. All the information from the grading process is subsequently transferred to the grading form for documentation. The grading process must be handled properly to select quality fruit and to remove defective units that show signs of noncompliance with the standard criteria. Fruit bunches are evaluated and classified based on standard criteria as set by the Malaysian Palm Oil Board (MPOB). The standard criteria are used by the MPOB to promote quality awareness among the mills, the plantations and the small holding sectors (Yusuf and Chan 2004). Fruit bunches are typically evaluated using visual examination based on standard criteria such as colour, number of detached or attached fruitlets, physical appearance and disease. Each criterion carries a different weight of importance in the evaluation. These weights are necessary and can be used to decide the most important criterion during the fruit evaluation. Generally, the ripeness of an oil palm bunch is determined by using the percentage of detached fruits per bunch (Siregar 1976). However, the colour of fruit also provides valuable information in estimating maturity and ripeness. Colour characteristics do not only indicate the original product quality, but also determine the efficiency of the manufacturing (Abdullah et al. 2004). Therefore, colour becomes an important feature for identifying the ripeness and maturity of oil palm fruit. Several studies available in the literature specifically focus on visual analysis in the evaluation of oil palm fruit. Abdullah et al. (2001) conducted a stepwise discriminant analysis for inspecting the quality features of oil palms through colour image analysis. Meanwhile, the correlation between the oil content and the
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colour of oil palm fruit has also been investigated (Rashid et al. 2002). Accordingly, Abdullah et al. (2004) focused on image acquisition technologies, using a machine vision system and computerised radar tomography to assess the physical properties of oil palm fruit. Alfatni et al. (2008) found that the ripeness of a fruit bunch could be classified into different categories of fruit bunches based on Red, Green, and Blue (RGB) colour intensity. In addition, Abbas et al. (2005) also investigated the feasibility of using moisture measurements to assess the quality of oil palm fruits. However, from the literature, it can be concluded that colour is the main characteristic that plays a pivotal role in determining the quality of the fruit. Currently, human graders are involved directly in the evaluation and grading process in the mills. Even though numerous studies (Abbas et al. 2005; Abdullah et al. 2001; Abdullah et al. 2004; Alfatni et al. 2008; Rashid et al. 2002) have been published regarding automating the grading process to accelerate sorting and evaluation, that kind of technology is still not implemented in Malaysian palm oil mills. For that reason, human grading still remains the most suitable method due to the high cost of advanced machine implementation. In practice, grading experts, whose capability and experience are needed to adequately grade fresh fruit bunches, inspect and grade oil palm fruits at a mill. The skill and experience of human graders are important, as the grading process involves expert visual evaluation. Consequently, accumulated knowledge is useful in the grading process, even though the evaluation is based on several quantitative and qualitative criteria that are influenced by the grader’s experiences and knowledge. Thus, the evaluation involves both accurate and inexact information, since the fruit grading evaluation depends upon subjective human judgments. The objective of this paper is to provide an estimation of weights of attributes by means of fuzzy regression. Moreover, this paper introduces a fuzzy hierarchy evaluation model to assist and improve the quality inspection process as well as to support the decision-making process in the palm oil industry. The remainder of this paper is organised as follows. Related research is reviewed briefly in Section 2. Section 3 explains two widely used methods, namely, AHP and TOPSIS, for comparison with our proposed method using real data. Section 4 describes the fuzzy hierarchical evaluation model. The fuzzy weight in the fuzzy hierarchical evaluation model is assessed by fuzzy regression analysis in Section 5. Section 6 discusses fuzzy hierarchical evaluation decision-making based on our model. Section 7 presents a real application of the model in the evaluation of oil palm grading, and Section 8 concludes this paper with some additional remarks.
2 Overview of related works In decision making process, as the problem grows more complex, decomposition of the problem becomes necessary. Decomposition is the process of dividing a problem into several small sub-problems and arranging them into a hierarchical structure, such as an Analytic Hierarchy Process (AHP). This type of structure can reduce the complexity of the problem, allowing better capture, description and understanding of realistic problems. AHP is a multiattribute approach in decision making that has been successfully applied to many real-world decision making problems (Saaty 1990). The
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AHP structures a multiattribute problem hierarchically, investigates the levels of the hierarchy separately and produces a result in rank order (Irfan and Nilsen 2006). The goal of AHP is to enable us to employ a number of pair-wise comparisons obtained using human judgment. However, many practical cases in a human preference model involve imprecise values, making it difficult to assign exact numerical values for comparison judgments (Irfan and Nilsen 2006; Zadeh 1998). This is due to factors such as incomplete and imprecise subjectivity, which tend to be present to some degree. Therefore, inexact elements included in the AHP decision analysis will add uncertainty and vagueness to the decision process. The Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) is another approach in multiattribute decision making. TOPSIS, which was introduced by Hwang and Yoon (1981), evaluates options geometrically. In this case, alternatives are chosen based on the shortest distance from the positive ideal solution and the longest distance from the negative ideal solution. TOPSIS defines an index called similarity, or relative closeness to the positive-ideal solution and remoteness from the negative-ideal solution. The alternative with the maximum similarity to the positiveideal solution will be selected with priority (Yoon and Hwang 1995). Fuzzy elements have also been introduced and examined in TOPSIS research in order to deal with fuzzy environments (Li 2007). Decision-making situations commonly involve complex, uncertain and imprecise information. Fuzzy decision-making has been tackled successfully to deal with vagueness in linguistics and expressing human knowledge and inference mechanisms in a natural way. Multiple criteria are considered in a decision process. A multicriteria analysis with fuzzy pairwise comparisons is presented in Deng (1999). Kreng and Wu (2007) demonstrated a comprehensive hierarchical framework by using a fuzzy AHP approach and a technique for determining weights for evaluating knowledge portal system development tools. Takahagi (2008) introduces an identification method for fuzzy measures using diamond pairwise comparisons. Meanwhile, Yeh and Chang (2008) presented a new fuzzy multicriteria decision-making approach for evaluating decision alternatives involving subjective judgments made by a group of decision makers. In this study, a pairwise comparison process was used to make comparative judgments, and a linguistic rating method was used to make absolute judgments. A hierarchical weighting method was developed to assess the weights of a large number of evaluation criteria by pairwise comparisons. Enea and Piazza (2004) show that better results can be achieved by considering all the information deriving from the constraints within fuzzy AHP in terms of certainty and reliability. Kuo et al. (2006) present an innovative method, namely, green fuzzy design analysis (GFDA). Their study involves simple and efficient procedures to evaluate product design alternatives based on environmental considerations using fuzzy logic. The hierarchical structure of environmentally-conscious design indices was constructed using the analytical hierarchy process (AHP). The fuzzy multi-attribute decision-making (FMADM) technique is then used to select the most desirable design alternative. On the other hand, Wang et al. (2008) claimed that the priority vectors determined by the extent analysis method do not represent the relative importance of decision criteria or alternatives and that the misapplication of the extent analysis method to fuzzy AHP problems may lead to a wrong decision. They further state that some useful
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decision information such as decision criteria and fuzzy comparison matrices are not considered. Therefore, the evaluation and analysis of the decision must be defined carefully to avoid misleading interpretations. Toyoura et al. (2004) and Watada and Pedrycz (2008) presented fuzzy regression analysis for treating the computation with words. This is essential in the assessment process of experts, who transform the linguistic variables of features and characteristics of an objective into the linguistic expression of the total assessment. A series of multivariate analyses have also been extensively examined and various means are presented for analyzing data in a fuzzy data environment (Watada 2005). Mehran et al. (2005) reviewed relevant articles on fuzzy regression and provided a simple approach to determine the coefficients of a fuzzy linear relationship. Meanwhile, Abdalla and Buckley (2007) applied a new fuzzy Monte Carlo method to a certain fuzzy linear regression problem to estimate the best solution. In this case, the best solution is a vector of triangular fuzzy numbers for the fuzzy coefficients in the model, which minimises one of two error measures. Conventional statistical regression and new fuzzy regression approaches can be used to find relationships among productivity, consumer satisfaction and profitability. In He et al. (2007), the traditional fuzzy linear regression model was applied, producing estimates for the impact coefficients that are consistent with the ordinary least squares results. They then proposed a revised fuzzy linear regression model that improves the goodness-of-fit. In addition, Divakaran and Terence (2005) examine the application of fuzzy sets and fuzzy measure theories to obtain subjective descriptions of indication importance for policy capturing. In their work, the subjective estimates of criteria weights were represented with fuzzy sets and fuzzy measures were applied to determine the importance of criteria and relationships. The study showed that the fuzzy approach yields results consistent with those of linear regression. Decision-making with multiple criteria usually supports the decision making process under numerous and conflicting evaluations. Apart from that, the decision-making process also involves knowledge from experts. As stated in the expert system methodology, knowledge acquisition is the process of obtaining and gathering information from human experts in a particular area and presenting it in an appropriate form implemented on a computer. The gathered information is then analysed to reveal key knowledge, concepts and relationships. The knowledge extracted from human intelligence can be utilised to support decision making without direct consultation from human experts (Ishak and Siraj 2002). The concept of knowledge extraction has been exploited in various applications, discovering and learning new patterns or knowledge. These patterns and rules can be used to guide decision making and forecast the effects of these decisions. For example, in farm management information systems, the agricultural knowledge base for farming information is used to help improve farmer knowledge as well as support the decision-making process (McCown 2002). Girard and Hubert (1999) also explain the effect of decision support and knowledge-based systems on enhancing agricultural decisions. In summary, the integration of different learning and adaptation techniques has in recent years contributed to a large number of new intelligent system designs for overcoming individual limitations and achieving synergistic effects through hybridisation or fusion of these techniques (Saaty 1990). Consequently, a hybrid system is much preferred to a single system, as the hybrid system has more capabilities derived from the multiple techniques adopted.
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3 Evaluation and selection A multicriteria decision-making problem usually requires decision makers to provide qualitative assessments of the performance of each alternative considering various attributes and to find the best solution among all feasible options. There are several techniques available to evaluate the alternatives based on numerous available data samples. Among these, AHP (Saaty 1990) is the most frequently used method because of its ability to evaluate complex multi-attribute alternatives and become a practical tool of multicriteria decision analysis. There has been extensive research in this area that has been successfully applied in real situations (Sugihara and Tanaka 2001). TOPSIS is also one of the most popular of the ideal point methods and is one of the best-known MADM methods (Li 2007). While the AHP concentrates on pairwise comparison judgment, the TOPSIS method is based on an aggregating function, which represents the closeness of the evaluation to the ideal solution. However, the evaluation conducted by the traditional AHP and TOPSIS methods does not consider the interval or fuzzy value. Therefore, in this paper, we selected to evaluate the alternatives and compare the results produced by fuzzy hierarchical evaluation method (FHEM) with interval values for evaluation.
3.1 AHP The AHP process is as follows (Saaty 1994): (1) Construct a pairwise comparison matrix with a scale of relative importance. The pairwise comparison matrix is as follows ⎡
1 ⎢ 1/a12 ⎢ A = [aii ] = ⎢ . ⎣ .. 1/a1m
a12 1 .. . 1/a2m
··· ..
. ···
⎤ a1m a2m ⎥ ⎥ .. ⎥ . ⎦ 1
where ai j = 1 and a ji = 1/ai j ; i, j = 1, 2, . . . , m. Find the relative normalized weight (w j ) of each attribute. Find the maximum eigen value, λ max max −m) Calculate the consistency index as C I = (λ (m−1) Obtain the random index (R I ) for the number of attributes used in the decision making. (6) Calculate the consistency ratio C R = CR II .
(2) (3) (4) (5)
The AHP value using a direct rating evaluation is computed as follows:
Rj =
K i=1
123
a ji wi , for j = 1, 2, . . . , m
(1)
A fuzzy regression approach to a hierarchical evaluation model
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where R j is the sample for the jth alternative, mis the number of alternatives, and K is the number of attributes; a ji denotes the score of the jth alternative related to the ith attribute; and wi denotes the weight of the ith attribute. 3.2 TOPSIS The steps in the general TOPSIS process are as follows (Yoon and Hwang 1995): Step 1: Compute a normalised decision matrix for the ranking. Assume A j is the sample for the jth alternative, j = 1, 2, . . . , n; Fi represents the ith attribute, i = 1, 2, . . . , k, and f ji is a value indicating the performance rating of each alternative solution with respect to each criterion Fi . The structure of the matrix can be expressed as the following:
A1 D = A2 .. . An
⎡ F1 F2 · · · Fk ⎤ f 11 f 12 · · · f 1k ⎢ f 21 f 22 · · · f 2k ⎥ ⎢ ⎥ ⎢ .. .. .. ⎥ .. ⎣ . . . . ⎦ f n1 f n2 · · · f nk
The normalised value r ji is calculated as: f ji r ji = n j=1
f ji2
,
(2)
where j = 1, 2, . . . , n; i = 1, 2, . . . , k Step 2: Calculate the weighted normalised decision matrix by multiplying the normalised decision matrix by its weights. Let wi denote the weight of the ith attribute. The weighted normalised value is calculated as follows: v j = wi r ji .
(3)
Step 3: Determine the positive ideal solution V + and the negative ideal solutionV − , respectively:
+ max v j | j ∈ J , min v j | j ∈ J V + = v+ j , . . . , vn =
− V + = v− min v j | j ∈ J , max v j | j ∈ J , . . . , v n = j
(4)
where J concerning with the positive criteria and J is concerning with the negative criteria.
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Step 4: Find the separation measure using the dimensional Euclidean distance. D + denotes the separation from the positive ideal, and D − is the separation from the negative ideal. The separation measures D + and D − of each alternative are given as follows: D+ j
=
D− j =
k i=1
k i=1
(v ji − vi+ )2 ,
j = 1, . . . , n
(5)
(v ji − vi− )2 ,
j = 1, . . . , n
(6)
Step 5: Calculate the relative closeness of the jth alternative to the ideal solution and rank the alternatives in descending order. The relative closeness of the alternative A j is defined as follows (Chen and Tsao 2008; Byun and Lee 2005; Yoon and Hwang 1995):
Cj =
D− j − D+ j + Dj
, 0 ≤ C j ≤ 1, j = 1, . . . , n
(7)
All alternatives are compared with the positive ideal solution and the negative ideal solution. Larger index values indicate better performance of the alternatives.
4 Fuzzy hierarchical evaluation model The fuzzy hierarchical evaluation model (FHEM) uses an importance scale as stated in conventional AHP method. However, straight forward rating is used in the FHEM instead of pairwise comparison of AHP. Ordinary AHP uses a 5 to 9-point scale for the level of importance to compare the criteria with each other. Meanwhile, triangular fuzzy numbers are used instead of crisp numbers to describe the fuzzy importance level. A triangular fuzzy number is denoted by A = (a, h), using central value a and width h. Table 1 shows the intensity of an importance scale for a crisp number (Saaty 1980) and a fuzzy number. A combination of crisp and fuzzy numbers is used based on the appropriateness for the criteria of the problem, and is assigned to the alternatives to measure their performance against each criterion. The mixture of crisp and fuzzy numbers can give flexibility and extension to an evaluation process, where a suitable judgment scale can be made that corresponds to the criteria. Assume we have K attributes and n samples. Use i to indicate an attribute number and j as a sample number. In order to build the hierarchical evaluation model, let us through the extension principle denote a judgment matrix by A = [a ji ]n×K and a fuzzy weight vector of criteria selection by W = [Wi ]1×K .
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Table 1 Intensity of importance scale used in fruit grading Intensity of importance Crisp value
Definition
Fuzzy value Notation
Membership function A = (a, h)
1˜ 2˜
(1,1)
Equal importance
(2,1)
Equal to moderately importance
3˜ 4˜
(3,1)
Moderate importance
(4,1)
Moderate to strong importance
5˜ 6˜
(5,1)
Strong importance
(6,1)
Strong to very strong importance
(7,1)
Very strong importance
8
7˜ 8˜
(8,1)
Very to extremely strong importance
9
9˜
(9,1)
Extreme importance
1 2 3 4 5 6 7
The total score vector R = [r j ]n×1 of alternatives can be calculated with the following expressions: R = [r j ] = A · WT Rj =
K
a ji · wi ,
(8)
i=1
where T is the transpose of matrix or vector. WhenA, B, C and D denotes fuzzy numbers, we have the following relations: µ AB+C D (T ) =
∨
T =u+v
µ AB (u) ∧ µC D (v)
and µ AB (T ) = ∨ µ A (u) ∧ µ B (v). T =uv
(9)
5 Fuzzy regression model A fuzzy regression model is built in terms of fuzzy numbers and all observed values expressing uncertainty in the system. Thus, a fuzzy regression model can also be called a possibilistic regression model (Tanaka and Watada 1988; Yabuuchi and Watada 1996; Watada 1994, 1996; Watada and Toyoura 2002). In other words, the fuzzy regression model aims to build a model that contains all observed data within the estimated fuzzy numbers. The fuzzy regression is written as follows: Y = [Y j ] = [A1 x j1 + A2 x j2 + · · · + An x jn ] = Axtj x j1 = 1; j = 1, 2, . . . n
(10)
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where regression coefficient Ai is a triangular-shaped fuzzy number Ai = (ai , h i ) with centre ai and width h i . In Eq. (10), x j is a value vector of all criteria for the j-th sample. According to the extension principle, we can rewrite Eq. (10) as follows: Y j = Axtj = (axtj , h|x j |t )
(11)
where |x j | = (|x j1 |, |x j2 |, . . . , |x j K |). The output of the fuzzy regression (10), whose coefficients are fuzzy numbers, results in a fuzzy number. The regression model with fuzzy coefficients can be described using the lower boundary axtj − h|x j |t , centre axtj and upper boundary axtj + h|x j |t . A sample (y j , x j )( j = 1, 2, . . . , n) is defined for the total evaluation with centre y j , width d j as a fuzzy number y j = (y j , d j ), and a value vector of all criteria x j , where the template membership function of fuzzy coefficients is set to L(α), and membership grade is α, which extends to a sample included in the regression model. The inclusion relation between the model and the samples should be written as follows: y j + L −1 (α)d j ≤ axtj + L −1 (α)h|x j |t y j − L −1 (α)d j ≥ axtj − L −1 (α)h|x j |t
(12)
In other words, the fuzzy regression model is built to contain all samples in the model. This problem results in a linear program (LP). Using the notations of observed data (y j , x j ), y j =(y j , d j ), x j =[x j1 , x j2 , . . . , x j K ] for j=1, 2, . . . , n and fuzzy coefficients Ai =(ai , hi ) for i=1, 2, . . . , K , the regression model can be mathematically written as the following LP problem: min
n
a,h j=1
h|x j |t
subject to y j + L −1 (α)d j ≤ axtj + L −1 (a)h|x j |t y j − L −1 (α)d j ≥ axtj − L −1 (a)h|x j |t ( j = 1, 2, . . . , n), h ≥ 0.
(13)
Solving the linear programming problem mentioned above, we have a fuzzy regression. This fuzzy regression contains all samples in its width and results in an expression of all possibilities that the samples embody, which the treated system should contain. It is possible in the formulation of the fuzzy regression model to treat non-fuzzy data with no width by setting the width h j to 0 in the above equations. 6 Fuzzy hierarchical decision making In this study, the general decision process of oil palm fruit grading is enhanced using a hierarchical structure and the fuzzy regression method. This decision-making process consists of five stages:
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selection of high-quality oil palm fruit bunches
Color
Attached Fruitlet
Detached Fruitlet
Surface
Condition
Sample #1
Sample #1
Sample #1
Sample #1
Sample #1
Sample #n
Sample #n
Sample #n
Sample #n
Sample #n
Fig. 1 Hierarchy model for oil palm grading
A. B. C. D. E.
Review related reference and information acquisition. Construct the fuzzy hierarchical evaluation structure. Determine weights using fuzzy regression. Evaluate the alternative samples. Execute decision making and analysis.
6.1 Review related reference and information acquisition The initial step in the decision framework is to review related references to accumulate the key pieces of knowledge in the study domain. With the advancement of technology, greater quantities of information and knowledge have been properly documented and published. These documents can be used as references. Furthermore, expert interviews and brainstorming can also be arranged in order to gain additional insight and validate the findings from published references. In addition, the findings from this step are useful for determining and decomposing the problem hierarchically. This kind of information gathering process is rather similar to the knowledge acquisition step in the expert system methodology. The preliminary study of the oil palm grading process was conducted by reviewing and extracting knowledge from published references consisting of books on oil palm fruit grading process guides, research papers, surveys and reports, which provided secondary information for this project. The information gathered was then represented using an appropriate knowledge model. The basic acquisition procedure consisted of locating each criterion for the grading process within the deterministic tables that contain key pieces of knowledge useful for the next process in this study. 6.2 Construction of the hierarchical estimation model The hierarchical evaluation model in this system consists of the total evaluation, criteria, and alternatives to be evaluated. The main objective was to select a stan-
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Table 2 Descriptive criteria used in oil palm fruit grading Criteria
Description
c1 : Color
Color of the fruitlets
c2 : Attached fruitlets
Number or percentage of attached fruitlets from the fruit bunch
c3 : Detached fruitlets
Number or percentage of detached fruitlets from the fruit bunch
c4 : Surface
External surface of the fruit bunch
c5 : Condition
Fruit bunch condition as a whole
dard quality of oil palm fruit bunches. Several criteria were considered during the process of inspection for quality. Figure 1 illustrates the elements in the evaluation process. 6.3 Weight determination using fuzzy regression Fuzzy regression analysis was used to model an expert evaluation structure. A fuzzy weight value for each criterion was used to build the fuzzy hierarchical structure for the total evaluation of oil palm fruits. Table 2 shows the weights and descriptions of each criterion. In this case study, 20 sample alternatives were used for the weight against each criterion. 6.4 Ranking the alternative samples In this analysis, 20 samples were analyzed in order to obtain the rank of alternatives among the samples. The result obtained from Eq. (13) is used for the input weights for evaluation ranking of oil palm fruit samples. The two evaluation methods compared here are the AHP and TOPSIS algorithms. The outcome from these methods is then scrutinised to evaluate the ranking of oil palm fruit samples. 6.5 Decision making and analysis The preference judgment of each criterion is given by the expert in a straightforward manner using the importance scale of AHP as stated in Table 1. The weights of each criterion are then decided by fuzzy regression instead of pairwise comparison matrix as used in the AHP model. 7 Illustrative example and discussion This section gives an example of FHEM. The data sample and total evaluation are presented in Table 3. The values for each criterion were assigned in a straightforward manner based on an intensity of importance scale, as stated in Table 1. For example, criterion c1 is assigned to 9, which represents the fact that colour has an extremely high importance for the selection of sample fruit. This means that in this case, the color
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Table 3 Data samples with total evaluation given by an expert
Sample yj = yj,dj c2 c1
c3
c4
c5
A1
(9,0.2)
9
5
9
5
5
A2
(9,0.1)
9
5
8
6
6
A3
(8,0.2)
8
8
5
4
4
A4
(5,0.1)
3
8
4
4
5
A5
(6,0.1)
5
8
4
6
7
A6
(7,0.2)
6
5
8
3
6
A7
(8,0.2)
7
7
2
3
3
A8
(8,0.1)
7
6
3
3
2
A9
(6,0.1)
5
7
3
5
5
A10
(5,0.1)
5
5
7
6
8
A11
(8,0.2)
7
5
7
5
3
A12
(7,0.1)
6
5
3
3
6
A13
(5,0.1)
4
8
3
6
6
A14
(5,0.1)
4
5
7
8
8
A15
(6,0.1)
5
3
6
4
8
A16
(7,0.2)
6
4
7
5
5
A17
(4,0.1)
3
3
8
6
6
A18
(5,0.1)
4
4
4
3
3
A19
(6,0.1)
5
3
7
5
2
A20
(8,0.1)
7
5
8
2
8
criterion in alternative 1 for example is more preferable or qualified to be selected as good quality fruit from expert opinion rather than other alternatives. The regression model (13) was applied to the dataset and the weight obtained as shown in Table 4, where ai and h i denote a centre value of weight and its width of criteria ci . Each weight is represented as ci = (ai , h i ), for i = 1, 2, . . . , 5. The evaluations c1 to c5 in Table 3 are the criteria obtained from the experts. From Table 4, the result shows that in the expert’s judgment, Color, Attached Fruitlet and Detached Fruitlet attributes are the most important, with weights of (0.925,0.000), (0.000, 0.224) and (0.075, 0.040), respectively. Other criteria of fruit characteristics were not strongly weighted. The Attached Fruitlet data indicate that this attribute is also important and covers values ranging from 0 to 0.224. This result indicates that experts should also stress attached fruitlet judgment. If, instead, the attached fruitlet showed a weak dominance, then the other criteria might represent strong dominance in the total evaluation. y j = (y j , d j ) is the total evaluation given by the expert. Even though the information can be used for ranking all the alternatives, they cannot provide the weight of each criterion towards the total evaluation in the sample set. Therefore, in this study we show how fuzzy regression can provide the estimated weight which can be used in future for predicting the total evaluation. The weight value yielded by the fuzzy regression model is helpful for assisting the grading process with minimal monitoring by human experts.
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Table 4 Weights of criteria
Center value
Width
a1 = 0.925
h 1 = 0.000
a2 = 0.000
h 2 = 0.224
a3 = 0.075
h 3 = 0.040
a4 = 0.000
h 4 = 0.000
a5 = 0.000
h 5 = 0.014
Table 5 Comparison of the expert evaluation and FHEM Rank
Sample
Expert evaluation, y j = (y j , d j )
Total evaluation by FHEM, y j = ( y˜ j , d j )
1
A1
(9,0.2)
(8.999, 1.48)
2
A2
(9,0.1)
(8.924, 1.44)
3
A3
(8,0.2)
(7.774, 1.99)
4
A20
(8,0.1)
(7.074, 1.44)
5
A11
(8,0.2)
(6.999, 1.40)
6
A8
(8,0.1)
(6.699, 1.46)
7
A7
(8,0.2)
(6.624, 1.65)
8
A6
(7,0.2)
(6.149, 1.44)
9
A16
(7,0.2)
(6.074, 1.18)
10
A12
(7,0.1)
(5.774, 1.24)
11
A10
(5,0.1)
(5.150, 1.40)
12
A19
(6,0.1)
(5.150, 0.95)
13
A15
(6,0.1)
(5.075, 0.91)
14
A5
(6,0.1)
(4.925, 1.95)
15
A9
(6,0.1)
(4.850, 1.69)
16
A14
(5,0.1)
(4.225, 1.40)
17
A18
(5,0.1)
(4.000, 1.06)
18
A13
(5,0.1)
(3.925, 1.91)
19
A17
(4,0.1)
(3.375, 0.99)
20
A4
(5,0.1)
(3.075, 1.95)
Furthermore, the FHEM showed the width of the decision, that is, the range of the evaluation. Therefore, we can see that A1 and A2 have similar evaluations and are not so distinguishable via FHEM, with widths of 1.48 and 1.44, compared to the difference between the two evaluations (8.99 vs. 8.924). In the same way, A3 and A20 have similar expert evaluations and are also not so distinguishable by FHEM, with widths of 1.99 and 1.44, compared to the difference between the two evaluations (7.774 vs. 7.074). As such, the width data indicate that FHEM can play a pivotal role in interpretation. The input data was observed from the experts. Therefore the obtained results show the experts judgments. From the results, a model of the total expert evaluation was obtained. After the weight value for each criterion was derived by means of fuzzy regression, the value was used to estimate the total evaluation based on the fuzzy hier-
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Table 6 Evaluation results obtained using three methods Ranking
Sample
AHP Preference (P j )
Sample
TOPSIS Preference (P j )
Sample
FHEM Preference (P j )
1
A1
0.078
A1
1.000
A1
(8.999, 1.48)
2
A2
0.078
A2
0.987
A2
(8.924, 1.44)
3
A3
0.068
A3
0.827
A3
(7.774, 1.99)
4
A20
0.062
A20
0.668
A20
(7.074, 1.44) (6.999, 1.40)
5
A11
0.061
A11
0.667
A11
6
A8
0.058
A8
0.66
A8
(6.699, 1.46)
7
A7
0.058
A7
0.658
A7
(6.624, 1.65)
8
A6
0.054
A6
0.503
A6
(6.149, 1.44)
9
A16
0.053
A16
0.502
A16
(6.074, 1.18)
10
A12
0.05
A12
0.497
A12
(5.774, 1.24)
11
A10
0.045
A10
0.338
A10
(5.150, 1.40)
12
A19
0.045
A19
0.338
A19
(5.150, 0.95)
13
A15
0.044
A15
0.336
A15
(5.075, 0.91)
14
A5
0.043
A5
0.333
A5
(4.925, 1.95)
15
A9
0.042
A9
0.332
A9
(4.850, 1.69)
16
A14
0.037
A14
0.177
A14
(4.225, 1.40)
17
A18
0.035
A18
0.168
A18
(4.000, 1.06)
18
A13
0.034
A13
0.166
A13
(3.925, 1.91)
19
A17
0.029
A17
0.075
A17
(3.375, 0.99)
20
A4
0.027
A4
0.026
A4
(3.075, 1.95)
archical evaluation model described in Section 5. The estimated results show that this model produces values that are highly similar to the expert evaluation values. Table 5 shows the tabulated results for actual and estimated values. Table 6 shows the evaluation result of the FHEM method compared with the AHP and TOPSIS methods. Let Pi (for i = 1, 2, . . . , n) represent the final preference of alternative Ai when all decision criteria are considered. We obtain the top four final ranking scores of alternatives using the FHEM method, as FHEM A1 FHEM A2 FHEM A3 FHEM A20 . Meanwhile, the AHP method produces AHP A1 AHP A2 AHP A3 AHP A20 and the TOPSIS method gives TOPSIS A1 TOPSIS A2 TOPSIS A3 TOPSIS A20 . Since the comparable methods do not involve the fuzzy weights, the center value of estimated weight of attributes produced by Eq. (13) is used. From the comparison, we see that the FHEM method achieves the same ranking of results as the AHP and TOPSIS methods. However, the ranking results obtained by the FHEM method show added flexibility with the introduction of the width to the evaluation. The width in this evaluation is important as it reflects natural human judgment, which tends to evaluate in interval or fuzzy values rather than crisp and precise judgments.
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8 Conclusions Human expertise is usually involved in decision-making. The judgment experience and knowledge of these experts is unique to each person. However, better understanding of this judgment knowledge, which can be represented by weights of criteria during a decision-making process, can be useful for facilitating the decision-making process with minimal evaluation input from human experts. Apart from that, the fuzzy hierarchical structure is also capable of considering uncertain values in the judgment evaluation. This uncertainty element is important, as the judgment evaluation strongly involves individual human preferences. Quality inspection of oil palm fruit bunches is vital for the production of palm oil. The work described in this paper reveals that fuzzy evaluation in a hierarchy can be effectively used to better facilitate the decision making process during the inspection of oil palm fruit bunch quality. Acknowledgments A. Nureize expresses her appreciation to the University Tun Hussein Onn Malaysia (UTHM) and the Ministry of Higher Education (MOHE) for her study leave, and, to the Malaysian Palm Oil Board (MPOB) for providing research data and discussion.
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