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 agri-cultural products and plays a pivotal role in agriagri-cultural 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 regres-sion 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. WatadaGraduate School of Information, Production and System, Waseda University, 2-7 Hibikino, Wakamatsu, Kitakyushu 808-0135, Fukuoka, Japan
e-mail: [email protected] J. Watada
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). High-quality oil palm fruit can improve the High-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 require-ments (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 subse-quently 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.
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 feasibil-ity of using moisture measurements to assess the qualfeasibil-ity 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 pro-cess 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 pub-lished 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 knowl-edge 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 evalua-tion model to assist and improve the quality inspecevalua-tion 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 Section2. Section3 explains two widely used methods, namely, AHP and TOPSIS, for comparison with our proposed method using real data. Section4describes the fuzzy hierarchical eval-uation model. The fuzzy weight in the fuzzy hierarchical evaleval-uation model is assessed by fuzzy regression analysis in Section5. Section6discusses fuzzy hierarchical eval-uation decision-making based on our model. Section7presents a real application of the model in the evaluation of oil palm grading, and Section8concludes this paper with some additional remarks.
2 Overview of related works
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 com-parison 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 byHwang and Yoon(1981), evaluates options geometrically. In this case, alterna-tives 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 positive-ideal 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 vague-ness 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 inDeng(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 sys-tem 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.
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) andWatada 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 multi-variate 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 deter-mine 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 tri-angular 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 satisfac-tion and profitability. InHe 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.
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 sam-ples. 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
A=[ai i]=
(2) Find the relative normalized weight (wj)of each attribute.
(3) Find the maximum eigen value,λmax
(4) Calculate the consistency index asC I =(λmax(m−−1)m)
(5) Obtain the random index (R I)for the number of attributes used in the decision making.
(6) Calculate the consistency ratioC R=C IR I.
The AHP value using a direct rating evaluation is computed as follows:
Rj =
K
i=1
whereRjis the sample for the jth alternative,mis the number of alternatives, andKis
the number of attributes;aj idenotes the score of the jth alternative related to theith
attribute; andwidenotes the weight of theith 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. AssumeAjis the sample for the jth alternative,j =1,2, . . . ,n;Firepresents theith attribute,i=1,2, . . . ,k, andfj i
is a value indicating the performance rating of each alternative solution with respect to each criterionFi. The structure of the matrix can be expressed as the following:
D=
The normalised valuerj iis calculated as:
rj i =
Calculate the weighted normalised decision matrix by multiplying the normalised decision matrix by its weights. Let wi denote the weight of theith attribute. The
weighted normalised value is calculated as follows:
vj =wirj i. (3)
Step 3:
Determine the positive ideal solutionV+and the negative ideal solutionV−, respec-tively:
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 measuresD+andD−of each alternative are given as follows:
D+j =
k
i=1(vj i−v
+
i )2, j =1, . . . ,n (5)
D−j =
k
i=1(vj i−v
−
i )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 alternativeAjis defined
as follows (Chen and Tsao 2008;Byun and Lee 2005;Yoon and Hwang 1995):
Cj =
D−j
D+j +D−j , 0≤Cj ≤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 byA=(a,h), using central valueaand widthh. Table1shows 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 haveKattributes andnsamples. Useito 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 byA = [aj i]n×K and a
Table 1 Intensity of importance scale used in fruit grading
Intensity of importance Definition
Crisp value Fuzzy value
Notation Membership functionA=(a,h)
1 1˜ (1,1) Equal importance
2 2˜ (2,1) Equal to moderately importance
3 3˜ (3,1) Moderate importance
4 4˜ (4,1) Moderate to strong importance
5 5˜ (5,1) Strong importance
6 6˜ (6,1) Strong to very strong importance
7 7˜ (7,1) Very strong importance
8 8˜ (8,1) Very to extremely strong importance
9 9˜ (9,1) Extreme importance
The total score vectorR= [rj]n×1of alternatives can be calculated with the
fol-lowing expressions:
R= [rj] =A·WT
Rj = K
i=1
aj i ·wi, (8)
where T is the transpose of matrix or vector. WhenA,B,C and D denotes fuzzy numbers, we have the following relations:
µA B+C D(T)= ∨
T=u+vµA B(u)∧µC D(v)
andµA B(T)= ∨
T=uvµA(u)
∧µB(v). (9)
5 Fuzzy regression model
A fuzzy regression model is built in terms of fuzzy numbers and all observed val-ues 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 = [Yj] = [A1xj1+A2xj2+ · · · +Anxj n] =Axtj xj1=1; j=1,2, . . .n
where regression coefficient Ai is a triangular-shaped fuzzy number Ai = (ai,hi)
with centreaiand widthhi. In Eq. (10),xj is a value vector of all criteria for thej-th
sample.
According to the extension principle, we can rewrite Eq. (10) as follows:
Yj =Axtj =(axtj,h|xj|t) (11)
where|xj| =(|xj1|,|xj2|, . . . ,|xj 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|xj|t, centre axtjand upper boundary axtj +h|xj|t. A sample
(yj,xj)(j = 1,2, . . . ,n) is defined for the total evaluation with centre yj, width dj as a fuzzy numberyj =(yj,dj), and a value vector of all criteriaxj, where the
template membership function of fuzzy coefficients is set toL(α), and membership grade isα, which extends to a sample included in the regression model. The inclusion relation between the model and the samples should be written as follows:
yj+L−1(α)dj ≤axtj +L
−1(α)h|x
j|t
yj−L−1(α)dj ≥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(yj,xj),yj=(yj,dj),xj=[xj1,xj2, . . . ,xj K]
for j=1,2, . . . ,nand fuzzy coefficientsAi=(ai,hi)fori=1,2, . . . ,K,the
regres-sion model can be mathematically written as the following LP problem:
min
a,h n
j=1
h|xj|t
subj ect t o
yj+L−1(α)dj ≤axtj+L−1(a)h|xj|t yj−L−1(α)dj ≥axtj−L−1(a)h|xj|t
(j=1,2, . . . ,n),
h≥0.
(13)
Solving the linear programming problem mentioned above, we have a fuzzy regres-sion. 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 widthhj to 0 in the above equations.
6 Fuzzy hierarchical decision making
Color Attached Fruitlet
Surface Detached
Fruitlet Sample #1
Sample #n
Sample #1 Sample #1 Sample #1
Sample #n Sample #n
Sample #n
selection of high-quality oil palm fruit bunches
Condition
Sample #1
Sample #n
Fig. 1 Hierarchy model for oil palm grading
A. Review related reference and information acquisition. B. Construct the fuzzy hierarchical evaluation structure. C. Determine weights using fuzzy regression.
D. Evaluate the alternative samples. E. 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 technol-ogy, greater quantities of information and knowledge have been properly documented and published. These documents can be used as references. Furthermore, expert inter-views 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 sec-ondary 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
stan-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. Figure1illustrates 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 Table1. 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
Table 3 Data samples with total evaluation given by an expert
Sample yj=yj,dj c1 c2 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 Table4, whereaiandhidenote a centre value of weight and its width of
crite-riaci. Each weight is represented asci =(ai,hi), fori =1,2, . . . ,5. The evaluations c1toc5in Table3are the criteria obtained from the experts. From Table4, 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 domi-nance, then the other criteria might represent strong dominance in the total evaluation.
yj =(yj,dj)is the total evaluation given by the expert. Even though the information
Table 4 Weights of criteria
Center value Width
a1=0.925 h1=0.000
a2=0.000 h2=0.224
a3=0.075 h3=0.040
a4=0.000 h4=0.000
a5=0.000 h5=0.014
Table 5 Comparison of the expert evaluation and FHEM
Rank Sample Expert evaluation,yj=(yj,dj) Total evaluation by FHEM,yj=(y˜j,dj)
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)
hier-Table 6 Evaluation results obtained using three methods
Ranking Sample AHP Sample TOPSIS Sample FHEM
Preference(Pj) Preference(Pj) Preference(Pj)
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)
5 A11 0.061 A11 0.667 A11 (6.999, 1.40)
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 Section5. The estimated results show that this model produces values that are highly similar to the expert evaluation values. Table5 shows the tabulated results for actual and estimated values.
Table6shows the evaluation result of the FHEM method compared with the AHP and TOPSIS methods. Let Pi (fori = 1,2, . . . ,n)represent the final preference of alternative Aiwhen all decision criteria are considered. We obtain the top four final ranking scores of alternatives using the FHEM method, asFHEMA1 ≻FHEMA2 ≻
FHEMA3≻FHEMA20. Meanwhile, the AHP method producesAHPA1≻AHPA2≻
AHPA3 ≻ AHPA20 and the TOPSIS method gives TOPSISA1 ≻ TOPSISA2 ≻
TOPSISA3 ≻ TOPSISA20. Since the comparable methods do not involve the fuzzy
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 under-standing 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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