A fuzzy regression approach to a hierarc (1)

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: nureize@uthm.edu.my
J. Watada
e-mail: junzow@osb.att.ne.jp

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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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107

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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109

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

111

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

(v ji − vi+ )2 ,

j = 1, . . . , n

(5)

k

(v ji − vi− )2 ,

j = 1, . . . , n

(6)

i=1

i=1

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

1̃

(1,1)

2

2̃

(2,1)

Equal importance
Equal to moderately importance

3

3̃

(3,1)

Moderate importance

4

4̃

(4,1)

Moderate to strong importance

5

5̃

(5,1)

Strong importance

6

6̃

(6,1)

Strong to very strong importance

7

7̃

(7,1)

Very strong importance

8

8̃

(8,1)

Very to extremely strong importance

9

9̃

(9,1)

Extreme importance

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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115

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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117

Table 3 Data samples with total evaluation given by an expert


c1
c2
Sample
yj = yj,dj

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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A. Nureize, J. Watada

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 = ( ỹ 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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119

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)

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 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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