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Application of fuzzy linear regression method for sensory evaluation of fried donut
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Application of fuzzy linear regression method for sensory evaluation of fried donut
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Zahra Sadaat Zolfaghari a , Mohebbat Mohebbi a,∗ , Marzieh Najariyan b a
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Department of Food Science and Technology, Ferdowsi University of Mashhad, PO Box 91775-1163, Iran Department of Mathematics, Ferdowsi University of Mashhad, PO Box 91775-1163, Iran
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a r t i c l e
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i n f o
a b s t r a c t
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Article history: Received in revised form 20 February 2013 Available online xxx
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Keywords: Fuzzy linear regression Sensory evaluation Linear programming Donut
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1. Introduction
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Sensory evaluation is a scientific discipline that is widely used to determine the quality of food products. But sensory characteristics cannot be quantified exactly; hence, the relationships among variables are not clear. In this paper, a Fuzzy linear regression was proposed to model the relationship between overall acceptance and sensory characteristics (aroma, surface color, porosity, hardness, oiliness, and flavor) of 36 different types of fried donuts. Modeling was done assuming that independent variables are crisp and coefficients are triangular fuzzy numbers. Coefficients were estimated considering 864 limits due to 36 samples, 12 evaluators (432 instance) and 2 limits for each sample. Between different states of fuzzy numbers (symmetrical, constant asymmetrical, increasing asymmetrical and decreasing asymmetrical) symmetrical fuzzy coefficient provided the best fitting of sensory data. This function showed aroma did not have any effect on overall acceptance, on the contrary, flavor exerted the strongest effect on desirability of donuts, increasing brownness of crust color yet decreasing oiliness reduced desirability of donuts. More porous and softer texture led to more acceptable products. © 2014 Elsevier B.V. All rights reserved.
Sensory evaluation is a scientific method to gather information about the evaluated product [1]. The principal uses sensory evaluation techniques are used in quality control, product development and research in food science [2]. Making decisions is based on sensory testing to gather valid and reliable data about the evaluated product. Sensory evaluation data are characterized by imprecision, inaccuracy and uncertainty [3]. A typical problem about analysis of vague data is that of assigning numbers to subjective perceptions or to linguistic variables. In fact, there are many cases where observations cannot be verified or quantified exactly and relationships among variables are not clear. Thus, one can only provide their approximate description, or intervals to enclose them [4]. Fuzzy approaches have been successfully applied in many experiments that involved fuzzy data. Zhou and Zeng [2] explained a fuzzy logic based method for analyzing sensory evaluation data of industrial products. Evaluation of sensory scores of green tea samples was conducted using fuzzy logic [3]. Alternatively, one of the
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recommended methods for analysis of linguistic data is fuzzy linear regression. Fuzzy linear regression was first introduced by Tanaka et al. [5–7], which is based on possibility theory and fuzzy set theory [8,9]. Uncertainty in this type of regression model becomes fuzziness, not randomness [10], while classic regression is based on probability theory and both the independent and dependent variables are real numbers. Statistical regressions have some disadvantages in the certain situations: if the number of observations is inadequate, if there is difficulty verifying that whether the error is normally distributed, and if there is vague relationship between the independent and dependent variables, if there is ambiguity associated with the event or if the linearity assumption is inappropriate. These are the very situations fuzzy regression was meant to address [11]. Parameter estimation of fuzzy linear regression (FLR) is commonly done under two factors: the degree of the fitting and the vagueness of the model which can be transferred into two approaches viz. (i) Linear programming and (ii) fuzzy least squares methods. FLR models for crisp input-fuzzy output data can be represented as follows: ˜0 + A ˜ 1 xi1 + · · · + A ˜ j xij + · · · + A ˜ m xim Y˜ i = f (x, A) = A
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∗ Corresponding author. Tel.: +98 5118795618 29x315; fax: +98 5118787430. E-mail addresses: [email protected] (Z.S. Zolfaghari), [email protected] (M. Mohebbi), [email protected] (M. Najariyan).
(1-1)
˜ i for i = 1, . . ., n are fuzzy coefficients, where Y˜ is the fuzzy output, A x = (x1 , x2 , . . ., xn ) is an n-dimensional crisp input vector and xij is the jth observed value of the ith input variable. A tilde character (∼)
http://dx.doi.org/10.1016/j.asoc.2014.03.010 1568-4946/© 2014 Elsevier B.V. All rights reserved.
Please cite this article in press as: Z.S. Zolfaghari, et al., Application of fuzzy linear regression method for sensory evaluation of fried donut, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.03.010
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is placed above the name of a fuzzy variable to distinguish a fuzzy variable from a crisp variable [12]. In this study sensory characteristics of fried donut including aroma, surface color, porosity hardness, oiliness and flavor were evaluated. Donut is a proper case study and it has distinctive sensory parameters. It is a sweet fried golden-brown snack with high energy value that is served as a convenient food. It represents one of the largest breakfast categories in the bakery products because it is portable and easy-to-eat. The donut market alone is a $3–4 billion business in the U.S. [13]. Therefore the study of sensory characteristics of donut plays an important role in the production of donut with higher quality in this market. Donuts are divided into two general classes: cake donuts (chemically leavened) and yeastraised donuts (requires fermentation and proofing time). The fat content of fried donut may reach up to 50% of the total weight [14,15]. Nowadays, what is abundantly observable is problem associated to lipid consumption and relevant diet problems like excess weight and coronary heart disease. This, in turn, leads to an increase in research corresponding to production of fried foods with lower oil content and the same quality. In this study, pre-baking process, hydrocolloid coating and partial substituting wheat flour with soy flour, were applied and sensorial characteristic of different formulations of donut were evaluated. Then, analysis of sensory data with new method according to fuzzy linear regression was carried out. To our knowledge there is no published data on application of fuzzy linear regression in sensory evaluation of foods.
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2. Materials and methods
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2.1. Donut preparation
60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84
Fig. 1. Donut preparation procedure. 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105
Donut ingredients including wheat flour, soy flour, sugar, yeast, butter, egg, milk powder, vanilla, lemon juice and salt were purchased from local market in Mashhad. Powder additives including flour, milk powder, vanilla and salt were sifted twice then mixing additives by an Electric Standard Mixer (Hugel, No. HG550TMEM) for 15 min. Dough was rested at 35 ◦ C for 45 min in a proofer (Irankhodsaz, Iran), rolled out to 1 cm thickness and cut with a manually donut cutter (inner diameter: 2.95 cm; outer diameter: 7.60 cm), then proofed at 35 ◦ C for 15 min after second proofing. Prebaking was done at 100 ◦ C for 20 min (SINMAG, 705E). Prebaked samples were then cooled at room temperature for 20 min. Coating the donuts with methyl cellulose solution (1%) and Gum tragacanth (1%) solution was accomplished. Finally, frying was carried out in a domestic thermostatically temperature-controlled fryer (Black & Decker, Type 01) at 150 ± 3 ◦ C for 6 min. Fried donuts were cooled and their surface oil was removed by paper towel for 30 min. Schematic of dough preparation is depicted in Fig. 1. Fig. 2 shows different formulations of prepared donut.
2.2. Sensory evaluations Sensory evaluation of fried donuts was carried out in 6 days. Judges were selected from students of Ferdowsi university of Mashhad, aged between 22 and 30 (6 male and 6 female). Evaluators were interested in sensory evaluation of donut. Scoring was carried out in a 5 point hedonic scale according to Table 1. Quality attributes including aroma, surface color, porosity, hardness, oiliness, and flavor of samples and overall acceptance. Each sample was randomly numbered and presented to panel members.
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3. Theory
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3.1. Fuzzy linear regression
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Assuming unclear relationship among sensory characteristics and overall acceptance, sensory evaluation data were modeled
Fig. 2. Different formulations of donuts.
Please cite this article in press as: Z.S. Zolfaghari, et al., Application of fuzzy linear regression method for sensory evaluation of fried donut, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.03.010
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Table 1 Scoring of sensory characteristics of donut.
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Sensory characteristic
Scoring levels
Aroma Surface color Porosity Hardness Oiliness Flavor Overall acceptance
Very good Bright Very good Hard Very oili Very good Very good
Good Somewhat bright Good Somewhat hard Oili Good Good
according to fuzzy linear regression with crisp independent variables and fuzzy coefficients. In this section, fuzzy numbers and their membership functions are defined. Consequently, limits and spreads increasing problems are described.
Fair Fair Fair Fair Fair Fair Fair
Bad Somewhat dark Bad Somewhat soft Somewhat dry Bad Bad
number and defined as, Y˜ = (f c (x), f 3 (x)) when fc (x) is the mode and f3 (x) is spread of TFN. Now we can write Eq. (1-1) as follows: f c (x) = a0 + a1 x1 + · · · + an xn f s (x) = s0 + s1 x1 + · · · + sn xn
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124 125 126 127 128 129 130 131
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133 134 135 136 137
138
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3.2. Definition of fuzzy numbers The fuzzy coefficients are assumed triangular fuzzy numbers (TFNs). A TFN can be represented with a triple à = (a, sL , sR ), where a, sR and sL are mode, right spread and left spread of triangular fuzzy numbers (TFN), respectively [16]. Consequently, the coefficients can be characterized by a membership function) Ã(x). If sR = sL = s, TFN is a symmetrical triangular fuzzy number and membership function is according to Eq. (2-1). Fig. 3 represented a symmetrical TFN.
Y˜ (x) =
⎧ a−x ⎨ 1− a−s≤x ≤a ⎩
s x−a 1− s
(2-1) a
And when sR = / sL , we have asymmetrical TFN. Its membership function denotes according Eq. (2-2). We can express the right spread according to the left spread, while its coefficient is a positive real number and it is named elongation coefficient. Fig. 4 shows an asymmetrical TFN.
Y˜ (x) =
⎧ a−x a − sL ≤ x ≤ a ⎨ 1− L ⎩
s x−a 1− KsL
(2-2)
a < x ≤ a + ksL
Very bad Dark Very bad Soft Dry Very bad Very bad
(2-3)
Y) can be defined according to Eqs. And membership function (˜ (2-3) and (2-1):
Y˜ =
⎧ (f c (x) − y) ⎪ f c (x) − f s (x) ≤ y ≤ f c (x) ⎨ 1 − − f s (x) c ⎪ ⎩ 1 − (y − f (x)
f s (x)
(2-4)
141
143
144
145 146
147
f c (x) < y ≤ f c (x) + f s (x)
3.4. FLR with asymmetric parameters
148
If Ãi (i = 0, 1, . . ., n) assumed asymmetrical fuzzy number, thus output will be asymmetrical fuzzy number Y˜ = (f c (x), fsL (x), fsR (x)) when fc (x) is the mode, fsL (x) is left spread and fsR (x) is right spread of TFN. Now fuzzy linear regression and fuzzy membership function are denoted as shown in Eqs. (2-5) and (2-6).
149 150 151 152 153
f c (x) = a0 + a1 x1 + · · · + an xn fsL (x) = s0L + s1L x1L + · · · + snL xnL fsR (x)
Y˜ =
=
s0R
+ s1R x1R
(2-5)
154
(2-6)
155
+ · · · + snR xnR
⎧ f c (x) − y ⎪ f c (x) − fSL (x) ≤ y ≤ f c (x) ⎨ 1 − f L (x) S
c ⎪ ⎩ 1 − y −Rf (x) f c (x) < y ≤ f c (x) + fSR (x)
fS (x)
3.3. FLR with symmetric parameters 3.5. Definition of limits
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If Ãi (i = 0, 1, . . ., n) is symmetrical fuzzy number and xi is crisp real number, then the output of Eq. (1-1), will be a triangular fuzzy
Fig. 3. Membership function of symmetrically TNF.
To determine the fuzzy parameters (Ãj ), according to formulation of fuzzy, regression objectives were done as the following linear programming limits [6,9].
Fig. 4. Membership function of asymmetrical TNF.
Please cite this article in press as: Z.S. Zolfaghari, et al., Application of fuzzy linear regression method for sensory evaluation of fried donut, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.03.010
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3.5.1. Degree of belonging (h) The first limit is degree of belonging (h). The constraints require that each observation yj have at least “h” ∈ [0,1] degree of belonging to ˜ Yj , which is equivalent to Y˜ j (yj )≥h,
∀j = 1, 2, . . ., m
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171
173
(II) Asymmetric:
(1 + k0 )ms0L +
s.t.
(1 − h)s0L + (1 − h)
i=1
(1 − h)s0 + (1 − h)
(k sL |x |) + a0 i=1 i i ji
[si
Z = m(s0L + s0R ) +
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178 179
180
1−
182 183
f c (xj ) − yj f s (xj ) yj − f c (xj )
j=1
f s (xj )
i=1 i=1
n
190
[(siL + siR )
j=1
|xij |]
(2-9)
j = 1, 2, . . ., m
c
j = 1, 2, . . ., m
(si |xji |) − a0 − (si |xji |) + a0 +
n
i=1
(ai xji )≥ − yj ,
i=1
(ai xji )≥yj ,
n
n
j = 1, 2, . . ., m
(2-11)
(1 − h)s0L + (1 − h)
i=1
(siL |xji |) − a0 −
i=1
(ki siL |xji |) + a0 +
n
(1 − h)s0 + (1 − h)
n
i=1
n
(ai xji )≥ − yj ,
i=1
(ai xji )≥ − yj ,
j = 1, 2, . . ., m
194
(2-15)
s.t.
(1 − h)s0 + (1 − h)
i=1
(si |xij |) − a0 −
(1 − h)s0 + (1 − h)
i=1
(si |xij |) + a0 +
[s i=1 i
|x |] j=1 ij
n n
si ≥0,
195
j = 1, 2, . . ., m 196 197
n i=1
+
n
(ai xji )≥ − yj ,
i=1
(ai xji )≥ − yj ,
j = 1, 2, . . ., m
(2-16)
198
j = 1, 2, . . ., m
where xij is the jth observed value of the ith input variable. As we had 36 different samples, 12 evaluators and 6 independent variables (aroma, color, porosity, hardness, oiliness and flavor) thus there were 432 instances and 864 limits. Minimizing the target function considering these limits was done according to linear programming method and using LINGO software (ver. 10). MATLAB (R2007b) program was used for defuzzification of coefficient with center of gravity method (COG), and then the best model was selected.
4.1. The goodness of fit measures
MSE =
i=1
(˜y − y)2
SSE n−k−1
201 202 203 204 205 206 207
209 210 211
212
In order to measure the power of model, some indices of fitting regression model were used through minimization of a distance function between the empirical fuzzy data and the model. This statistic measures the total deviation of the response values from the fit to the response values. It is also called the summed square of residuals and is usually labeled as SSE. MSE is the mean square error or the residual mean square. An MSE value closer to 0 indicates a fit that is more useful for prediction. In particular, MSE were calculated according to relations (3-1) and (3-2) [4] SSE =
200
208
As described previously, target function considering the defined limits was minimized per different values of h and fuzzy parameters were calculated.
m
199
213 214 215 216 217 218 219 220 221
(3-1)
222
(3-2)
223
where n is the count of instances and k is the count of variables. 224
2ms0 + 2
n i=1
(ai xij )≥ − yj , j = 1, 2, . . ., m
i=1
(ai xij )≥yj ,
n
(2-12)
j = 1, 2, . . ., m
i = 1, . . ., n.
where xij is the jth observed value of the ith input variable. Asymmetric: Assuming sR = ksL Eq. (2-9) leads to Eq. (2-13)
n
Z = (1 + k0 )ms0L +
n
193
j = 1, 2, . . ., m
By replacing (2-5) and sR = ksL in above relations we have:
j = 1, 2, . . ., m
m
Min
Z = m(s0L + s0R ) + 191
m
i=1
187
189
(2-8)
Our problem is to find out the fuzzy parameters using linear programming:
s0 ≥0,
fsR (xj )
(2-14) ≥h → (1 − h)fsR (xj ) + f c (xj )≥yj
i = 1, . . ., n.
Minimizing the target function considering limits leads to these equations is another limit of the model (2-10): By replacing (2-3) in (2-10) we have:
n
f (xj ) − yj
j = 1, 2, . . ., m
|xij |]
(siL |xji |) − a0 −
|xi j|]
≥h → (1 − h)f (xj ) + f (xj )≥yj
(1 − h)s0 + (1 − h)
188
si ≥0,
j=1
(2-10)
n
c
≥h → (1 − h)fsL (xj ) − f c (xj )≥ − yj
4. Result and discussion
s
(1 − h)s0 + (1 − h)
186
[(1 + ki )si
n n
≥h → (1 − h)f s (xj ) − f c (xj )≥ − yj
184
185
i=1
In order to find an appropriate fuzzy regression model the target function should be minimized when “h” has an acceptable degree of belonging to ˜ Yj . Limits in the symmetric and asymmetric condition of TFNs are calculated according to following instruction. Symmetric: 1−
181
m L
Min
i=1
177
n
(I) Symmetric: Z = 2ms0 + 2
fsL (xj )
192
Asymmetrical fuzzy parameters can be obtained from the solution of the following linear programming problem (2-16):
3.5.2. Target function (Z) The sum of TFN spreads is defined as a target function (Z). Target function is defined in symmetric and asymmetric condition of TFNs as follows:
172
176
1−
The proposed model should have an acceptable degree of belonging to ˜ Yj in the spread of [0,1]. if “h” is nearer to 1, the proposed model will be more reliable.
n m
175
1−
f c (xj ) − yj
(2-7)
s0 ≥0,
167
Then we have:
m
[(siL + siR ) i=1
n
i=1
[(1 + ki )siL
|x |] j=1 ij
m
j=1
|xij |]
(2-13)
The minimum value of MSE shows minimum error of regression model and power of model.
225 226
4.2. Symmetric
227
The best regression model considering symmetric fuzzy coefficient, was obtained when h = 0.8 (Table 2) and it isdenoted as
Please cite this article in press as: Z.S. Zolfaghari, et al., Application of fuzzy linear regression method for sensory evaluation of fried donut, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.03.010
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S0
S1
S2
S3
S4
S5
S6
Z
MSE
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.9191 1.0340 1.1818 1.3787 1.6545 2.0681 2.7575 4.1363 8.2727
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0.1464 0.1647 0.1883 0.2196 0.2636 0.3295 0.4393 0.6590 1.3181
0.0606 0.0681 0.0779 0.0909 0.1090 0.1363 0.1818 0.2727 0.5454
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
625.989 704.238 804.844 938.984 1126.782 1408.477 1877.97 2816.955 5633.909
0.9161 0.9255 0.911 0.8835 0.8297 0.7556 0.5243 0.3767 0.784
h
ac0
ac1
ac2
ac3
ac4
ac5
ac6
0.1–0.9
0.9
0
−0.0181
0.1227
0.0727
−0.0363
0.5272
follows (Eq. (3-3)):
230
Yˆ = [0.9, 4.1363] + [0, 0]x1 + [−0.0181, 0]x2 + [0.1227, 0.659]x3 + [0.0727, 1.3181]x4 + [−0.0363, 0]x5 + [0.5272, 0]x6
231
234 235
236
Y = 0.2963 − 0.0181x2 + 0.122x3 + 0.0778x4
237
(3-4)
4.3. Asymmetric
1. Constant values of k Table 3 shows the coefficients when k was assumed constant 2. Increasing trend of ki Table 4, asymmetrical fuzzy parameters considering increasing trend of ki 3. Decreasing trend of ki Table 5, asymmetrical fuzzy parameters considering decreasing trend of ki
(3-3)
Table 3 Asymmetrical fuzzy parameters considering constant value of k.
0.5 1.5 2 2.5
Ã0
Ã1
Ã2
Ã3
Ã4
240 241 242 243
244
In this section, coefficients were considered asymmetrical triangular fuzzy numbers. Finding the best regression model, different states of elongation coefficient (k) were assumed and according to values of MSE the best model was selected.
In this model x1 , x2 , x3 , x4 , x5 , x6 are aroma, surface color, porosity, hardness, oiliness and flavor, respectively. After defuzzification of Eq. (3-3), the best symmetric regression model was obtained
ki
238
239
Shayannejad [18] compared fuzzy regression method and Artificial Neural Networks (ANN) to the determination of potential evapotranspiration. Results of this study demonstrate fuzzy regression method with symmetric TFNs had better fitness.
232 233
(Eq. (3-4))
− 0.0363x5 + 0.5272x6
h: degree of belonging to ˜ Yj ; si : spread of TFN; ai : mode of TFN; Z: value of target function; MSE: mean squared error.
229
5
Ã5
Ã6
s01
ac0
s11
ac1
s21
ac2
s31
ac3
s41
ac4
s51
ac5
s61
ac6
5.51 3.30 2.75 2.36
1.17 0.73 0.62 0.54
0 0 0 0
0 0 0 0
0 0 0 0
−0.01 −0.01 −0.01 −0.01
0.87 0.52 0.43 0.37
0.16 0.09 0.07 0.06
0.36 0.21 0.18 0.15
0.09 0.06 0.05 0.04
0 0 0 0
−0.03 −0.03 −0.03 −0.03
0 0 0 0
0.52 0.52 0.52 0.52
Z
MSE
5633.909 5633.909 5633.909 5633.909
1.2631 0.8402 1.5175 2.2254
ki : constant elongation coefficient; Ãi : fuzzy coefficients of regression model; s1i : left spread of asymmetrical TFN; aci : mode of asymmetrical TFN; Z: value of target function; MSE: mean squared error.
Table 4 Asymmetrical fuzzy parameters considering increasing trend of ki . k0
k1
k2
k3
k4
k5
k6
s01
s11
s21
s31
s41
s51
s61
Z
MSE
0.5 0.8 0.3
0.8 1.1 0.6
1.1 1.4 0.9
1.4 1.7 1.2
1.7 2 1.5
2 2.3 1.8
2.3 2.6 2.1
5.51 4.59 6.36
0 0 0
0 0 0
0.54 0.48 0.59
0.20 0.18 0.21
0 0 0
0 0 0
5633.9 5633.9 5633.9
0.89 2.37 0.42
k0
k1
k2
k3
k4
k5
k6
ac0
ac1
ac2
ac3
ac4
ac5
ac6
0.5 0.8 0.3
0.8 1.1 0.6
1.1 1.4 0.9
1.4 1.7 1.2
1.7 2 1.5
2 2.3 1.8
2.3 2.6 2.1
1.17 0.99 1.34
0 0 0
−0.01 −0.01 −0.01
0.10 0.08 0.11
0.05 0.05 0.06
−0.036 −0.036 −0.036
0.52 0.52 0.52
ki : elongation coefficient; s1i : left spread of asymmetrical TFN; aci : mode of asymmetrical TFN; Z: value of target function; MSE: mean squared error. Table 5 Asymmetrical fuzzy parameters considering decreasing trend of ki . k0
k1
k2
k3
k4
k5
k6
s01
s11
s21
s31
s41
s51
s61
Z
MSE
2.1 2.3 2.6
1.8 2 2.3
1.5 1.7 2
1.2 1.4 1.7
0.9 1.1 1.4
0.6 0.8 1.1
0.3 0.5 0.8
2.67 2.51 2.30
0 0 0
0 0 0
0.6 0.55 0.49
0.29 0.26 0.23
0 0 0
0 0 0
5633.9 5633.9 5633.9
3.00 4.20 5.96
k0
k1
k2
k3
k4
k5
k6
ac0
ac1
ac2
ac3
ac4
ac5
ac6
2.1 2.3 2.6
1.8 2 2.3
1.5 1.7 2
1.2 1.4 1.7
0.9 1.1 1.4
0.6 0.8 1.1
0.3 0.5 0.8
0.61 0.57 0.52
0 0 0
−0.0181 −0.0181 −0.0181
0.11 0.10 0.09
0.08 0.07 0.06
−0.036 −0.036 −0.036
0.53 0.53 0.53
ki : elongation coefficient; s1i : left spread of asymmetrical TFN; aci : mode of asymmetrical TFN; Z: value of target function; MSE: mean squared error.
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Table 6 Ranking of donut samples.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Soy flour addition (%)
Raising agent
Pre-backing
Coating
Score
5% 10% 10% 10% 10% 5% 0% 0% 10% 5% 5% 10% 5% 0% 10% 0% 0% 0% 5% 0% 0% 5% 0% 10% 5% 5% 0% 10% 10% 10% 5% 5% 0% 10% 5% 10%
Cake donut Yeast-raised donut Yeast-raised donut Yeast-raised donut Cake donut Cake donut Yeast-raised donut Yeast-raised donut Cake donut Cake donut Yeast-raised donut Cake donut Yeast-raised donut Cake donut Cake donut Cake donut Yeast-raised donut Cake donut Cake donut Cake donut Yeast-raised donut Cake donut Cake donut Yeast-raised donut Yeast-raised donut Cake donut Yeast-raised donut Cake donut Cake donut Yeast-raised donut Yeast-raised donut Yeast-raised donut Cake donut Cake donut Yeast-raised donut Yeast-raised donut
Control Control Control Control Control Control Control Control Control Control Control Control Control Control Pre-backed Control Control Pre-backed Pre-backed Control Pre-backed Pre-backed Pre-backed Pre-backed Control Pre-backed Pre-backed Pre-backed Pre-backed Pre-backed Pre-backed Pre-backed Pre-backed Pre-backed Pre-backed Pre-backed
Uncoated GT coated MC coated Uncoated Uncoated GT coated MC coated GT coated GT coated MC coated MC coated MC coated GT coated MC coated MC coated Uncoated Uncoated GT coated Uncoated GT coated MC coated GT coated Uncoated GT coated Uncoated MC coated GT coated Uncoated GT coated MC coated GT coated Uncoated MC coated Uncoated Uncoated MC coated
1.5288 1.5380 1.5659 1.6200 1.6407 1.6526 1.6540 1.6583 1.6632 1.6697 1.6962 1.7249 1.7450 1.8129 1.8253 1.8389 1.8470 1.8614 1.8658 1.8757 1.8777 1.9120 1.9157 1.9176 1.9336 1.9564 1.9708 1.9728 2.0068 2.0205 2.0216 2.0792 2.0832 2.1125 2.1339 2.1343
GT coated: coated donuts with Gum tragacanth solution; MC coated: coated donuts with methyl cellulose solution; control: fried donut without pre-baking; pre-backed: fried donut with 2 min per-backing process.
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According to Table 4 decreasing trend of ki , leads to increasing MSE and reducing the quality of the model.
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K0 = 0.3,
k1 = 0.6,
k2 = 0.9,
k3 = 1.2,
k4 = 1.5,
k6 = 2.1
flavor. These results indicate that increasing the brownness of surface color decreased overall acceptance, while increasing oiliness of donuts, increased overall acceptance and flavor had the strongest effect on the acceptability of donuts.
282 283 284 285
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Yˆ = [1.3454, 6.3636] + [0, 0]x1 + [−0.01818, 0]x2
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+ [0.1107, 0.5991]x3 + [0.0618, 0.2118]x4
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+ [−0.0363, 0]x5 + [0.5272, 0]x6
4.4. Ranking of donut samples according to fuzzy regression model (3-5)
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After defuzzification of Eq. (3-5), the best regression model with asymmetrical coefficient was obtained as follow:
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Y = −0.1393 − 0.0181x2 + 0.1515x3 + 0.0959x4
265
268
− 0.0363x5 + 0.5272x6
(3-6)
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According to the MSEs of model (3-4) and (3-6), it could be found that asymmetrical coefficients did not improve regression model. On the contrary, another study on the reasons of prosperity of water-user cooperatives in Iran [19] examined symmetric and asymmetric TFNs and the best model was obtained when TFNs were asymmetric and ki had perpendicular and lateral increasing trend. Also Mohammadi nad Taheri [20] applied fuzzy regression to modeling Pedotransfer Functions. Asymmetric TFNs and constant ki showed the best fit in this study [20]. Consequently Eq. (3-4) shows the best fit of sensory evaluation data. As per the opinion of judges regarding the general quality attributes for donut sample, aroma is the least important one compared to crust color, oiliness and
For the ranking of samples in general, fuzzy linear regression model was used. Overall acceptance values obtained from Eq. (3-4). A comparison of values showed that the most desirable sample was cake donut with 5% soy flour without coating. Cake donut was produced by chemical leavening agent. On the other hand, pre-backed donut with 0% soy flour addition and leavened by yeast had lowest quality acceptance compared to other samples (Table 6). Ranking of samples showed pre-baking process have a negative effect on overall acceptance of donut samples and it is not suitable for the preparation of fried donut. 5. Conclusion In this study we proposed a fuzzy linear regression model for analyzing sensory evaluation data. This new promising approach of “Fuzzy Regression” is discussed which is capable of handling such a situation. In this model, not a clear relationship between overall acceptance and sensory characteristics was assumed, and then the best model was calculated considering crisp independent variables, dependent fuzzy variables and fuzzy coefficients. Results showed that symmetrical fuzzy parameter provided the best fitting
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of sensory data. According to this model, surface color had a negative effect on overall acceptance. Overall acceptance increased with samples Oiliness. Flavor had the strongest effect and aroma had no effect on overall acceptance. This method was more efficient and appropriate than classic statistic methods in modeling sensory data. It could be generalized in sensory evaluation analysis and when the data cannot be recorded or collected precisely.
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Uncited reference
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[17]. References [1] L. Martinez, Sensory evaluation based on linguistic decision analysis, Int. J. Approx. Reason. 44 (2007) 148–164. [2] B. Zhou, X. Zeng, L. Koelll, Y. Ding, A fuzzy logic based method for analyzing sensory evaluation data, in: Second IEEE International Conference on Intelligent Systems, 2004, pp. 178–183. [3] V.R. Sinija, H.N. Mishra, Fuzzy analysis of sensory data for quality evaluation and ranking of instant green tea powder and granules, Food Bioprocess Technol. 4 (2008) 408–416. [4] P. D’Urso, T. Gastaldi, A least-squares approach to fuzzy linear regression analysis, Comput. Stat. Data Anal. 34 (2000) 427–440. [5] H. Tanaka, S. Uejima, K. Asai, Fuzzy linear regression model International Congress on Applied Systems and Cybernetics, vol. VI, Acapulco, Mexico, 1980, pp. 2933–2938.
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[6] H. Tanaka, S. Uejima, K. Asai, Linear regression analysis with fuzzy model, IEEE Trans. Syst. Man Cyber. 12 (1982) 903–907. [7] P. D’Urso, Linear regression analysis for fuzzy/crisp input and fuzzy/crisp output data, Comput. Stat. Data Anal. 42 (2003) 47–72. [8] H.F. Wang, R.C. Tsaur, Insight of a fuzzy regression model, Fuzzy Set Syst. 112 (2000) 355–369. [9] Y.H.O. Chang, B.M. Ayyub, Fuzzy regression methods – a comparative assessment, Fuzzy Set Syst. 119 (2001) 187–203. [10] M.S. Yang, H.H. Liu, Fuzzy least-squares algorithms for interactive fuzzy linear regression models, Fuzzy Set Syst. 135 (2003) 305–316. [11] A.F. Shapiro, Fuzzy regression and the term structure of interest rates revisited, AFIR (2004) 1–17. [12] J. Lu, R. Wang, An enhanced fuzzy linear regression model with more flexible spreads, Fuzzy Set Syst. 160 (2009) 2505–2523. [13] K.J. Tan, G.S. Mittal, Physicochemical properties changes of donuts during vacuum frying, Int. J. Food Prop. 9 (2006) 85–98. [14] T. Funami, M. Funami, T. Tawada, Y. Nakao, Decreasing oil uptake of doughnuts during deep-fat frying using curdlan, J Food Sci. 64 (1999) 883–888. [15] E.J. Pinthus, P. Weinberg, I.S. Saguy, Criterion for oil uptake during deep-fat frying, J Food Sci. 58 (1993) 204–422. [16] G.G. Rodriguez, A. Blanco, A. Colubi, M.A. Lubiano, Estimation of a simple linear regression model for fuzzy random variables, Fuzzy Set Syst. 160 (2009) 357–370. [17] C.G. Wen, C.S. Lee, Development of a cost function for wastewater treatment systems with fuzzy regression, Fuzzy Set Syst. 106 (1999) 143–153. [18] M. Shayannejad, S.J. Sadatinejad, H. Fahmi, Determination of potential evapotranspiration using the fuzzy regression method, IR-WRR 3 (2008) 9–19. [19] A.M. Amini, M. Khayati, the reasons of prosperity of water-user cooperatives in Iran, Agric. Econ. Dev. 53 (2006) 69–91. [20] J. Mohammadi, S.M. Taheri, Modeling pedotransfer functions using fuzzy regression, J. Crop Prod. Process. 9 (2005) 51–61.
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Contents lists available at ScienceDirect
Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
Application of fuzzy linear regression method for sensory evaluation of fried donut
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Zahra Sadaat Zolfaghari a , Mohebbat Mohebbi a,∗ , Marzieh Najariyan b a
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b
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Department of Food Science and Technology, Ferdowsi University of Mashhad, PO Box 91775-1163, Iran Department of Mathematics, Ferdowsi University of Mashhad, PO Box 91775-1163, Iran
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a r t i c l e
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i n f o
a b s t r a c t
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Article history: Received in revised form 20 February 2013 Available online xxx
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Keywords: Fuzzy linear regression Sensory evaluation Linear programming Donut
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1. Introduction
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Sensory evaluation is a scientific discipline that is widely used to determine the quality of food products. But sensory characteristics cannot be quantified exactly; hence, the relationships among variables are not clear. In this paper, a Fuzzy linear regression was proposed to model the relationship between overall acceptance and sensory characteristics (aroma, surface color, porosity, hardness, oiliness, and flavor) of 36 different types of fried donuts. Modeling was done assuming that independent variables are crisp and coefficients are triangular fuzzy numbers. Coefficients were estimated considering 864 limits due to 36 samples, 12 evaluators (432 instance) and 2 limits for each sample. Between different states of fuzzy numbers (symmetrical, constant asymmetrical, increasing asymmetrical and decreasing asymmetrical) symmetrical fuzzy coefficient provided the best fitting of sensory data. This function showed aroma did not have any effect on overall acceptance, on the contrary, flavor exerted the strongest effect on desirability of donuts, increasing brownness of crust color yet decreasing oiliness reduced desirability of donuts. More porous and softer texture led to more acceptable products. © 2014 Elsevier B.V. All rights reserved.
Sensory evaluation is a scientific method to gather information about the evaluated product [1]. The principal uses sensory evaluation techniques are used in quality control, product development and research in food science [2]. Making decisions is based on sensory testing to gather valid and reliable data about the evaluated product. Sensory evaluation data are characterized by imprecision, inaccuracy and uncertainty [3]. A typical problem about analysis of vague data is that of assigning numbers to subjective perceptions or to linguistic variables. In fact, there are many cases where observations cannot be verified or quantified exactly and relationships among variables are not clear. Thus, one can only provide their approximate description, or intervals to enclose them [4]. Fuzzy approaches have been successfully applied in many experiments that involved fuzzy data. Zhou and Zeng [2] explained a fuzzy logic based method for analyzing sensory evaluation data of industrial products. Evaluation of sensory scores of green tea samples was conducted using fuzzy logic [3]. Alternatively, one of the
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recommended methods for analysis of linguistic data is fuzzy linear regression. Fuzzy linear regression was first introduced by Tanaka et al. [5–7], which is based on possibility theory and fuzzy set theory [8,9]. Uncertainty in this type of regression model becomes fuzziness, not randomness [10], while classic regression is based on probability theory and both the independent and dependent variables are real numbers. Statistical regressions have some disadvantages in the certain situations: if the number of observations is inadequate, if there is difficulty verifying that whether the error is normally distributed, and if there is vague relationship between the independent and dependent variables, if there is ambiguity associated with the event or if the linearity assumption is inappropriate. These are the very situations fuzzy regression was meant to address [11]. Parameter estimation of fuzzy linear regression (FLR) is commonly done under two factors: the degree of the fitting and the vagueness of the model which can be transferred into two approaches viz. (i) Linear programming and (ii) fuzzy least squares methods. FLR models for crisp input-fuzzy output data can be represented as follows: ˜0 + A ˜ 1 xi1 + · · · + A ˜ j xij + · · · + A ˜ m xim Y˜ i = f (x, A) = A
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∗ Corresponding author. Tel.: +98 5118795618 29x315; fax: +98 5118787430. E-mail addresses: [email protected] (Z.S. Zolfaghari), [email protected] (M. Mohebbi), [email protected] (M. Najariyan).
(1-1)
˜ i for i = 1, . . ., n are fuzzy coefficients, where Y˜ is the fuzzy output, A x = (x1 , x2 , . . ., xn ) is an n-dimensional crisp input vector and xij is the jth observed value of the ith input variable. A tilde character (∼)
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Please cite this article in press as: Z.S. Zolfaghari, et al., Application of fuzzy linear regression method for sensory evaluation of fried donut, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.03.010
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is placed above the name of a fuzzy variable to distinguish a fuzzy variable from a crisp variable [12]. In this study sensory characteristics of fried donut including aroma, surface color, porosity hardness, oiliness and flavor were evaluated. Donut is a proper case study and it has distinctive sensory parameters. It is a sweet fried golden-brown snack with high energy value that is served as a convenient food. It represents one of the largest breakfast categories in the bakery products because it is portable and easy-to-eat. The donut market alone is a $3–4 billion business in the U.S. [13]. Therefore the study of sensory characteristics of donut plays an important role in the production of donut with higher quality in this market. Donuts are divided into two general classes: cake donuts (chemically leavened) and yeastraised donuts (requires fermentation and proofing time). The fat content of fried donut may reach up to 50% of the total weight [14,15]. Nowadays, what is abundantly observable is problem associated to lipid consumption and relevant diet problems like excess weight and coronary heart disease. This, in turn, leads to an increase in research corresponding to production of fried foods with lower oil content and the same quality. In this study, pre-baking process, hydrocolloid coating and partial substituting wheat flour with soy flour, were applied and sensorial characteristic of different formulations of donut were evaluated. Then, analysis of sensory data with new method according to fuzzy linear regression was carried out. To our knowledge there is no published data on application of fuzzy linear regression in sensory evaluation of foods.
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2. Materials and methods
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2.1. Donut preparation
60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84
Fig. 1. Donut preparation procedure. 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105
Donut ingredients including wheat flour, soy flour, sugar, yeast, butter, egg, milk powder, vanilla, lemon juice and salt were purchased from local market in Mashhad. Powder additives including flour, milk powder, vanilla and salt were sifted twice then mixing additives by an Electric Standard Mixer (Hugel, No. HG550TMEM) for 15 min. Dough was rested at 35 ◦ C for 45 min in a proofer (Irankhodsaz, Iran), rolled out to 1 cm thickness and cut with a manually donut cutter (inner diameter: 2.95 cm; outer diameter: 7.60 cm), then proofed at 35 ◦ C for 15 min after second proofing. Prebaking was done at 100 ◦ C for 20 min (SINMAG, 705E). Prebaked samples were then cooled at room temperature for 20 min. Coating the donuts with methyl cellulose solution (1%) and Gum tragacanth (1%) solution was accomplished. Finally, frying was carried out in a domestic thermostatically temperature-controlled fryer (Black & Decker, Type 01) at 150 ± 3 ◦ C for 6 min. Fried donuts were cooled and their surface oil was removed by paper towel for 30 min. Schematic of dough preparation is depicted in Fig. 1. Fig. 2 shows different formulations of prepared donut.
2.2. Sensory evaluations Sensory evaluation of fried donuts was carried out in 6 days. Judges were selected from students of Ferdowsi university of Mashhad, aged between 22 and 30 (6 male and 6 female). Evaluators were interested in sensory evaluation of donut. Scoring was carried out in a 5 point hedonic scale according to Table 1. Quality attributes including aroma, surface color, porosity, hardness, oiliness, and flavor of samples and overall acceptance. Each sample was randomly numbered and presented to panel members.
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3. Theory
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3.1. Fuzzy linear regression
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Assuming unclear relationship among sensory characteristics and overall acceptance, sensory evaluation data were modeled
Fig. 2. Different formulations of donuts.
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Table 1 Scoring of sensory characteristics of donut.
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Sensory characteristic
Scoring levels
Aroma Surface color Porosity Hardness Oiliness Flavor Overall acceptance
Very good Bright Very good Hard Very oili Very good Very good
Good Somewhat bright Good Somewhat hard Oili Good Good
according to fuzzy linear regression with crisp independent variables and fuzzy coefficients. In this section, fuzzy numbers and their membership functions are defined. Consequently, limits and spreads increasing problems are described.
Fair Fair Fair Fair Fair Fair Fair
Bad Somewhat dark Bad Somewhat soft Somewhat dry Bad Bad
number and defined as, Y˜ = (f c (x), f 3 (x)) when fc (x) is the mode and f3 (x) is spread of TFN. Now we can write Eq. (1-1) as follows: f c (x) = a0 + a1 x1 + · · · + an xn f s (x) = s0 + s1 x1 + · · · + sn xn
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124 125 126 127 128 129 130 131
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133 134 135 136 137
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3.2. Definition of fuzzy numbers The fuzzy coefficients are assumed triangular fuzzy numbers (TFNs). A TFN can be represented with a triple à = (a, sL , sR ), where a, sR and sL are mode, right spread and left spread of triangular fuzzy numbers (TFN), respectively [16]. Consequently, the coefficients can be characterized by a membership function) Ã(x). If sR = sL = s, TFN is a symmetrical triangular fuzzy number and membership function is according to Eq. (2-1). Fig. 3 represented a symmetrical TFN.
Y˜ (x) =
⎧ a−x ⎨ 1− a−s≤x ≤a ⎩
s x−a 1− s
(2-1) a
And when sR = / sL , we have asymmetrical TFN. Its membership function denotes according Eq. (2-2). We can express the right spread according to the left spread, while its coefficient is a positive real number and it is named elongation coefficient. Fig. 4 shows an asymmetrical TFN.
Y˜ (x) =
⎧ a−x a − sL ≤ x ≤ a ⎨ 1− L ⎩
s x−a 1− KsL
(2-2)
a < x ≤ a + ksL
Very bad Dark Very bad Soft Dry Very bad Very bad
(2-3)
Y) can be defined according to Eqs. And membership function (˜ (2-3) and (2-1):
Y˜ =
⎧ (f c (x) − y) ⎪ f c (x) − f s (x) ≤ y ≤ f c (x) ⎨ 1 − − f s (x) c ⎪ ⎩ 1 − (y − f (x)
f s (x)
(2-4)
141
143
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145 146
147
f c (x) < y ≤ f c (x) + f s (x)
3.4. FLR with asymmetric parameters
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If Ãi (i = 0, 1, . . ., n) assumed asymmetrical fuzzy number, thus output will be asymmetrical fuzzy number Y˜ = (f c (x), fsL (x), fsR (x)) when fc (x) is the mode, fsL (x) is left spread and fsR (x) is right spread of TFN. Now fuzzy linear regression and fuzzy membership function are denoted as shown in Eqs. (2-5) and (2-6).
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f c (x) = a0 + a1 x1 + · · · + an xn fsL (x) = s0L + s1L x1L + · · · + snL xnL fsR (x)
Y˜ =
=
s0R
+ s1R x1R
(2-5)
154
(2-6)
155
+ · · · + snR xnR
⎧ f c (x) − y ⎪ f c (x) − fSL (x) ≤ y ≤ f c (x) ⎨ 1 − f L (x) S
c ⎪ ⎩ 1 − y −Rf (x) f c (x) < y ≤ f c (x) + fSR (x)
fS (x)
3.3. FLR with symmetric parameters 3.5. Definition of limits
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If Ãi (i = 0, 1, . . ., n) is symmetrical fuzzy number and xi is crisp real number, then the output of Eq. (1-1), will be a triangular fuzzy
Fig. 3. Membership function of symmetrically TNF.
To determine the fuzzy parameters (Ãj ), according to formulation of fuzzy, regression objectives were done as the following linear programming limits [6,9].
Fig. 4. Membership function of asymmetrical TNF.
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3.5.1. Degree of belonging (h) The first limit is degree of belonging (h). The constraints require that each observation yj have at least “h” ∈ [0,1] degree of belonging to ˜ Yj , which is equivalent to Y˜ j (yj )≥h,
∀j = 1, 2, . . ., m
168 169 170
171
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(II) Asymmetric:
(1 + k0 )ms0L +
s.t.
(1 − h)s0L + (1 − h)
i=1
(1 − h)s0 + (1 − h)
(k sL |x |) + a0 i=1 i i ji
[si
Z = m(s0L + s0R ) +
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180
1−
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f c (xj ) − yj f s (xj ) yj − f c (xj )
j=1
f s (xj )
i=1 i=1
n
190
[(siL + siR )
j=1
|xij |]
(2-9)
j = 1, 2, . . ., m
c
j = 1, 2, . . ., m
(si |xji |) − a0 − (si |xji |) + a0 +
n
i=1
(ai xji )≥ − yj ,
i=1
(ai xji )≥yj ,
n
n
j = 1, 2, . . ., m
(2-11)
(1 − h)s0L + (1 − h)
i=1
(siL |xji |) − a0 −
i=1
(ki siL |xji |) + a0 +
n
(1 − h)s0 + (1 − h)
n
i=1
n
(ai xji )≥ − yj ,
i=1
(ai xji )≥ − yj ,
j = 1, 2, . . ., m
194
(2-15)
s.t.
(1 − h)s0 + (1 − h)
i=1
(si |xij |) − a0 −
(1 − h)s0 + (1 − h)
i=1
(si |xij |) + a0 +
[s i=1 i
|x |] j=1 ij
n n
si ≥0,
195
j = 1, 2, . . ., m 196 197
n i=1
+
n
(ai xji )≥ − yj ,
i=1
(ai xji )≥ − yj ,
j = 1, 2, . . ., m
(2-16)
198
j = 1, 2, . . ., m
where xij is the jth observed value of the ith input variable. As we had 36 different samples, 12 evaluators and 6 independent variables (aroma, color, porosity, hardness, oiliness and flavor) thus there were 432 instances and 864 limits. Minimizing the target function considering these limits was done according to linear programming method and using LINGO software (ver. 10). MATLAB (R2007b) program was used for defuzzification of coefficient with center of gravity method (COG), and then the best model was selected.
4.1. The goodness of fit measures
MSE =
i=1
(˜y − y)2
SSE n−k−1
201 202 203 204 205 206 207
209 210 211
212
In order to measure the power of model, some indices of fitting regression model were used through minimization of a distance function between the empirical fuzzy data and the model. This statistic measures the total deviation of the response values from the fit to the response values. It is also called the summed square of residuals and is usually labeled as SSE. MSE is the mean square error or the residual mean square. An MSE value closer to 0 indicates a fit that is more useful for prediction. In particular, MSE were calculated according to relations (3-1) and (3-2) [4] SSE =
200
208
As described previously, target function considering the defined limits was minimized per different values of h and fuzzy parameters were calculated.
m
199
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(3-1)
222
(3-2)
223
where n is the count of instances and k is the count of variables. 224
2ms0 + 2
n i=1
(ai xij )≥ − yj , j = 1, 2, . . ., m
i=1
(ai xij )≥yj ,
n
(2-12)
j = 1, 2, . . ., m
i = 1, . . ., n.
where xij is the jth observed value of the ith input variable. Asymmetric: Assuming sR = ksL Eq. (2-9) leads to Eq. (2-13)
n
Z = (1 + k0 )ms0L +
n
193
j = 1, 2, . . ., m
By replacing (2-5) and sR = ksL in above relations we have:
j = 1, 2, . . ., m
m
Min
Z = m(s0L + s0R ) + 191
m
i=1
187
189
(2-8)
Our problem is to find out the fuzzy parameters using linear programming:
s0 ≥0,
fsR (xj )
(2-14) ≥h → (1 − h)fsR (xj ) + f c (xj )≥yj
i = 1, . . ., n.
Minimizing the target function considering limits leads to these equations is another limit of the model (2-10): By replacing (2-3) in (2-10) we have:
n
f (xj ) − yj
j = 1, 2, . . ., m
|xij |]
(siL |xji |) − a0 −
|xi j|]
≥h → (1 − h)f (xj ) + f (xj )≥yj
(1 − h)s0 + (1 − h)
188
si ≥0,
j=1
(2-10)
n
c
≥h → (1 − h)fsL (xj ) − f c (xj )≥ − yj
4. Result and discussion
s
(1 − h)s0 + (1 − h)
186
[(1 + ki )si
n n
≥h → (1 − h)f s (xj ) − f c (xj )≥ − yj
184
185
i=1
In order to find an appropriate fuzzy regression model the target function should be minimized when “h” has an acceptable degree of belonging to ˜ Yj . Limits in the symmetric and asymmetric condition of TFNs are calculated according to following instruction. Symmetric: 1−
181
m L
Min
i=1
177
n
(I) Symmetric: Z = 2ms0 + 2
fsL (xj )
192
Asymmetrical fuzzy parameters can be obtained from the solution of the following linear programming problem (2-16):
3.5.2. Target function (Z) The sum of TFN spreads is defined as a target function (Z). Target function is defined in symmetric and asymmetric condition of TFNs as follows:
172
176
1−
The proposed model should have an acceptable degree of belonging to ˜ Yj in the spread of [0,1]. if “h” is nearer to 1, the proposed model will be more reliable.
n m
175
1−
f c (xj ) − yj
(2-7)
s0 ≥0,
167
Then we have:
m
[(siL + siR ) i=1
n
i=1
[(1 + ki )siL
|x |] j=1 ij
m
j=1
|xij |]
(2-13)
The minimum value of MSE shows minimum error of regression model and power of model.
225 226
4.2. Symmetric
227
The best regression model considering symmetric fuzzy coefficient, was obtained when h = 0.8 (Table 2) and it isdenoted as
Please cite this article in press as: Z.S. Zolfaghari, et al., Application of fuzzy linear regression method for sensory evaluation of fried donut, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.03.010
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G Model
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Z.S. Zolfaghari et al. / Applied Soft Computing xxx (2014) xxx–xxx Table 2 Symmetrical fuzzy parameters at different values of h. h
S0
S1
S2
S3
S4
S5
S6
Z
MSE
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.9191 1.0340 1.1818 1.3787 1.6545 2.0681 2.7575 4.1363 8.2727
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0.1464 0.1647 0.1883 0.2196 0.2636 0.3295 0.4393 0.6590 1.3181
0.0606 0.0681 0.0779 0.0909 0.1090 0.1363 0.1818 0.2727 0.5454
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
625.989 704.238 804.844 938.984 1126.782 1408.477 1877.97 2816.955 5633.909
0.9161 0.9255 0.911 0.8835 0.8297 0.7556 0.5243 0.3767 0.784
h
ac0
ac1
ac2
ac3
ac4
ac5
ac6
0.1–0.9
0.9
0
−0.0181
0.1227
0.0727
−0.0363
0.5272
follows (Eq. (3-3)):
230
Yˆ = [0.9, 4.1363] + [0, 0]x1 + [−0.0181, 0]x2 + [0.1227, 0.659]x3 + [0.0727, 1.3181]x4 + [−0.0363, 0]x5 + [0.5272, 0]x6
231
234 235
236
Y = 0.2963 − 0.0181x2 + 0.122x3 + 0.0778x4
237
(3-4)
4.3. Asymmetric
1. Constant values of k Table 3 shows the coefficients when k was assumed constant 2. Increasing trend of ki Table 4, asymmetrical fuzzy parameters considering increasing trend of ki 3. Decreasing trend of ki Table 5, asymmetrical fuzzy parameters considering decreasing trend of ki
(3-3)
Table 3 Asymmetrical fuzzy parameters considering constant value of k.
0.5 1.5 2 2.5
Ã0
Ã1
Ã2
Ã3
Ã4
240 241 242 243
244
In this section, coefficients were considered asymmetrical triangular fuzzy numbers. Finding the best regression model, different states of elongation coefficient (k) were assumed and according to values of MSE the best model was selected.
In this model x1 , x2 , x3 , x4 , x5 , x6 are aroma, surface color, porosity, hardness, oiliness and flavor, respectively. After defuzzification of Eq. (3-3), the best symmetric regression model was obtained
ki
238
239
Shayannejad [18] compared fuzzy regression method and Artificial Neural Networks (ANN) to the determination of potential evapotranspiration. Results of this study demonstrate fuzzy regression method with symmetric TFNs had better fitness.
232 233
(Eq. (3-4))
− 0.0363x5 + 0.5272x6
h: degree of belonging to ˜ Yj ; si : spread of TFN; ai : mode of TFN; Z: value of target function; MSE: mean squared error.
229
5
Ã5
Ã6
s01
ac0
s11
ac1
s21
ac2
s31
ac3
s41
ac4
s51
ac5
s61
ac6
5.51 3.30 2.75 2.36
1.17 0.73 0.62 0.54
0 0 0 0
0 0 0 0
0 0 0 0
−0.01 −0.01 −0.01 −0.01
0.87 0.52 0.43 0.37
0.16 0.09 0.07 0.06
0.36 0.21 0.18 0.15
0.09 0.06 0.05 0.04
0 0 0 0
−0.03 −0.03 −0.03 −0.03
0 0 0 0
0.52 0.52 0.52 0.52
Z
MSE
5633.909 5633.909 5633.909 5633.909
1.2631 0.8402 1.5175 2.2254
ki : constant elongation coefficient; Ãi : fuzzy coefficients of regression model; s1i : left spread of asymmetrical TFN; aci : mode of asymmetrical TFN; Z: value of target function; MSE: mean squared error.
Table 4 Asymmetrical fuzzy parameters considering increasing trend of ki . k0
k1
k2
k3
k4
k5
k6
s01
s11
s21
s31
s41
s51
s61
Z
MSE
0.5 0.8 0.3
0.8 1.1 0.6
1.1 1.4 0.9
1.4 1.7 1.2
1.7 2 1.5
2 2.3 1.8
2.3 2.6 2.1
5.51 4.59 6.36
0 0 0
0 0 0
0.54 0.48 0.59
0.20 0.18 0.21
0 0 0
0 0 0
5633.9 5633.9 5633.9
0.89 2.37 0.42
k0
k1
k2
k3
k4
k5
k6
ac0
ac1
ac2
ac3
ac4
ac5
ac6
0.5 0.8 0.3
0.8 1.1 0.6
1.1 1.4 0.9
1.4 1.7 1.2
1.7 2 1.5
2 2.3 1.8
2.3 2.6 2.1
1.17 0.99 1.34
0 0 0
−0.01 −0.01 −0.01
0.10 0.08 0.11
0.05 0.05 0.06
−0.036 −0.036 −0.036
0.52 0.52 0.52
ki : elongation coefficient; s1i : left spread of asymmetrical TFN; aci : mode of asymmetrical TFN; Z: value of target function; MSE: mean squared error. Table 5 Asymmetrical fuzzy parameters considering decreasing trend of ki . k0
k1
k2
k3
k4
k5
k6
s01
s11
s21
s31
s41
s51
s61
Z
MSE
2.1 2.3 2.6
1.8 2 2.3
1.5 1.7 2
1.2 1.4 1.7
0.9 1.1 1.4
0.6 0.8 1.1
0.3 0.5 0.8
2.67 2.51 2.30
0 0 0
0 0 0
0.6 0.55 0.49
0.29 0.26 0.23
0 0 0
0 0 0
5633.9 5633.9 5633.9
3.00 4.20 5.96
k0
k1
k2
k3
k4
k5
k6
ac0
ac1
ac2
ac3
ac4
ac5
ac6
2.1 2.3 2.6
1.8 2 2.3
1.5 1.7 2
1.2 1.4 1.7
0.9 1.1 1.4
0.6 0.8 1.1
0.3 0.5 0.8
0.61 0.57 0.52
0 0 0
−0.0181 −0.0181 −0.0181
0.11 0.10 0.09
0.08 0.07 0.06
−0.036 −0.036 −0.036
0.53 0.53 0.53
ki : elongation coefficient; s1i : left spread of asymmetrical TFN; aci : mode of asymmetrical TFN; Z: value of target function; MSE: mean squared error.
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Table 6 Ranking of donut samples.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Soy flour addition (%)
Raising agent
Pre-backing
Coating
Score
5% 10% 10% 10% 10% 5% 0% 0% 10% 5% 5% 10% 5% 0% 10% 0% 0% 0% 5% 0% 0% 5% 0% 10% 5% 5% 0% 10% 10% 10% 5% 5% 0% 10% 5% 10%
Cake donut Yeast-raised donut Yeast-raised donut Yeast-raised donut Cake donut Cake donut Yeast-raised donut Yeast-raised donut Cake donut Cake donut Yeast-raised donut Cake donut Yeast-raised donut Cake donut Cake donut Cake donut Yeast-raised donut Cake donut Cake donut Cake donut Yeast-raised donut Cake donut Cake donut Yeast-raised donut Yeast-raised donut Cake donut Yeast-raised donut Cake donut Cake donut Yeast-raised donut Yeast-raised donut Yeast-raised donut Cake donut Cake donut Yeast-raised donut Yeast-raised donut
Control Control Control Control Control Control Control Control Control Control Control Control Control Control Pre-backed Control Control Pre-backed Pre-backed Control Pre-backed Pre-backed Pre-backed Pre-backed Control Pre-backed Pre-backed Pre-backed Pre-backed Pre-backed Pre-backed Pre-backed Pre-backed Pre-backed Pre-backed Pre-backed
Uncoated GT coated MC coated Uncoated Uncoated GT coated MC coated GT coated GT coated MC coated MC coated MC coated GT coated MC coated MC coated Uncoated Uncoated GT coated Uncoated GT coated MC coated GT coated Uncoated GT coated Uncoated MC coated GT coated Uncoated GT coated MC coated GT coated Uncoated MC coated Uncoated Uncoated MC coated
1.5288 1.5380 1.5659 1.6200 1.6407 1.6526 1.6540 1.6583 1.6632 1.6697 1.6962 1.7249 1.7450 1.8129 1.8253 1.8389 1.8470 1.8614 1.8658 1.8757 1.8777 1.9120 1.9157 1.9176 1.9336 1.9564 1.9708 1.9728 2.0068 2.0205 2.0216 2.0792 2.0832 2.1125 2.1339 2.1343
GT coated: coated donuts with Gum tragacanth solution; MC coated: coated donuts with methyl cellulose solution; control: fried donut without pre-baking; pre-backed: fried donut with 2 min per-backing process.
257 258
According to Table 4 decreasing trend of ki , leads to increasing MSE and reducing the quality of the model.
259
K0 = 0.3,
k1 = 0.6,
k2 = 0.9,
k3 = 1.2,
k4 = 1.5,
k6 = 2.1
flavor. These results indicate that increasing the brownness of surface color decreased overall acceptance, while increasing oiliness of donuts, increased overall acceptance and flavor had the strongest effect on the acceptability of donuts.
282 283 284 285
260 261
Yˆ = [1.3454, 6.3636] + [0, 0]x1 + [−0.01818, 0]x2
262
+ [0.1107, 0.5991]x3 + [0.0618, 0.2118]x4
263
+ [−0.0363, 0]x5 + [0.5272, 0]x6
4.4. Ranking of donut samples according to fuzzy regression model (3-5)
264
266
After defuzzification of Eq. (3-5), the best regression model with asymmetrical coefficient was obtained as follow:
267
Y = −0.1393 − 0.0181x2 + 0.1515x3 + 0.0959x4
265
268
− 0.0363x5 + 0.5272x6
(3-6)
269 270 271 272 273 274 275 276 277 278 279 280 281
According to the MSEs of model (3-4) and (3-6), it could be found that asymmetrical coefficients did not improve regression model. On the contrary, another study on the reasons of prosperity of water-user cooperatives in Iran [19] examined symmetric and asymmetric TFNs and the best model was obtained when TFNs were asymmetric and ki had perpendicular and lateral increasing trend. Also Mohammadi nad Taheri [20] applied fuzzy regression to modeling Pedotransfer Functions. Asymmetric TFNs and constant ki showed the best fit in this study [20]. Consequently Eq. (3-4) shows the best fit of sensory evaluation data. As per the opinion of judges regarding the general quality attributes for donut sample, aroma is the least important one compared to crust color, oiliness and
For the ranking of samples in general, fuzzy linear regression model was used. Overall acceptance values obtained from Eq. (3-4). A comparison of values showed that the most desirable sample was cake donut with 5% soy flour without coating. Cake donut was produced by chemical leavening agent. On the other hand, pre-backed donut with 0% soy flour addition and leavened by yeast had lowest quality acceptance compared to other samples (Table 6). Ranking of samples showed pre-baking process have a negative effect on overall acceptance of donut samples and it is not suitable for the preparation of fried donut. 5. Conclusion In this study we proposed a fuzzy linear regression model for analyzing sensory evaluation data. This new promising approach of “Fuzzy Regression” is discussed which is capable of handling such a situation. In this model, not a clear relationship between overall acceptance and sensory characteristics was assumed, and then the best model was calculated considering crisp independent variables, dependent fuzzy variables and fuzzy coefficients. Results showed that symmetrical fuzzy parameter provided the best fitting
Please cite this article in press as: Z.S. Zolfaghari, et al., Application of fuzzy linear regression method for sensory evaluation of fried donut, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.03.010
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of sensory data. According to this model, surface color had a negative effect on overall acceptance. Overall acceptance increased with samples Oiliness. Flavor had the strongest effect and aroma had no effect on overall acceptance. This method was more efficient and appropriate than classic statistic methods in modeling sensory data. It could be generalized in sensory evaluation analysis and when the data cannot be recorded or collected precisely.
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Please cite this article in press as: Z.S. Zolfaghari, et al., Application of fuzzy linear regression method for sensory evaluation of fried donut, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.03.010
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