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RANDOMIZED BLOCK DESIGN IN FUZZY ENVIRONMENTS
A. Kirthik VairaMariappan1 and Manigandan Palanisamy2
1Department of Statistics, Government Arts College, Dharmapuri, Tamilnadu, India
maryaaindia@gmail.com 2Department of Statistics, Periyar University, Salem, Tamilnadu, India srmanigandan95@gmail.com
Abstract
On the basis of the statistics, ANOVA also provides a method of data analysis that is motivated by consideration of the experimental design or Design of Experiment (DOE). Experimental design plays an essential part in statistical analysis and data interpretation. One factor of criteria forms the basis of a one-way classification. Two factors or two criteria form the basis of two-way classification. Innovations and creations require experimentation as their foundation. Replication, randomization, and local control are the three fundamental tenets of experimental designs, which are used to determine the cause and effect of interactions. The error of any treatment can be isolated and any number of treatments may be omitted from the analysis without complicating it. The data provided in this study are vague and need an extended version of the RBD to investigate these vague observations. The simplest of all designs based on the principles of randomization and replication are Completely Randomized Designs (CRD). When the experimental materials aren't uniform in some circumstances. Divide the experimental region into smaller, homogeneous blocks in RBD. The treatment is applied at random to each block, and each block is reproduced. Since uncertainty is a common feature of all real-world issues and denotes fuzziness and unpredictability, Randomized Block Design has long been widely used in the agricultural and industrial sectors. It is therefore impossible to avoid using statistical RBD analysis with fuzzy observations. The objective of this study was to develop the problem of a Randomized Block Design (RBD) test for Triangular Fuzzy Numbers (TFN) is discussed in this paper. However, in a scenario that is actual, the underlying relationship is not a clear-cut function of a particular form; it has some ambiguity or imprecision. The estimated numbers are very similar to the actual ones. This approach may generally be used for any real-time triangle fuzzy number calculation. In this proposed methodology, it is obvious that if the value of the observed fuzzy test statistics is similar to real numbers in the testing crisp hypotheses, then fuzzy RBD is very sensitive for making the determinations as to whether to accept or reject the fuzzy null hypotheses and also debates the application of the method for example.
Keywords: RBD, Fuzzy RBD, TFN, Decision Rule
1. Introduction
The Completely Randomized Design (CRD) was simple because the principle of local control was not used, and experimental material was assumed to be homogeneous, but it is noted that the experimental material is not absolutely homogeneous. A fertility gradient in one direction is often present in agricultural field experiments. The simple method of regulating the variability of the
A. Kirthik VairaMariappan and Manigandan Palanisamy RT&A, No 3 (74) RANDOMIZED BLOCK DESIGN IN FUZZY ENVIRONMENTS_Volume 18, September 2023
experimental material in such a situation consists of stratifying or grouping the entire experimental
area into relatively homogeneous strata or subgroups (called blocks) perpendicular to the fertility
gradient direction. These blocks are so designed that plots are homogeneous within a block, and
heterogeneous between blocks. In other words, inside a block there might be less variation, and the
main difference or variation between blocks. It should be held in mind that for an effective blocking
of the content, familiarity with the design of experimental units is important. The method of dividing
experimental material into a number of blocks gives rise to a design known as RBD that can be
described as an arrangement of t treatments in r blocks such that each treatment takes place exactly
once in each block. Fuzzy set theory [23] was extended to several areas that need to handle
ambiguous and unclear data. These areas include estimated logic, decision-making, optimization,
power, etc.
The sample findings are crisp in conventional statistical research, and a statistical test leads to the binary decision. Many authors have studied the statistical theories that are evaluated in fuzzy environments using the fuzzy set theory principles introduced by Zadeh [24]. Chachi et al. [4] are proposing a new approach to the issue of evaluating statistical hypotheses. As a fuzzy subset of the real line, Dubois and Prade [6] identified some of the fuzzy numbers. Mikihiko Konishi et al. [16] suggested an Analysis of Variance (ANOVA) for the fuzzy interval data using the definition of the fuzzy set. Wu [21, 22] introduced hypothesis testing of a single factor ANOVA model for fuzzy data by solving optimization problems using the h-level and the notions of pessimistic degree and optimistic degree. The two-factor ANOVA test were analysed by Gajivaradhan and Parthiban [8] using an alpha cut interval method for trapezoidal fuzzy numbers. A bootstrap approach to the multi-sample test of means with imprecise data was suggested by Gil et al. [10]. When both the theories and the available data are fuzzy, Arefi and Taheri [2] formed the testing hypothesis. Filzmoser and Viertl [7], proposed to test hypotheses on the fuzzy p value with fuzzy data. Nakama et al. [15] Discuss derive statistical tests that are ideal for testing the null hypotheses, and develop a bootstrap scheme to estimate the p values of the test statistics observed. Ahmed et, al. [1], proposed, a new dimension of the methodology involving a fuzzy regression approach to RBD introduced, which is involving qualitative predictor variables under consideration on multiple linear regression. The idea from this research will be a useful thread for establishing comprehensive connectivity between RBD and regression. The researchers concluded that fuzzy MLR can predict much better compared to MLR itself. Mariappan and Pachamuthu [14] suggested the statistical testing of hypotheses for fuzzy CRD using TFN. Magno do N et, al. [13], developed a fuzzy model that could predict weight loss as a function of the rapid cooling of table grapes in different plastic film bags. Modelling was performed using three types of plastic film bags (micro-perforated, macro-perforated, and non-perforated) at three levels of palletization (lower, intermediate, and upper), arranged in an experimental design in randomized blocks, in a 3 * 3 factorial scheme, with three blocks. The influence of the relative humidity and amplitude of humidity on the variable weight loss percentage of the Arra 15 grape variety was measured. The average percentage error of the fuzzy model was 9.78%. The intermediate level alone showed an error of 4.02%. Thus, the developed fuzzy model provided a good prediction of the weight loss of table grapes. The classical RBD model for TFN is analyzed in this paper using a numerical example.
2. Preliminaries 2.1 Triangular Fuzzy Number
A triangular fuzzy number A is a fuzzy number fully specified by tribles (a,b,c) such that a < b < c with membership function defined as
Ju-Xx) =
0 if x < a
x — a
b — a
x — с
if a < x < b ifb < x < c
b — с 0 if x > c
where a is the indicates of lower point, b is the indicates of centre point and c is the indicates of upper point.
The triangular fuzzy number is represented diagramatically as
Figure 1: Triangular Fuzzy Numbers The form of a fuzzy interval number can be expressed as a triangular fuzzy number follows:
A =
;0 < h, r < 1
{(b-a)r + a} ;{(b-c)h + c}
where r is the level of pessimistic and h is the level of optimistic of the fuzzy numbers
A = (a,b,c).
3. Statistical Analysis of RBD
Through proving the local control (blocking) measure in the design, an increase in CRD can be obtained. One such design is entirely RBD. ANOVA technique for two-way data classification is applicable to the RBD layout experiment. The data obtained from the experiment is graded by two factors namely treatments and blocks according to different levels. For RBD, the linear model is described by
y = n + a + bj + e jj; i = 1,2,..., t;j = 1,2,..., b where, y is the observations corresponding to the ith treatment and jth block, /j is the
general mean effect which is fixed, a is the fixed effect due to the ith treatment, b is the fixed effect
th
due to the j block and e is the random error effect. To determine whether the factor level means
J ij
/ equal or not. The following testing hypotheses are known as
H0 : /=/=/=/ against H1 : not all / are equal
tb , tb
■th
Let ^Tj = y = G be the grand total of tb observations, ^Ty = y. = Tt be the i
'J V
tb ■ th 2 treatment total, "ST y = v = b is the J block total and also cf = G_ . Then, the various sum of
Z^-Tv y.J J J kr
squares, mean sum of squares and F Ratio listed given below:
A. Kirthik VairaMariappan and Manigandan Palanisamy RT&A, No 3 (74) RANDOMIZED BLOCK DESIGN IN FUZZY ENVIRONMENTS_Volume 18, September 2023
SST = QSST =Xy2 - G2 which has (tb-1) df, SSBTR = QSSBr = - G which has
ij kr ij r kr
tb
(t -1) df, SSB = QSSB — - — which has (b -1) df, SSE = QSSE = QSST - QSSBTR - QSSBB
ij k kr
SSE
which has (t - 1)(b -1) df , MSBTR = SSBTR, MSBB = SSBB , MSE = ■
(t -1) MSBB (b -1) (t- 1)(b-1)
= MSBTR and F = MSBB .
BTR MSE BB MSE
In the ANOVA table, all these values are referred to and inferences are drawn.
Table 1: ANOVA Table for Crisp RBD
SV
df
SS
MSS
Total
(tb -1)
Qs.
F Ratio
Between Treatments (t -1) qssbtr MSBTR f f btr
Between Blocks (b -1) qssbb MSBB f 1 bb
Experimental Error (t-1)(b-1) qsse MSE -
3.1 Decision Rule of Between Treatments and Between Blocks
The decision rules of F test to accept or reject between treatments and between blocks at <2% significance level the null hypothesis and alternative hypothesis. Suppose that if F < F, [where
F is the tabulated value for (t -1), (t - 1)(b -1) and (b—1), (t - 1)(b -1) degrees of freedom, and
F is the calculated value], then the null hypothesis is rejected. Otherwise, alternative
hypothesis H0 is rejected.
3.2 Fuzzy Analysis of RBD
The triangular fuzzy approach to the fuzzy statistical analysis of RBD. Throughout this case, the data recorded as well as the observations are regarded as TFN. Below is the mathematical general linear model:
yij=fi + ai+bj+ey\ i = \,2,...,t,j = \,2,...,b
In the fuzzy interval RBD models, the general linear model of classical RBD is classified; the fuzzy lower and upper level models are regarded as: y!. = (jli)] + (cl )'r + (bj)' + (¿^)' and
yfj =<J2)h +(«,)* +<pj% inwhich (Xj)f and iyvfh is the observation corresponding
to the ith level of factor A and j'h level of factor B . (/~ty'r and (ju)" is the general mean effect
which is fixed. and (atis the fixed effect due to the ith level of factor A. (p~)Lr anc' (^)a
is the fixed effect due to the j'h level of factor B . (¿'^)' and (£i;)', is the random error effect
which is independent identically distributed (iid) with mean is 0 and constant variance is <j2;
i = 1,2,___, t and j = 1,2.,...,b . After this, to test the lower and upper level model and the fuzzy
null hypotheses and fuzzy alternative hypotheses respectively, utilizing classical RBD methods. The simplest the fuzzy null hypotheses HQ : ju1= ju2=... = jur against the fuzzy alternative hypotheses This implies the following two sets (Lower and Upper levels)
of hypotheses are given below.
3.3 Fuzzy Hypotheses of Lower and Upper Level Models
The fuzzy null hypotheses of between treatment and between block is fl'. \ J^ = ¿¿f =... = against the fuzzy alternative hypotheses between treatment and between block is Hf : * ¿2 * - * Mr ■
The fuzzy null hypotheses of between treatment and between block is Hq : jii[ = jli2 =... = /"/ against the fuzzy alternative hypotheses of between treatment and between block is fl\ : /7,' ^ /j'2 ^ ... ^ /7' •
In TFN pessimistic and optimistic for the fuzzy lower and upper level models from the null hypothesis of acceptance or rejection direction levels. Through the use of triangular fuzzy lower and upper levels formulas are (b — atJ)r + dy where 0 < i < t;0 < j < b and
(( — cij )h + where 0 < i < t; 0 < j < b . (Note that rL = 1 and hU = 1, centre level). Then the required formula for lower level of fuzzy RBD is given below:
ssbtr-
ssbbL = =
MSBTRL =
[fön [Œ2>n
i. 'h t
j=i ssbtr
tb
, SSEt = SS'f ' - SSBTRr - SSBBt
ssel
, MSBBl = SSBB and MSEL = ■ (t -1) r (b -1) r (t - i)(b -
~ MSBTR and (p ^ MSBB
0 Bm)r - MSB K BB)r MSE
In the lower level of ANOVA table for fuzzy RBD table, all these values are represented and fuzzy decision rule is drawn.
Table 2: ANOVA Table for Lower Level of Fuzzy RBD
sy df ss mss f — Ratio
Between Treatments (t -1) ssbtrL MSBTRL BTR ) r
Between Blocks (b -1) ssbbL msbbL (Fbb ))
Experimental Error (t-1)(p-1) sseL mseL -
Total
(tb -1)
sstl
3.2 Fuzzy Decision Rule of (Lower and Upper levels) Between Treatments and Between Blocks
Suppose that if FT < Fc, [where FT is the tabulated value for (t — 1), (t — 1)(b — 1) and (b — 1), (t — 1)(b — 1) degrees of freedom, and Fc is the calculated value (using 3.1)], then the fuzzy null hypotheses of lower level for /7' and fuzzy null hypotheses of upper level for //' is rejected for 0 < rL < rT where 0 < rT < 1 and 0 < hU < hj, where 0 < \ < 1. Otherwise, fuzzy alternative hypotheses of lower level for //' and fuzzy alternative hypotheses of upper level for //,' is rejected for 0 < hU < hj. where 0 < h ^ 1.
The proposed classical technique for evaluating RBD model fuzzy hypotheses with fuzzy
A. Kirthik VairaMariappan and Manigandan Palanisamy RT&A, No 3 (74) RANDOMIZED BLOCK DESIGN IN FUZZY ENVIRONMENTS_Volume 18, September 2023
data is illustrated with an example below.
e = mc2 (1)
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4. Applications
In our study, to collect the yields of primary data groundnut varieties at Omalur, Salem District of Tamil Nadu. Three replicates of various groundnut varieties (TMV 2, TMV 7, VRI 2) in kilograms and four yields of (Y1, Y2, Y3, Y4). Via an RBD, with four replications of groundnut in kilograms for yields per plot, three varieties of crops are tested, the layout being TFN due to certain work friction is given as below.
Table 3: Table for Classical RBD using TFN
Varieties of Yields in kilograms
Groundnut Y1 Y2 Y3 Y4
TMV 2 56,58,60 54,58,62 53,56,59 54,58,62
TMV 7 58,60,62 53,58,63 56,59,62 57,60,63
VRI 2 59,62,65 58,60,62 57,60,63 57,59,61
To test if there is any substantial difference between the production of the groundnut varieties in the yields in kilograms per plot. Let //, be the mean number of yields in kilograms per plots for the ith varieties of groundnut. Now, the null hypothesis, h : /7, = /72 = /7, = /74 and the alternative hypothesis, h : not all jû. 's are equal.
H : To test whether groundnut varieties do not vary significantly with respect to yields. Hl : To test if groundnut varieties vary significantly with respect to yields. Let us consider the lower-level model is given below
4.1. Lower Level Model
Table 4: Table for Upper Level Model
Varieties of Yields in kilograms
Groundnut Y1 Y2 Y3 Y4
TMV 2 2r + 56 4r + 54 3r + 53 4r + 54
TMV 7 2r + 58 5r + 53 3r+56 3r + 57
VRI 2 3r + 59 2r + 58 3r + 57 2r + 57
SSTL = 10r2 - 30r + 46, SSBTR = 1.5r2 -10.5r + 24.5 SSBBL = 2.67r2 -10.67r + 12.67, SSE^ = 5.83r2 - 8.83r + 8.83
A. Kirthik VairaMariappan and Manigandan Palanisamy RT&A, No 3 (74) RANDOMIZED BLOCK DESIGN IN FUZZY ENVIRONMENTS_Volume 18, September 2023
MSBTRL = 0.75r2 -5.25r +12.25,MSBBL = 0.89r2 -3.56r + 4.22, MSEL = 0.97r2 — 1.47r + 1.47 ~ L _ 0.75r2 -5.25r+ 12.25 ~ L _ 0.89r2-3.56r + 4.22 C^W, - o.97r2 — 1.47r +1.47 ' C ^" 0.97r2-1.47r+ 1.47
4.2. Fuzzy Decision Rule of Between Treatments
If F'- >Ft, for all r;0<r <1 where FT -5.14 is the F table value of a at 5% level of significance with (2,6) df then, the fuzzy null hypotheses //,' is rejected for the r; 0 < r < 1. Thus, the disparity between the treatments is substantial. Therefore, groundnut varieties vary greatly in yields.
4.3. Fuzzy Decision Rule of Between Blocks
If F'- <Ft, for all r;0<r <1 where FT =4.16 is the F table value of a at 5% level of significance with (3,6) df then, the fuzzy null hypotheses //,' is accepted for the r; 0 < r < 1. Furthermore, the difference between treatments is not significant. Therefore, the groundnut varieties are not substantially different in terms of yields.
Let us consider the upper level model is given below
4.4. Upper Level Model
Table 4: Table for Upper Level Model
Varieties of Yields in kilograms
Groundnut Y1 Y2 Y3 Y4
TMV 2 -2h + 60 -Ah + 62 - 3h + 59 -Ah + 62
TMV 7 -2h + 62 - 5h + 63 - 3h + 62 - 3h + 63
VRI 2 - 3h + 65 —2h + 62 - 3h + 63 -2h + 61
Likewise, upper level models of ambiguous RBD use formula and table, thus avoiding calculation.
4.5. Fuzzy Decision Rule of Between Treatments
If F > P! , for all h:0<h <1 where /% =5.14 is the F table value of a at 5% level of
t h ~ j-
significance with (2,6) df then, the fuzzy null hypotheses n'u is accepted for the /?;()</?<!.
Consequently, the disparity between treatments is not significant. Therefore, with regard to yields in kilograms, groundnut varieties do not vary significantly.
4.6. Fuzzy Decision Rule of Between Blocks
If FT > Fjf, for all h;0 < h <1 where FT =8.14 is the Ftable value of a at 5% level of
significance with (6,3) df then, the null hypothesis of the n'u is accepted for the /?;()</?<!.
Therefore, the discrepancy between treatments is not significant. Consequently, with regard to yields in kilograms, groundnut varieties do not vary significantly.
A. Kirthik VairaMariappan and Manigandan Palanisamy RT&A, No 3 (74)
RANDOMIZED BLOCK DESIGN IN FUZZY ENVIRONMENTS_Volume 18, September 2023
Therefore, so because fuzzy null hypotheses between treatments //('j and f['u of the lower level data is rejected and upper level data is accepted for all r; 0 < r < 1 and h; 0 < h < 1 (note that null hypotheses of accepted or rejected at r = 1 and h — 1, that is the centre level), the between blocks of fuzzy null hypothesis Hq and //,' of the lower and upper level data is accepted for all r; 0 < r < 1 and h; 0 < h < 1 the fuzzy null hypotheses HQ of the fuzzy RBD model is accepted and rejected for all r; 0 < r < 1 and h; 0 < h < 1. Thus, we conclude that four yields of groundnut kilograms are equal if r; 0 < r < 1 and h; 0 < h < 1.
5. Conclusion
A statistical test of the hypothesis for RBD model using TFN for fuzzy data is suggested in this study. Can make a decision on the fuzzy RBD model hypothesis based on the hypothesis determinations of two crisp RBD models. Since our fuzzy test is rather than standard significance tests, it appears to be a useful method in circumstances with imprecise data and also extend the crisp RBD to LSD, BIBD, PBIBD for fuzzy environments.
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