STRIP-PLOT ANALYSIS FOR THE CONSTRUCTION OF COMPLETE TRIPARTITE AND CUBIC GRAPHS Текст научной статьи по специальности «Строительство и архитектура»

STRIP-PLOT ANALYSIS FOR THE CONSTRUCTION OF COMPLETE TRIPARTITE AND CUBIC GRAPHS

V. Saranya

Research Scholar, Department of Statistics, Periyar University, Salem, Tamil Nadu, India

s ar anya88st at@gmail.com

S. Kavitha

Assistant Professor, Department of Statistics, Periyar University, Salem, Tamil Nadu, India

pustatkavitha@gmail.com

Abstract

The Strip-Plot Design (SPD) is plays an important role in the complete block designs and also using the agricultural, medical and industry fields. SPD is best suited for a two-factor experiment that has more treatments than can be accommodated by a complete block design. In a SPD, one factor is assigned to the horizontal strip plot, one factor is assigned to the vertical -strip plot and one factor is interaction plot. Also, few experimental materials may be rare while other test items may be available in altering doses of other therapeutic factors, which may be expensive or time-consuming. One of the main features of SPD involves three types of experimental errors: row - strip plot error, coloum - strip plot error and interaction plot error. Experimenting across processing steps is essential for studying the interaction of factors where certain factors come from one step and others arrive from the other. The strip-plot design is a very efficient design for investigating multiple-step processes in terms of both resources and time. Strip-plot designs are economical when the factors are hard to change and the process under research has three discrete stages. When we want to study interactions between factors where some factors are from one step and other factors from another step, it is important to conduct experiments across processing steps. The approach is flexible because it can handle experimental design problems involving factors acting at different levels, unlike the existing method. Graphs are widely used representations of both natural and human-made structures. Graph theory can be used to investigate "things that are connected to other things. "Fits nearly everywhere. Some tough problems become easier to solve when they are represented graphically. We reviewed the agricultural field yield of the strip-plot design and early work on the design of industrial strip-plot design in this paper. We have also described the model of strip-plot design. We, therefore, advise experimenters to ensure that their strip-plot designs contain a sufficient number of rows and columns so that valid statistical inference is possible. A bipartite graph is one in which the edges can be divided into two sets without going into sets. A complete bipartite graph is a bipartite graph that is completed. The complete tripartite graph in which the edges can be divided into three set without going into sets. The cubic graph is a graph in which all vertices have degree three. This paper describes the construction and Statis tical Analysis of SPD using some particular types of graphs is discussed through numerical examples.

Keywords: Strip -plot design, complete tripartite graph, cubic graph

1. Introduction

India is the third-largest producer of cotton in the world. Cotton grows well in drier parts of the, black soil, red soil and alive soil of the Deccan plateau. It requires high temperature, light rainfall or irrigation, 210 frost-free days and bright sunshine for its growth. It is a Kharif crop and requires 6 to 8 months to mature. The challenge is developing design organizations that meet quality and cost criteria. Every attempt at agricultural science research includes the design of experiments. Suppose to investigate more than one factor simultaneously in a single experiment, which is called the factorial experiment of the design.

Some factors to be tested need bigger plots, and others require smaller plots. Different plots are required in such cases, and the resulting design is known as split plot design (SPD). In 1925, Fisher developed this design for the purpose of agricultural experiments. The cost of the experiment can often be reduced by avoiding complete randomization.

The strip- plot design (SPD) is essential in complete block designs and applications in agriculture, medicine, and industry. One component is assigned to the horizontal strip plot, one to the vertical strip plot, and one to the interaction plot in an SPD.

Graph theory is one of the fastest-growing sciences. Graphs in their applications, are commonly used to represent distinct objects and the relationship between these objects. The visual representation of a graph is the declaration of an object vertex, while the relationship between objects is expressed as an edge. In recent years, graph theory has established itself as an important mathematical tool in various subjects, from available research and chemistry to genetics and linguistics and from electrical engineering and geography to sociology and architecture in its own right. At the same time is mathematical to discipline in its own right. Peter Horak et al. [1] have focused on this result is a special case of a general conjecture made by Erdos and NeSetiil: For each d > 3, the edge set of a graph of maximum degree d can always be partitioned into [5d2/4] subsets, each of which induces a matching. Raymond Greenlaw and Rossella Petreschi [2] have developed a new algorithm is presented for cubic graphs.

Arden Miller [3] has focused on using statistical experimental designs Strip-Plot Configurations of Fractional Factorials. George A. et al. [4] have discussed the strip-plot design for two-step processes. Elizabeth J. et al. [5] have reviewed recent developments and provided guidelines for using the Decomposition of complete tripartite graphs into gregarious 4-cycles. Heidi Arnouts et al. [6] have focused on the Strip-plot experiments, and the cost of experimentation can often be reduced by forgoing complete randomization. Antal Ivanyi et al. [7] have developed an exchange algorithm for tripartite graphs with given degree set. Abdollah Khodkar [8] has discussed the signed edge domination numbers of complete tripartite graphs. Sheikh Rashid et al. [9] has discussed the study of cubic graphs with its application and introduced certain concepts, including cubic graphs, internal cubic graphs, and external cubic graphs, and illustrate these concepts by examples. Velimor D. et al. [10] have presented the procedure for complete tripartite graphs with spanning maximal planar subgraphs.

Peter Bradshaw [11] has focused on vertex-disjoint triangles as a "tratching." The problem of finding a tratching that covers all vertices of a tripartite graph can be shown to be NP-complete using a reduction from the three-dimensional matching problem. K Nisa et al. [11] have discussed the Analysis of variance for strip plot design with missing values: bias correction of the mean squares. Hossein Rashmansloua et al. [13] discussed about cubic graphs with novel application and define the direct product. we introduce the notion of complete cubic graphs and present some properties of self-complementary cubic graphs. Peter Goos [14] has reviewed recent developments and provided guidelines for using the fish patty experiment: a strip-plot look. This paper discussed a statistical analysis of SPD using complete tripartite and cubic graphs with a numerical example.

2. Preliminaries

2.1 Strip - Plot Design

In strip plot design, each block is divided into several vertical and horizontal strips depending on the levels of the individual factors. Therefore, the Analysis of strip plot design is carried out in three parts. The first part is the vertical strip analysis, the second part is the horizontal strip analysis, and the third is the interaction analysis.

2.2 Complete Tripartite Graph

A complete tripartite graph is a set of vertices split into three disjoint sets such that no two graph vertices within the same set are adjacent and every vertex in one set is adjacent to every vertex in the other two sets. If the three sets contain p, q, and r graph vertices, a complete tripartite graph.

2.3 Cubic Graph

In the mathematical field of graph theory, a cubic graph is one in which all vertices have degree three. In other words, a cubic graph is a three-regular graph. Cubic graphs are also called trivalent graphs.

Yijk is observation corresponds to the kth level of factor (A), jth level of factor (A) and ith replication. |j the general mean effect.

Tj is ith block effect, A is the jth level of factor A, B is the kth level of factor B.

is the interaction between jth level factor A and kth level factor B, the error components.

and are independently and normally distributed with means zero and

respective variance , and .

In statistical analysis, separate estimates of error are obtained for the main effects of the factors A and B and their interaction A.B. Thus, three mean error squares will be applicable for testing the significance of the main results of the characteristics and their interaction separately.

The vertical strip plot for the first factor, the horizontal strip plot for the second factor, and the vertical and horizontal bars in the interaction strip plot for the interaction between two factors are always perpendicular to each other. The correlation plot is very small and primarily illustrates the interaction between the two design factors. As a result, we may say that correlation is assessed more precisely in strip plot design. This is an outline of the variance analysis table:

• Correction factor (C.F.) = —

• Total sum of square (SST) =

• Replication sum of square (SSR) = — - C

• Horizontal factor sum of square (S.S. (H.F.)) = — -C

• Horizontal factor error sum of square (SSEa) = -- -S -

3. Statistical Analysis of Strip - Plot Design

The linear model for strip-plot design is

Y =y +x+ ßj + (xß)ij+Yk +(xY)ik+(. +£ i=1,2...r, j=1,2...v, k=1,2...n

(1)

• Vertical factor sum of square (S.S. (V.F.)) = --C

• Vertical factor error sum of square (SSEb) = -- -SSR-

• Interaction effect sum of square = -- - SSA-

• Interaction error sum of square (SSEc) = SST- (All other sum of square)

Table 1: ANOVA table for strip - plot design

Sv Df Ss Mss F-Ratio

R. (R) (r-1) SSR

H.F. (A) (a-1) SSA

H.F.E. (a) (r-1)(a-1) SSEa -

V.F. (B) (b-1) SSB

V.F.E.(b) (r-1)(b-1) SSEb -

I.E. (AB) (a-1)(b-1) SSAB

I.E. (c) (r-1)(a-1)(b-1) SSEc -

4. Construction of Strip - Plot Design using Graphs

4.1. Method for Construction for Tripartite Graph

• Let us consider the horizontal strip, vertical strip, and intersection plots as vertex set Q. This vertex set P can be divided into three subsets: Q1, Q2, and Q3.

• Then the replication is considered as the first subset Q1, variety as the second subset Q2, and Soils as the third subset Q3.

• Now consider the first (replication) vertex (Ri) of the first subset, and then Ri is connected to all the vertices of the second and third subset through edges.

• Next, consider the second replication vertex (R2). It's connected to all the vertices of the second and third subsets through the edges.

• Similarly, all the remaining replication vertices of the first subset are connected to all the vertices of the second and third subsets through the corresponding edges.

• Finally, we get the complete tripartite graph for the vertical strip, horizontal strip, and intersection plots.

4.1.1 Application

In our study, to collect the yields of primary data on cotton cultivation varieties at Salem District of Tamilnadu. Three replicates of various cotton varieties (LRA(P.T.), Supriya, Surabhi) in kilograms and three Soil (Black, Red, and Alive). The four replications of Cotton cultivation in kilograms for

yields per plot, three varieties of crops are tested, the layout being Strip plot design data is given below.

Table 2: Replication wise data for yield of cotton (kg/ha)

Replication R1 R2 R3 R4

Variety Soil(S1)

V1 3328 3258 3400 3128

V2 3220 3150 3115 3015

V3 2850 2800 2700 2625

Soil(S2)

V1 2814 2750 2915 2963

V2 2656 2655 2500 2700

V3 2515 2514 2415 2400

Soil(S3)

V1 3050 3118 3250 3150

V2 2950 3000 3065 2950

V3 2650 2750 2950 2800

Table 3: Replication xvariety for horizondal factor

V1 V2 V3 Replication Total

R1 9192 8826 8015 26033

R2 9126 8805 8064 25995

R3 9565 8680 8003 26248

R4 9241 8665 7825 25731

Variety Total 37124 34976 31907 104007

The complete tripartite graph construction method for horizontal - strip plot is given below.

• From the above table 3 vertex is fixed as Q, which is divided into three subsets, the figure 1 shows that Qi (replication), Q2 (variety) and Q3 (soils).

• The figure 2 shows that first replication vertex (Ri) connected to all the vertices of variety (V1, V2 and V3) through the edge values 9192(Y1), 8826(Y2), and 8015(Y3).

Figure 1: Graph of subsets Figure 2: Graph for first replication (R1)

• The figure 3 shows that second replication vertex (R2), and it is connected to all the vertices of variety (V1, V2, and V3) through the edge values 9126(Y1), 8805(Y2), and 8064(Y3).

• Similarly, the figure 4 shows that third and fourth replication vertices (R3 and R4) are connected to all the vertices of variety (V1, V2 and V3) through the

corresponding edge values. (Yi, Y2 and Y3) 9565, 8680, and 8003 (Yi, Y2 and Y3)9241, 8665, and 7825.

Figure 3: Graph for second replication (R2) Figure 4: Graph for third and fourth replication (R3 and R4)

• The figure 5 shows that complete tripartite graph of variety and replication for the

horizontal - strip plot.

Figure 5: Graph for complete tripartite graph of horizondal - strip plot

Table 4: Replication x soils for vertical factor

S1 S2 S3 Replication Total

R1 9398 7885 8650 26033

R2 9208 7919 3868 25995

R3 9215 7830 9203 26248

R4 8768 8063 8900 25731

Soils Total 36589 31797 35621 104007

The construction method of the complete tripartite graph for vertical - strip plot is given below

• From the above table 4 that first replication vertex (Ri). The figure 6 shows that first replication vertex is connected to all soils (Si, S2 and S3) through the values 9398, 7985, and 8650(Yi, Y2 and Y3).

• The figure 7 shows that second replication vertex (Ri). The second replication vertex is connected to all Soils (Si, S2 and S3) through the values 9208, 79i9, and 8868 (Yi, Y2 and Y3).

Figure 6: Graph for first replication (R1) Figure 7: Graph for second replication (R2)

• Similarly, the figure 8 shows that third and fourth replication vertices (R3 and R4) are connected to all the vertices of soils (S1, S2 and S3) through the corresponding edge values. 9215, 7830 and 9203 (Y1, Y2 and Y3) 8768, 8063 and 8900 (Y1, Y2 and Y3).

• The figure 9 shows that complete tripartite graph for replication and soils vertical - strip plot.

Figure 8: Graph for third and fourth replication (R3 and R4) Figure 9: Complete tripartite graph of vertical

Strip - plot

Table 5: Variety x soils for interaction plot

S1 S2 S3 Variety Total

V1 13114 11442 12568 37124

V2 12500 10511 11965 34976

V3 10975 9844 11088 31907

Soils Total 36589 31797 35621 104007

The construction method of complete tripartite graph for interaction plot are given below

• The above table 5 that first variety vertex (V1). The first variety vertex is connected to all soils (S1, S2 and S3) through the values 13114, 11442 and12568 (Y1, Y2 and Y3).

Figure 10: Graph for first variety (Vi) Figure 11: Graph for second and third variety(ViandVs)

• Similarly, the figure ii shows that second and third verities vertexes (V2 and V3) are connected to all the Soils (Si, S2 and S3)., through the corresponding values i2500,i05ii and ii965(Yi, Y2 and Y3) 10975, 9844 and 11088 (Yi, Y2 and Y3).

• The figure i2 shows that complete tripartite graph for the variety and soil interaction plot.

Figure 12: Complete tripartite graph for interaction plot

• The figure i3 shows that complete tripartite graph for replication and variety,

replication and soils, and variety and soils.

Figure 13: Complete tripartite graph for horizondal, vertical and interaction strip plot

Compute the correction factor and sum of squares as

• Correction factor (C.F.) = 300484890.3

• Total sum of square (SST) = 2490006.7 Compute the sum of squares for the horizontal analysis:

• Replication sum of square (SSR) = i4996.256

• Horizontal factor sum of square (S.S. (H.F.)) = ii45826.5

• Horizontal factor error sum of square (SSEa) = 40929.8 Compute the sum of squares for the vertical analysis:

• Vertical factor sum of square (S.S. (V.F.)) = i070090.6

• Vertical factor error sum of square (SSEb) = ii8i9i.7 Compute the sum of squares for the interaction analysis:

• Interaction effect sum of square = 5970i.4

• Interaction error sum of square (SSEc) = 40.27i

Table 6: ANOVA for strip plot design

Sv D.f Ss Mss F-Ratio P-Value

Replication 3 14996.256 4998.752 1.36467 0.26720020

Variety(A) 2 1145826.5 5722913.25 83.9847 -

Error(Ea) 6 40929.8 6821.633 - -

Soils(B) 2 1070090.6 535040.33 27.161314 0.00100000

Error(Eb) 6 118191.7 19698.617 - -

Interaction(AxB) 4 59701.4 14925.35 4.44747 0.01958176

Error (Ec) 12 40271 3355.916 - -

Total 35 - - - -

The table value of replication and variety is greater than the calculated values. So the null hypothesis is accepted. There is no significant difference between the four replications and the three varieties. The table value of soils is greater than the calculated value. So the null hypothesis is accepted. There is no significant difference between the three soil levels. The table value of the interaction effect is also more important than the calculated value. So the null hypothesis is accepted.

There is no significant difference between the interaction effects. The P-value of the above experiment is more significant than the 5% significance level. Therefore the null hypothesis is accepted. There is no significant difference that occurred in the above experiment.

4.2 Method for Construction of Cubic Graph

• Let us consider the horizontal-strip plot, vertical-strip plot, and interaction plot factors as vertex set Q. Then the elements are divided into two subsets, Qi and Q2.

• Then the replication is considered the first subset Qi and variety as the second subset Q2.

• Now consider the first (replication) vertex Ri of the first subset and then Ri is connected to all the vertices of the second subset through edges.

• Next, consider the second replication vertex R2 it is connected to all the vertices of the second subset through the edges.

• Similarly, all the remaining replication vertices of the first subset are connected to all the vertices of the second subset through the corresponding edges.

• Finally, we get the cubic graph for horizontal, vertical, and interaction plots.

4.2.1 Application

In our study, to collect the kilometers of primary data on petrol two-wheeler brands at Salem District of Tamilnadu. Three replicates of various two-wheeler brands (Honda, Tvs, Suzuki), in kilometers and three route way of (Hillstration, City, Highways). The four replications of petrol in kilometers per litter, three brands of kilometres are tested, and the layout being Strip plot design data is given below.

Table 7: Day wise for kilometres of petrol

Days D1 D2 D3

Brand Route(R1)

B1 30 31 31

B2 35 34 34

B3 33 32 33

Route(R2)

B1 35 36 37

B2 42 40 41

B3 37 38 39

Route(R3)

B1 50 51 50

B2 57 55 56

B3 54 53 54

Table 8: Days x brand for horizondal factor

B1 B2 B3 Days Total

D1 115 134 124 373

D2 118 129 123 370

D3 118 131 126 375

Brand Total 351 394 373 1118

The construction method of cubic graph for horizontal-strip plot is given below.

• From the above table 8 vertex is fixed as Q, which is divided into two subsets, the figure i4 shows that Qi (days) and Q2 (brand).

• The figure i5 shows that first day vertex (Di). The first days vertex is connected to all brand (Bi, B2 and B3) through the values ii5(Yi), i34(Y2) i24(Y3).

Figure 14: Graph of subsets Figure 15: Graph of first day (Di)

Similarly, the figure i6 shows that second and third day vertex (D2 and D3). The second and third days vertex is connected to all brand (Bi, B2 and B3) through the values ii8, i29, and i23(Yi, Y2 and Y3), ii8, i3i and i26 (Yi, Y2 and Y3). The figure i7 shows that cubic graph for days and brand.

0

&

Table 9: Days x route for vertical factor

R1 R2 R3 Days Total

D1 98 114 161 373

D2 97 114 159 370

D3 98 117 160 375

Route Total 293 345 480 1118

The construction method of the cubic graph vertical -strip plot is given below.

• From the above table 9 vertex is fixed as Q, which is divided into two subsets, the figure 18 shows that Qi (days) and Q2 (route).

• The figure i9 shows that first day vertex (Di). The first days vertex is connected to all Route (Ri, R2 and R3) through the values 98(Yi), ii4(Y2), i6i(Ya).

Figure 18: Cubic graph for subset Figure 19: Cubic graph for first day (Di)

• Similarly, the figure 20 shows that second and third day vertex (D2 and D3). The second and third day vertex is connected to all routes (Ri, R2 and R3), through the values 97, ii4 and i59(Yi, Y2 and Y3), 98, ii7 and i60 (Yi, Y2, and Y3).

• The figure 2i shows that cubic graph for days and route.

Figure 20: Cubic graph for second and third days (D2 and D3) Figure 21: Cubic graph for vertical - strip plot

Table 10: Brand x route for Interaction factor

R1 R2 R3 Brand Total

B1 92 108 151 351

B2 103 123 168 394

B3 98 114 161 373

Route Total 293 345 480 1118

The construction method of the cubic graph for the Interaction strip plot is given below.

• From the above table i0 vertex is fixed as Q, which is divided into two subsets, the figure 22 shows that Qi (brand) and Q2 (route).

• The figure 23 shows that first vertex (Ri). The first route vertex is connected to all brand (Bi, B2 and Bs) through the values 92(Yi), i08(Y2), i5i(Y3).

Figure 22: Cubic graph for subset

Figure 23: Cubic graph for first brand (Bi)

Similarly, the figure 24 shows that second and third route vertex (R2 and R3). The second and third route vertex is connected to all brand (B1, B2, and Bs), through the values i03, i23, and i68 (Yi, Y2 and Y3), 98, ii4, and i6i (Yi, Y2 and Y3). The figure 25 shows that cubic graph for route and brand.

Figure 24: Cubic graph for second and third brand (Biand B3)

Compute Compute

Compute Compute

Figure 25: Cubic graph for interaction Strip - plot

the correction factor and sum of squares as Correction factor (C.F.) = 46293.48i48 Total sum of square (SST) = 2i88.5i852 the sum of squares for the horizontal analysis: Replication sum of square (SSD) = i.4074 Horizontal factor sum of square (S.S. (H.F.)) = i02.7407 Horizontal factor error sum of square (SSEa) = 6.3704 the sum of squares for the vertical analysis:

Vertical factor sum of square (S.S. (V.F.)) = 2070.2963 Vertical factor error sum of square (SSEb) =i.43096 the sum of squares for the interaction analysis: Interaction effect sum of square = 4.i477 Interaction error sum of square (SSEc) = 2.37456

0

0

Table 11: ANOVA for strip plot design

Sv D.f Ss Mss F-Ratio P-Value

Replication 2 1.4074 0.7037 2.263180 0.12596233

Brand(A) 2 102.7407 51.3704 32.25563 -

Error(Ea) 4 6.3704 1.5926 - -

Route(B) 2 2070.2963 1035.14815 2795.884156 0.00000000

Error(Eb) 4 1.48096 0.37024 - -

Inter action(AxB) 4 1.48096 1.036925 3.4934472 0.04526749

Error (Ec) 8 4.1477 0.29682 - -

Total 26 2.37456 - - -

The table values of replication and brand method are more significant than the calculated values. So the null hypothesis is accepted. There is no significant difference between the three replications and the three-route method. The table value of the route method is greater than the calculated value. So the null hypothesis is accepted. There is no significant difference between the three route methods. The table value of the interaction effect is also more effective than the calculated value. So the null hypothesis is accepted. There is no significant difference between the interaction effects.

The P-value of the above experiment is more significant than the 5% significance level. Therefore the null hypothesis is accepted. There is no significant difference that occurred in the above experiment.

5. Conclusion

Many real-world experiments deviate from textbook examples and sometimes involve multiple types of structures. Running agricultural and industrial tests in strip plot analysis is an effective method to reduce costs. The strip-plot design is the most efficient design in terms of both the resources required and the time required to study multi-step processes. This paper describes the construction and analysis of strip-plot analysis using some particular type of graphs through numerical examples from different fields, the hypothesis testing is compared by the strip-plot ANOVA method with the software using the method. When comparing the results of these methods, they produce the same results. Here some particular type of graphs is used to construct the SPD. In the future, there is an idea to expand this procedure to other experimental designs, such as Split-Split Plot Designs, Incomplete Block Designs etc.

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List of Abbreviation

Sv - Sources of variance

D.f - Degrees of freedom

Ss - Sum of squares

Mss - Mean sum of squares

R. (R) - Replication (R)

H.F. (A) - Horizontal Factor(A)

H.F.E. (a) - Horizontal Factor Error (a) V.F. (B) - Vertical Factor(B)

V.F.E. (b) - Vertical Factor Error (b)

I.E. (AB) - Interaction Effect (AB) I.E.(c) - Interaction Error(c).