AN OPTIMUM RESOURCE SCORE ESTIMATION METHOD USING BIPARTITE GRAPH MODEL AND SINGLE NODE SYSTEMATIC SAMPLING Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

AN OPTIMUM RESOURCE SCORE ESTIMATION METHOD USING BIPARTITE GRAPH MODEL AND SINGLE NODE SYSTEMATIC SAMPLING

DEEPIKA RAJORIYA and D. SHUKLA

Dr. Harisingh Gour University, Sagar (M.P.) 470003 [email protected], [email protected]

Abstract

Consider a graphical population of vertices (nodes) and edges, where edges are connected with vertices to form a Bipartite graph. A complete Bipartite graph has vertices that can be partitioned into two subsets such that no edge has both endpoints in the same subset, and every possible edge connected to vertices in different subsets is a part of graph. In real life, there may hundreds of cities where at least one possible way exists reaching source to destinations. Several tourist places and small towns are the examples where the road transportation is available between origin and destination and these roads constitute Bipartite graph when they are like edges. The travel needs resource consumption who could be measured through resource-score. Walking at the hill station needs more energy consumption than at the plane area. This paper suggests an example to estimate the resource consumption by the values of score. Further, paper proposes a sample based methodology for calculating the average resource consumption between a pair of small town (city) and tourist place. Bipartite graph is used as a model tool. A single-node systematic sampling procedure is proposed under the Bipartite graph setup which is found useful for solution. The suggested estimation strategy is optimum at specific choice of parametric values. For quick selection, ready-reckoner tables are prepared who provide immediate optimum choice of constant. Results are numerically supported by the empirical study and proved by the calculation of confidence intervals.

Keywords: Graph; Bipartite Graph; Estimator; Bias; Mean Squared Error (MSE); Optimum Choice, Confidence intervals (C.I.).

1. Introduction

Traveling from one place to another, for pleasure and entertainment, is an integral part of human life. It is a diversified international phenomenon essential for recreation, interaction, exchange of thoughts and participation in events. Travel agencies offer world tour plans and holiday packages with maximum facilities at minimum cost. Private and public sector organizations, around the world, offer special leave to their employees for tour and travel event including financial support. For economists and scientists challenges are to get optimum estimates of traveling cost using appropriate models between any pair of tourist places. It is rather more difficult to pick up significant cost variables to incorporate in a cost model. Travel by road, by train or by flight need different types of resource consumption like time, fuel, energy, money etc. In a contribution of Moons[2], some equations for computing travel cost are suggested. More explicitly, travel cost methods are for estimation of economic use values associated with ecosystem or site who are frequently used for recreation. Such are important for valuation of economic benefits, useful in planning and decision making (see Ecosystem Valuation [3]) like :

(a) change in access cost of a recreational location,

(b) elimination of an existing tourist site from list,

(c) addition of new sites in tour plan lists.

The logical basis of a travel cost methods is cost and time, incurred to visit a site. This is analogous to estimating desire of people to pay for money to purchase goods based on quantities and prices (see Ecosystem Valuation[3]). Computing methodologies for travel expense, as suggested in [2] are 1 to 5 listed below:

1. Travel Time

2. Fuel Cost =

t

8 distance(i) i=1 averagespeed(i)

fuel cost(EURO/car per km)*distance(km) number of passengers

0 T , , , total car usage cost(EURf/car pe rkm)*distance(km)

3. ToM cur usage costs = -g „umber of pJenger -—

4. Total calculated costs = fuel cost + travel cost * 3

5. Total calulated cost = Total car usage cost + Trave time * 3.

While travel by road, distinction may be among four wheelers, for example small car, gas car, petrol car and diesel car. Table 1.1 presents the calculated costs derived from [2].

Table 1.1 Car Type and Cost (see [2])

S. No. Car Type Small gasoline Big gasoline Small diesel Big diesel

1. Cylinder capacity(in cm3) < 1600 > 1600 < 2000 > 2000

2. Fuel consumption(liter/km) 0.0716 0.0771 0.0534 0.0758

3. Price of fuel(EURO/liter) 0.89 0.89 0.61 0.61

4. Fuel cost(EURO/liter) 0.064 0.068 0.033 0.046

5. Total car-usage cost(EURO/liter) 0.33 0.52 0.25 0.38

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2. Mathematical Setup for Travel Cost

Let A, B, C, D are four locations at distances $1 and $2 scattered apart geographically. While reaching from A to B, the resource consumption is $1 whereas from C to D it is $2. Figure 2.1 reveals graphical relationship to travel from origin A and C to destinations B and D with distances and resource consumptions.

0'

o7

Fig 2.1 Resource consumption and distance for travel

Average resource consumption is <t = < + 2 which is an unknown quantity but useful for those who work as travel advisors. Government employees, in many countries, are entitled for availing the Leave Travel Concession (LTC) repeatedly in a block period of few years. They need to prepare and place an estimate of the likely expenditure to the government department for prior sanction of the advance money before the start of journey. This requires calculation of resource consumption in different segments of likely expenditure relating to travel plan. Table 2.1 presents an example of calculation of resource scores.

Table 2.1 Resource Score Traveling Plan from A to B and C to D

Segments A to B Resource Score(0-25) C to D Resource Score(0-25)

Fare(Bus/Train/Air) X11 X12

Food X21 X22

Boarding/Loading X31 X32

Local Convenience X41 X42

Energy Consumption X51 X52

Time Consumption X61 X62

Class-intervals for resource scores are as per perception like very low (0-5), low(5-10), medium(10-15), high(15-20), very high(20-25). Energy and time consumption in walking at hill station are higher than at cities located in plain area. Let Xj be the resource score of ith segment of jth travel plan (i = 1,2,3,4,5,6, j = 1,2) as per table 2.1. Average resource score of jth plan is:

^6=1 xij

* ;

6

In this study <$>j is taken as a variable of main interest for different travel plans and packages and needs to be estimated regularly on sample basis over time domain. While travel plans are large in number, Oj are useful unknown parameters for travel agents to prepare the estimate of travel advance money for employees of private or public sectors where tour packages are part of privilege of the annual salary.

3. Graphical Model for <&j Estimation

Let there are two groups of vertices, each having well connected edges. Every vertex of first group is connected to all vertices of other group once and only once. This constitutes a Bipartite graph and the mean-edge-length estimation using sampling techniques can be used with graph as a model tool for obtaining solution.

To extend further, assume that first group has even number of vertices, divided into two

A

C

subgroups such that each contains equal number of vertices. First subgroup is of main interest while second contains well known correlated information, for example distances between cities. Figure 2.1 is a Bipartite graph structure useful as a model for average travel cost estimation problem considered in this paper.

c3

Hi Hz Ha Hi Hs He

Gi C2

Fig 3.1 Bipartite Type Graph Structure Remark 1. If the vertices of any graph G (say) can be split into two disjoint subsets V1 and V2 in such a way that each edge in the graph joins a vertex in Vi to a vertex in V2, and there are no edge in graph G that connect two vertices in V1 or two vertices in V2, then the graph is called Bipartite graph. For example, every tree structure is a bipartite graph (see [1]). Remark 2. Suppose a graph G contains x vertices in the first set and y vertices in the second set. A complete Bipartite graph is a graph in which each vertex in the first set is joined with every single vertex in the second set (see fig 3.1 and [8] [14] [15][18] [19]).

3.1. Example of Bipartite Graph Application (as per fig 3.1)

Let there are six tourist places (destinations) and four origins. For each origin, the travel connectivity exists to all tourist places. Four origins form the first group and six tourist places constitute the second group. First group is further divided into two subgroups, each of two vertices (origins). Edge lengths of first subgroup reveal the total resources likely to consume to travel from origin to the tourist places. Second subgroup is a prototype of the first subgroup where edges represent travel distances who are well known through the travel booklets. Larger distance traveled leads to higher consumption of resources and corresponding requirement of more travel cost. The focus of this study is to know about what an average amount of resource consumption likely to occur while traveling between any pair of origins and tourist places.

Table 3.1 Node-Edge Matrix for Bipartite Graph (Fig 3.1)

/1 /2 /3 /4 /5 /6 row total'

/1 i 1 1 1 1 1 6

/2 i 1 1 1 1 1 6

/3 1 1 1 1 1 1 6

/4 1 1 1 1 1 1 6

36.

Table 3.1 displays the edge connectivity with vertices of groups where unit number denotes existence of an edge.

3.2. General Bipartite Graph Model

Let k origins (vertices) of even number denoted as {y!v f2, f3, ...Fk } divided into groups G1 and G2 each strictly of size t = \. The G1 contains half of vertices (denoted as cities) while G2 has remaining who are prototype of G1(like taxi-stand of the same city in G1).

G1 : {f^^f3,...Ft] (1)

G2 : {v't+v vt+2, u't+3,...u'k} (2)

Further, let there exist r tourist places (vertices) marked as third group G3 .

G3 : {vi, V2, V3,...Vr} (3)

Each vertex of Gi and G2 is connected to every vertex of G3 one and only once which constitutes a model like a Bipartite graph among groups G1, G2 and G3. Similarity of identities exist in sequential pair {vi, v't+1}, {vi, Vt+2}, {v3, vt+3},-.{Vt, vk! just as same origins or same objects or same organizations.

Assume eij denotes edge with weights connecting to Ith vertex of G1 to the jth vertex of G3 and ej the edge with weights of Ith vertex of G2 to the jth vertex of G3 (i = 1,2,3,..., t, j = 1,2,3,...,r). Further, k and r both are large integers and r > k holds for an even k. Moreover, eij represents weights as resource consumption score (unknown and to estimate) while e'ij are the known weights in advance like road distances. The problem undertaken in this paper is to estimate the average amount of resource consumption likely to occur between any pair of vertices of G1 and G3 with the help of edge weights prototype pair of vertices of G2 and G3 where information are priorly known.

Table 3.2 General Node-Edge Matrix

Group (Gi) Group (G3)

Node Vi Vz V3 ......... Vr Total

1 1 1 1 r

H'z 1 1 1 1 r

1 1 1 1 r

V-'t 1 1 1 1 r

Group (Gz) ^'t+i 1 1 1 1 r

V't+z 1 1 1 1 r

V' t+3 1 1 1 1 r

V-'k 1 1 1 1 r

In table 3.2, if an edge exists between (i, j)th pair then ej = eij = 1 else zero everywhere.

4. Single-Node-Systematic Random Sampling

A Single-Node Systematic Sampling scheme without replacement is proposed as under:

Step I For given r, even k, t = |, groups G1, G2, G3, large k and r in population, construct population matrix as stated in table 3.2. Vertex-edge connectivity structure constitutes a model like a Bipartite graph.

Step II Assign labels to vertices in G1 and G2 like 1,2,3,..., t, t + 1, t + 2, t + 3, ...k.

Step III Draw one vertex randomly from label 1,2,3,..., t (table 3.2). Assume it is Ith vertex (l = 1,2,3,..., t) from G1. Using systematic sampling concept, the unit labeled (l + t) is automatically selected in the sample from G2. The resultant is single node (vertex) systematic random selection of units (v[, v'i+t) from G1 and G2, each having r edges.

Step IV Note down weights, marked as edge-lengths, of vi and vl+t connecting to all vertices of G3. Sampled weight values of r edges are as under:

vi : (seij, j = 1,2,3,..., r) sample from G1, G3 (4)

Fi+t : (se'1+t, j = 1,2,3,..., r) sample from G2, G3 (5)

Step V Apply an appropriate estimation procedure to obtain the estimate of unknown

parameter (average resource consumption score).

5. Estimation

For a large Bipartite graph, the population parameters (means) are:

1 t r

A1 = 1 EE ej (6)

,l i=1j=1

1 t r

A2 = 1 EE e'ij (7)

rt i=1 j=1

Using step I to step V of Single-Node-Systematic Sampling, the two sample means are:

1 r

§1 = 1 E sej (8)

j=1

1 r

§2 = 1 E se'j (9)

j=1

where sej and sej denote weights like edge-lengths appeared in sample. For very small numbers h1, h2, \h11 < 1, \h2\ < 1, one can use large sample approximations discussed in [5],[6],[7].

§1 = A1 (1 + h1) §2 = A2(1 + h2) with E(h1) = E(h2) = 0 (10)

t _ 1 t _ 1 t _ 1

E(h1) = 1-1 (C*e)2, E(h2) = (C*ei)2, E(hh2) = p(C*e)(C*ei) (11)

(C* e) = (12)

(C* ei) = (13)

A2

(C* eei) = (14)

(A1.A2)

1

tr

(Se)2 = EE (ej - A1 )2 (15)

i=1 j=1

1 t r

(se )2 = EE (e'j - A2 )2 (16)

- i=1 j=1

1

(See')

tr

rt 1

EE (eij - *1)(4 - A)

(17)

i=1 j=1

Seei

[(Se)(Sei)]

(18)

Equation (5.13) is correlation coefficient between eij and eij in Bipartite population.

5.1. Proposed method of estimation

Deriving motivation from [9],[10], [12], [14], [16], [17] and [18], estimation strategy E is proposed as under:

E = ¿1 f1 (A"2, & )] [f2 (A"2, ¿2)]-1 (19)

where

f1 (A_2, ¿2) = [(A + C)A2 + gBS~2]

f2(A_2, ¿2) = [(A + gB)A2 + C&]

and A = (P - 1)(P - 2); B = (P - 1)(P - 4)(P - 5); C = (P - 2)(P - 3)(P - 4); g = r = 1; 0 < P < to.

The proposed method of estimation is in accordance with Shukla et al. [9], [10],[12] with change in the term B which is now cubic in this paper but was considered quadratic in earlier contributions.

Theorem 1. Proposed method (5.14) could be expressed as

E = A1 Proof. E

(1 + h1) + (A+gB+C) {h2 + h1 h2 - A+Cfc} = ¿1 [f1 (A2, ¿2 )] [f2 (A_2, ¿2)]-1

Using (5.5), |h11 < 1, |h21 < 1

f1(A_2, ¿2 ) = [(A + C)A2 + gBA2 (1 + h2 )]

f2(A_2, ¿2 ) = [(A + gB)A2 + CA2 (1 + h2 )] The (5.15) expressed using (5.5) as

f1 (A2,¿2) = A"2 [(A + gB + C) + gB(h2)] = A2 [(A + gB + C)]

1 +

gBh2

(A + gB + C)J

(20) (21)

(22)

Since |h21 < 1, therefore | (A+?BB+C) I < 1, due to V g > 0, P > 0 For (5.16), the expansion of (1 + x)-1 is used when |x| < 1 as under:

[f2(A2, ¿2)] -1 = (A2)-1 [(A + gB + C) + Ch2]-1

(A2)-1 [(A + gB + C)]-1

1 +

Ch2

(A + gB + C)J

1

P

A_1 (1 + h1)

Then, E = A1 (1 + h1) gBh2

1 +

gBh2

1 +

(A + gB + C)J

(A + gB + C)J Ch2

1 -

1 +

+

Ch2

(A + gB + C)J

C2h22

C3h32

(A + gB + C) (A + gB + C)2 (A + gB + C)3'

A_1(1 + h)

+A_1 (1 + h1)

1 -

Ch2

+

C2h22

C3h32

(A + gB + C) (A + gB + C)2 (A + gB + C)3'

gBh2

gBCh22

+

gBC2h2

(A + gB + C) (A + gB + C)2 (A + gB + C)3

The terms [(h1)u(h2)v] could be ignored for (u + v) > 2, u, v = 0,1,2,3,4,5... because of their low impact in ultimate estimate of parameter due to \h1\ < 1, \h2\ < 1, and

1

(A+gB+C) Then above reduces to:

< 1 (see [9] [10] [12])

(1 + h1) +

- C)h2 (gB - C)Ch2

- C)h1 h2

(A + gB + C) (A + gB + C)2 (A + gB + C)

(1 + h1) +

- C)

(A + gB + C)

h2 + h1 h2 -

Ch2

(A + gB + C)

(23)

(24)

Remark 3. Define Z = [(A+°b+c)].

Theorem 2. Using theorem 5.1, the bias of the proposed method is Bias(E) = B(E) = A1 (^^) [p(Ce)(CeJ) - {a+Cb+qC)2'_

where p denotes correlation coefficients between ei and ei at the Bipartite graph population level.

Proof. Let E(.) denotes the expected value and E(h1) = 0; E(h2) = 0 (as in theorem 5.1 and using equation (5.5));

Bias(E) = B(E) = E[E - A1 ] (see [5],[6],[7]) =E

E A 1 A 1

A1 |(1 + h1) + Z (^h2 + h1 h2 - (A+%C)) } - A_1

A J h1 + Z i h2 + h1h2 - (A+gBi+C) )

theorem 5.1, remark 5.1

E(h1) + z{E(h2) + E(h1 h2) - (AC+CB++C) } Z{E(h1h2) -

[using (5.5), (5.6)]

Bias(E) = A1 Z() ^p(Ce)(Ce') - (A+CB+C)(Ce'[using (5.5), (5.6)] ■

Theorem 3. The proposed methodology is almost unbiased for a given pair of (g, R) when following equation is satisfied at suitable choice of P R(A + gB) + C(R - 1)= 0, where R = pC), g = 1

Proof. Bias(E) = 0 =

[p(Ce)(Ce') - (A^j

For given ( g, R) , re-arranging above, one can get equation

R(A + gB) + C(R - 1)= 0

1

1

1

Theorem 4. For a given pair of (g, R), on maximum of three values of P, the proposed strategy E is almost unbiased.

Proof. Condition of unbiasedness is R(A + gB) + C(R — 1) = 0 This could be expressed in the power of P as:

(R - 1)P3 + [R + gR + g] P2 + [23R + 299R - 26] P - [22R + 20gR - 24] = 0 (25)

Above equation is of degree three in P and has maximum of three roots at which E is almost unbiased. ■

Corollary 1. Few of maximum three roots of (5.20) may imaginary or overlapping depending upon given (g, R).

Corollary 2. The best choice of P, among maximum of three, is that bearing lowest mean squared error.

Theorem 5. The mean squared error of E under (5.5) to (5.13) and using (5.20) is

MSE[E] = (A1 )2(^) [(C*e)2 + Z2(C*ei)2 + 2Zp(C*e)(C*ei)] Proof. MSE[E] = E[E - A1 ]2 (see [], [], [])

A1 {(1 + h1) + ((A+-B+C) + ."} - A1 using theorem 5.1 and ignoring terms [(h1)u(h2)v],

E

u, v = 0,1,2,3,4 of higher order (u + v) > 2 as adopted in theorem 5.1 = (A1 )2E [h1 + Zh2]2

= (A)2 [E(h2) + (Z)2E(h2) + (2Z)E(h1h2)]

MSE[E] = (A)2(-) [(C*e)2 + Z2(C*ei)2 + 2Zp(C*e)(C*ei)] using (5.6). ■

Theorem 6. For a given pair of (g, R), the minimum mean squared error occurs at suitable

choice of P when RA + (R + 1)gB + (R - 1)C = 0 is satisfied.

Proof. Differentiating MSE[E], in theorem 5.5, with respect to the term Z and equating to

zero, one can get,

2Z = - 2

p(C*e)

-2R

_(CV )_

^ (gB - C) + R(A + gB + C) = 0

RA + (R + 1)gB + (R - 1)C = 0 (26)

Theorem 7. For given (g, R), the maximum three values of P exist at which MSE[E] is minimum.

Proof. The condition for minimum MSE is RA + (R + 1)gB + (R - 1)C = 0 which could be expressed in the power of P as:

P3[R(g + 1) + (g - 1)] - P2[2R(5g + 4) + (10g - 9)] + P[R(29g + 23) + (29g - 26)] -

2[R(10g + 11) + 2(5g - 6)]= 0

Above equation is of degree three in P, therefore, has at most the three roots. ■

Corollary 3. One can get three values of P, using theorem 5.7, having the minimum MSE. The best P is that having the lowest bias.

6. Best Selection Procedure of P for Almost Optimum MSE Strategy

Step I Find P for given pair of (g, R) producing almost unbiased strategy.

Step II Plot the range of P over the X-axis obtained by step I.

Step III Find P for given pair of (g, R) producing minimum MSE.

Step IV Plot the range of P over X-axis under step III.

Step V The overlapping range is the best choice of P for efficient estimation.

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7. Empirical Study

Consider six tourist places (^i, ^3, ^4, ^5, ^6), two remote small towns (cities) (^1, ^2). The is prototype of and ^4 is prototype of ^2 like taxi stands of and cities respectively. The weight of edges e'^ are assumed known like distances while weight of edges are resource consumption scores likely during traveling by road and unknown.

Table 7.1 Population

Tourist Places U1 U2 U3 U4 U5 U6

U\ en = 09 £12 = 13 £13 = 21 £14 = 15 £15 = 14 £16 = 17

Cities and u2 £21 = 15 e22 = 07 £23 = 16 £24 = 19 £25 = 11 £26 = 14

Prototype U3 e31 = 04 e3 2 = 06 ¿33 = 11 £34 = 08 £35 = 07 £3 6 = 09

U4 e41 = 07 e4 2 = 03 €f43 = 08 £44 = 09 £45 = 05 £46 = 06

Table 7.3 First Sample (Random selection of )

Tourist Places Ui U2 U3 U4 U5 U6

Cities and eii = 09 £12 = 13 £13 = 21 £14 = 15 £15 = 14 £16 = 17

Prototype u3 e31 = 04 e3 2 = 06 £33 = 11 £34 = 08 £35 = 07 £3 6 = 09

Table 7.4 Second Sample (Random selection of u2)

Tourist Places Ui U2 U3 U4 U5 U6

Cities and u2 £21 = 15 e22 = 07 £23 = 16 £24 = 19 £25 = 11 £26 = 14

Prototype u4 e41 = 07 e4 2 = 03 6'43 = 08 £f4 4 = 09 £45 = 05 £46 = 06

Table 7.5 Bipartite Graph Population Parameters [table 7.1]

S.No. Parameter Value Description/Equation no.

1. rt 12 Population size

2. t 2 Sample size

3. Ä 1 14.25 Using (5.1)

4. Ä2 6.91 Using (5.2)

5. Se 3.95 Using (5.10)

6. S^ 2.27 Using(5.11)

7. C* e 0.27 Using (5.7)

8. C* ef 0.32 Using (5.8)

9. P 0.98 (5.13)

10. R 0.8285 Using Theorem (5.6)

11. r 6 Large Integer

Table 7.6 Almost Unbiased Choice of P for given (g, R)

S.No. R g Choice of P Bias MSE

1. 0.8285 0.5 P1 = 0.61 -0.0001 0.1308

2. 0.8285 0.5 P2 = — — —

3. 0.8285 0.5 P3 = -- — —

Table 7.7 Choice of P for Minimum MSE for given (g, R)

S.No. R g Choice of q MSE Bias

1. 0.8285 0.5 P1(opt) = 0.8830 0.0462 0.0100

2. 0.8285 0.5 P2(opt) = — —

3. 0.8285 0.5 P3(opt) = — —

Remark 4. As per data of table 7.1, and calculation of table 7.6 and table 7.7 when R=0.8285, g=0.5, the most suitable range of P is P= 0.61 to P=0.8830 to produce the best estimate of average resource consumption score using proposed E. This range of consistent P provides almost unbiased optimal estimate of average resource score.

Table 7.8 Comparison with Particular Cases of Proposed E (Using PRE)

S.No. Choice of P Bias(E) (Theorem 5.2) MSE(E) (Theorem 5.5) Comparison (PRE)

1. At P=1 0.0220 0.1001 53.1468%

2. At P=2 0.1066 6.1768 99.24407%

3. At P=3 0.0533 3.2824 98.5711%

4. At P=4 0.0000 1.3049 96.4058%

5. At P=5 -0.0212 0.4958 90.5405%

Where, Percentage Relative Efficiency (PRE) at Popt = 0.8830 _ [MSE(E)p=1,2,3,4,5] - [MSE(E)p(opt)] = MSE(E)p=i,2,3,4,5

Remark 5. As per table 7.8, for given pair of (g,R)=(0.5, 0.8285), the best choice of P is P = 0.8830 which provides lowest bias and minimum MSE for the proposed strategy E in light of given data set of table 7.1.

Remark 6. The proposed strategy E is constantly better over particular cases at P= 1, 2, 3, 4, 5 as evident from table 7.8 using PRE.

Table 7.9 Calculation for First Sample Parameter and Confidence Interval (C.I.)

¿1 ¿2 (Se)2 (se)2 (Ce ) (ce ) (see') P C.I.

14.8333 7.5 16.1660 5.9000 0.2711 0.3238 3.1144 0.3189 at P =0.61 (unbiasedness) [12.47, 17.18]

- - - - - - - at P =0.883 (Minimum MSE) [12.11, 17.47]

Table 7.10 Calculation for Second Sample Parameter and Confidence Interval (C.I.)

¿1 ¿2 (Se)2 (se )2 (Ce) (ce) (see< ) P C.I.

13.6666 6.3333 17.4666 4.6666 0.3058 0.3410 2.9888 0.3310 at P =0.61 (unbiasedness) [11.28,16.04]

- - - - - - - at P= 0.883 (minimum MSE) [11.03,16.30]

Remark 7. ¿1 and ¿2 are in (5.3) and (5.4), and others are defined as (Ce) = f

(c. ) = ^ (cee' ) = M

(Se)2 = — Ej=1 (Se„ - ¿1)

2

(se')2 = — j (s£jj - ¿2)2

(See') = r-1 E|=i 4=1 (Bey - ¿1)^. - ¿2)

/ _ see/

P = [(Se )(Se')]

Remark 8. Let P[A] denotes the probability of event A then a 95% confidence interval in defined

as:

P

(E)p - 1.96MSE(E)p, (E)p + 1.96^ MSE(E)p

0.95

This interval indicates that there is 95% chance, the true value of population mean lies in the range P |"(E)p - 1.96 VMSE(E)p, (E)p + 1.96 VMSE(E)p this interval will be the optimal confidence interval wit'

respect to unbiasedness as the case may be)

0.95. If the value of P = P0ft then 1 respect to minimum MSE (or with

Table 7.11 Ready Reckoner for P Value for Unbiasedness for given (g, R)

S.No. R g Choice of P Bias MSE S.No. R g Choice of P Bias MSE

1. 0.2 0.3 P1 = 4.5641 -0.0182 0.7089 46. 1.2 0.3 P1 = 1.2549 0.0016 0.7877

2. 0.2 0.3 P2 = 4 0.0000 1.3049 47. 1.2 0.3 P2 = 4 0.0000 1.3049

3. 0.2 0.3 P3 = -- - - 48. 1.2 0.3 P3 = -- - -

4. 0.2 0.6 P1 = 0.45410 -0.0202 0.6601 49. 1.2 0.6 P1 = 1.335 0.0682 0.7085

5. 0.2 0.6 P2 = 4 0.0000 1.3049 50 1.2 0.6 P2 = 4 0.0000 1.3049

6. 0.2 0.6 P3 = -- - - 51. 1.2 0.6 P3 = -- - -

7. 0.2 0.9 P1 = 4.5264 -0.0224 0.6019 52. 1.2 0.9 P1 = 1.0900 0.0675 0.6823

8. 0.2 0.9 P2 = 4 0.0000 0.1304 53. 1.2 0.9 P2 = 4 0.0000 0.1304

9. 0.2 0.9 P3 = -- - - 54. 1.2 0.9 P3 = -- - -

10. 0.4 0.3 P1 = 5.2910 -0.0209 0.4150 55. 1.4 0.3 P1 = 1.3560 0.1407 2.2086

11. 0.4 0.3 P2 = 4 0.0000 1.3049 56. 1.4 0.3 P2 = 4 0.0000 1.3049

12. 0.4 0.3 P3 = -- - - 57. 1.4 0.3 P3 = -- - -

13. 0.4 0.6 P1 = 5.3389 -0.0195 0.4573 58. 1.4 0.6 P1 = 1.2048 0.1366 1.9884

14. 0.4 0.6 P2 = 4 0.0000 1.3049 59. 1.4 0.6 P2 = 4 0.0000 1.3049

15. 0.4 0.6 P3 = -- - - 60. 1.4 0.6 P3 = -- - -

16. 0.4 0.9 P1 = 5.4050 -0.0177 0.5174 61. 1.4 0.9 P1 = 1.1430 0.1345 1.9025

17. 0.4 0.9 P2 = 4 0.0000 1.3049 62. 1.4 0.9 P2 = 4 0.0000 1.3049

18. 0.4 0.9 P3 = -- - - 63. 1.4 0.9 P3 = -- - -

19. 0.6 0.3 P1 = 7.0810 -0.0144 0.2561 64. 1.6 0.3 P1 = 1.4121 0.2341 4.3451

20. 0.6 0.3 P2 = 4 -0.0000 1.3049 65. 1.6 0.3 P2 = 4 -0.0000 1.3049

21. 0.6 0.3 P3 = -- - - 66. 1.6 0.3 P3 = -- - -

22. 0.6 0.6 P1 = 12.0001 -0.0085 0.5738 67. 1.6 0.6 P1 = 1.2490 0.2259 3.9084

23. 0.6 0.6 P2 = 4 0.0000 1.309 68. 1.6 0.6 P2 = 4 0.0000 1.309

24. 0.6 0.6 P3 = -- - - 69. 1.6 0.6 P3 = -- - -

25. 0.6 0.9 P1 = 0.0001 -0.0044 0.8903 70. 1.6 0.9 P1 = 1.1781 0.2232 3.7648

26. 0.6 0.9 P2 = 4 0.0000 1.3049 71. 1.6 0.9 P2 = 4 0.0000 1.3049

27. 0.6 0.9 P3 = -- - - 72. 1.6 0.9 P3 = -- - -

28. 0.8 0.3 P1 = 0.0001 0.0027 0.1015 73. 1.8 0.3 P1 = 1.4480 0.3497 7.1827

29. 0.8 0.3 P2 = 4 0.0000 1.3049 74. 1.8 0.3 P2 = 4 0.0000 1.3049

30. 0.8 0.3 P3 = -- - - 75. 1.8 0.3 P3 = -- - -

31. 0.8 0.6 P1 = 0.6354 -0.0020 0.1752 76. 1.8 0.6 P1 = 1.2800 0.3391 6.5374

32. 0.8 0.6 P2 = 4 0.0000 1.3049 77. 1.8 0.6 P2 = 4 0.0000 1.3049

33. 0.8 0.6 P3 = -- - - 78. 1.8 0.6 P3 = -- - -

34. 0.8 0.9 P1 = 0.8201 -0.0015 0.1456 79. 1.8 0.9 P1 = 1.2030 0.3331 6.2609

35. 0.8 0.9 P2 = 4 0.0000 1.3049 80. 1.8 0.9 P2 = 4 0.0000 1.3049

36. 0.8 0.9 P3 = -- - - 81. 1.8 0.9 P3 = -- - -

37. 1.0 0.3 P1 = 0.9008 0.0158 0.0592 82. 2.0 0.3 P1 = 1.4738 0.4928 10.8517

38. 1.0 0.3 P2 = 4 0.0000 1.3049 83. 2.0 0.3 P2 = 4 0.0000 1.3049

39. 1.0 0.3 P3 = -- - - 84. 2.0 0.3 P3 = -- - -

40. 1.0 0.6 P1 = 0.9800 0.0186 0.0766 85. 2.0 0.6 P1 = 1.3020 0.4704 9.7348

41. 1.0 0.6 P2 = 4 0.0000 1.3049 86. 2.0 0.6 P2 = 4 0.0000 1.3049

42. 1.0 0.6 P3 = -- - - 87. 2.0 0.6 P3 = -- - -

43. 1.0 0.9 P1 = 0.9500 0.0109 0.0470 88. 2.0 0.9 P1 = 1.2220 0.4678 9.4728

44. 1.0 0.9 P2 = 4 0.0000 1.3049 89. 2.0 0.9 P2 = 4 0.0000 1.3049

45. 1.0 0.9 P3 = -- - - 90. 2.0 0.9 P3 = -- - -

Table 7.12 Ready Reckoner for P Value for Minimum MSE for given (g, R)

S.No. R g Choice of P Bias MSE S.No. R g Choice of P Bias MSE

1. 0.2 0.3 P1 = 4.5001 -0.0171 0.7561 46. 1.2 0.3 P1 = 1.1452 0.0396 0.2993

2. 0.2 0.3 P = -- - - 47. 1.2 0.3 P2 = -- - -

3. 0.2 0.3 P3 = -- - - 48. 1.2 0.3 P3 = -- - -

4. 0.2 0.6 P1 = 4.5005 -0.0196 0.6861 49. 1.2 0.6 P1 = 1.0743 0.0408 0.2994

5. 0.2 0.6 P2 = -- - - 50. 1.2 0.6 P2 = -- - -

6. 0.2 0.6 P3 = -- - - 51. 1.2 0.6 P3 = -- - -

7. 0.2 0.9 Pi = 0.1964 -0.0052 0.7840 52. 1.2 0.9 P1 = 1.0495 0.0410 0.2969

8. 0.2 0.9 P2 = -- - - 53. 1.2 0.9 P2 = -- - -

9. 0.2 0.9 P3 = -- - - 54. 1.2 0.9 P3 = -- - -

10. 0.4 0.3 P1 = 5.5014 -0.0203 0.3744 55. 1.4 0.3 P1 = 1.2340 0.0621 0.6498

11. 0.4 0.3 P2 = -- - - 56. 1.4 0.3 P2 = -- - -

12. 0.4 0.3 P3 = -- - - 57. 1.4 0.3 P3 = -- - -

13. 0.4 0.6 P1 = 0.2300 -0.0052 0.3826 48. 1.4 0.6 P1 = 1.1270 0.0644 0.6452

14. 0.4 0.6 P2 = -- - - 59. 1.4 0.6 P2 = -- - -

15. 0.4 0.6 P3 = -- - - 60. 1.4 0.6 P3 = -- - -

16. 0.4 0.9 P1 = 0.6381 -0.0057 0.3830 61. 1.4 0.9 P1 = 1.1201 0.0991 1.2263

17. 0.4 0.9 P2 = -- - - 62. 1.4 0.9 P2 = -- - -

18. 0.4 0.9 P3 = -- - - 63. 1.4 0.9 P3 = -- - -

19. 0.6 0.3 P1 = 0.0001 0.0023 0.1094 64. 1.6 0.3 P1 = 1.2928 0.0888 1.1370

20. 0.6 0.3 P2 = -- - - 65. 1.6 0.3 P2 = -- - -

21. 0.6 0.3 P3 = -- - - 66. 1.6 0.3 P3 = -- - -

22. 0.6 0.6 P1 = 0.6900 -0.0009 0.1416 67. 1.6 0.6 P1 = 1.1664 0.0928 1.1367

23. 0.6 0.6 P2 = -- - - 68. 1.6 0.6 P2 = -- - -

24. 0.6 0.6 P3 = -- - - 69. 1.6 0.6 P3 = -- - -

25. 0.6 0.9 P1 = 0.8501 -0.0001 0.1098 70. 1.6 0.9 P1 = 1.0990 0.0756 0.8146

26. 0.6 0.9 P2 = -- - - 71. 1.6 0.9 P2 = -- - -

27. 0.6 0.9 P3 = -- - - 72. 1.6 0.9 P3 = -- - -

28. 0.8 0.3 P1 = 0.7102 0.0093 0.0473 73. 1.8 0.3 P1 = 1.3360 0.1203 1.7742

29. 0.8 0.3 P2 = -- - - 74. 1.8 0.3 P2 = -- - -

30. 0.8 0.3 P3 = -- - - 75. 1.8 0.3 P3 = -- - -

31. 0.8 0.6 P1 = 0.8863 0.0081 0.0477 76. 1.8 0.6 P1 = 1.1970 0.1259 1.7725

32. 0.8 0.6 P2 = -- - - 77. 1.8 0.6 P2 = -- - -

33. 0.8 0.6 P3 = -- - - 78. 1.8 0.6 P3 = -- - -

34. 0.8 0.9 P1 = 0.9301 0.0079 0.0476 79. 1.8 0.9 P1 = 1.1395 0.1282 1.7790

35. 0.8 0.9 P2 = -- - - 80. 1.8 0.9 P2 = -- - -

36. 0.8 0.9 P3 = -- - - 81. 1.8 0.9 P3 = -- - -

37. 1.0 0.3 P1 = 0.9700 0.0198 0.0831 82. 2.0 0.3 P1 = 1.3690 0.1567 2.5605

38. 1.0 0.3 P2 = -- - - 83. 2.0 0.3 P2 = -- - -

39. 1.0 0.3 P3 = -- - - 84. 2.0 0.3 P3 = -- - -

40. 1.0 0.6 P1 = 0.9801 0.0187 0.0767 85. 2.0 0.6 P1 = 1.2215 0.1638 2.5523

41. 1.0 0.6 P2 = -- - - 86. 2.0 0.6 P2 = -- - -

42. 1.0 0.6 P3 = -- - - 87. 2.0 0.6 P3 = -- - -

43. 1.0 0.9 P1 = 1.001 0.0223 0.1022 88. 2.0 0.9 P1 = 1.1585 0.1671 2.5642

44. 1.0 0.9 P2 = -- - - 89. 2.0 0.9 P2 = -- - -

45. 1.0 0.9 P3 = -- - - 90. 2.0 0.9 P3 = -- - -

Fig 7.1 Three Dimensional Graph for Almost Unbiasedness

2 * w

R g

P

oo ^

Fig 7.2 Three Dimensional Graph for Optimum MSE 7.1. About given (g, R)

To observe that g = j, where t is size of the group G1 who is known and fixed before the draw of random sample using the single-node-systematic sampling procedure. The quantity R is R

* e

_ p(C e) as described in theorem 5.6. The R is a ratio of C*e and C*e' who are coefficients of

(C* e')

variations and remain almost stable over time occasions. Therefore, R could be guessed from the past data or by past reports or by the current sample. While minor fluctuations in R, the table 7.13 helps to find out most suitable values of P.

Table 7.13 Best Choice of P Under Fluctuations of R (independent of g)

Range of R Best range of P for unbiasedness Best range of P for opt. MSE

0 < R < 0.6 0 < P < 5.4 0 < P < 5.5

0.6 < R < 1.0 0 < P < 0.95 0.71 < P < 1.001

1.0 < R < 1.6 1.09 < P < 1.7 1.0 < P < 1.29

1.6 < R < 2.0 1.22 < P < 1.45 1.15 < P < 1.36

6

4

2

0

R

Note 7.2 The table 7.13 is independent of variations of g which is created using the Ready-reckoner tables 7.11 and 7.12. However, for any given (g, R), one can choose the best P using Ready-reckoner table 7.11 and 7.12.

Note 7.3 The most suitable value of P,for almost unbiased estimation, is (P = 0.10 to P = 5.4) for all (g,0 < g < 1) and for all R, (0 < R < 2) (see table 7.13).

Note 7.4 The most suitable value of P for low MSE is (P = 0.001 to P = 5.5) for all g and

0 < R < 2 (see table 7.13)

Note 7.5 The general recommended P is P G (0.1,5.5) where one can get low bias and low MSE by the proposed strategy, whatever be the g, (0 < g < 1) and whatever be R, 0 < R < 2 .

Note 7.6 This is beauty of the proposed strategy E because it is now independent to the known pair (g, R) to produce good estimate of average consumption of score in a Bipartite graph population when user has chosen 0.1 < P < 5.5.

Note 7.7 The figure 7.2 presents three dimensional aspect of choice of P for given pair (g, R). The X- axis has R, Z- axis has g and Y- axis has calculated value of P.

On recapitulation, the content of the paper has a sample based methodology for evaluating the average resource consumption scores with the help of Bipartite graph. A single node systematic sampling procedure is suggested in the content which is a graph sampling based procedure, useful for parameter estimation in places where similar situation exists. This procedure opens up avenues for further researches in the area of sampling where population is synonymous to the graphical structure. The Bipartite graph is used for getting solution of the problem of estimation of the travel resource consumption parameter. An estimation strategy is proposed and its properties are derived. It is proved that they are bias and MSE controlled both, at the same time due to cubic equation. The proposed also converts to an almost unbiased optimum strategy at some appropriate choice of P.

The main difficulty occurs with the strategy is the selection of suitable value of constant incorporated in its structure. Two Ready-reckoner tables have been prepared who are useful for the quick selection of constant for give pair of ( g, R) values. These tables provide to users, the population independency just as to utilize only (g, R), irrespective of the other population characteristics. Whatever may the distance and resource consumption, if (g, R) are given (or guessed or estimated or calculated from past data) ,the bias and MSE can be predicted through these Ready-reckoner tables who made easy to apply the suggested. It can be used for developing computer algorithms and softwares for enhancing the applications.

Two systematic samples are taken into account and their computed confidence intervals are (12.47 - 17.18), (12.11 - 17.47) for first sample , (12.28 - 16.04), (11.03 - 16.30) for second sample. All intervals are catching the true values of average resource consumption which is 14.25. The interval based prediction is sound enough indicating the efficiency of the suggested strategy. For 0 < R < 2 , a table has been developed, for the ease of users, towards rapid selection of best value of constant P whatever be the value of g and whatever be the population values in terms of distance and resource consumption scores . The general recommendation for best choice of P is the range (0.1 - 5.5) irrespective of other population features and dependencies. Bipartite graph is used as a model tool for developing the single node systematic sampling procedure who can be extended further using other kinds of estimation procedures exiting in the concerned literature.

8. Conclusion

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