Introduction

Determining the technological value of cotton fibre is an interesting field of textile research. The quality of final yarn is largely influenced (up to 80%) by the characteristics of raw cotton [1]. However, the level to which various fibre properties influence yarn quality is diverse, and also changes depending on the yarn manufacturing technology. Besides, a cotton may have conflicting standards in terms of different quality criteria. Therefore, the ranking or grading of cotton fibres in terms of different quality criteria will certainly not be the same. This will make the situation more complex, and applying multiple-criteria decision-making (MCDM) methods can probably deliver a plausible solution. The solution must produce an index of technological value or overall quality of cotton fibre, and the index should incorporate all the important fibre parameters. The weights of the fibre parameters should be commensurate with their importance on the final yarn quality.

Traditionally, three fibre parameters have been used to determine the quality value of cotton fibre. These are grade, fibre length and fibre fineness. The development of fibre testing instruments such as the High Volume Instrument (HVI) and the Advanced Fibre Information System (AFIS) has revolutionised the concept of fibre testing. With the HVI it is pragmatically possible to determine most of the quality characteristics of a cotton bale within two minutes. Based on the HVI results, composite indexes such as the fibre quality index (FQI) and spinning consistency index (SCI) can be used to determine the technological value of cotton; this can play a pivotal role in an engineered fibre selection programme [2-3].

In this paper, a new method of determining the technological value of cotton using a multiplicative analytic hierarchy process (multiplicative AHP) of the MCDM method is postulated. The technological value of cotton was also determined by the three traditional methods, namely the fibre quality index (FQI), the spinning consistency index (SCI) and the premium-discount index (PDI). The ranking of cotton fibres produced by these four methods was compared with the ranking of final yarn tenacity, and a rank correlation analysis was carried out.

Overview of MCDM and AHP

Multiple Criteria Decision Making is a well-known branch of Operations Research (OR), which deals with decision problems involving a number of decision criteria and a finite number of alternatives. Various MCDM techniques, such as the weighted sum model (WSM), the weighted product model (WPM), the analytic hierarchy process (AHP), the revised AHP, the technique for order preference by similarity to an ideal solution (TOPSIS), and elimination and choice translating reality (ELECTRE), can be used in engineering decision-making problems, depending upon the complexity of the situation [48] The Analytic Hierarchy Process (AHP), introduced by Saaty [9-12], is one of the most frequently discussed methods of MCDM. Although some researchers [13-16] have raised concerns over the theoretical basis of AHP, it has proven to be an extremely useful decision-making method. The reason for AHP’s popularity lies in the fact that it can handle the objective as well as subjective factors, and the criteria weights and alternative scores are elicited through the formation of a pair-wise comparison matrix, which is the heart of the AHP.

Details of AHP methodology

Step 1:

Develop the hierarchical structure of the problem. The overall objective or goal of the problem is positioned at the top of the hierarchy, and the decision alternatives are placed at the bottom. Between the top and bottom levels are found the relevant attributes of the decision problem such as criteria and sub-criteria. The number of levels in the hierarchy depends on the complexity of the problem.

Step 2:

Generate relational data for comparing the alternatives. This requires the decision maker to formulate pair-wise comparison matrices of elements at each level in the hierarchy relative to each activity at the next, higher level. In AHP, if a problem involves M alternatives and N criteria, then the decision maker has to construct N judgment matrices of alternatives of M x M order and one judgment matrix of criteria of N x N order. Finally, the decision matrix of M x N order is formed by using the relative scores of the alternatives with respect to each criterion. In AHP, the relational scale of real numbers from 1 to 9 and their reciprocals are used to assign preferences in a systematic manner. When comparing two criteria (or alternatives) with respect to an attribute in a higher level, the relational scale proposed by Saaty [9-12] is used. The scale is shown in Table 1.

Table 1. The fundamental relational scale for pair-wise comparisons

Intensity of importance on an absolute scale	Definition	Explanation
1	Equal importance	Two activities contribute equally to the objective.
3	Moderate importance of one over another	Experience and judgment slightly favour one activity over another.
5	Essential or strong importance	Experience and judgment strongly favour one activity over another.
7	Very strong importance	An activity is strongly favoured and its dominance is demonstrated in practice.
9	Extreme importance	The evidence favouring one activity over another is of the highest possible order of affirmation.
2, 4, 6, 8	Intermediate values between two adjacent judgment	When compromise is needed.
Reciprocals	If activity p has one of the above numbers assigned to it when compared with activity q, then q has the reciprocal value when compared with p.

Step 3:

In this step, the relative importance of different criteria with respect to the goal of the problem and the alternative scores with respect to each of the criteria is determined. For N criteria, the size of the comparison matrix (C1) will be N x N, and the entry cij will denote the relative importance of criterion i

1

with respect to the criterion j. In the matrix, cij = 1 if when i = j and c = .

ji

c

ij

 1 c ... c ⎤

12 1N

⎢⎥

c 1 ... c

21 2 N

C₁ =^{ }

 ... ... 1 ...  ⎢

⎢cc ... 1 ⎥

 N1 N 2 

The relative weight or importance of the i th criteria (Wi) is determined by calculating the geometric mean (GM) of the i th row, and then normalising the geometric means of the rows of the above matrix. This can be represented as follows:

1

 ^N ⎫^N GM

GM =_ cij and W = ⁱ (1)

i ∏

iN  j=1 

_∑GM_i

i=1

^C3

Then, matrix C3 and C4 are calculated such that C = C x C and C = , where

3124 ^C2

C =[WW ... W ]^T .

2 12 N

The principal eigen vector (λmax) of the original pair-wise comparison matrix (C1) is calculated from the average of matrix C4. To check the consistency in a pair-wise comparison judgment, the consistency index (CI) and consistency ratio (CR) are calculated from the following equations:

λ_max − N CI

CI = and CR = (2) N −1 RCI

where RCI is the random consistency index; its value could be obtained from Table 2. If the value of CR is 0.1 or less, then the judgment is considered to be consistent and acceptable. Otherwise the decision maker has to make some changes in the entry of the pair-wise comparison matrix.

Table 2. RCI values for different numbers of alternative (M)

M	1	2	3	4	5	6	7	8	9
RCI	0	0	0.58	0.90	1.12	1.24	1.32	1.41	1.45

Similarly, N numbers of pair-wise comparison matrices (one for each criterion) of MxM order are formed, where each alternative is pitted against all of its competitors, and pair-wise comparison is made with respect to each of the decision criterion. The eigen vector of each of these ‘N’ matrices represents the alternative performance scores in the corresponding criterion and from a column of the final decision matrix. The decision matrix appears as follows:

Criterion

C1 C2 C3 … CN Alternative W1 W2 W3 … WN

A1	a11	a12	a13	…	a1N
A2	a21	a22	a23	…	a2N
A3	a31	a32	a33	…	a3N
…	…	…	…	…	…
AM	aM1	aM2	aM3	…	aMN

M Here _∑a_ij =1 i=1

Step 4:

In this step, the final priority of all the alternatives is determined considering the alternative scores (aij) in each criteria and the weight of the corresponding criteria (Wj) using the following equation.

N

_AHP =max_∑a W ij. j for i = 1,2,3, …..M (3) j=1

Multiplicative AHP

Several researchers have criticized the AHP due to the ranking inconsistencies associated with it [1316]. The AHP does not necessarily demonstrate the transivity property; i.e. if A1>A2 and A2>A3 then A1>A3. To overcome these problems, Barzilai & Lootsma [17] have proposed a multiplicative variant of AHP. In this approach, the relative performance of the i th alternative in terms of the j th criterion, i.e. aij of decision matrix and criteria weights Wj, are not processed according to formula 1. Instead, the following formula is used:

⎡ ^N ⎡a ⎤

A Kj ^Wj

R^K = (4)

 ⎥∏ 

⎣^AL  j=1⎢⎣^aLj ⎥

where aKj and aLj are the scores of alternative K and L on the j th criterion, and Wj is the weightage of

⎡A 

the j th criterion. If R >1 then AK is more desirable than AL (in the maximisation case). One can

⎢^K ⎥

A

⎣L ⎦

use a variant of the above formula to compute the preference values, which can in turn be used to rank alternatives:

N W

j

Pi () (5)

⁼∏^aij j=1

The ranking produced by the multiplicative AHP is very robust and immune to ranking inconsistencies. Moreover, the ranking produced by this method remains independent of the method of normalisation. Therefore, the test results obtained from the HVI or AFIS can be used directly in equation 5 without any normalization [18]. Here the Wj will have a negative sign for the cost criterion, which has a negative impact on the overall objective.

Traditional Models to Determine the Technological Values of Cotton

Fibre quality index (FQI)

This is probably the most widely used method to determine the technological value of cotton [19-22]. The main reason for its popularity may be attributed to the simplicity of the equation used. Several variants of the FQI model are available. In this work we have used the following form of FQI proposed by the South Indian Textile Research Association [21].

...

LURFS M

FQI = (6) FF

where L is 2.5% span length, UR is the uniformity ratio, FS is the fibre bundle tenacity, M is the maturity coefficient, and FF is the fibre fineness (micronaire). If the HVI mode of fibre testing is used, then the above expression is changed as follows:

..

UHMLUI FS

FQI_HVI = (7) FF

where FQIHVI is the HVI quality index, UHML is the upper half mean length and UI is the uniformity index.

Spinning consistency index (SCI)

This is a calculation for predicting the overall quality and spinnability of the cotton fibre. It is chiefly used to gain within and between lay-down consistencies of major cotton properties. The regression equation of SCI uses most of the individual HVI measurements, and it is based on the five-year crop average of U. S. Upland and Pima cotton. The regression equation [23] used to calculate SCI is as follows:

SCI =−414.67 +2.9 FS +49.17 UHML +4.74 UI −9.32FF +0.65Rd +0.36( +b) (8)

where Rd is the reflectance degree and +b is the yellowness of cotton fibre.

Premium-Discount Index (PDI)

This method was proposed by Mogazhy et al. [24]. It includes the development of a multiple regression equation relating fibre properties and yarn strength, the determination of the percentage contribution of fibre properties to yarn strength, the selection of a reference set of cotton properties, the determination of a difference factor between the fibre property and the reference set, and finally the development of a premium-discount formula. The regression equation of the following form is developed from the available fibre and yarn data.

Yarn Tenacity = C +C FS +C FE +C UHML +C UI +C SFC +C FF (9)

1234 56 7

where C1, C2, …..C7, are the regression coefficients, FE is the fibre breaking elongation in percentage, and SFC is the short fibre content as measured by AFIS. The regression coefficients are dependent on the scales of measurement, and therefore cannot be used as a measure of their importance. To overcome this problem, ‘β’ coefficients of the variables are determined using the standardised variables in the regression equation. The relative contribution of the i th fibre property can be determined by the following equation:

B

i 2 (10)

C_i % =100( _N )R

∑^Bi

i=1

where Bi is the ‘β’ coefficient of the i th variable, N is the number of HVI fibre properties and R² is the coefficient of determination. The reference set is expressed in terms of the average and standard deviation of a fibre property. In the next step, the relative difference factor for each cotton fibre is determined by the following equation:

(x −µ)

ⁱⁱ (11)

^Di ^{= σ}i

where xi is the i th fibre property of a cotton, µi and σi are the overall average and standard deviation of all the cottons in the i th property. Now, based on the percentage contribution of fibre property Ci% and the difference factor Di, the premium-discount index (PDI) could be calculated using the following equation:

N =_∑(._ii ) (12)

PDI CD

i=1

Here the sign of the product in the summation should follow the sign of the variable as obtained in the regression equation.

Material and Methods

Data collection and analysis

Each year the International Textile Centre (USA) conducts a crop study for different varieties of cotton. The results of the crop study of 1997 and 1998, which includes 33 sets of fibre and yarn data for two different yarn counts (22 Ne and 30 Ne), were used in our investigation. We ranked the 33 cotton fibres according to their FQI, SCI, PDI and multiplicative AHP (MIAHP) values. We also ranked the 33 cottons according to the final yarn tenacity. Separate rankings were obtained for 22 Ne and 30 Ne. The difference between the two rankings (fibre quality ranking and yarn tenacity ranking) was calculated to measure the rank correlation coefficient between them by using the following equation.

6_∑d²

(13)

=−

^Rs ¹2

( −1)

MM

where Rs is the rank correlation, d is the absolute difference between the two rankings, and M is the total number of alternatives (33). The summary statistics of fibre properties are given in Table 3.

Table 3. Summary statistics of cotton fibre properties

Fibre Properties	Minimum	Maximum	Mean	Standard deviation
Fibre bundle tenacity, cN/tex	26.5	34.0	29.05	1.477
Fibre elongation, %	5.3	6.9	6.27	0.458
UHML, inch	0.97	1.20	1.06	0.047
Uniformity index	79.2	83.2	81.57	0.971
Short Fibre Content	5.6	18.4	9.77	3.043
Micronaire	3.1	5.0	4.23	0.453

Hierarchy formulation for multiplicative AHP

The goal or objective of the present investigation is to determine the technological value of cotton, which should reflect the achievable level of yarn quality (yarn strength). In general, the cotton fibre criteria of this problem can be classified under three headings, namely tensile properties, length properties and fineness properties. Tensile properties can be divided into two sub-criteria, fibre bundle tenacity (FS) and elongation (FE). Similarly, UHML, UI and SFC are the relevant sub-criteria of length properties to be considered here. Fineness is solely represented by the micronaire (FF) value of cotton. At the lowest level of the hierarchy, there are 33 cotton fibre alternatives, which should be ranked according to their technological value. The schematic representation of the problem is depicted in Figure 1.

FS

Cotton 2 … Cotton 32

Figure 1. Hierarchical structure of cotton fibre quality http://www.autexrj.org/No2-2005/0126.pdf 76

Cotton 33

Determination of criteria weights

With respect to the overall objective problem, the pair-wise comparison matrix of three criteria is given in Table 4. Here the comparisons are made according to Saaty’s scale given in Table 1.

Table 4. Pair-wise comparison matrix of criteria with respect to objective

Criteria	Tensile	Length	Fineness	GM	Normalized GM
Tensile	1	1/2	3	1.145	0.309
Length	2	1	5	2.154	0.581
Fineness	1/3	1/5	1	0.406	0.110

It can be inferred from Table 4 that tensile properties moderately predominate over the fineness properties. However, the length properties demonstrate a strong preponderance over the fineness properties. The dominance of length properties over the tensile properties is between equal to moderate. The normalised GM column of Table 4 indicates that the length properties of cotton fibres have the most dominant influence with a relative weight of 0.581. The relative weights of tensile and fineness properties are 0.309 and 0.110 respectively. For the measurement of consistency of judgment, the original matrix is multiplied by the weight vector to obtain the product as shown below:

 1 1/2 3⎤⎡0.309 ⎤⎡0.930