Modelling, optimization and decision making techniques in designing of functional clothing
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Indian Journal of Fibre & Textile Research
Vol. 36, December 2011, pp. 398-409
Modelling, optimization and decision making techniques in designing of
functional clothing
Abhijit Majumdara & Surya Prakash Singhb
a b
Department of Textile Technology, Department of Management Studies, Indian Institute of Technology, New Delhi 110 016, India
and
Anindya Ghosh
Government College of Engineering and Textile Technology, Berhampore 742 101, India
Functional clothing are actually engineered textiles as they require to meet the stringent performance characteristics
rather than the aesthetic properties. Therefore, the trial and error approach of product design does not seem to be a viable
way for functional clothing. It needs more potent approaches of modelling, optimization and decision making so that the
design and functional requirements of clothing can be met with acceptable tolerance. This paper provides a brief outline of
various techniques of modelling, optimization and decision making intended for designing of functional clothing. In the
modelling part, regression and artificial neural network approaches have been discussed with the examples of thermal
property and water repellency modelling. Subsequently, linear programming and genetic algorithm techniques have been
invoked in the optimization part. Optimization of ultraviolet radiation protective clothing is taken up as a case study. Finally,
multi-criteria decision making techniques have been explained with the hypothetical example of selection of best body
armour vest for defense applications.
Keywords: Artificial neural network, Decision making technique, Functional clothing, Genetic algorithm,
Linear programming, Modeling technique, Regression
1 Introduction material often invokes the scientific knowledge of
Functional clothing are flexible materials decision making. Moreover, textile structures can be
consisting of a network of natural or synthetic fibres of various types. For example, fabrics can be made by
and it is designed to be practically useful rather than using weaving, knitting, nonwoven and braiding
attractive. Functional clothing has to fulfill various technologies and each of these technologies produces
requirements in terms of strength, modulus, a structure which is distinct from the rest. A woven
antibacterial activity, moisture management, heat structure is preferred for soft body armour whereas
resistance, electromagnetic radiation protection, water knitted fabrics and nonwoven assemblies are having
repellence and so on, depending on the domain of competitive edges in sports clothing and face masks
applications. Functional clothing are commonly used respectively. Modelling techniques are needed to
in sports, protection and medical applications. In stark understand the intricate relationships between various
contrast with normal apparels, functional clothing has structural attributes and functional properties of
to fulfill the performance requirements with accuracy clothing. In most of the practical cases, functional
and precision. Therefore, material selection, clothing have to fulfill multiple design or performance
engineering design optimization, structure-property requirements. For example, ultraviolet protective
modelling and performance evaluation have to be clothing should have a certain level of air
done systematically so that the clothing meet the permeability so that the wearer does not suffer from
requirements. Several materials may be available discomfort. Body armour should have high impact
which can fulfill the specification with different resistance and low bending stiffness so that the soldier
degree of satisfaction. Therefore, choosing the best can move with unconstrained agility. Fulfilling
multiple performance requirements by choosing
______________ proper input variables often needs optimization
a
To whom all the correspondence should be addressed.
E-mail: majumdar@textile.iitd.ac.in techniques. This paper presents a brief outline ofMAJUMDAR et al.: DESIGNING OF FUNCTIONAL CLOTHING 399
modelling, optimization and decision making systems yˆi = a + bx …(2)
which can be used for designing of functional
clothing. where a and b are the regression constants.
2 Modelling Systems ∂E ∂E
To calculate the values of a and b,and are
Model is a simplistic representation of some real ∂a ∂b
phenomenon. Models are often used to simulate the calculated and equated with zero which finally yield
performance of a product at various conditions and the following normal equations:
thus the trial and error involved in product design can
be obviated to certain extent. Mathematical models n n
are very popular in scientific fraternity as they are ∑ y = na + b∑ x
i i =1
…(3)
derived from the basic principles of science.
However, the performance of the mathematical n n n
models is often marred due to the simplified ∑ xy = a∑ x + b∑ x 2
…(4)
assumptions used while developing the models. i i =1 i =1
Statistical regression models are very easy to develop
using the experimental data. Prediction accuracy of The estimate of a and b can be obtained by solving
regression models is generally good provided the the system of normal equations and consequently the
proper form of functional relationship has been used. value of dependent variable ( yˆi ) can be predicted
In recent years, artificial neural network (ANN) has from the given value of independent variable (x)
become very popular due to its excellent prediction within the experimental range. Polynomial, power,
accuracy. In the following part of the paper, logarithmic and exponential are the popularly used
regression and ANN models have been discussed. forms of nonlinear models.
2.1 Regression Models
Regression models are very popular to establish the y = a + bx + cx 2 + ..... + kx10 ( Polynomial )
relationship between the dependent and independent y = ax b ( Power )
variables using experimental data. The number of y = a log bx ( Logarithmic)
dependent variable is only one, whereas the number
bx
of independent variable may be more than one. In y = ae ( Exponential )
most of the cases a linear form of relationship …(5)
between the variables is modeled and it is known as
linear regression. If the number of independent 2.2 Artificial Neural Network (ANN) Model
variable exceeds one then the model is called Artificial neural network (ANN) works by
multiple linear regression. The underlying principle mimicking the principles of biological nervous
of developing a regression model revolves around system2,3. Therefore, the elements of ANN are
the minimization of error function as defined analogical with the components of biological neurons.
below1: ANN is used in cases where huge number of
experimental data is available but the complex
n functional relationship between the variables is
E = ∑ ( yi − yˆi ) 2 …(1) unknown. A typical multilayer neural network is
i =1
shown in Fig. 1. The ANN model consists of at least
where E is the error function i.e. the squared three layers, each composed of certain number of
difference between the actual value of dependent neurons or mathematical processing elements. One or
variable ( yi ) and the predicted value of the dependent more hidden layers can be placed between the input
and output layers. All the input variables form the
variable ( yˆi ); and n, the number of experimental input layer. The variables to be modeled are placed in
observations. the output layer. The number of hidden layers and the
If the relationship between the dependent and number of neurons in hidden layers vary depending
independent variables is linear, then the following on the complexity of the function to be modelled.
equation can be written: Each neuron receives inputs from the neurons of the400 INDIAN J. FIBRE TEXT. RES., DECEMBER 2011
Fig. 1—Artificial neural network model
previous layer and these signals are multiplied by
Fig. 2—Fuzzy set of strong fibre
some numerical values or weights (analogical with
synapse strength of biological neuron). The weighted
Testing of a fibre x, whether it is strong or
inputs are then summed up and passed through a
otherwise, using the characteristic function χ is shown
transfer function or activation function (analogical
below:
with membrane potential of biological neuron),
which converts the output to a fixed range of values. 1, if x > 5
The output of transfer function is then transmitted to χ A ( x) = …(7)
the neurons of next layer. This process is continued 0, if x ≤ 5
and finally the predicted value of the output is
obtained. Initially, ANN starts with random
combination of weights connecting various neurons A fuzzy set is an extension of a classical set. If X is
and therefore the error is generally very high. The the universe of discourse and its elements are denoted
connection weights are then optimized using some by x, then a fuzzy set A in X is defined as a set of
mathematical algorithm so that the error function is ordered pairs, as shown below:
minimized. This process is known as training.
Various algorithms are available to train the ANN A = {x, µ A ( x)| x ∈ X } …(8)
and back-propagation algorithm is the most popular
among the existing algorithms. Details of back- where µA(x) is the membership function of x in A.
propagation algorithm can be found in published This can be extended to define the fuzzy set of
literature4. strong fibre as shown below:
2.3 Fuzzy Logic
A = {(4.0, 0.0), (4.5,0.5), (5.0,1.0)} …(9)
Fuzzy logic is an extension of crisp logic. It was
developed by Prof. Lotfi A. Zadeh at University of
California at Barkley, USA in 1965 (ref. 5). Fuzzy It implies that the belongingness to the fuzzy set of
logic is useful in imprecision handling as it is based strong fibre at 4.0, 4.5 and 5.0 gpd is 0, 0.5 and 1
on approximation rather than exactness. In crisp logic, respectively. This has been represented pictorially in
such as binary logic, variables are true or false, i.e. Fig. 2.
1 or 0. In fuzzy logic, a fuzzy set contains elements
with partial membership ranging from 0 to 1 to define Once the fuzzy sets are chosen, the membership
uncertainty for classes that do not have clearly function form for each set should be decided.
defined boundaries. For each input and output Membership function converts the input from 0 to 1,
variable of a fuzzy inference system (FIS), the fuzzy indicating the belongingness of the input to a fuzzy
sets are created by dividing the universe of discourse set. Membership function can have various forms,
into a number of sub-regions, named in linguistic such as triangle, trapezoid, sigmoid and Gaussian6,7.
terms like high, medium, and low. A classical set of The linguistic terms are then used to establish fuzzy
strong fibre (tenacity more than 5 gpd) may be rules which relate input fuzzy sets with output fuzzy
expressed as follows: sets. A fuzzy rule base consists of a number of fuzzy
if-then rules each of them has an antecedent part
A = {x | x > 5} …(6) (if part) and a consequent part (then part). ForMAJUMDAR et al.: DESIGNING OF FUNCTIONAL CLOTHING 401
example, in the case of two-input and single-output
fuzzy system, it could be expressed as follows:
If x is Ai and y is Bi then z is Ci …(10)
where x, y and z are the variables representing two
inputs and one output; Ai, Bi and Ci, the linguistic fuzzy
sets of x, y and z respectively. The output of each rule
is also a fuzzy set. All the output fuzzy sets are
aggregated into a single fuzzy set. Finally, the resulting
set is resolved to a crisp number by “defuzzification”.
2.4 Applications of Modelling Systems
There are numerous examples where regression
and ANN models have been used to predict the
properties of functional clothing8-10. Majumdar11
predicted the thermal conductivity of various knitted
Fig. 3—ANN model for predicting the thermal conductivity of
structures made from bamboo-cotton blended yarns knitted fabrics11
using ANN model. Knitted structure type (single
jersey, rib and interlock), yarn count, bamboo fibre %,
fabric thickness and areal density were used as inputs
as shown in Fig. 3. Out of 27 samples, 22 were used
for the training of ANN and remaining five samples
were used for the testing. The correlation coefficient
between actual and predicted values of thermal
conductivity was higher than 0.95 for both the Treatment time (s)
training and testing data. The mean absolute error was
Fig. 4—Effect of time and gas flow rate on water repellence12
lower than 3%. The author also analyzed the
developed model and found that finer yarns with gas flow rate (SLM) increases the water repellence
higher % of bamboo fibre produces lower thermal capability of fabrics as represented in Fig. 4.
conductivity. It was also revealed that volume
porosity is the key parameter which determines the 3 Optimization Systems
thermal conductivity of knitted fabrics. Optimization is a quantitative approach to produce
In another work, water repellence behaviour of the overall best results by choosing the proper combinations
plasma treated disposable surgical garments was of variables. In other words, problems that seek to
modelled by Allan et al.12 by using ANN. Cotton minimize or maximize a mathematical function
fabrics were treated with hexafluoroethane (C2F6) by involving a set of variables, subject to a set of
varying three process conditions namely power level, constraints, are classified as optimization problems13.
treatment time and gas flow rate (litres per minute). The mathematical function to be minimized or
The water repellency behaviour of treated fabrics was maximized is known as objective function. The other
measured objectively by image processing technique conditions to be fulfilled are termed as constraints. If the
and denoted by a parameter called final area index objective function as well as the constraints are linear
(FAI). Three ANN models were developed in stages functions of variables then the problem is called linear
with different numbers of training data. The final optimization problem. If the objective function or any of
model was developed with 80 samples which resulted the constraint equations involves nonlinearity then it is
mean error of 3.27 FAI and R2 of 0.79. However, classified as nonlinear optimization. A classification of
ANN model always underestimated the FAI value at optimization problem is shown in Fig. 5.
the optimum process conditions. This may be due to 3.1 Linear Programming
the fact that most of the training data belonged to the Linear programming is the simplest optimization
lower values of FAI. The effect of three process technique which attempts to maximize or minimize a
conditions was also investigated with the help of linear function of decision variables. The values of the
trained ANN model. Higher treatment time, power and decision variables are chosen such that a set of402 INDIAN J. FIBRE TEXT. RES., DECEMBER 2011
Fig. 5—Classification of optimization problem
restricting conditions is satisfied. Linear programming
involving only two decision variables can be solved
by using graphical method. However, iterative Fig. 6—Optimum point of constrained linear programming
Simplex method is used to solve linear programming problem
problem involving three or more decision variables.
Linear programming is very commonly used to solve
the product mix problem of manufacturing industries.
An example has been presented here for the
understanding of the readers.
Let, two sizes of functional clothing namely M and
L are being manufactured in an industry which aims at
maximization of overall profit. Profit per unit sales is
Rs. 5000 and 10,000 for sizes M and L respectively.
Besides, the machine hour requirement per unit
production is 2 and 2.5 for sizes M and L respectively.
The company must produce at least 10 functional
clothing in a day to meet the market demand. The
stated facts can be converted to a linear programming Fig. 7—Pareto optimal front for UPF and air permeability
problem, as shown below:
will provide maximum ultraviolet protection factor
Objective function: Maximize : 5000 M + 10000 L (UPF) but minimum air permeability. On the other
…(11) hand, there will be another design which will provide
minimum UPF but maximum air permeability.
2 M + 2.5L ≤ 24 Between these two extreme designs, infinite number
Subject to: …(12)
M + L ≥ 10 of designs will exist which are of some compromise
between UPF and air permeability. This set of trade-
After solving the above linear programming off designs is known as a Pareto set. The curve
problem, it is found that the maximum profit of the created by plotting objective one (UPF) against
industry will be Rs. 90,000 per day provided it objective two (air permeability) for the best designs is
manufactures 2 and 8 units of functional clothing of known as Pareto frontier. None of the solutions in
sizes M and L respectively. A graphical representation of Pareto front is better than the other, i.e. any one of
this linear programming problem is depicted in Fig. 6. them is an acceptable solution. The choice of one
design solution over other exclusively depends upon
3.2 Multi-objective Optimization and Goal Programming the requirement of the process engineer. Majumdar
Adding multiple objectives to an optimization et al.14 developed Pareto optimal front for UPF and air
problem increases the computational complexity. For permeability of cotton woven fabrics as depicted in
example, if the design of ultraviolet protective Fig. 7. The optimal design fronts are different for
clothing has to be optimized which will provide good various yarn linear densities. It is observed that for a
air permeability then these two objectives conflict and fabric having UPF value of 30, the air permeability
a trade-off is needed. There will be one design which will be better if it is woven using 20 Ne weft yarns.MAJUMDAR et al.: DESIGNING OF FUNCTIONAL CLOTHING 403
Goal programming technique is often used to solve
the multi-objective optimization problems. In goal
programming, a numeric goal is established for each
goal function or constraint. The objective function
minimizes the weighted sum of undesirable deviations
from the respective goals. The example given in the
previous section can be converted to a goal
programming problem assuming that the profit goal of
the organization is Rs. 90000.
5000 M + 10000 L + d1− − d1+ = 90000
2 M + 2.5 L + d 2 − − d 2 + = 24 …(13)
− + Fig. 8—Function having local and global minima
M + L + d3 − d 3 = 10
survival. Over successive generations, the population
− + − ‘evolves’ toward an optimal solution. Genetic
Minimise = w d + w2 d 2 + w3 d3
1 1 …(14)
algorithm can be applied to solve a variety of
optimization problems where the objective function is
where w1, w2 and w3 are the weights assigned to the
discontinuous, non-differentiable, stochastic or highly
deviational variables.
non-linear. An elaborate description of GA can be
3.3 Genetic Algorithm (GA) found in published literature3,8,15.
The genetic algorithm (GA) is an unorthodox
3.4 Simulated Annealing (SA)
search method based on natural selection process for
The SA is a useful meta-heuristic for solving hard
solving complicated optimization problems. John
combinatorial optimization problems and the QAP in
Holland15 of University of Michigan developed it in
particular. It was first introduced by Kirkpatrick
the early 1970s. Unlike conventional derivative based
et al.16. The SA is a step-by-step method which could
optimization that requires differentiability of the
be considered as an improvement of the local
function to be optimized, GA can handle functions
optimization algorithm. The local optimization
with discontinuities or piece-wise segments. Besides,
algorithm proceeds by generating, at each iteration, a
gradient based optimization algorithms can get stuck
solution in the neighbourhood of the previous one. If
in local minima or maxima as they rely on the slope
the value of criterion corresponding to the new
of the function. Genetic algorithm overcomes this
solution is better than the previous one, the new
problem. The following function is having local and
solution is selected, otherwise it is rejected. The SA
global minima (Fig. 8):
algorithm terminates either when it is no longer
f ( x) = ( x − 1)( x − 2)( x − 3)( x − 4)( x − 5)( x − 6) …(15) possible to improve the solution or the maximum
number of trials decided by the user is reached. The
Gradient based optimization, while searching for main drawback of the local optimization algorithm is
the global minima, may get stuck at 3.5 which is that it terminates at a local minimum which depends
actually local minima. However, GA is certain to find on the initial solution and may be far from the global
out the global minima of the function at 1.34. minimum.
To perform the optimization task, GA maintains a The SA algorithm avoids entrapment in a local
population of points called ‘individuals’, each of optimum. The difference with the local optimization
which is a potential solution to the optimization is that a solution A0 derived from a solution A is not
problem. Generally, the individuals are coded with a only accepted if A0 is better than A but it may also be
string of binary numbers. The GA repeatedly modifies accepted if A0 is worse than A. Boltzmann’s law is
the population of individual solutions using selection, used to determine this acceptance probability that is
crossover and mutation operators. At each step, the given as P(accept)= e-∆z/bt, where b is Boltzmann’s
genetic algorithm selects individuals from the current constant and t (TI < t < TF, where TI and TF are the
population (parents) and uses them to produce initial and final temperatures respectively) is the given
children for the next generation, which competes for parameter called the temperature which changes over404 INDIAN J. FIBRE TEXT. RES., DECEMBER 2011
time according to some cooling schedule, and (c) Geometric function: Nk = Nk-1/a, where ‘a’ is
∆z = z(A) - z(A) >=0. This is known as the constant less than 1 and k = 0, 1, . . . ,Q;
Metropolis acceptance rule which implies that (d) Logarithmic function: Nk = Constant/log (Tk),
(i) the smaller the increase of the ∆z value, the more where k = 0, 1, . . . ,Q; and
likely the new solution is selected, and (ii) the lower (e) Exponential function: Nk = (Nk-1)1/a, where ‘a’ is
the value of ‘t’ and greater the number of trials ‘Q’, constant less than 1 and k = 0, 1, . . . ,Q.
the less likely the new solution is selected.
3.4.3 Cooling (Annealing) Schedule
The basic algorithm of SA is given as follows: Temperature is used to compute the acceptance
Step 1— Randomly, select the initial solution ‘i’ as a probability of a solution which is worse than the
starting solution for SA. previous one. The few functions for updating the
Step 2— Choose an initial temperature TI > 0. temperature are as follows:
Step 3—Choose the temperature updating function (a) Arithmetic function tk+1 = tk - constant, k = 0, 1, . .
i.e. annealing (or cooling) schedule. . ,Q;
Step 4— Choose the epoch length function. (b) Geometric function tk+1 = α.tk where k = 0, 1, . . . ,
Step 5—Set temperature change counter t = 0 and Q, t0 = TI (initial temperature) constant, and α < 1;
epoch length counter l = 0. (c) Logarithmic function tk = constant/log(k+2),
Step 6—Generate Solution A0 in the neighbourhood where k = 0, 1, . . . ,Q;
of A by exchanging two facilities. (d) Inverse function tk+1 = tk/ (1+α.tk), where k = 0, 1, .
Step 7—Calculate ∆z = z(A) - z(A). . . ,Q, t0=TI (initial temperature) constant, αMAJUMDAR et al.: DESIGNING OF FUNCTIONAL CLOTHING 405
exists, i.e. a move is allowed even if a new solution s′
from the neighbourhood of the current solution s is
worse than the current one.
Naturally, the return to the locally optimal
solutions previously visited is to be forbidden to avoid
cycling. TS is based on the methodology of
prohibitions, some moves are "frozen" (become
"tabu") from time to time. More formally, the TS
algorithm starts from an initial solution s° in S. The
process is then continued in an iterative way −
moving from a solution s to a neighbouring one s′. At Fig. 9— Pseudo code for generic ACO procedure.
each step of the procedure, a subset of the
neighbouring solutions of the current solution is pheromone. More the ants will take the left path,
considered, and the move to the solution that higher the pheromone trail is. This fact will be
improves the objective function value f is chosen. increased by the evaporation stage.
Naturally, s′ must not necessary be better than s; if This principle of communicating ants has been
there are no improving moves, the algorithm chooses used as a framework for solving combinatorial
the one that least increases the objective function [a optimization problems. Figure 1 presents the generic
move is performed to the neighbour s′ even if f(s′) > ant colony algorithm. The first step consists mainly in
f(s)]. In order to eliminate an immediate returning to the initialization of the pheromone trail. In the
the solution just visited, the reverse move must be iteration step, each ant constructs a complete solution
forbidden. This is done by storing the corresponding to the problem according to a probabilistic state
solution (move) (or its "attribute") in a memory transition rule. The state transition rule depends
[called a tabu list (T)]. The tabu list keeps information mainly on the state of the pheromone. Once all ants
on the last h = | T | moves which have been done generate a solution, a global pheromone updating rule
during the search process. Thus, a move from s to s′ is is applied in two phases— an evaporation phase
considered as tabu if s′ (or its "attribute") is contained where a fraction of the pheromone evaporates, and a
in T. This way of proceeding hinders the algorithm reinforcement phase where each ant deposits an
from going back to a solution reached within the last amount of pheromone which is proportional to the
h steps. The pseudo-code for the standard (pure) tabu fitness of its solution. This process is iterated until
search paradigm is presented in Fig. 1. More details algorithm satisfies stopping criteria.
on the fundamentals and principles of TS are found in 3.7 Applications of Optimization Systems
literature19,20. Srivastav22 attempted to optimize the polyester-
cotton woven fabric parameters (weft count,
3.6 Ant Colony Optimization (ACO)
picks/cm, and % of polyester in weft) so that air
The ants based algorithm has been introduced by permeability (AP) is maximized and the ultraviolet
Maniezzo and Colorni21 which is based on the protection factor (UPF) meets the minimum
principle of simple communication, an ant group is requirement. Thirty-six woven fabric samples were
able to find the shortest path between any two points. prepared using three different levels of weft yarn
During their trips a chemical trial (pheromone) is left count, three different levels of picks per cm values
on the ground. The role of this trail is to guide other and four different levels of polyester fibre % in the
ants towards the target point. For one ant, the path is weft yarn. Linear regression equations were
chosen according to the quantity of pheromone. developed for relating AP and UPF with the
Furthermore, this chemical substance has a decreasing
independent variables (x weft count in tex, y
action over time, and the quantity left by one ant
picks per cm and z polyester fibre % in weft). The
depends on the amount of food found and the number
optimization problem is shown below:
of ants using this trail. As illustrated in Fig. 9, when
facing an obstacle, there is an equal probability for Maximize AP =173.618-1.363x -5.396y +0.056z …(16)
every ant to choose the left or right path. As the left Subject to
trail is shorter than the right one and so required less
travel time, it will end up with higher level of UPF + -16.856 + 0.368 x + 0.749y + 0.068z ≥ 14 …(17)406 INDIAN J. FIBRE TEXT. RES., DECEMBER 2011
15 ≤ x ≤30, 16≤ y ≤24, 0≤ z ≤100 …(18) 4.1 Analytic Hierarchy Process (AHP)
AHP was developed by Saaty23-26. In AHP a pair-
The optimization problem was solved using linear wise comparison matrix of attributes is constructed
programming technique. The values of x, y and z were using a nine point scale of relative importance. An
found to be 30, 17.5 and 100. One validation fabric attribute compared to itself or with any other attribute
sample was then woven using the solution parameters. having the same importance is assigned the value 1.
The functional properties of the validation fabric Thus, the right diagonal of pair-wise comparison
sample showed reasonably good agreement with the matrix is comprised only 1. The numbers 3, 5, 7 and 9
targeted properties. The deviation in air permeability correspond to verbal judgments of ‘moderate
and UPF was lower than 1 unit and 3 unit importance’, ‘strong importance’, ‘very strong
respectively. importance’ and ‘absolute importance’ respectively.
For N decision criteria, the size of the comparison
4 Multi-criteria Decision Making (MCDM) matrix will be N × N and the entry cij will denote the
Multi-criteria decision making (MCDM) is a relative importance of criterion i with respect to
branch of operations. It is useful when several criterion j. In the matrix, cij = 1 if when i = j and
alternatives are to be evaluated or ranked with respect 1
to the overall goal based on numerable decision c ji = . A typical pair-wise comparison matrix (C1)
cij
criteria.
The three main steps of MCDM are as follows: of criteria is shown below:
1 c12 ... c1N
(i) Determine the goal, relevant criteria and c 1 ... c2 N
alternatives of the decision problem. C1 = 21
(ii) Ascertain numerical weights (or scores) to ... ... 1 ...
relative importance of criteria.
(iii) Process alternative scores to determine the cN 1 cN 2 ... 1
ranking of each alternative. The principle eigen vector of the above matrix
represents the relative weights of the decision criteria.
Various MCDM techniques such as weighted sum The relative weight of the ith criteria (Wi) is determined
model (WSM), weighted product model (WPM), by calculating the geometric mean of the i th row (GMi)
analytic hierarchy process (AHP), TOPSIS, and of the above matrix and then normalizing the geometric
elimination and choice translating reality (ELECTRE) means of rows. This can be represented as follows:
can be used in engineering decision making problems,
1
depending upon the nature and complexity of
situation. AHP is one of the most popular methods of N N GM i
GM i = ∏ cij and Wi = N …(20)
MCDM. The reason behind the popularity of AHP j =1
lies in the fact that it can handle objective as well as ∑ GM ii =1
subjective attributes, and the criteria weights and
alternative scores are calculated trough the formation Similarly, N numbers of pair-wise comparison
of pair-wise comparison matrix, which is the heart of matrices, one for each criterion, of M x M order are
the AHP. The total number of pair-wise comparisons formed where each alternative is compared with each
in a decision making problem, having M alternatives other. The eigen vector of each of these ‘N’ matrices
and N criteria, can be expressed by the following represents the alternative performance scores in the
equation: corresponding criterion and from a column of the
N ( N − 1) M ( M − 1) final decision matrix as shown in Table 1.
+ N. ...(19) M
2 2
This may be unmanageable where a huge number
Here, ∑a
i =1
ij =1 …(21)
of decision criteria and alternatives are involved. The The final priority of all the alternatives is
TOPSIS is more potent in handling the tangible calculated by considering the alternative scores (aij) in
attributes and there is no limit in terms of number of each criterion and the weight of the corresponding
criteria or alternative. criterion (Wj) using the following equation:MAJUMDAR et al.: DESIGNING OF FUNCTIONAL CLOTHING 407
Table 1— Decision matrix of AHP The normalized matrix is then converted to
Alternatives Decision criteria weighted normalized matrix by multiplying each
C1 C2 C3 … CN
column of the normalized decision matrix with the
associated criteria weight. Hence, an element vij of
(W1) (W2) (W3) … (WN) weighted normalized matrix is represented as follows:
A1 a11 a12 a13 … a1N
vij = rij .W j ...(24)
A2 a21 a22 a23 … a2N
The weights of decision criteria can be determined
A3 a31 a32 a33 … a3N
by the AHP, which has been explained in the previous
… … … … … … section. The next step produces the positive ideal (A*)
AM aM1 aM2 aM3 … aMN and negative ideal (A-) solutions in the following
manner:
N
AAHP = max ∑ aij.W j for i = 1,2,3, …..M …(22) A* = {(max vij / j ∈ J ), (min vij / j ∈ J ') for i = 1, 2,3,....M }
j =1
= {v1 *, v2 *,.....vN *} …(25)
Alternative with the maximum score is the most
preferred one and vice versa. A− = {(min vij / j ∈ J ), (max vij / j ∈ J ') for i = 1, 2,3,....M }
4.2 Technique for Order Preference by Similarity to Ideal = {v1− , v2 − ,....., vN − } …(26)
Solutions (TOPSIS)
TOPSIS was developed by Hwang and Yoon27. where
The basic philosophy of this method is that the J = { j = 1, 2,...., N / j associated with benefit or positive criteria}
selected alternative should have the shortest and
geometrical distance from the best possible solution J ' = { j = 1, 2,...., N / j associated with cost or negative criteria}
and longest distance from the worst possible solution. For the benefit criteria, the decision maker prefers
First, the relevant objective or goal, decision criteria
the maximum value among the alternatives.
and alternatives of the problem are identified. Then Therefore, A* indicates the positive ideal solution.
the decision matrix is formulated based on the Similarly, A- indicates the negative ideal solution. The
information available regarding the problem. If the separation distances of each alternative from A* and
number of alternatives is M and the number of criteria A- are calculated using the following expressions.
is N, then the decision matrix having an order of
M × N can be represented as follows: 0.5
N
Si = ∑ (vij − v j * ) 2 and
*
a11 a12 ... a1N
j =1
a a22 ... a2 N
= 21
0.5
DMxN N
... ... ... ... Si = ∑ (vij − v j − )2 , i = 1, 2,..., M
−
…(27)
j =1
aM 1 aM 2 ... aMN
where Si* and Si- are the separation distances of
where an element aij of the decision matrix represents alternative i from A* and A- respectively.
the actual value of the i th alternative in terms of j th Finally, the relative closeness (Ci*) value, to the
criteria. The decision matrix is converted to ideal solution, is determined for each alternative using
normalized decision matrix, so that the scores the following equation; the value of Ci* lies within the
obtained in different scales or units become range 0 - 1:
comparable. An element rij of the normalized decision
matrix can be calculated using the following equation: Si −
Ci* = …(28)
aij ( Si * + Si − )
rij = 0.5
…(23)
M 2
∑ (aij ) The alternative having the maximum Ci* is the best
i =1 and vice versa.408 INDIAN J. FIBRE TEXT. RES., DECEMBER 2011
4.3 Application of Decision Making Systems Table 2— Pair-wise comparison matrix of decision criteria
An application of AHP system has been Parameter Impact Comfort Cost Geometric Normalized
demonstrated with a hypothetical example of body resistance score mean geometric mean
armour selection based on three decision criteria Impact 1 3 5 2.46 0.64
namely impact resistance, comfort score and cost. The resistance
impact resistance of body armour is characterized by Comfort 1/3 1 3 1 0.26
the V50 speed at which the bullet has equal probability score
to pierce the vest or to be stopped by the vest. Cost 1/5 1/3 1 0.41 0.10
Comfort score has been taken as an overall index Table 3— Features of body armours
representing the flexibility, thermal resistance and Alternatives Impact resistance Comfort score Cost, Rs
moisture vapour transmission of the body armour. V50, m/s
Higher V50 speed is a desirable or benefit criterion and A1 450 1000 40,000
A2 500 1500 50,000
so is the comfort score. However, price of the vest is a A3 475 800 45,000
negative or cost criterion and lower value is desirable. A4 400 2000 45,000
Table 2 shows the pair-wise comparison matrix of Ideal 500 2000 40000
three decision criteria based on the perception of Worst 400 750 50,000
decision maker. Here numerical scores has been given
Table 4— Normalized features of bullet proof body armours
as per Saaty’s23 nine point scale as described in
Alternatives Impact Comfort Cost
section 4.1. Impact resistance has moderate dominance resistance (0.64) score (0.26) (0.10)
over the comfort and comfort has moderate
A1 0.9 0.5 1.0
dominance over the cost. Cost has the least influence A2 1.0 0.75 0.8
on decision as the high impact resistance and greater A3 0.95 0.40 0.89
comfort are imperative for body armours. After A4 0.8 1 0.89
calculating the normalized geometric mean of rows, it
has been found that the weights of impact resistance, the pair-wise comparison matrix (Table 2) and see
comfort score and cost are 0.64, 0.26 and 0.10 how the ranking of alternatives are responding. This is
respectively. The scores of four alternatives known as sensitivity analysis.
(A1 - A4) in three decision criteria are shown in Table 3.
Table 4 shows the normalized scores of alternatives. 5 Conclusion
The scores have been normalized using the following Various modelling, optimization and decision
expressions: making techniques have been discussed in this paper
with suitable examples pertaining to functional
Score clothing. These techniques are very frequently used in
Normalized score= (For a benefit criterion)
Maximum score manufacturing and service industries. Unfortunately,
these techniques have seldom received any attention
Minimum score
Normalized score= (For a cost criterion) in traditional textile industry. As the quality
Score requirement for the functional clothing are very
stringent, these modelling, optimization and decision
The weighted score of four alternatives can be
making techniques with sound mathematical
calculated as follows.
foundation are very important for functional clothing
Score A1 = 0.64 × 0.9 + 0.2 × 0.5 + 0.1 × 1 = 0.776 industries. It is pertinent to mention here that in recent
years some very powerful modelling and optimization
Score A 2 = 0.64 × 1 + 0.2 × 0.75 + 0.1 × 0.8 = 0.87 tools like support vector machine, simulated
Score A 3 = 0.64 × 0.95 + 0.2 × 0.4 + 0.1 × 0.89 = 0.777 annealing, particle swarm optimization and ant colony
Score A 4 = 0.64 × 0.8 + 0.2 × 1 + 0.1 × 0.89 = 0.801 optimization have been developed. Researches are
being done to amalgamate multiple modelling and
Here, alternative A2 is the most preferred one optimization tools so that they become more powerful
although it is not the best in comfort and cost criteria. and complement each other. It is expected that these
In contrast, alternative A1 is least preferred alternative emerging systems will be embraced by the functional
although it is the cheapest among the alternatives. The clothing industry to solve the complex problems
decision maker can further change the scores given in related to design and manufacturing.MAJUMDAR et al.: DESIGNING OF FUNCTIONAL CLOTHING 409 References 13 Sarkar R A & Newton C S, Optimization Modelling: A 1 Johnson R A & Gupta C B, Probability and Statistics for Practical Approach (CRC press, London), 2007. Engineers (Pearson Education, New Delhi), 2009. 14 Majumdar A, Kothari V K, Mondal A & Hatua P, 2 Kartalopoulos S V, Understanding Neural Networks and Optimization of woven fabric parameters for ultraviolet Fuzzy Logic: Basic Concepts and Applications (Prentice-Hall radiation protection, paper presented at the 39th Textile of India Pvt. Ltd., New Delhi), 2000. Research Symposium, New Delhi, December 2010. 3 Rajasekaran S & Pai G A V, Neural Networks, Fuzzy Logic 15 Holland J L, Genetic algorithms and their optimal allocation and Genetic Algorithms: Synthesis and Applications of trials, SIAM J Comput, 2(2) (1973) 88-105. (Prentice-Hall of India Pvt. Ltd., New Delhi), 2003. 16 Kirkpatrick S, Gelatt C D & Vecchi M P, Optimization by 4 Rumelhart D E, Hinton G & Williams R J, Learning Internal Simulated Anealing, Science, 220 (1983) 671-680. Representations by Error propagation, in Parallel Distributed 17 Van Laarhovern P J M & Aarts E H L, Simulated Annealing: Processing, Vol 1 (MIT Press, Cambridge), 1986, 318-362. Theory and Application (Springer Verlag Publication, The 5 Zadeh L A, Fuzzy sets, Information and Control, 8 (1965) Netherlands), 1987. 338-353. 18 Dowsland K A, Some experiments with simulated annealing 6 Berkan R C & Trubatch S L, Fuzzy Systems Design techniques for packing problems, Eur J Operational Res, 68 Principles (Standard Publishers Distributors, New Delhi), (1993) 389-399. 2000. 19 Glover F, Tabu Search: Part I, ORSA J Comput, 1 (1989) 7 Klir G J & Yuan B, Fuzzy Sets and Fuzzy Logic: Theory and 190–206. Applications (Prentice-Hall of India Pvt. Ltd., New Delhi), 20 Glover F, Tabu Search: Part II, ORSA J Comput, 2 (1990) 2000. 4–32. 8 Majumdar A, Soft computing in fibrous materials 21 Maniezzo V & Colorni A, The ant system applied to the engineering, Text Prog, 3 (2011) 1-95. quadratic assignment problem, IEEE Transaction on 9 Lee I, Kosko B & Anderson W F, Modelling gunshot bruises Knowledge and Data Engineering, 11 (5) (1999) 769-778. in soft body armour with an adaptive fuzzy system, IEEE 22 Srivastava N, Ultraviolet radiation protection by woven Transactions on Systems, Man, and Cybernetics, Part B: fabrics, M.Tech thesis, Indian Institute of Technology, Delhi, Cybernetics, 35 (2005) 1374-1390. India, 2011. 10 Ramaiah G B, Chennaiah R Y & Satyanarayanarao G K, 23 Saaty T L, The Analytic Hierarchy Process (McGraw-Hill Investigation and modelling on protective textiles using International, New York), 1980. artificial neural networks for defense applications, Mater Sci 24 Saaty T L, How to make a decision: The Analytic Hierarchy Eng B, 168 (2010) 100-105. Process, Eur J Operational Res, 48 (1990)9-26. 11 Majumdar A, Modelling of thermal conductivity of knitted 25 Saaty T L, Priority setting in complex problems, IEEE fabrics made from bamboo fibre blended yarns using Transaction on Eng. Management, EM 30, 3 (1983) 140-155. artificial neural network, J Text Inst, 102 (2010) 752-762. 26 Saaty T L, Axiomatic foundation of the Analytic Hierarchy 12 Allan G, Fotheringham A & Weedall P, The use of plasma Process, Management Sci, 32 (7) (1983) 841-855. and neural modelling to optimise the application of a 27 Hwang C L & Yoon K, Multiple Attribute Decision Making: repellent coating to disposable surgical garments, AUTEX Methods and Applications ( Springer-Verlag, New York), Res J, 2(2) (2002) 64-68. 1981.
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