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
Functional clothing are flexible materials
consisting of a network of natural or synthetic fibres
and it is designed to be practically useful rather than
attractive. Functional clothing has to fulfill various
requirements in terms of strength, modulus,
antibacterial activity, moisture management, heat
resistance, electromagnetic radiation protection, water
repellence and so on, depending on the domain of
applications. Functional clothing are commonly used
in sports, protection and medical applications. In stark
contrast with normal apparels, functional clothing has
to fulfill the performance requirements with accuracy
and precision. Therefore, material selection,
engineering design optimization, structure-property
modelling and performance evaluation have to be
done systematically so that the clothing meet the
requirements. Several materials may be available
which can fulfill the specification with different
degree of satisfaction. Therefore, choosing the best
______________
a
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E-mail: [email protected]
material often invokes the scientific knowledge of
decision making. Moreover, textile structures can be
of various types. For example, fabrics can be made by
using weaving, knitting, nonwoven and braiding
technologies and each of these technologies produces
a structure which is distinct from the rest. A woven
structure is preferred for soft body armour whereas
knitted fabrics and nonwoven assemblies are having
competitive edges in sports clothing and face masks
respectively. Modelling techniques are needed to
understand the intricate relationships between various
structural attributes and functional properties of
clothing. In most of the practical cases, functional
clothing have to fulfill multiple design or performance
requirements. For example, ultraviolet protective
clothing should have a certain level of air
permeability so that the wearer does not suffer from
discomfort. Body armour should have high impact
resistance and low bending stiffness so that the soldier
can move with unconstrained agility. Fulfilling
multiple performance requirements by choosing
proper input variables often needs optimization
techniques. This paper presents a brief outline of
MAJUMDAR et al.: DESIGNING OF FUNCTIONAL CLOTHING
modelling, optimization and decision making systems
which can be used for designing of functional
clothing.
2 Modelling Systems
Model is a simplistic representation of some real
phenomenon. Models are often used to simulate the
performance of a product at various conditions and
thus the trial and error involved in product design can
be obviated to certain extent. Mathematical models
are very popular in scientific fraternity as they are
derived from the basic principles of science.
However, the performance of the mathematical
models is often marred due to the simplified
assumptions used while developing the models.
Statistical regression models are very easy to develop
using the experimental data. Prediction accuracy of
regression models is generally good provided the
proper form of functional relationship has been used.
In recent years, artificial neural network (ANN) has
become very popular due to its excellent prediction
accuracy. In the following part of the paper,
regression and ANN models have been discussed.
2.1 Regression Models
Regression models are very popular to establish the
relationship between the dependent and independent
variables using experimental data. The number of
dependent variable is only one, whereas the number
of independent variable may be more than one. In
most of the cases a linear form of relationship
between the variables is modeled and it is known as
linear regression. If the number of independent
variable exceeds one then the model is called
multiple linear regression. The underlying principle
of developing a regression model revolves around
the minimization of error function as defined
below1:
n
E = ∑ ( yi − yˆi ) 2
…(1)
i =1
where E is the error function i.e. the squared
difference between the actual value of dependent
variable ( yi ) and the predicted value of the dependent
variable ( yˆi ); and n, the number of experimental
observations.
If the relationship between the dependent and
independent variables is linear, then the following
equation can be written:
399
yˆi = a + bx
…(2)
where a and b are the regression constants.
∂E
∂E
and
are
∂a
∂b
calculated and equated with zero which finally yield
the following normal equations:
To calculate the values of a and b,
n
n
i
i =1
∑ y = na + b∑ x
n
n
…(3)
n
∑ xy = a∑ x + b∑ x
i =1
i
2
…(4)
i =1
The estimate of a and b can be obtained by solving
the system of normal equations and consequently the
value of dependent variable ( yˆi ) can be predicted
from the given value of independent variable (x)
within the experimental range. Polynomial, power,
logarithmic and exponential are the popularly used
forms of nonlinear models.
y = a + bx + cx 2 + ..... + kx10
( Polynomial )
y = ax b
( Power )
y = a log bx
( Logarithmic)
y = ae
bx
( Exponential )
…(5)
2.2 Artificial Neural Network (ANN) Model
Artificial neural network (ANN) works by
mimicking the principles of biological nervous
system2,3. Therefore, the elements of ANN are
analogical with the components of biological neurons.
ANN is used in cases where huge number of
experimental data is available but the complex
functional relationship between the variables is
unknown. A typical multilayer neural network is
shown in Fig. 1. The ANN model consists of at least
three layers, each composed of certain number of
neurons or mathematical processing elements. One or
more hidden layers can be placed between the input
and output layers. All the input variables form the
input layer. The variables to be modeled are placed in
the output layer. The number of hidden layers and the
number of neurons in hidden layers vary depending
on the complexity of the function to be modelled.
Each neuron receives inputs from the neurons of the
INDIAN J. FIBRE TEXT. RES., DECEMBER 2011
400
Fig. 1—Artificial neural network model
previous layer and these signals are multiplied by
some numerical values or weights (analogical with
synapse strength of biological neuron). The weighted
inputs are then summed up and passed through a
transfer function or activation function (analogical
with membrane potential of biological neuron),
which converts the output to a fixed range of values.
The output of transfer function is then transmitted to
the neurons of next layer. This process is continued
and finally the predicted value of the output is
obtained. Initially, ANN starts with random
combination of weights connecting various neurons
and therefore the error is generally very high. The
connection weights are then optimized using some
mathematical algorithm so that the error function is
minimized. This process is known as training.
Various algorithms are available to train the ANN
and back-propagation algorithm is the most popular
among the existing algorithms. Details of backpropagation algorithm can be found in published
literature4.
2.3 Fuzzy Logic
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
logic is useful in imprecision handling as it is based
on approximation rather than exactness. In crisp logic,
such as binary logic, variables are true or false, i.e.
1 or 0. In fuzzy logic, a fuzzy set contains elements
with partial membership ranging from 0 to 1 to define
uncertainty for classes that do not have clearly
defined boundaries. For each input and output
variable of a fuzzy inference system (FIS), the fuzzy
sets are created by dividing the universe of discourse
into a number of sub-regions, named in linguistic
terms like high, medium, and low. A classical set of
strong fibre (tenacity more than 5 gpd) may be
expressed as follows:
A = {x | x > 5}
…(6)
Fig. 2—Fuzzy set of strong fibre
Testing of a fibre x, whether it is strong or
otherwise, using the characteristic function χ is shown
below:
if x > 5
1,

χ A ( x) = 
…(7)
0,
if x ≤ 5

A fuzzy set is an extension of a classical set. If X is
the universe of discourse and its elements are denoted
by x, then a fuzzy set A in X is defined as a set of
ordered pairs, as shown below:
A = {x, µ A ( x)| x ∈ X }
…(8)
where µA(x) is the membership function of x in A.
This can be extended to define the fuzzy set of
strong fibre as shown below:
A = {(4.0, 0.0), (4.5,0.5), (5.0,1.0)}
…(9)
It implies that the belongingness to the fuzzy set of
strong fibre at 4.0, 4.5 and 5.0 gpd is 0, 0.5 and 1
respectively. This has been represented pictorially in
Fig. 2.
Once the fuzzy sets are chosen, the membership
function form for each set should be decided.
Membership function converts the input from 0 to 1,
indicating the belongingness of the input to a fuzzy
set. Membership function can have various forms,
such as triangle, trapezoid, sigmoid and Gaussian6,7.
The linguistic terms are then used to establish fuzzy
rules which relate input fuzzy sets with output fuzzy
sets. A fuzzy rule base consists of a number of fuzzy
if-then rules each of them has an antecedent part
(if part) and a consequent part (then part). For
MAJUMDAR 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
structures made from bamboo-cotton blended yarns
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
training and testing data. The mean absolute error was
lower than 3%. The author also analyzed the
developed model and found that finer yarns with
higher % of bamboo fibre produces lower thermal
conductivity. It was also revealed that volume
porosity is the key parameter which determines the
thermal conductivity of knitted fabrics.
In another work, water repellence behaviour of the
plasma treated disposable surgical garments was
modelled by Allan et al.12 by using ANN. Cotton
fabrics were treated with hexafluoroethane (C2F6) by
varying three process conditions namely power level,
treatment time and gas flow rate (litres per minute).
The water repellency behaviour of treated fabrics was
measured objectively by image processing technique
and denoted by a parameter called final area index
(FAI). Three ANN models were developed in stages
with different numbers of training data. The final
model was developed with 80 samples which resulted
mean error of 3.27 FAI and R2 of 0.79. However,
ANN model always underestimated the FAI value at
the optimum process conditions. This may be due to
the fact that most of the training data belonged to the
lower values of FAI. The effect of three process
conditions was also investigated with the help of
trained ANN model. Higher treatment time, power and
Fig. 3—ANN model for predicting the thermal conductivity of
knitted fabrics11
Treatment time (s)
Fig. 4—Effect of time and gas flow rate on water repellence12
gas flow rate (SLM) increases the water repellence
capability of fabrics as represented in Fig. 4.
3 Optimization Systems
Optimization is a quantitative approach to produce
overall best results by choosing the proper combinations
of variables. In other words, problems that seek to
minimize or maximize a mathematical function
involving a set of variables, subject to a set of
constraints, are classified as optimization problems13.
The mathematical function to be minimized or
maximized is known as objective function. The other
conditions to be fulfilled are termed as constraints. If the
objective function as well as the constraints are linear
functions of variables then the problem is called linear
optimization problem. If the objective function or any of
the constraint equations involves nonlinearity then it is
classified as nonlinear optimization. A classification of
optimization problem is shown in Fig. 5.
3.1 Linear Programming
Linear programming is the simplest optimization
technique which attempts to maximize or minimize a
linear function of decision variables. The values of the
decision variables are chosen such that a set of
INDIAN J. FIBRE TEXT. RES., DECEMBER 2011
402
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
Simplex method is used to solve linear programming
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
problem, as shown below:
Objective function: Maximize : 5000 M + 10000 L
…(11)
Subject to:
2 M + 2.5L ≤ 24
M + L ≥ 10
…(12)
After solving the above linear programming
problem, it is found that the maximum profit of the
industry will be Rs. 90,000 per day provided it
manufactures 2 and 8 units of functional clothing of
sizes M and L respectively. A graphical representation of
this linear programming problem is depicted in Fig. 6.
3.2 Multi-objective Optimization and Goal Programming
Adding multiple objectives to an optimization
problem increases the computational complexity. For
example, if the design of ultraviolet protective
clothing has to be optimized which will provide good
air permeability then these two objectives conflict and
a trade-off is needed. There will be one design which
Fig. 6—Optimum point of constrained linear programming
problem
Fig. 7—Pareto optimal front for UPF and air permeability
will provide maximum ultraviolet protection factor
(UPF) but minimum air permeability. On the other
hand, there will be another design which will provide
minimum UPF but maximum air permeability.
Between these two extreme designs, infinite number
of designs will exist which are of some compromise
between UPF and air permeability. This set of tradeoff designs is known as a Pareto set. The curve
created by plotting objective one (UPF) against
objective two (air permeability) for the best designs is
known as Pareto frontier. None of the solutions in
Pareto front is better than the other, i.e. any one of
them is an acceptable solution. The choice of one
design solution over other exclusively depends upon
the requirement of the process engineer. Majumdar
et al.14 developed Pareto optimal front for UPF and air
permeability of cotton woven fabrics as depicted in
Fig. 7. The optimal design fronts are different for
various yarn linear densities. It is observed that for a
fabric having UPF value of 30, the air permeability
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
−
1 1
+
Minimise = w d + w2 d 2 + w3 d3
−
…(14)
where w1, w2 and w3 are the weights assigned to the
deviational variables.
3.3 Genetic Algorithm (GA)
The genetic algorithm (GA) is an unorthodox
search method based on natural selection process for
solving complicated optimization problems. John
Holland15 of University of Michigan developed it in
the early 1970s. Unlike conventional derivative based
optimization that requires differentiability of the
function to be optimized, GA can handle functions
with discontinuities or piece-wise segments. Besides,
gradient based optimization algorithms can get stuck
in local minima or maxima as they rely on the slope
of the function. Genetic algorithm overcomes this
problem. The following function is having local and
global minima (Fig. 8):
f ( x) = ( x − 1)( x − 2)( x − 3)( x − 4)( x − 5)( x − 6) …(15)
Gradient based optimization, while searching for
the global minima, may get stuck at 3.5 which is
actually local minima. However, GA is certain to find
out the global minima of the function at 1.34.
To perform the optimization task, GA maintains a
population of points called ‘individuals’, each of
which is a potential solution to the optimization
problem. Generally, the individuals are coded with a
string of binary numbers. The GA repeatedly modifies
the population of individual solutions using selection,
crossover and mutation operators. At each step, the
genetic algorithm selects individuals from the current
population (parents) and uses them to produce
children for the next generation, which competes for
survival. Over successive generations, the population
‘evolves’ toward an optimal solution. Genetic
algorithm can be applied to solve a variety of
optimization problems where the objective function is
discontinuous, non-differentiable, stochastic or highly
non-linear. An elaborate description of GA can be
found in published literature3,8,15.
3.4 Simulated Annealing (SA)
The SA is a useful meta-heuristic for solving hard
combinatorial optimization problems and the QAP in
particular. It was first introduced by Kirkpatrick
et al.16. The SA is a step-by-step method which could
be considered as an improvement of the local
optimization algorithm. The local optimization
algorithm proceeds by generating, at each iteration, a
solution in the neighbourhood of the previous one. If
the value of criterion corresponding to the new
solution is better than the previous one, the new
solution is selected, otherwise it is rejected. The SA
algorithm terminates either when it is no longer
possible to improve the solution or the maximum
number of trials decided by the user is reached. The
main drawback of the local optimization algorithm is
that it terminates at a local minimum which depends
on the initial solution and may be far from the global
minimum.
The SA algorithm avoids entrapment in a local
optimum. The difference with the local optimization
is that a solution A0 derived from a solution A is not
only accepted if A0 is better than A but it may also be
accepted if A0 is worse than A. Boltzmann’s law is
used to determine this acceptance probability that is
given as P(accept)= e-∆z/bt, where b is Boltzmann’s
constant and t (TI < t < TF, where TI and TF are the
initial and final temperatures respectively) is the given
parameter called the temperature which changes over
404
INDIAN J. FIBRE TEXT. RES., DECEMBER 2011
time according to some cooling schedule, and
∆z = z(A) - z(A) >=0. This is known as the
Metropolis acceptance rule which implies that
(i) the smaller the increase of the ∆z value, the more
likely the new solution is selected, and (ii) the lower
the value of ‘t’ and greater the number of trials ‘Q’,
the less likely the new solution is selected.
The basic algorithm of SA is given as follows:
Step 1— Randomly, select the initial solution ‘i’ as a
starting solution for SA.
Step 2— Choose an initial temperature TI > 0.
Step 3—Choose the temperature updating function
i.e. annealing (or cooling) schedule.
Step 4— Choose the epoch length function.
Step 5—Set temperature change counter t = 0 and
epoch length counter l = 0.
Step 6—Generate Solution A0 in the neighbourhood
of A by exchanging two facilities.
Step 7—Calculate ∆z = z(A) - z(A).
Step 8—If ∆z < 0 Then replace A by A’ else go to
Step 10.
Step 9— If random (0, 1) < exp (-∆z/ bt) then A’ = A.
Step 10—Repeat steps 7 to 10 until l = Q (maximum
number of trials for which the temperature is
‘t’).
Step 11—Calculate the next temperature as per the
temperature change function taken at step 3
and repeat steps 6 to 11 for the next
temperature.
Step 12—Repeat these steps until the stopping criteria
becomes true.
The SA procedure chosen has to set the following
factors: the initial temperature, the epoch length, the
cooling (annealing) schedule and the termination
criterion.
3.4.1 Initial Temperature
Kirkpatrick et al.16 proposed a large initial
temperature so that essentially all the solutions are
accepted at the first stage of the SA process with a
probability of P = 0.8.
3.4.2 Epoch Length
Let Nk be the epoch length (i.e. the number of trials
to be performed with the same temperature value).
Some commonly used functions are as follows:
(a) Constant function: Nk = Constant, where k = 0, 1, .
. . ,Q;
(b) Arithmetic function: Nk = Nk-1 + Constant, where
k = 0, 1, . . . ,Q;
(c) Geometric function: Nk = Nk-1/a, where ‘a’ is
constant less than 1 and k = 0, 1, . . . ,Q;
(d) Logarithmic function: Nk = Constant/log (Tk),
where k = 0, 1, . . . ,Q; and
(e) Exponential function: Nk = (Nk-1)1/a, where ‘a’ is
constant less than 1 and k = 0, 1, . . . ,Q.
3.4.3 Cooling (Annealing) Schedule
Temperature is used to compute the acceptance
probability of a solution which is worse than the
previous one. The few functions for updating the
temperature are as follows:
(a) Arithmetic function tk+1 = tk - constant, k = 0, 1, . .
. ,Q;
(b) Geometric function tk+1 = α.tk where k = 0, 1, . . . ,
Q, t0 = TI (initial temperature) constant, and α < 1;
(c) Logarithmic function tk = constant/log(k+2),
where k = 0, 1, . . . ,Q;
(d) Inverse function tk+1 = tk/ (1+α.tk), where k = 0, 1, .
. . ,Q, t0=TI (initial temperature) constant, α<<t0,
and tk = constant/(1+k).
3.4.4 Stopping Criteria
A few of the tests described in the literature are
given below:
(a) a given total number of iterations have been
performed;
(b) if the previously defined number of acceptance for
a given number of trials has not been obtained;
(c) if the given final temperature is not reached;
(d) if there is no improvement for a number of
iterations.
More details about the simulated annealing are
available in literature17,18.
3.5 Tabu Search (TS)
The TS originates from the local search
technique19,20. However, the TS go beyond the basic
structure of the local search approach and enable to
escape local optima. TS-based algorithms continue
the search even if a locally optimal solution is
encountered. Therefore, TS is a process of chains of
moves from one local optimum to another. The best
local optimum found during this process is regarded
as a result of TS. Thus, TS is an extended local search
approach. Consequently, it explores much larger part
of the solution space when comparing with local
search. Hence, TS provides more room for
discovering high quality solutions than the local
search. The key idea of TS is allowing climbing
moves when no improving neighbouring solution
MAJUMDAR et al.: DESIGNING OF FUNCTIONAL CLOTHING
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
each step of the procedure, a subset of the
neighbouring solutions of the current solution is
considered, and the move to the solution that
improves the objective function value f is chosen.
Naturally, s′ must not necessary be better than s; if
there are no improving moves, the algorithm chooses
the one that least increases the objective function [a
move is performed to the neighbour s′ even if f(s′) >
f(s)]. In order to eliminate an immediate returning to
the solution just visited, the reverse move must be
forbidden. This is done by storing the corresponding
solution (move) (or its "attribute") in a memory
[called a tabu list (T)]. The tabu list keeps information
on the last h = | T | moves which have been done
during the search process. Thus, a move from s to s′ is
considered as tabu if s′ (or its "attribute") is contained
in T. This way of proceeding hinders the algorithm
from going back to a solution reached within the last
h steps. The pseudo-code for the standard (pure) tabu
search paradigm is presented in Fig. 1. More details
on the fundamentals and principles of TS are found in
literature19,20.
3.6 Ant Colony Optimization (ACO)
The ants based algorithm has been introduced by
Maniezzo and Colorni21 which is based on the
principle of simple communication, an ant group is
able to find the shortest path between any two points.
During their trips a chemical trial (pheromone) is left
on the ground. The role of this trail is to guide other
ants towards the target point. For one ant, the path is
chosen according to the quantity of pheromone.
Furthermore, this chemical substance has a decreasing
action over time, and the quantity left by one ant
depends on the amount of food found and the number
of ants using this trail. As illustrated in Fig. 9, when
facing an obstacle, there is an equal probability for
every ant to choose the left or right path. As the left
trail is shorter than the right one and so required less
travel time, it will end up with higher level of
405
Fig. 9— Pseudo code for generic ACO procedure.
pheromone. More the ants will take the left path,
higher the pheromone trail is. This fact will be
increased by the evaporation stage.
This principle of communicating ants has been
used as a framework for solving combinatorial
optimization problems. Figure 1 presents the generic
ant colony algorithm. The first step consists mainly in
the initialization of the pheromone trail. In the
iteration step, each ant constructs a complete solution
to the problem according to a probabilistic state
transition rule. The state transition rule depends
mainly on the state of the pheromone. Once all ants
generate a solution, a global pheromone updating rule
is applied in two phases— an evaporation phase
where a fraction of the pheromone evaporates, and a
reinforcement phase where each ant deposits an
amount of pheromone which is proportional to the
fitness of its solution. This process is iterated until
algorithm satisfies stopping criteria.
3.7 Applications of Optimization Systems
Srivastav22 attempted to optimize the polyestercotton woven fabric parameters (weft count,
picks/cm, and % of polyester in weft) so that air
permeability (AP) is maximized and the ultraviolet
protection factor (UPF) meets the minimum
requirement. Thirty-six woven fabric samples were
prepared using three different levels of weft yarn
count, three different levels of picks per cm values
and four different levels of polyester fibre % in the
weft yarn. Linear regression equations were
developed for relating AP and UPF with the
independent variables (x weft count in tex, y
picks per cm and z polyester fibre % in weft). The
optimization problem is shown below:
Maximize
AP =173.618-1.363x -5.396y +0.056z …(16)
Subject to
UPF + -16.856 + 0.368 x + 0.749y + 0.068z ≥ 14 …(17)
406
15 ≤ x ≤30, 16≤ y ≤24, 0≤ z ≤100
INDIAN J. FIBRE TEXT. RES., DECEMBER 2011
…(18)
The optimization problem was solved using linear
programming technique. The values of x, y and z were
found to be 30, 17.5 and 100. One validation fabric
sample was then woven using the solution parameters.
The functional properties of the validation fabric
sample showed reasonably good agreement with the
targeted properties. The deviation in air permeability
and UPF was lower than 1 unit and 3 unit
respectively.
4 Multi-criteria Decision Making (MCDM)
Multi-criteria decision making (MCDM) is a
branch of operations. It is useful when several
alternatives are to be evaluated or ranked with respect
to the overall goal based on numerable decision
criteria.
The three main steps of MCDM are as follows:
(i)
Determine the goal, relevant criteria and
alternatives of the decision problem.
(ii) Ascertain numerical weights (or scores) to
relative importance of criteria.
(iii) Process alternative scores to determine the
ranking of each alternative.
Various MCDM techniques such as weighted sum
model (WSM), weighted product model (WPM),
analytic hierarchy process (AHP), TOPSIS, and
elimination and choice translating reality (ELECTRE)
can be used in engineering decision making problems,
depending upon the nature and complexity of
situation. AHP is one of the most popular methods of
MCDM. The reason behind the popularity of AHP
lies in the fact that it can handle objective as well as
subjective attributes, and the criteria weights and
alternative scores are calculated trough the formation
of pair-wise comparison matrix, which is the heart of
the AHP. The total number of pair-wise comparisons
in a decision making problem, having M alternatives
and N criteria, can be expressed by the following
equation:
N ( N − 1)
M ( M − 1)
+ N.
2
2
...(19)
This may be unmanageable where a huge number
of decision criteria and alternatives are involved. The
TOPSIS is more potent in handling the tangible
attributes and there is no limit in terms of number of
criteria or alternative.
4.1 Analytic Hierarchy Process (AHP)
AHP was developed by Saaty23-26. In AHP a pairwise comparison matrix of attributes is constructed
using a nine point scale of relative importance. An
attribute compared to itself or with any other attribute
having the same importance is assigned the value 1.
Thus, the right diagonal of pair-wise comparison
matrix is comprised only 1. The numbers 3, 5, 7 and 9
correspond to verbal judgments of ‘moderate
importance’, ‘strong importance’, ‘very strong
importance’ and ‘absolute importance’ respectively.
For N decision criteria, the size of the comparison
matrix will be N × N and the entry cij will denote the
relative importance of criterion i with respect to
criterion j. In the matrix, cij = 1 if when i = j and
1
c ji = . A typical pair-wise comparison matrix (C1)
cij
of criteria is shown below:
 1 c12 ... c1N 
c
1 ... c2 N 
C1 =  21
 ... ... 1 ... 


cN 1 cN 2 ... 1 
The principle eigen vector of the above matrix
represents the relative weights of the decision criteria.
The relative weight of the ith criteria (Wi) is determined
by calculating the geometric mean of the i th row (GMi)
of the above matrix and then normalizing the geometric
means of rows. This can be represented as follows:
1
GM i
 N  N
GM i = ∏ cij  and Wi = N
 j =1 
∑ GM i
…(20)
i =1
Similarly, N numbers of pair-wise comparison
matrices, one for each criterion, of M x M order are
formed where each alternative is compared with each
other. 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 as shown in Table 1.
M
Here,
∑a
ij
=1
…(21)
i =1
The final priority of all the alternatives is
calculated by considering the alternative scores (aij) in
each criterion and the weight of the corresponding
criterion (Wj) using the following equation:
MAJUMDAR et al.: DESIGNING OF FUNCTIONAL CLOTHING
Table 1— Decision matrix of AHP
Alternatives
Decision criteria
C1
C2
C3
…
CN
(W1)
(W2)
(W3)
…
(WN)
A1
a11
a12
a13
…
a1N
A2
a21
a22
a23
…
a2N
A3
a31
a32
a33
…
a3N
…
…
…
…
…
…
AM
aM1
aM2
aM3
…
aMN
N
AAHP = max ∑ aij.W j for i = 1,2,3, …..M
…(22)
The normalized matrix is then converted to
weighted normalized matrix by multiplying each
column of the normalized decision matrix with the
associated criteria weight. Hence, an element vij of
weighted normalized matrix is represented as follows:
vij = rij .W j
...(24)
The weights of decision criteria can be determined
by the AHP, which has been explained in the previous
section. The next step produces the positive ideal (A*)
and negative ideal (A-) solutions in the following
manner:
A* = {(max vij / j ∈ J ), (min vij / j ∈ J ') for i = 1, 2,3,....M }
j =1
= {v1 *, v2 *,.....vN *}
Alternative with the maximum score is the most
preferred one and vice versa.
DMxN
 a11
a
=  21
 ...

 aM 1
a12
a22
...
aM 2
= {v1− , v2 − ,....., vN − }
rij =
M
2
 ∑ (aij ) 
 i =1

0.5
…(26)
where
J = { j = 1, 2,...., N / j associated with benefit or positive criteria}
and
J ' = { j = 1, 2,...., N / j associated with cost or negative criteria}
For the benefit criteria, the decision maker prefers
the maximum value among the alternatives.
Therefore, A* indicates the positive ideal solution.
Similarly, A- indicates the negative ideal solution. The
separation distances of each alternative from A* and
A- are calculated using the following expressions.
0.5
N

Si = ∑ (vij − v j * ) 2  and
 j =1

*
... a1N 
... a2 N 
... ... 

... aMN 
0.5
N

Si = ∑ (vij − v j − )2  , i = 1, 2,..., M
 j =1

−
where an element aij of the decision matrix represents
the actual value of the i th alternative in terms of j th
criteria. The decision matrix is converted to
normalized decision matrix, so that the scores
obtained in different scales or units become
comparable. An element rij of the normalized decision
matrix can be calculated using the following equation:
aij
…(25)
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
Solutions (TOPSIS)
TOPSIS was developed by Hwang and Yoon27.
The basic philosophy of this method is that the
selected alternative should have the shortest
geometrical distance from the best possible solution
and longest distance from the worst possible solution.
First, the relevant objective or goal, decision criteria
and alternatives of the problem are identified. Then
the decision matrix is formulated based on the
information available regarding the problem. If the
number of alternatives is M and the number of criteria
is N, then the decision matrix having an order of
M × N can be represented as follows:
407
…(23)
…(27)
where Si* and Si- are the separation distances of
alternative i from A* and A- respectively.
Finally, the relative closeness (Ci*) value, to the
ideal solution, is determined for each alternative using
the following equation; the value of Ci* lies within the
range 0 - 1:
Ci* =
Si −
( Si * + Si − )
…(28)
The alternative having the maximum Ci* is the best
and vice versa.
INDIAN J. FIBRE TEXT. RES., DECEMBER 2011
408
4.3 Application of Decision Making Systems
An application of AHP system has been
demonstrated with a hypothetical example of body
armour selection based on three decision criteria
namely impact resistance, comfort score and cost. The
impact resistance of body armour is characterized by
the V50 speed at which the bullet has equal probability
to pierce the vest or to be stopped by the vest.
Comfort score has been taken as an overall index
representing the flexibility, thermal resistance and
moisture vapour transmission of the body armour.
Higher V50 speed is a desirable or benefit criterion and
so is the comfort score. However, price of the vest is a
negative or cost criterion and lower value is desirable.
Table 2 shows the pair-wise comparison matrix of
three decision criteria based on the perception of
decision maker. Here numerical scores has been given
as per Saaty’s23 nine point scale as described in
section 4.1. Impact resistance has moderate dominance
over the comfort and comfort has moderate
dominance over the cost. Cost has the least influence
on decision as the high impact resistance and greater
comfort are imperative for body armours. After
calculating the normalized geometric mean of rows, it
has been found that the weights of impact resistance,
comfort score and cost are 0.64, 0.26 and 0.10
respectively. The scores of four alternatives
(A1 - A4) in three decision criteria are shown in Table 3.
Table 4 shows the normalized scores of alternatives.
The scores have been normalized using the following
expressions:
Normalized score=
Score
(For a benefit criterion)
Maximum score
Normalized score=
Minimum score
(For a cost criterion)
Score
The weighted score of four alternatives can be
calculated as follows.
Score A1 = 0.64 × 0.9 + 0.2 × 0.5 + 0.1 × 1 = 0.776
Score A 2 = 0.64 × 1 + 0.2 × 0.75 + 0.1 × 0.8 = 0.87
Score A 3 = 0.64 × 0.95 + 0.2 × 0.4 + 0.1 × 0.89 = 0.777
Score A 4 = 0.64 × 0.8 + 0.2 × 1 + 0.1 × 0.89 = 0.801
Here, alternative A2 is the most preferred one
although it is not the best in comfort and cost criteria.
In contrast, alternative A1 is least preferred alternative
although it is the cheapest among the alternatives. The
decision maker can further change the scores given in
Table 2— Pair-wise comparison matrix of decision criteria
Parameter
Impact Comfort Cost Geometric Normalized
resistance score
mean geometric mean
Impact
resistance
Comfort
score
Cost
Alternatives
A1
A2
A3
A4
Ideal
Worst
1
3
5
2.46
0.64
1/3
1
3
1
0.26
1/5
1/3
1
0.41
0.10
Table 3— Features of body armours
Impact resistance Comfort score
V50, m/s
450
1000
500
1500
475
800
400
2000
500
2000
400
750
Cost, Rs
40,000
50,000
45,000
45,000
40000
50,000
Table 4— Normalized features of bullet proof body armours
Alternatives
Impact
Comfort
Cost
resistance (0.64)
score (0.26)
(0.10)
A1
A2
A3
A4
0.9
1.0
0.95
0.8
0.5
0.75
0.40
1
1.0
0.8
0.89
0.89
the pair-wise comparison matrix (Table 2) and see
how the ranking of alternatives are responding. This is
known as sensitivity analysis.
5 Conclusion
Various modelling, optimization and decision
making techniques have been discussed in this paper
with suitable examples pertaining to functional
clothing. These techniques are very frequently used in
manufacturing and service industries. Unfortunately,
these techniques have seldom received any attention
in traditional textile industry. As the quality
requirement for the functional clothing are very
stringent, these modelling, optimization and decision
making techniques with sound mathematical
foundation are very important for functional clothing
industries. It is pertinent to mention here that in recent
years some very powerful modelling and optimization
tools like support vector machine, simulated
annealing, particle swarm optimization and ant colony
optimization have been developed. Researches are
being done to amalgamate multiple modelling and
optimization tools so that they become more powerful
and complement each other. It is expected that these
emerging systems will be embraced by the functional
clothing industry to solve the complex problems
related to design and manufacturing.
MAJUMDAR et al.: DESIGNING OF FUNCTIONAL CLOTHING
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