3.3 Defuzzification Mathematical Approaches
3.3.1 The Parametric Nonlinear Programming Technique
The PNLP technique is a general procedure used to tackle the uncertain environment in queueing systems. This can be done by formulating the conventional queueing models into fuzzy queueing models. Kao et al. (1999) were the first researchers to attempt extending the queueing models under an uncertain environment. This technique is used to construct membership functions according to the adoption of four basic fuzzy queues, which represent single channel models. Numerous studies have focused on fuzzy queueing models which are much more realistic model forms than the conventional queueing models (Yang & Chang, 2015; Bagherinejad & Pishkenari, 2016). Within the context of the conventional queueing theory, it is required that the arrival rates and the service rates follow a certain distribution.
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The general mathematical steps for this technique must consider at least two parameters with one server which follows the pattern of the basic elements in queueing models arrival rates and service rates (Chen, 2004a, 2004b, and 2005). According to Zadeh‟s extension principle (Zadeh, 1978) and the α-cut approach, the construction of the membership functions and constraints which relate to the PM of queueing models (Kao et al., 1999) is given as follows:
( , )( ) sup min . ( ), ( ) / ( , ) . f A S A S a x s y z a s z f a s
(3.31) where, ,A S represents the fuzzy numbers of basic elements, which are arrival rates and service rates, respectively,
( , )( ) f A S z
denotes the membership function of the PM based on Zadeh‟s extension principle (i.e.,W W L and q, S, q LS),
( )
A a
denotes the membership function for arrival rates, a is in the set, A with a
membership value of
A( )a
(i.e., the set A can be represented for all elementsaA
),( )
S s
denotes the membership function for service rates, s is in the set, S with a membership value of S( )s (i.e., the set S can be represented for all elements
sS
),( , )
z f a s represents the membership function of PM, where a represents the crisp arrival rates and s represents the crisp service rates.
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The PNLP technique is used to construct the membership function f A S( , ) to derive the α-cuts of f A S( , ). According to (3.31), we need either A( )a and S( )s
or A( )a and S( )s , such that z f a s( , ), to satisfy f A S( , )( )z . This can be accomplished by using the PNLP technique for the former case for finding the lower bound (LB) and upper bound (UB) of the α-cut of f A S( , )( )z :
( ) min ( , ) f LB f a s (3.32) s.t. LBA( ) a UBA( ) , sS( ) , UBf( ) max ( , )f a s (3.33) s.t. LBA( ) a UBA( ) , sS( ) , and for the latter case:
( ) min ( , ) f LB f a s (3.34) s.t. LBs( ) s UBs( ) , aA( ) , Hence, the upper bound is:
( ) max ( , ) f
UB f a s
(3.35) s.t. LBs( ) s UBs( ) ,
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aA( ) ,
where, A( ) and S( ) represent the functions of crisp sets of the arrival rates and
service rates, respectively, where,
is a value representing the level in the closed interval of [0,1]. This means aA( ) and sS( ) can be replaced by( ), ( )
A A
a LB UB and s LBS( ) ,UBS( ) . Therefore, Equations (3.32) to (3.35) are the same. Then, it can be rewritten as:
( ) min ( , ) f LB f a s (3.36) s.t. LBA( ) a UBA( ) , LBs( ) s UBs( ) , ( ) max ( , ) f UB f a s (3.37) s.t. LBA( ) a UBA( ) , LBs( ) s UBs( ) . where, ( ), ( ) A A LB UB
represents the lower bound and upper bound of the crisp values interval for arrival rates,
( ), ( )
S S
LB UB
represents the lower bound and upper bound of the crisp values interval for service rates.
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From
LBf( ) ,UBf( ) (0,1
, the membership function of f A S( , ) can be constructed as a pair of mathematical programs to find the optimal solutions as this closed interval. If both LBf( ) and UBf( ) are invertible with respect to α, then a left shape function -1( )
( ) f
LS z LB and right shape function 1 ( )
( ) f
RS z UB can be formulated as the membership function f A S( , ):
1 2 2 3 ( , ) ( ), ( ) ( ), 0, f A S LS z z z z z RS z z z z otherwise (3.38) such that 1 3 2 2 ( ) ( ) 0, ( ) ( ) 1. LS z RS z LS z RS z where, 1
,
2z z
andz
3 denote the main points for each value of fuzzy PM (i.e.,( ) ( ) ( )
,
,
i i i s q qW
W
L
and ( )i sL ), for i=1, 2 representing Class One and Class Two, respectively. From different possibilities of the α-cut, when the formulation is relatively simple, a closed formula can be derived by taking the inverse of its α-cuts. It can be extracted by substituting α-values to obtain different crisp values of each performance measure.
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Generally, the PNLP technique consists of several steps which are considered for developing most conventional queueing models. Therefore, these steps can be presented as:
Step 1: Input the fuzzy numbers of basic elements which are the arrival rates and service rates represented by membership functions, such as TpFn or TrFn.
Step 2: Derive the membership functions for the PM of conventional queueing model by using Zadeh‟s extension principle and the α-cut approach.
Step 3: Before deriving the membership functions of these performance measures, determine the constraints according to fuzzy numbers (i.e., the arrival rates and service rates) via LB and UB by using the α-cut approach. Determine the cases through these basic elements in the procedure to derive the fuzzy PM (i.e.,
,
,
q s q
W W L
and L ). sStep 4: Substitute a pair of LB and UB of fuzzy numbers through cases according to Equations (3.36) and (3.37).
Step 5: Simplify these equations of each LB and UB of membership function for each value of performance measures. Then substitute the α-cut interval as closed interval [0,1] to obtain different crisp values for each value of PM. On the other hand, substitute the values of α-cut interval into the basic elements which are arrival rates and service rates to obtain crisp values.
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Step 6: Plot these values as closed intervals represented by the membership functions of each performance measure based on the type of membership function chosen. Figure 3.7 shows these general steps for this technique:
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Figure 3.7. General Procedure of the PNLP Technique (Kao et al., 1999)
Figure 3.7 shows the steps of the PNLP technique considering all parameters inside the fuzzy queues. It involves both triangular or trapezoidal fuzzy numbers between
0,1 and the α-cut approach. Consequently, the results of this technique are characterized to derive different membership functions of performance measures of the queueing model.The main advantage of this technique is the acceptance of more than two basic elements, such as arrival rates and service rates in queueing models (Ke et al., 2007; Yang & Chang, 2015). This technique strongly represents all types of queueing
Derive the membership functions for the performance measures
Determine the constraints from fuzzy numbers via LB and UB by using the α-cut approach Substitute a pair of LB and UB of fuzzy numbers
to the performance measures
Simplify the LB and UB of the membership function for each values of PM and substituting α-cut to
obtain crisp values
Input fuzzy numbers of arrival rates and service rates represented by membership functions
Start End Step 1 Step 2 Step 3 Step 4 Step 5
Plot the membership functions of the performance measures
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models and all PM related to the estimation of the classical queueing models under uncertain environments (Wang & Wang, 2005). Therefore, it is appropriate to use in this research for multiple channel queueing models with priority discipline. This technique allows a range of different crisp values to be accepted inside the closed intervals to obtain the performance measures. Furthermore, it determines the core value inside closed intervals which is known as the optimal value based on membership function represented as one core from the triangular membership function or two cores from the trapezoidal membership function (Jin et al., 2002). We further adopted an alternative queueing model technique in manufacturing industries to provide more conclusive analysis to queueing problems under an uncertain environment.