ORIGINAL ARTICLE

Robust clustering approach for the shortest path problemon finite capacity fuzzy queuing network

M. Aminnayeri & P. Biukaghazadeh

Received: 6 December 2010 /Accepted: 24 October 2011 /Published online: 15 November 2011# Springer-Verlag London Limited 2011

Abstract This paper introduces a new model in the field offuzzy queuing networks based on the clustering and finitecapacity concepts. The proposed model includes fuzzyqueuing systems which are located at the nodes of thenetwork. Arc lengths, interarrival times, and service timesare all fuzzy triangular fuzzy numbers. In order to find theshortest path on this network, queuing systems should betransformed to waiting times. The waiting times of eachsystem are calculated by a conditional transformation. Onthe other hand, a robust fuzzy method is proposed forclustering of the arriving customers to the network.Robustness of this method prevents noisy data to affectresults. Outputs of the clustering reduce shortest pathcalculations drastically. Based on a simulation process, thefuzzy queuing network is reduced to a deterministicnetwork. A robust simulated annealing is designed for thisnetwork to find the shortest path. Numerical results showedthat the clustering process is successful in eliminatingoutliers, and could be addressed as an efficient method. Theconvergence and solution time of the algorithm is reason-ably better in comparison with published methods. Severalexperiments are conducted to compare the proposedmethod with corresponding researches.

Keywords Fuzzy c-means . Simulated annealing . Robust .

Queuing network

1 Introduction

Queuing theory is used in many researches of appliedmathematics, while recent works focus on network ofqueues. This paper introduces a new model for the queuingnetworks. Several concepts such as dynamic networks,robust fuzzy clustering, queuing systems, simulation, short-est path models, and optimization are used in this paper todescribe a new model. Hence, a preliminary knowledge inthe mentioned areas is required to understand the model ofthis research.

The next section presents a literature review on relatedresearches. Section 3 describes the proposed model indetail. Section 4 is devoted to the validation of the model.This section has three main subsections: (a) clusteringanalysis, (b) SA analysis, and (c) performance analysis.Conclusions and recommendations for future researches arepresented in the last section.

2 Literature review

Queuing networks are often studied to describe thebehavior of queues under different conditions. The analysiscould determine the effect of various variables on theperformance of the network. Because of the important roleof these systems in the evaluation and prediction ofproduction and manufacturing systems, queuing networkmodels have been extensively studied in recent years [1–3].While more realistic models were required, finite capacityqueuing systems became important [4–7]. This type ofqueuing network has been designed to indicate populationconstraints and capacity resources [5]. Arrival of customersto the server is stopped when a queue reaches its maximumcapacity. Various researches have been made to define the

M. Aminnayeri (*) : P. BiukaghazadehDepartment of Industrial Engineering,Amirkabir University of Technology,424 Hafez Avenue,1591634311, Tehran, Irane-mail: mjnayeri@aut.ac.ir

Int J Adv Manuf Technol (2012) 61:745–755DOI 10.1007/s00170-011-3719-2

behavior of the real world systems in blocking queues suchas [8] and [9]. Calculations of blocking queues in thenetworks could be approximate or exact. Exact solutionscould evaluate passage time distribution, queue length, andaverage performance indices [10]. However, queuingnetwork models with blocking in its general form do nothave an exact solution, and simulation or approximationshould be used [11–13]. The most common blocking typesare as follows:

& Blocking after service: A customer could not enter thefull capacity queue j at a node, and it should wait in theserver of the node i until the completion of the servicein the node j.

& Blocking before service: Service of customers at thequeue of the node i is blocked until the queue in thenode j could accept customers.

& Repetitive service blocking: If the queue of the node jdenies customer entrance due to full capacity, then theservice of the customer in the node i should be repeatedindependently until the node j could accept thecustomer.

Other blocking mechanisms in queuing networks arediscussed in detail in [8] and [9].

Among deterministic queuing networks, non-deterministicqueuing networks are also discussed. It has significantimportance to describe the parameters of the network byuncertain variables. These uncertainties may be modeledthrough vast combinations of parameters. A technique foruncertainty of parameters is to use intervals and relatedapproaches. Researches such as [14] and [15] introducedintervals for input parameters of the queuing networks. Themean value analysis of interval parameters for queuingnetwork models is presented in [16]. Histogram-basedcharacterization for queuing networks is also discussed insome researches such as [16] and [17]. The main difficulty isthe approximation of the solutions, which result in unrealisticconditions for some networks.

In order to characterize a varying risk for a parameter,varying interval widths could be used. Fuzzy numbersallow us such characterization of risk levels for continuousparameter intervals [18]. Models which characterized withfuzzy numbers are quite convenient for the evaluation.Based on the α-cuts of input parameters, an existinginterval could be adapted to perform fuzzy numbers asinput parameters. In the proposed model of this paper, asimulation process is applied to transform the uncertainparameters of queuing systems to deterministic parameters.Prior to this simulation, a robust fuzzy clustering forcustomers should be carried out.

Clustering is the method of assignment of a datasetinto similar characterized groups. Each point of thedataset is exactly restricted to one cluster in conven-

tional clustering methods. Zadeh [19] proposed thefuzzy sets at 1965. Based on the researches of Zadeh,the partial membership to each cluster could be defined.The partial membership for clustering has been widelystudied so far [20–22]. The most discussed fuzzyclustering method is fuzzy c-means algorithm, which ismainly studied in last decades [23]. In this method, hardclusters are replaced with fuzzy clusters. Hence, eachpiece of the dataset may belong to several clusters,having different membership degrees. For a given set of ndata, X=x1, x2, …, xn the objective function of the fuzzyc-means is as:

Ofcm U ;Cð Þ ¼X

nk¼1

Xvi¼1 uij

� �md2 xk ; cið Þ ð2:1Þ

where d (xk, ci) is the distance between the center of theith cluster and data xk. In this objective function, uij is themembership function of the xk in the ith cluster. Finally, nand v denotes the number of data patterns and clusters,respectively. Minimizing the objective function could beperformed through iterative calculation of the followingterms:

uik ¼ 1PV

j¼1dikdjk

� � 1m�1

ð2:2Þ

ci ¼Pn

k¼1 uikð ÞmxkPnk¼1 uikð Þm ð2:3Þ

In formulas 2.1, 2.2, and 2.3, the following conditionsshould be satisfied:

uik 2 0; 1½ �; 8 i;

Xvi¼1 uik ¼ 1; 8 k;

0 <X

nk¼1 uik < n:

Hence, the conventional approach clusters the data basedon the intensity value. To overcome difficulties of c-meansmethod, Yang and Ko [24] presented fuzzy c-numbersclustering algorithm. However, [24] also assumed noise-free datasets for the evaluation of the algorithm. Most of thementioned algorithms assume that the dataset is not affectedby noise or outliers [25]. However, in real applications,these assumptions are often false. To overcome thegeneration of the misleading clusters, robust algorithmsare proposed. Dave [26] proposed robust fuzzy c-means forthe first time. In this method, noise is denoted by noise

746 Int J Adv Manuf Technol (2012) 61:745–755

cluster. The noise has a constant distance δ from all theobjects. Presence of the noise modifies the objectivefunction as follows:

Orfcm U ;Cð Þ ¼X

nk¼1

Xvi¼1 uij

� �md2 xk ; cið Þ þ

Xnk¼1u

m»j:d

2

ð2:4Þ

where u»j ¼ 1�Pvi¼1 uij is the membership function of the

xj to the noise cluster. The value of δ in this method has animportant role in the success of the method. If δ is assumedtoo large, then this method reduces to simple fuzzy c-meansand outlier data should be included in the clusters. If δ isassumed too small, a lot of data could be eliminated asoutliers. It is suggested that the value of the δ should beselected based on the scale of the data [27, 28].

Beside these methods, many researches have been madeconcerning the treatment of outlier or noise data [26, 28–31]. Table 1 indicates some of the researches concerningrobust fuzzy clustering methods.

The main gap between this research and the previousresearches could be described as:

& Clustering method to reduce the calculations& Finite capacity fuzzy queues on the network& Robust clustering and optimization methods which

eliminates outliers

In order to apply a clustering method to the arrivingcustomers, a modified robust fuzzy c-means, MRFCM,is designed. A transformation on the fuzzy queues is usedto compute the waiting time of the queues. Finally,through an optimization method, the shortest path for thenetwork is computed. The procedure of modeling andsolution of the problem will be described in the nextsection.

3 The proposed model

The model includes three main phases:

& Phases I: Robust fuzzy clustering& Phases II: Transformation and simulation& Phases III: Optimization

At the first phase of the model, arriving customers areclustered based on their required service times. To eliminatethe effect of the noise and outliers, a robust method isdesigned. Based on the results of this phase, transformationand simulation processes are carried out in phase II. To finda shortest path on the finite capacity queuing network, anoptimization method is used in phase III.

3.1 Phase I: robust fuzzy clustering

The proposed clustering algorithm is a robust fuzzyalgorithm. Let X=x1, x2, …, xn be the set of n data. Let cidenote the center of the ith cluster. The proposed algorithmminimizes the following objective function:

Orfcm U ;Cð Þ ¼X

nk¼1

Xvi¼1 uij

� �md2 xk ; cið Þ

þX

nk¼1u

m»j:d

2

� aX

vi¼1

Xnk¼1uij

h i2ð3:1Þ

Subject to:X

vi¼1 uij ¼ 1; for j 2 1; 2; . . . ; nf g ð3:2Þ

First term of 3.1 is common in almost all fuzzy clusteringmethods. The compactness and the shape of clusters arecontrolled by this term. The second term of 3.1 is used toeliminate the noise and outliers. Indeed, a noise cluster is

Table 1 Comparison of fuzzyclustering methods Method Description Researches

Metric approach Inserting robust properties inobjectives and metrics

D’Urso [37]

Kersten [38]

Hathaway et al. [39]

Leski [40]

Possibilistic approach Outliers have small membershipdegrees to all clusters

Krishnapuram and Keller[41, 42]

Noise approach Outliers have special cluster Dave [43]

Dave and Krishnapuram [26]

Dave and Sen [28]

Semi-fuzzy approach A two-level conditional membershipfunction is assigned to outliers

Selim and Ismail [44]

Influence weighting approach During the clustering procedureweights are modified for each datum

Keller [31]

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devoted to this type of input data. The third term controls thenumber of clusters. This term allows the procedure to assignoptimized number of clusters to the whole data. Byminimizingthe collection of these terms simultaneously, the input data isclustered in an optimized number of clusters, while the noiseand outliers have the least possible effect on this procedure.

The sum of the memberships of each piece of data to acluster indicates the cardinality of that cluster which isdefined as:

Ni ¼X

nj¼1 uij

� � ð3:3Þ

At each iteration of the minimization procedure, theclusters with the cardinality less than a threshold areeliminated. Hence, a few of clusters remain for final iterationsof the algorithm. The parameter α which plays the role ofbalancing the first and the third terms of 3.1 is defined as [30]:

aðkÞ ¼ h0 exp � kt

� �Pnk¼1

Pvi¼1 uij

� �md2 xk ; cið ÞPv

i¼1

Pnk¼1 uij

� �2 ð3:4Þ

The robust fuzzy clustering method is utilized forclustering of the arriving customers. The customers areclustered based on their required service times. In subsec-tion 4.3, this procedure is discussed in detail.

3.2 Phase II: transformation and simulation

In order to find the shortest path on a network of fuzzy queues,each of the queues should be transformed to its related waitingtime. Biukaghazadeh and Fatemi Ghomi [32] proposed atransformation function for the fuzzy simple queues. In thissubsection, an advanced transformation of this type ispresented to tackle with blocking queues on the network.

The parametric programs are used for α-cuts of the fuzzyservice and interarrival times as [32]:

lp að Þ ¼ min p a; sð Þsubject to:

lA að Þ � a � uA að Þ

lS að Þ � s � uS að Þ

up að Þ ¼ max p a; sð Þsubject to:

lA að Þ � a � uA að Þ

lS að Þ � s � uS að Þ

Based on the results of these programs and byapplying inverse transformation, the membership func-tions of the queues are computed. A simulation processtransforms the fuzzy waiting time to a deterministicwaiting time. The waiting time for these queues withblocking is as:

W ¼s2

aif n < k and s < a

ns if n � k and s < a

1 if s > a

8>>><>>>:

ð3:5Þ

where the service time is indicated by s and interarrivaltime is presented by a. In this notation, n is used toshow the number of customers in system and k is used todefine the capacity of the queue. In order to find ashortest path on this network, a mathematical model isdeveloped. Notations used in the mathematical model areas follows:

a the confidence level(i, j) the arc between the node i and the node jA the set of arcs in the networkxijkl the decision variable for the arc between the

node i and the node j of the kth customerwhen l customers are in node j

ck the center of kth clustersj the service time of the server of the node jKj the capacity of the queue jlj the number of customers in node j

The mathematical program is as:

min T

subject to:

Cr T x; xð Þ � T� � a ð3:6Þ

Xi;jð Þ2Axijkl ¼ 0 ck > 1:1sj ð3:7Þ

Xi;jð Þ2Axijkl ¼ 0 lj > Kj ð3:8Þ

X1;jð Þ2Ax1jkl �

Xj;1ð Þ2Axj1kl ¼ 1 ð3:9Þ

Xi;jð Þ2Axijkl �

Xj;ið Þ2Axjikl ¼ 0 2 � i � n� 1 ð3:10Þ

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Xn;jð Þ2Axnjkl �

Xj;nð Þ2Axjnkl ¼ �1 ð3:11Þ

xij 2 0; 1f g; 8 i; jð Þ 2 AConstraint 3.6 guarantees the desired credibility level.

Constraint 3.7 allows appropriate customers for a definedserver. Customers who have a service time smaller thanthe service time of the node j are considered as

appropriate customers. A 10% allowance is set torepresent a more realistic condition. Constraint 3.8 blockscustomers for a full capacity queue. Constraints 3.9, 3.10,and 3.11 are used to select an arc from the first node to thelast node.

In order to solve this model, a simulation process isused. Computations of the proposed model with analyticmethods are almost impossible in most cases. Hence, asimulation method is used for estimation of the values. Thefuzzy simulation process introduced in [33] is used asfollows:

Step 1 Let e=0.Step 2 Set the number of samples, M.Step 3 From the set, Θ generate a uniformly distributed

θk, such that Pos {θk}≥ε for k=1, 2, …, M. In thisdefinition, ε is a sufficiently small number.

Step 4 Let vk=Pos {θk}.Step 5 Based on the result of step 4, let a ¼ T x; x q1ð Þð Þ ^ . . .

^T x; x qMð Þð Þ; b ¼ T x; x q1ð Þð Þ _ . . . _ T x; x qMð Þð Þ.Step 6 Generate a variable such as r from [a, b].Step 7 e ¼ eþ Cr T x; xð Þ � rf g, if r≥ 0. Such as

Cr T x; xð Þ � rf g ¼ 1=2 Max1�k�M vk T x; x qkð Þð Þj �f�

rg þ Min1�k�M 1� vk T x; x qkð Þð Þj < rf g� �

:

Step 8 e ¼ e� Crf T x; xð Þ � rg, if r<0. Such asCr T x; xð Þ � rf g ¼ 1=2 Max

1�k�M vk T x; x qkð Þð Þj �f�rg:þ Min

1�k�M 1� vk T x; x qkð Þð Þj > rf g� �:

Step 9 Steps 6 to 8 are repeated M times.Step 10 Calculate E T x; xð Þ½ � ¼ a _ 0þ b ^ 0þ e: b�ð

aÞ=M .Step 11 Return E [T(x, x)].

Table 2 Robust simulated annealing

The procedure of modified SA

T ← T0

Generate starting path pc

Repeat

Repeat

Generate pn N (pc)

If T ≥ 1/

ΔE ← (E(pn) – E(pc))

u ← U [0: 1]

If u < Pa (ΔE)

pc ← pn

End if

End if

Until temperature reduction

Reduce T such as: Tk = T0 / ln k

Until termination condition

Table 3 Membership functionsof datasets for modified robustfuzzy clustering

Vector Value Memberships for A1 Memberships for A2 Memberships for A3

u1 u2 u3 u1 u2 u3 u1 u2 u3

x1 1 0.91 0.01 0 0.89 0.01 0 0.88 0.005 0

x2 2 0.91 0.02 0 0.90 0.01 0 0.90 0.008 0

x3, x4, x5 3 0.94 0.04 0.001 0.91 0.03 0 0.91 0.03 0.01

x6 4 0.92 0.05 0.01 0.88 0.03 0.005 0.87 0.04 0.01

x7 5 0.44 0.46 0.02 0.40 0.40 0.01 0.38 0.39 0.02

x8 6 0.03 0.82 0.04 0.03 0.75 0.01 0.03 0.77 0.03

x9, x10, x11, x12 7 0.01 0.95 0.04 0.01 0.86 0.02 0.02 0.82 0.04

x13, x14 8 0.003 0.88 0.09 0.002 0.81 0.07 0.01 0.74 0.05

x15 9 0.001 0.52 0.43 0.001 0.43 0.41 0.01 0.41 0.40

x16 10 0.001 0.06 0.84 0 0.02 0.83 0.006 0.04 0.85

x17 11 0.001 0.02 0.89 0 0.01 0.91 0.003 0.03 0.90

x18, x19, x20 12 0 0.01 0.95 0 0.002 0.86 0.003 0.02 0.88

x21 17 0 0.002 0.06 0.003 0.02 0.05

x22, x23 20 0.001 0.02 0.03

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3.3 Optimization

Simulated annealing (SA) is known as a metaheuristicmethod. The process of heating and cooling of a solid ismimicked by this algorithm. SA was introduced in 1953[34], but it was not applied to a problem until 1983 [35].The model is handled by SA as follows:

& Configuration: The nodes are numbered i=1, 2, …, nand each node has its corresponding service time si.Configurations are defined as permutation of the nodeswhich are visited in the path.

& Rearrangement: The moves in iterations are of twotypes: (1) some of nodes of a path are replaced with thesame nodes in opposite order; (2) a section of the pathis replaced with a randomly selected order of nodes.

& Objective function: E is defined as the total length ofthe path. After the simulation process, the objectivefunction reduces to:

E ¼ L ¼Xn

i¼1xijkl Lijkl þ sj þ wj

� �; for all node j on the path

ð3:12Þ

& Annealing: First rearrangements are randomly generated.These rearrangements are used to define the range ofvalues of ΔE. The starting value of T should not besmaller than the largest value of ΔE. The newlygenerated value of T is held constant for 10N successfulrearrangements, or 100N reconfigurations. The proce-dure stops when there are no significant improvements.

The designed algorithm eliminates noise data. Toperform this refinement, a lower bound on the temperatureis defined. On the other hand, the temperature decreases by

Fig. 1 Scatter plots of thedatasets

Fig. 2 Number of clusters

Table 4 Successful solutions

Method Number of runs

10 100 300

Neurogenetic 45.1% 53.3% 56.7%

ACO 52.8% 59.9% 61.5%

PSO 42.2% 48.9% 54.6%

GA 49.3% 56.4% 58%

RSA 63.7% 68.9% 76.2%

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a logarithmic equation. The procedure of the algorithm isshown in Table 2:

4 Experimental results

This section has three subsections. At the first subsection ofexperiments, the robustness of the proposed clusteringalgorithm will be discussed. Noise data will be includedin some of datasets to evaluate the efficiency of theproposed algorithm. At the second subsection, the efficiencyof the robust SA (RSA) will be examined. Comparisons toother publish methods will be made to evaluate the proposedalgorithm. At the last subsection of this section, the overallperformance of the proposed algorithm, from its initial to finalstage, will be discussed.

4.1 Clustering analysis

In order to show the robustness of the proposed clusteringmethod, three datasets A1, A2, and A3 are used [36]. All ofthese datasets include one-dimensional vectors. The A1 isnoise-free, where other two datasets are noisy. A1 is the setof 20 interarrival times shown in Table 3. A2 is the union ofthe A1 and an outlier. A3 is the union of A1 and 3 outliers.Datasets are shown in Fig. 1.

In these experiments, three clusters are assumed.Observations for the A1 are in Table 3. As there is nooutlier in this dataset, the centers of the clusters are in theirexpected locations. The center of each cluster is shown witha filled circle in the scattered plot. The notation vi denotesthe center of clusters.

The scatter plots of the datasets show the effect of outlierson clustering procedures. The proposed algorithm, MRFCM,is compared to fuzzy c-means [23] (FCM) and robust fuzzy c-means [26] (RFCM). It is clear that for noise-free dataset,these methods have similar performances. Centers of theclusters are the same for this dataset. For the dataset A2, alittle difference between centers of clusters of the methodscould be observed. The MRFCM is not attracted significantlyby outlier data, while the attraction for RFCM and FCMmethods by the outlier is considerable. By increasing outliers,FCM and RFCM methods attract more heavily toward theoutliers. The attraction of the proposed algorithm is littleenough to address it as a robust method. Table 3 indicates themembership functions of the data to each cluster. At the lasttwo rows of the table, the outlier data is placed. It is obviousthat the membership functions of these data are small. Hence,the outliers are not considered in the mentioned clusters.

Fig. 3 Convergence of the algorithms

Table 5 Average solution times (seconds)

Method Number of runs

10 100 300

Neurogenetic 3.26 3.01 2.82

ACO 3.57 3.45 3.18

PSO 3.45 3.36 3.12

GA 2.99 2.76 2.53

RSA 2.34 2.18 2.09

Fig. 4 Queuing network

Table 6 Arc specifications

Arc Specification Arc Specification Arc Specification

(1, 2) (2, 3, 4) (4, 3) (2, 3, 5) (7, 11) (1, 4, 5)

(1, 3) (3, 5, 6) (4, 7) (3, 4, 5) (7, 12) (2, 3, 4)

(1, 4) (2, 3, 5) (4, 8) (4, 5, 6) (8, 10) (3, 5, 7)

(2, 5) (4, 6, 7) (5, 9) (4, 6, 7) (9, 11) (2, 3, 4)

(3, 2) (3, 4, 5) (6, 9) (1, 2, 4) (10, 12) (4, 6, 8)

(3, 5) (3, 5, 7) (7, 8) (3, 4, 5) (11, 13) (1, 2, 3)

(3, 6) (4, 5, 6) (7, 9) (2, 4, 7) (12, 11) (3, 4, 7)

(3, 7) (2, 3, 6) (7, 10) (3, 5, 6) (12, 13) (2, 4, 6)

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A crucial parameter of the clustering algorithms is thenumber of the clusters. This parameter is controlled in theproposed algorithm by the third term of the 3.1. Severaldatasets are examined to ensure the performance of thisfunction. The average number of clusters during the processof the data clustering is presented in Fig. 2.

Figure 2 presents average number of clusters for 500individual runs. It could be observed that because of theproper value of α, the average number of clusters decreasesrapidly. The largest slope of the trend line is between 2 and5. Hence, the algorithm converges after about 5 iterationsefficiently.

4.2 RSA analysis

In this research, the RSA is compared to neurogenetic, antcolony optimization (ACO), particle swarm optimization(PSO), and genetic algorithm (GA). The parameters of theRSA are as: T0=1,000, Tfinal=10, Tk=T0/ln k, Pa=e

–ΔE/T.The parameters of the neurogenetic algorithm are as: GAportion is 90%, and six solutions are fed from GA, pm=0.7,learning rate=0.05, initial weights=1. Parameters for ACOare set as: ρ=0.2, δ=0.1, q0=0.9, q1=0.9, α=2, β=1, c=10,τ0=0.6. PSO parameters are as: c1=0.80, c2=0.80, wmin=0.35, wmax=0.88. Particles are in the range of [0, 1]randomly at initial time. Each particle starting velocity iszero. The parameters of the GA are as: pop-size=40,iteration=500, simulation=500, pc=0.5, pm=0.6.

The number of successful solutions in an algorithm is animportant factor to evaluate its efficiency. To examine thisfactor, several randomly generated networks are utilized.Table 4 indicates the results.

The last row of the table shows that RSA has a betterperformance in finding optimal solutions. The proposedalgorithm has the largest percentage of successful solutionsfor all number of runs. Although among other algorithmsACO has a better performance, it seems weaker comparedto the RSA. In order to compare ACO, SA, and RSA inmore details, a convergence test is performed. Theparameters of these algorithms are set as predefined values.Figure 3 presents the convergence test results.

RSA converges faster than ACO and SA. After 100iterations, RSA almost reaches the optimal value, whileACO and SA reach this value after almost twice number ofiterations. Hence, the proposed algorithm converges effi-ciently.

At the last experiments of this subsection, solutiontimes of algorithms are compared. Table 5 indicates theresults.

Results of the algorithm runs showed that the solutiontime of the RSA is dominant for all number of runs.Therefore, the RSA could be addressed as the efficientalgorithm to solve the mentioned queuing network.

4.3 The performance analysis

In this subsection, a medium-scaled network is consideredfor discussion. Arriving customers are clustered based ontheir required service times by MRFCM method. Theclustered customers are entered to the queuing network.Based on the proposed transformation, the mathematicalmodel, and the simulation procedure, the network isevaluated in various times. RSA algorithm is utilized tofind the shortest path on such a network. Figure 4 indicatesthis network. Arc specifications and queuing parameters arein Tables 6 and 7, respectively.

Interarrival time of the nodes depends on the arcwhich is crossed to arrive to the node. Hence, theseparameters are not entered in Table 7. Prior to clusteringand simulation, each of nodes should be transformed tocorresponding waiting times. Based on the parametricprogramming and the proposed transformations, the wait-ing times of the nodes are calculated. The results arepresented in the last column of Table 7. Hence, a networkwith fuzzy length of arcs and nodes are encountered in therest of the problem.

Arriving customers are clustered based on the requiredservice times. The interarrival times and clusters of thesubsection 4.1 are also used in this subsection. Table 3indicates the clusters and the membership functions. Insteadof time wasting calculations on each customer, clusters areevaluated. Hence, the clustering reduces calculationsdrastically. In the example of subsection 4.1, instead of 20individual customers, 3 clusters are used for calculations.

At the final part of this subsection, the network isscanned continuously to show the performance of theoptimization method. Capacity of the queuing systems isassumed to be two customers at this experiment. Figure 5a–l show the network status by crossing clusters of customers.

Table 7 Specifications of queues in the nodes

Node Service time Waiting time

2 (12, 13, 15) (12.7, 13.3, 14.1)

3 (1, 2, 3) (3.6, 3.9, 4.5)

4 (13, 15, 17) (13.8, 14.5, 15.4)

5 (2, 5, 8) (4.9, 5.7, 6.6)

6 (2, 3, 5) (3.1, 3.7, 4.1)

7 (11, 14, 15) (12.8, 13.9, 14.4)

8 (12, 14, 16) (13.6, 14.1, 14.8)

9 (3, 4, 5) (5.2, 5.8, 6.5)

10 (14, 15, 16) (13.5, 14.6, 15.1)

11 (1, 3, 4) (3.2, 3.6, 4.4)

12 (12, 16, 17) (14.2, 14.7, 15.6)

13 (13, 14, 15) (12.5, 13, 14.2)

752 Int J Adv Manuf Technol (2012) 61:745–755

Two main results could be observed in Fig. 5:

& Arcs which their destination node is blocked should beeliminated from the network because of constraint 3.8in the model.

& The second and third clusters of customers tend to crossfrom a specific path, while the first cluster of customerscrosses from other path. This difference arises becauseof different required service times of these clusters.Constraint 3.7 manages this prevention.

The resulted shortest paths for the clusters are as:

& First cluster path: 1–3–7–11–13. The shortest pathlength: 30.

& Second cluster path: 1–4–8–10–5–13. The shortest pathlength: 62.

& Third cluster path: 1–4–8–10–5–13. The shortest pathlength: 92.

It could be observed that for a same path crossed bythe second and third clusters, the length of the shortestpath differs. This is because of different required servicetimes for these two clusters. The other cause is theaddition of the waiting time to the length of the pathfor the third cluster.

5 Conclusion

In this paper, a new model in fuzzy queuing networkswas introduced. The proposed model integrated fuzzyrobust clustering with queuing networks and shortest

Fig. 5 Network scan from a–l

Int J Adv Manuf Technol (2012) 61:745–755 753

path problems. The main contributions of this paper areas:

& A modified robust fuzzy c-means method was designed.This method is noise-resistant and has a better perfor-mance compared to other methods. Experimental resultsshowed that attraction of the clusters to noisy data areless than the other methods.

& A new model of fuzzy queuing network was proposed.In this network, queues have a finite capacity and utilizea blocking mechanism.

& RSA was proposed. Based on setting a lower bound onthe temperature, the noisy data are eliminated.

Numerical examples proved that the performanceof RSA is better compared to other publishedmethods.

The following suggestions may be pursued for furtherresearches:

& Applying advanced prevention mechanisms on thequeues.

& Studies on the scalability of the networks, and design ofmodels for large networks.

& Applying different structures such as trees instead ofnetworks.

Fig. 5 (continued)

754 Int J Adv Manuf Technol (2012) 61:745–755

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