adapted markovian model to control reliability assessment...

Scientia Iranica E (2013) 20(6), 2224{2237

Sharif University of TechnologyScientia Iranica

Transactions E: Industrial Engineeringwww.scientiairanica.com

Adapted Markovian model to control reliabilityassessment in multiple AGV manufacturing system

H. Fazlollahtabar� and S.G. Jalali NainiFaculty of Industrial Engineering, Iran University of Science and Technology, Tehran, P.O.Box 13114-16846, Iran.

Received 20 July 2012; received in revised form 23 November 2012; accepted 14 May 2013

KEYWORDSAutomatedmanufacturingsystems;Reliability assessment;Markovian model.

Abstract. This paper presents a Markovian model for Flexible Manufacturing Systems(FMSs). The model considers two features of automated exible manufacturing systemsequipped with the Automated Guided Vehicle (AGV), namely the reliability of machinesand the reliability of AGVs in a multiple AGV jobshop manufacturing system. Performancemeasure is a critical factor used to judge the e�ectiveness of a manufacturing system. Thestudies in the literature did not compare Markovian and neural networks, especially inthe reliability modeling of an advanced manufacturing system, considering AGVs. Thecurrent methods for modeling the reliability of a system involve determination of systemstate probabilities and transition states. Since the failure of the machines and AGVs couldbe considered in di�erent states, a Markovian model is proposed for reliability assessment.Also, a neural network model is developed to point out the di�erence in the accuracy of theMarkovian model in comparison with the neural network. The optimization objectives inthe proposed model are maximizing the total reliability of machines in shops in the wholejobshop system and maximizing the total reliability of the AGVs. The multi-objectivemathematical model is optimized using an analytic hierarchy process.

c 2013 Sharif University of Technology. All rights reserved.

1. Introduction

Traditional manufacturing has relied on dedicatedmass-production systems to achieve high productionvolumes at low costs. As living standards improveand the demands for new consumer goods rise, man-ufacturing exibility gains prominence as a strategictool for rapidly changing markets. Flexibility, however,cannot be properly incorporated in the decision-makingprocess, if it is not well de�ned and measured in aquantitative manner. Flexibility in its most rudimen-tary sense is the ability of a manufacturing systemto respond to changes and uncertainties associatedwith the production process [1-3]. A comprehensive

*. Corresponding author. Tel. +98 21 77 240 540E-mail addresses: [email protected] (H. Fazlollahtabar);[email protected] (S.G. Jalali)

classi�cation of eight exibility types was proposed inBrowne et al. [4].

Flexible Manufacturing Systems (FMS) are cru-cial for modern manufacturing to enhance the produc-tivity involved with high product proliferation [5]. Asone of the critical components of the FMS, the exibleMaterial Handling System (MHS) plays a strategic rolein the implementation of the FMS [6]. According toTompkins et al. [7], about 20-50% of the total produc-tion cost is spent on material handling. This makes thesubject of material handling increasingly important. Inaddition, all the complexity of manufacturing is passedon to the MHS. Therefore, the exible MHS has beenvital for improving the FMS, to ful�ll the requirementsof high product proliferation.

Automated Manufacturing Systems (AMS),which are equipped with several CNC machinesand the AGV-based material handling system, are

H. Fazlollahtabar and S.G. Jalali Naini/Scientia Iranica, Transactions E: Industrial Engineering 20 (2013) 2224{2237 2225

designed and implemented to gain the automationand e�ciency of production. To make use of allfeatures of AMS, the planning in the AMS decision-making process is critical, because the planningdecision has inuence on the subsequent decisionprocesses such as scheduling, dispatching, etc. Theplanning in automated manufacturing systems canbe characterized as being online and short-termnatured to respond to frequently changing productionorder. Given a production order, manufacturingplanning function is responsible to establish a planby decomposing the production task into a set ofsubtasks. An analysis of AMS dealing with changingdemand can be found in [8]. An extensive review ofthe loading problem for an FMS can be found in [9].An early stochastic programming approach to addressthe short-term production planning for an FMS canbe found in [10].

Automated Guided Vehicle System (AGVS) be-comes popular in many industrial �elds because of its

exibility, reliability, safety, and also its contributionto the increase of productivity and the improvementof housekeeping. But, the performance of a materialhandling system is signi�cantly inuenced by severaloperating policies. One of the important operatingpolicies is the positioning strategy of the idle vehicleson the guide path [11,12].

In most manufacturing systems, decision makingis worked out at several stages of design, planningand operation. The role of performance modeling issigni�cant in advanced manufacturing systems fromeconomic viewpoints. However, events such as machinebreakdown, changes in part type and volume, toolreplacement, raw material and other short interrup-tions are e�ective on the desired performance of amanufacturing system. This problem is critical dueto its impacts on the capacity of the system [13].Researches on the automated manufacturing systemsimply that the machine failure is the major problemin analyzing system performance, in comparison withother factors like raw material, equipment, softwareand workers [14]. Therefore, reliability considerationsshould be taken into account for manufacturing sys-tem analysis. Researchers who studied this probleminclude [15-21].

Since manufacturing systems experience di�erentfailure states, considering these states in modeling areliability problem is of importance. The best wayfor considering system states in modeling is to employMarkovian property. Reibman [22] stated that theproblem in estimating the probability of failure indi�erent states is vital for reliability computations.

The increasing demand for the reliability assess-ment in manufacturing systems, under several ran-dom parameters, has been investigated by severalapproaches facilitating the computations of probability

estimations. According to the following brief literaturereview, studies to compare Markovian and neuralnetworks are few; especially, modeling the reliability ofan advanced manufacturing system, considering AGVs,is also rare.

2. Literature review

The improvement of safety in the process industries isrelated to the assessment and reduction of risk in a cost-e�ective manner. Kan�cev and �Cepin [23] addressedthe trade-o� between risk and cost related to standbysafety systems. An age-dependent unavailability modelthat integrated the e�ects of the Test and Maintenance(T&M) activities, as well as component ageing, wasdeveloped and represented the basis for calculatingrisk. The repair \same-as-new" process was consideredregarding the T&M activities. Costs were expressedas a function of the selected risk measure. The time-averaged function of the selected risk measure wasobtained from a probabilistic safety assessment, i.e. thefault tree analysis. This function was further extendedwith the inclusion of additional parameters related toT&M activities, as well as ageing parameters relatedto component ageing. In that sense, a new model ofsystem unavailability, incorporating component ageingand T&M costs, was presented. The testing strategywas also addressed. Sequential and staggered testingswere compared. The developed approach was appliedon a standard safety system in nuclear power plantalthough the method was applicable to standby safetysystems that were tested and maintained in otherindustries as well.

The problem of selecting a suitable maintenancepolicy for repairable systems and for a �nite timeperiod was presented by Marquez and Heguedas [24].Since the late seventies, examples of models assessingcorrective and preventive maintenance policies overan equipment life cycle have been existed in theliterature. However, there are not too many contri-butions regarding real implementation of these modelsin the industry, considering realistic timeframes andfor repairable systems. Modeling this problem nor-mally requires the representation of di�erent correctiveand/or preventive actions that could take place atdi�erent moments, driving the equipment to di�erentstates with di�erent hazard rates. An approach topattern the system under �nite periods of time has beenthe utilization of semi-Markovian probabilistic models,allowing later a maintenance policy optimization, usingdynamic programming. These models are very exibleto represent a given system, but they are also complexand therefore very di�cult to handle when the numberof system possible states increases. Marquez andHeguedas [24] explored the trade-o� between exibilityand complexity of these models, and presented a

2226 H. Fazlollahtabar and S.G. Jalali Naini/Scientia Iranica, Transactions E: Industrial Engineering 20 (2013) 2224{2237

comparison in terms of model data requirements versuspotential bene�ts obtained from the model.

In the Generalized Renewal Process (GRP) relia-bility analysis for repairable systems, the Monte Carlo(MC) simulation method, instead of numerical method,is often used to estimate the model parameters becauseof the complexity and the di�culty of developing amathematically tractable probabilistic model. Wangand Yang [25] proposed a nonlinear programmingapproach to estimate the restoration factor for theKijima type GRP model I, as well as the modelII, based on the conditional Weibull distribution forrepairable systems, using negative log-likelihood as anobjective function, and adding inequality constraintsto model parameters. The method minimized thenegative log-likelihood directly, and avoided solving thecomplex system of equations. Three real and di�erenttypes of �eld failure data sets with time truncation forNC (Numerical Control) machine tools were analyzedby the proposed numerical method. The samplingformulas of failure times for the GRP models I andII were derived, and the e�ectiveness of the proposedmethod was validated with the MC (Monte Carlo)simulation method.

Ke et al. [26] considered a multi-repairmen prob-lem comprising of M operating machines with W warmstandbys (spares). Both operating and warm standbymachines were subject to failures. With a coverageprobability c, a failed unit was immediately detected,and attended by one of R repairmen if available. Ifthe failed unit was not detected with the probability 1-c, the system would have entered an unsafe state, andmust have been cleared by a reboot action. The repair-men were also subject to failures which result in service(repair) interruptions. The failed repairman resumedservice after a random period of time. In addition, therepair rate depended on the number of failed machines.The entire system was modeled as a �nite-state Markovchain, and its steady state distribution was obtained bya recursive matrix approach. The major performancemeasures were evaluated based on this distribution.Under a cost structure, the authors proposed to use theQuasi-Newton method and probabilistic global searchLausanne method to search for the global optimalsystem parameters.

Nowadays, Voice over Internet Protocol (VoIP)has become an evolutionary technology in telecommu-nications. Hence it is very important to study andenhance its dependability attributes. An analyticaldependability model for VoIP was proposed by Guptaand Dharmaraja [27]. The study was focused onanalyzing the combined e�ects of resource degradationand security breaches on the Quality of Service (QoS)of VoIP, to enhance its overall dependability. As apreventive maintenance policy to prevent or postponesoftware failures, which cause resource degradation,

software rejuvenation was adopted. The dependabilitymodel was analyzed using semi-Markov process, whichcaptures the e�ects of non-Markovian nature of thetime spent at various states of the system. The steady-state, as well as the time-dependent analysis of thedependability model, was previously presented.

Zhou et al. [28] presented a maintenance optimiza-tion method for a multi-state series-parallel system,considering economic dependence and state-dependentinspection intervals. The objective function consideredin the paper was the average revenue per unit timecalculated, based on the semi-regenerative theory andthe Universal Generating Function (UGF). A new algo-rithm, using the stochastic ordering, was also developedin the paper to reduce the search space of maintenancestrategies and to enhance the e�ciency of optimizationalgorithms. A numerical simulation was presented inthe study to evaluate the e�ciency of the proposedmaintenance strategy and optimization algorithms.

A reliability assessment for Hard Disk Drives(HDDs) is important yet di�cult for manufacturers.Motivated by the fact that the particle accumulationin HDDs, which accounts for most HDD catastrophicfailures, is contributed to the internal and externalsources, a counting process with two arrival sourceswas proposed by Ye et al. [29] to model the particlecumulative process in HDDs. The model successfullyexplained the collapse of traditional ALT approachesfor accelerated life test data. Parameter estimation andhypothesis tests for the model were developed and il-lustrated with real data from a HDD test. A simulationstudy was conducted to examine the accuracy of largesample normal approximations that were used to testthe existence of the internal and external sources.

An R out of N repairable system consisting ofN independent components is operating if at least Rcomponents are functioning. The system fails wheneverthe number of good components decreases from R to R-1. A failed component is sent to a repair facility havingseveral repairmen. Lifetimes of working componentsare i.i.d random variables having an exponential distri-bution. Repair times are i.i.d random variables havinga phase type distribution. Both cold and warm stand-by systems are considered. Barron et al. [30] presentedan algorithm deriving recursively in the number ofrepairmen the generator of the Markov process. Thenthey derived formulas for the point availability, limitingavailability, distribution of the down time and up time.Numerical examples were given for various repair timedistributions. The numerical examples showed thatthe availability is not very sensitive to the repair timedistribution, while the mean up time and the meandown time might be very sensitive to the repair timedistributions.

According to the brief reviewed literature, studiesto compare Markovian and neural networks are few.


Especially, modeling the reliability of an advancedmanufacturing system, considering AGVs, is also rare.

3. Statement of the problem

Here, a jobshop manufacturing system having multipleAGVs for material handling purpose is considered.In each shop, several machines perform the partprocessing according to a process plan. To transferthe parts among di�erent shops, AGVs are employed.The reliability of the whole manufacturing system isconcerned with the reliability of machines in shops andthe reliability of AGVs. The failure of the machinesand AGVs could be considered in di�erent states. Thefailure occurred for machines are:

� Amateur operator;� Equipment de�ciency;� Inappropriate part speci�cations.Also, the failures of AGVs are due to:

� Carrier overload;� Guide path fracture.

Using the Markovian property, we can con�gurethe transition diagram and the corresponding matrix.The result of the Markovian process is the failureprobability for machines and AGVs. These proba-bilities are applied in reliability computations. Forreliability, �rst we conceptualize di�erent scenariosexisting in the proposed manufacturing system. Theshops are in parallel since the parts are disseminatedthrough the system, according to the process plan.The sequence of machines in a shop may be importantor not, i.e. the part processing in a shop shouldbe performed sequentially on the machines, or thesequence is not important and parallel machining ispossible. Therefore, two separate cases of series andparallel should be modeled. AGVs are in series, and ifone AGV breaks down then the whole system shouldwait until the AGV is repaired or taken out of thesystem.

The aim of the decision maker is to maximize theperformance of the whole system. To achieve the aim,two objectives, namely maximizing the total reliabilityof machines in shops in the whole jobshop system, andmaximizing the total reliability of AGVs, should beinvestigated. Also, for the economic viewpoint of thesystem performance, the third objective is to minimizethe total repair cost in the system. As a unit (machineor AGV) in the system is broken down, the repairshould be performed on it to prepare it for functioning.

The aims of conducting this study are:p

Developing a reliability assessment methodology forAGV-based manufacturing systems;

pAnalyzing and including fault sources in machine-AGV state modeling in manufacturing systems;pMarkovian modeling for reliability assessment of amachine-AGV manufacturing system;pComparing the Markovian reliability assessmentwith the neural network method.

3.1. Model with reliabilityIt is necessary to incorporate reliability into the modelto ensure the level of service for each machine in eachshop and the AGVs. For modeling reliability, theapproach of Ball and Lin [31] is adopted and furtherextended.

The reliability is de�ned as the probability thatthe system works until time t. If a machine in a shop isbroken down, it can be regarded as a failure. A desiredlevel of reliability can be achieved by limiting the failureprobabilities. This approach for handling reliability iscalled the method of chance constraints in the contextof mathematical programming. The use of chanceconstraints in a vehicle routing problem was illustratedin Stewart and Golden [32]. Carbone [33] used chanceconstraints for selecting multiple facilities under nor-mally distributed demands. The model minimized anupper bound on the total demand-weighted distancewhile ensuring that the constraint was satis�ed withspeci�ed chance or probability. Shiode and Drezner [34]used a similar approach in a competitive locationproblem on a tree network.

It is assumed that the reliability of each ma-chine type and the AGV are independently due toexponential processes. Also, J is the total types ofmachines, i.e. drilling machines, turning machines andbending machines (three machine types). We discussthe reliability-based model as follows:

Rj(t) :The probability that machine type jth worksuntil time t;

R(t)system =8>>>>>>>:

1� JQj=1

(1�Rj(t))!;

when machines in eachshop are in parallel case

JQj=1

Rj(t)

!;

when machines in eachshop are in series case

(1)

In our proposed problem, AGVs are series and themachine types in each shop may be in parallel or seriescases, and the shops are parallel, i.e. a compositesystem is con�gured. Therefore, the reliability of thesystem is as follows:0@1� JY

j=1

(1�Rj(t))1A � �; (2)


where � is the lower bound for a desirable reliabilityof the system until time t. As previously assumed,the reliability of each machine type and AGV areindependently due to exponential distribution:

Rj(t) = e�t�j ; (3)

where �j is the exponential parameter for machine typeor AGV breakdown. Then,0@1� JY

j=1

(1� e�t�j )1A � �: (4)

It is obvious that to obtain a higher level of reliability,more cost is incurred to the system. Hence, a costfunction (Cj(t)) is de�ned to keep machine type jthreliable until time t. For the whole system, we have:

JXj=1

Cj(t): (5)

4. Mathematical formulation

In this section, we construct the proposed failure statediagrams and matrices for machines and AGVs, usingthe Markov system separately. A Markov system is asystem that can be in one of the several (numbered)states, and can pass from one state to another by eachtime step, according to �xed probabilities. If a Markovsystem is in state i, there is a �xed probability, pij ,of it going into state j the next time step, and pij iscalled a transition probability. A Markov system canbe illustrated by means of a state transition diagram,which is a diagram showing all the states and transitionprobabilities. The entries in each row add up to 1.

First, we con�gure the machines' state diagram.As stated before, the machines may be broken downin three states, namely (a) amateur operator, (b)equipment de�ciency and (c) inappropriate part speci-�cations. Note that the states refer to the breakdownstate causes, i.e. the machine is working or it isbroken down due to the failure states such as amateuroperator, equipment de�ciency and inappropriate partspeci�cations. In another word, since we are modelingthe reliability of the system, considering di�erent statechanges, it is common in Markovian computations tomonitor the state transition, while all are the causes ofbreakdown. The state transition diagram for machinesis shown in Figure 1.

As a result, the corresponding transition matrixPij is,

Pij =

24 1� �� " � "� 1� � � � � 1� � � �

35 ; (6)

Figure 1. The state transition diagram for machines.

where �, �, , �, " and � are the transition probabilitiesfrom the three states given in Figure 1. Using the prob-ability transition matrix and the limiting probability,we obtain each state's occurrence probability as follows:

�j =3Xi=1

�ipij for j = 1; 2; 3; (7)

3Xj=1

�j = 1: (8)

Using these probabilities, we can compute the relia-bility of each state that helps us to assess the totalreliability of the system.

We also can compute the long run probability foreach state, using the steady state distribution givenbelow:�

A B C�24 1� �� " � "� 1� � �

� � 1� � � �

35=�A B C

�; (9)

having A+B + C = 1.The same computations exist for AGVs di�erent

failure state, while we stated 2 states, i.e. we have twostate probabilities and a 2�2 transition matrix.

Now, for reliability we have:

R (t) = 1� F (t) ; (10)where F (t) is the failure probability computed aboveas states' probabilities. Note that, we can computethe reliability in two cases, �rst for current state, andsecond for steady state. The numerical comparison ofthe two could be interesting.

Having the current state of the system by theMarkovian model and by the means of neural net-work, we can compute the steady state probabilities.


Next, we review the arti�cial neural network andthe backpropagation neural network for our proposedwork. The reason is to �nd the di�erence between theaccuracy of the two methods and determine the moste�ective one. It is obvious that the neural network canbe more e�cient due to its using past data in trainingstage.

The aim to compute the steady state probabilityand reliability is to obtain an estimation of the systemavailability for long run planning horizon. Therefore, itis signi�cant for a decision maker to determine steadystate reliability, using the corresponding probability,accurately.

4.1. Arti�cial neural networkNeural networks are being widely used in many �elds ofstudy. This could be attributed to the fact that thesenetworks attempt to model the capabilities of humanbrains. Since the last decade, neural networks havebeen used as a theoretically sound alternative to tra-ditional statistical models. Although Neural Networks(NNs) originated from mathematical neurobiology, therather simpli�ed practical models currently in use havemoved steadily towards the �eld of statistics. A numberof researchers have illustrated the connection of neuralnetworks to traditional statistical models. For exam-ple, Gallinari et al. [35] presented analytical resultsestablishing a link between discriminant analysis andmultilayer perceptrons (MLP) used for classi�cationproblems. Cheng and Titterington [36] made a detailedanalysis and comparison of various neural networkmodels with traditional statistical models. Theyshowed strong associations of feed-forward neural net-works with discriminant analysis and regression mod-els, and unsupervised networks such as self-organizingneural networks with clustering. Neural networks arebeing used in areas of prediction and classi�cation,areas where regression models and related statisticaltechniques have traditionally been used. Ripley [37]discusses the statistical aspects of neural networks andclassi�es neural networks as one of the classes of exiblenonlinear regression models. Warner and Misra [38]present a comparison between regression analysis andneural network computation in terms of notation andimplementation. They also discuss when it wouldbe advantageous to use a neural network model inplace of a parametric regression model, as well assome of the di�culties in implementation. Vach etal. [39] presented a comparison between feed-forwardneural networks and logistic regression. The conceptualsimilarities and discrepancies between the two methodsare also analyzed.

Arti�cial neural networks have been applied suc-cessfully to many manufacturing and engineering areas.Zhengrong et al. [40] used quadratic regressions toassess the results of neural network for improving the

e�ciency of fermentation process development. Theresults show that di�erent sizes of neural nets within acertain range give an equally good prediction by usingthe \stopping training" technique, while quadraticregressions are sensitive to the size of the data sets.Smith and Mason [41] mentioned that regression andneural network modeling methods have become twocompeting empirical model-building methods. Theycompared the predictive capabilities of NNs and re-gression methods in manufacturing cost estimationproblems.

4.1.1. The backpropagation neural networkThe backpropagation algorithm trains a given feed-forward multilayer neural network for a given set ofinput patterns with known classi�cations. When eachentry of the sample set is presented to the network, thenetwork examines its output response to the sampleinput pattern. The output response is then comparedto the known and desired output and the error value iscalculated. Based on the error, the connection weightsare adjusted. The backpropagation algorithm is basedon Widrow-Ho� delta learning rule in which the weightadjustment is done through mean square error of theoutput response to the sample input. The general stepsof backpropagation are given below:

1. Propagate inputs forward in the usual way, i.e. alloutputs are computed using sigmoid thresholding ofthe inner product of the corresponding weight andinput vectors. All outputs at stage n are connectedto all inputs at stage n+ 1;

2. Propagate errors backwards by apportioning themto each unit according to the amount of the errorthe unit is responsible for.

We now discuss how to develop the stochasticbackpropagation algorithm for the general case. Thefollowing notations and de�nitions are needed:~xj Input vector for unit j (xji = ith input

to the jth unit);~wj Weight vector for unit j (wji = weight

on xji);zj = ~wj :~xj The weighted sum of inputs for unit j;oj Output of unit j (oj = �(zj));tj Target for unit j;Downstream(j)Set of units whose immediate inputs

include the output of j;Output Set of output units in the �nal layer.

Since we are updated after each training example,we can simplify the notation somewhat by assumingthat the training set consists of exactly one example,and so the error can simply be denoted by E.

We want to calculate @E@wji corresponding to eachinput weight, wji, of each output unit, j. Note �rst


that since zj is a function of wji , regardless of wherein the network unit j is located,

@E@wji

=@E@zj

:@zj@wji

=@E@zj

:xji; (11)

furthermore, @E@zj is the same, regardless of which inputweight of unit j we are trying to update. So, we denotethis quantity by �j .

Consider the case when j is an output unit. Weknow that:

E =12

Xk2Outputs

(tk � �(zk))2: (12)

Since the outputs of all units k 6= j are independentof wji, we can then drop the summation and considerjust the contribution to E by j, and we call it �j :

�j =@E@zj

=@@zj

12

(tj � oj)2 = �(tj � oj)@oj@zj=�(tj � oj) @@zj �(zj)=�(tj � oj)(1� �(zj))�(zj)

= �(tj � oj)(1� oj)oj : (13)Thus:

�wji = �� @E@wji = ��jxji: (14)

Now, consider the case when j is a hidden unit.Like before, we make the following two importantobservations:

1. For each unit k downstream from j, zk is a functionof zj .

2. The contribution to error by all units l 6= j, in thesame layer as j, is independent of wji.

We want to calculate @E@wij for each input weight,wji, for each hidden unit j. Note that wji inuencesjust zj which inuences oj which inuences zk, 8k 2Downstream(j), each of which inuences E. So, wecan write:

@E@wji

=X

k2Downstream(j)@E@zk

:@zk@oj

:@oj@zj

:@zj@wji

=X

k2Downstream(j)@E@zk

:@zk@oj

:@oj@zj

:xji: (15)

Again, note that all the terms, except xji in Eq. (15),are the same regardless of which input weight of unitj we are trying to update. Like before, we denote this

common quantity by �j . Also, note that @E@k �k,@k�ojwkj

and @oj@zj = oj(1� oj). Substituting them in Eq. (13).�j =

Xk2Downstream(j)

@E@zk

:@zk@oj

:@oj@zj

=X

k2Dowbnstream(j)�k:wkj :oj(1� oj); (16)

we obtain:

�k = oj(1� oj) Xk2Downstream(j)

�k:wkj : (17)

To adapt the backpropagation algorithm on our pro-posed model, consider the failure causes for machinesand AGVs as inputs and the current state failure prob-ability of machines and AGVs as outputs. We trainthe network collecting data in di�erent time periodsand compute the importance weight for each inputresulting the corresponding output. A con�gurationof the proposed neural network is shown in Figure 2.

We are now in a position to state the backpropa-gation algorithm formally.

Algorithm 1: Formal statement of stochasticbackpropagation.

(Training examples, �, ni, nh, no)

Each training example is of the form�~x;~t�

, where ~x

is the input vector and ~t is the target vector, � is thelearning rate (e.g., 0.05), ni, nh and no are the numberof input, hidden and output nodes, respectively. Inputfrom unit i to unit j is denoted by xji and its weightis denoted by wji. Create a feed-forward network withni inputs, nh hidden units and no output units.

Initialize all the weights to small random values (e.g.,between -0.05 and 0.05).

Figure 2. A con�guration of the proposed neuralnetwork.


While termination condition is not met DoFor each training example

�~x;~t�

,

1. Input the instance ~x and compute the output ou ofevery unit.

2. For each output unit k, calculate,

�k = ok(1� ok)(tk � ok): (18)3. For each hidden unit h, calculate,

�h = oh(1� oh) Xk2Downstream(h)

�k:wkh: (19)

4. Update each network weight wji as follows:

wji wji + �wji: (20)where,

�wji = ��jxji: (21)

This way, we can compare the performance ofbackpropagation neural network and limiting distribu-tion model for computing the steady state probabilities,using the current state probabilities.

4.2. Mathematical optimizationHere, the mathematical optimization model is given.As stated before, we aim to maximize the total re-liability of machines in shops in the whole jobshopsystem, and maximize the total reliability of the AGVs.Since the reliability model is stochastic, one may thinkabout simulation study. But considering multiple-objectives and especially including cost factors in theform of a composite mathematical function is di�cultand requires tiring and complicated simulation e�orts.Also, as we will present further, we considered several0.1 integer variables which are easier to be modelledmathematically.

4.2.1. Maximizing total reliability of machinesThe following mathematical notations are employed tomodel this maximization problem:

Mathematical notations:

k Index for machines, k = 1; :::;K:l Index for jobs, l = 1; :::; L:m Index for shops, m = 1; :::;M:Rk Reliability of machine k:

�kl =

(1 if machine k process job l:0 otherwise:

'km =

(1 if machine k in shop m is chosen:0 otherwise:

The mathematical model:

MaxXm

Xk

(Rk:'km); (22)

s.t.Xk

'km:�kl = 1; 8l;m; (23)

'km 2 f0; 1g ; 8k;m: (24)4.2.2. Maximizing total reliability of AGVsThe following mathematical notations are employed tomodel this maximization problem:

Mathematical notations:n Index for AGVs, n = 1; :::; N:Rn Reliability of AGV n:

&nm =

(1 if AGV n can service shop m:0 otherwise:

�n =

(1 if AGV n is chosen:0 otherwise:

The mathematical model:

MaxXn

Rn:�n; (25)

s.t.Xn

�n:&nm = 1; 8m; (26)

�n 2 f0; 1g ; 8n: (27)As a result, a multi-objective mathematical model iscon�gured as follows:

MaxXm

Xk

(Rk:'km); (28)

MaxXn

Rn:�n; (29)

s.t.Xk

'km:�kl = 1; 8l;m; (30)Xn

�n:&nm = 1; 8m; (31)Xk

'km = 1; 8m; (32)

�n 2 f0; 1g ; 8n; (33)'km 2 f0; 1g ; 8k;m: (34)


Next, an approach to optimize the proposed multi-objective model is given. We use objectives weighingmethod to integrate and optimize the model.

4.3. Analytic Hierarchy Process (AHP) formulti-objective optimization

To weight the objectives, we take a multi-criteriadecision-making approach. Multi-Criteria Decision-Making (MCDM), dealing primarily with the problemsof evaluation or selection, is a rapidly developing areain operations research and management science. AHPis a technique of considering data or information fora decision in a systematic manner. It is mainlyconcerned with a way of solving decision problems withuncertainties in multiple criteria characterization. It isbased on three principles: constructing the hierarchy,priority setting, and logical consistency. We applyAHP to weight the objectives.

Construction of the hierarchyA complicated decision problem, composed of variousattributes of an objective, is structured and decom-posed into sub-problems (sub-objectives, criteria, al-ternatives, etc.), within a hierarchy.

Priority settingThe relative \priority" given to each element in thehierarchy is determined by pair-wise comparisons of thecontributions of elements at a lower level in terms ofthe criteria (or elements) with a causal relationship.In AHP, multiple paired comparisons are based ona standardized comparison scale of nine levels (seeTable 1).

Let C = fc1; :::; cng be the set of criteria. Theresult of the pair-wise comparisons on n criteria canbe summarized in an n � n evaluation matrix A inwhich every element aij is the quotient of weights ofthe criteria, as shown below:

A = (aij) ; i; j = 1; :::; n: (35)

The relative priorities are given by the eigenvector (w)

Table 1. Scale of relative importance.

Intensityof importance

De�nitionof importance

1 Equal2 Weak3 Moderate4 Moderate plus5 Strong6 Strong plus7 Very strong or demonstrated8 Very, very strong9 Extreme

corresponding to the largest eigenvalue(�max) as:

Aw = �maxw: (36)

When pair-wise comparisons are completely consistent,the matrix A has rank 1 and �max = n. In that case,weights can be obtained by normalizing any of the rowsor columns of A.

The procedure described above is repeated for allsubsystems in the hierarchy. In order to synthesizethe various priority vectors, these vectors are weightedwith the global priority of the parent criteria andsynthesized. This process starts at the top of thehierarchy. As a result, the overall relative prioritiesto be given to the lowest level elements are obtained.These overall, relative priorities indicate the degreeto which the alternatives contribute to the objec-tive. These priorities represent a synthesis of thelocal priorities, and reect an evaluation process thatpermits integration of the perspectives of the variousstakeholders involved.

Consistency checkA measure of consistency of the given pair-wise com-parison is needed. The consistency is de�ned by therelation between the entries of A; that is, we say Ais consistent if aik = aij . ajk for all i; j; k. Theconsistency index (CI) is:

CI =(�max � n)

(n� 1) : (37)The �nal Consistency Ratio (CR), on the basis ofwhich one can conclude whether the evaluations aresu�ciently consistent, is calculated to be the ratio ofthe CI and the random Consistency Index (RI):

CR =CIRI

: (38)

The value 0.1 is the accepted upper limit for CR.If the �nal consistency ratio exceeds this value, theevaluation procedure needs to be repeated to improveconsistency. The measurement of consistency can beused to evaluate the consistency of decision-makers aswell as the consistency of all the hierarchies.

We are now ready to give an algorithm for com-puting objective weights using the AHP. The followingnotations and de�nitions are used:n Number of criteria;i Number of objectives;p Index for objectives, p = 1 or 2;d Index for criteria, 1 � d � D;Rpd The weight of pth item with respect to

dth criterion;wd The weight of dth criterion.


Algorithm 2: OWAHP (compute objectiveweights using the AHP)

Step 1: De�ne the decision problem and the goal.Step 2: Structure the hierarchy from the top through

the intermediate to the lowest level.Step 3: Construct the objective-criteria matrix using

Steps 4 to 8 using the AHP. (Steps 4 to 6 areperformed for all levels in the hierarchy.)

Step 4: Construct pair-wise comparison matrices foreach of the lower levels for each element in thelevel immediately above by using a relativescale measurement. The decision-maker hasthe option of expressing his or her intensityof preference on a nine-point scale. If twocriteria are of equal importance, a value of 1is set for the corresponding component in thecomparison matrix, while a value of 9 indicatesan absolute importance of one criterion overthe other (Table 1 shows the measurementscale).

Step 5: Compute the largest eigenvalue by the relativeweights of the criteria and the sum taken overall weighted eigenvector entries correspondingto those in the next lower level of the hierar-chy.

Analyze pair-wise comparison data usingthe eigenvalue technique. Using these pair-wise comparisons, estimate the objectives.The eigenvector of the largest eigenvalue ofmatrix A constitutes the estimation of relativeimportance of the attributes.

Step 6: Construct the consistency check and performconsequence weights analysis as follows:

A = (aij) =

266641 w1w2 : : :

w1wnw2

w1 1 : : :w2wn

......

. . .wnw1

wnw2 1

37775Note that if the matrix A is consistent (that is,aik = aij :ajk, for all i; j; k = 1; 2; :::; n), thenwe have (the weights are already known):

aij =wiwj; i; j = 1; 2; :::; n: (39)

If the pair-wise comparisons do not includeany inconsistencies, then �max = n. The moreconsistent the comparisons are, the closer thevalue of computed �max is to n. Set theConsistency Index (CI) (which measures theinconsistencies of pair-wise comparisons) tobe:

CI =(�max � n)

(n� 1) ;

Table 2. The objective-criteria matrix.

C1 C2 ... CdObjective 1 R11 R12 ... R1dObjective 2 R21 R22 ... R2d

Table 3. The criteria-criteria pair-wise comparisonmatrix.

C1 C2 ... Cd WdCriteria 1 1 a12 ... a1d w1Criteria 2 1/a12 1 ... a2d w2

......

......

......

Criteria d 1/a1d 1/a2d ... 1 wd

and let the Consistency Ratio (CR) be:

CR = 100�CIRI

�;

where n is the number of columns in A and RIis the random index, being the average of theCI obtained from a large number of randomlygenerated matrices.

Note that RI depends on the order ofthe matrix, and a CR value of 10% or less isconsidered acceptable.

Step 7: Form the objective-criteria matrix as speci�edin Table 2.

Step 8: As a result, con�gure the pair-wise comparisonfor criteria-criteria matrix as in Table 3.

The wd are gained by a normalizationprocess. The wd are the weights for criteria.

Step 9: Compute the overall weights for the objec-tives, using Tables 2 and 3, as follows:

= Total weight for objective 1

= R11 � w1 +R12 � w2 + :::+R1d � wd; 0 = Total weight for objective 2

= R21 � w1 +R22 � w2 + :::+R2d � wd;(40)

where + 0 = 1. Thus the integratedobjective function is formed as:

Max

:Xm

Xk

(Rk:'km) + 0:Xn

(Rn:�n)

!5. Computational results

Here, a numerical example is worked out to implythe e�ectiveness and applicability of the proposed


model. Using machines' (amateur operator, equipmentde�ciency, and inappropriate part speci�cations) andAGVs' (carrier overload and guide path fracture) fail-ure probability transition matrices, and by the meansof limiting probabilities, the states occurrence proba-bilities for each machine or AGV can be computed.

Also, note that the number of machines is 10,number of shops is 4, number of AGVs is 4, and numberof jobs to be processed on a product is 8. Since eachsof the failure states mentioned causes breakdown ofthe system, the failure states are parallel. Using theseprobabilities, we can compute the reliability of eachstate, using Eqs. (1) and (2) which helps us to assessthe total reliability of the system as follows:

Reliability (machine 1): 0.0124Reliability (machine 2): 0.0303Reliability (machine 3): 0.0435Reliability (machine 4): 0.0054Reliability (machine 5): 0.0641Reliability (machine 6): 0.00398Reliability (machine 7): 0.0287Reliability (machine 8): 0.00703Reliability (machine 9): 0.00239Reliability (machine 10): 0.0197Reliability (AGV 1): 0.0753Reliability (AGV 2): 0.0639Reliability (AGV 3): 0.0458Reliability (AGV 4): 0.389

As we stated before, another way to computethe steady state probabilities is backpropagation neu-ral network. The input of the system is given bya one dimensional vector, and the output is givenby a two/three dimensional matrix. To facilitatethe computations of backpropagation neural network,MATLAB 7.1 user interface, NNtool, is applied. Afeedforward network is programmed with one input,ten hidden units with logistics activation function,and two/three outputs. Using the MATLAB 7.1 userinterface NNtool, we insert the data and performthe required settings to train the data to obtain anappropriate pattern. Then, using the pattern, wecan approximate the output of the proposed neuralnetwork. We used pseudo data for the neural networkanalysis.

Outputs:Reliability (machine 1): 0.0133Reliability (machine 2): 0.031Reliability (machine 3): 0.045Reliability (machine 4): 0.0063Reliability (machine 5): 0.0628Reliability (machine 6): 0.00377Reliability (machine 7): 0.0281Reliability (machine 8): 0.00723

Reliability (machine 9): 0.00243Reliability (machine 10): 0.0184Reliability (AGV 1): 0.0761Reliability (AGV 2): 0.0652Reliability (AGV 3): 0.0463Reliability (AGV 4): 0.396

Clearly, backpropagation computations areslightly di�erent from the steady state equation ones.Since neural network employs several training datasets to adapt the appropriate pattern, its results aremore valid. A graphical comparison for the machinesreliability computations, using two approaches, isshown in Figure 3. The machines that process di�erentjobs are shown in Table 4. The AGVs that can servicedi�erent shops are shown in Table 5.

Here, to solve the proposed multi-objective model,AHP, as explained in Section 4.3, is employed. Theintegration weights for the objectives are 0.43 and 0.57,respectively. Optimizing the above linear program inMATLAB 7.1, we obtain 1123.45, as the integrated

Figure 3. Graphical comparison for the machinesreliability computations.

Table 4. The machines that process di�erent jobs.

�kl l 1 2 3 4 5 6 7 8k

1 0 1 0 0 0 1 0 02 0 0 0 0 1 0 1 13 1 1 0 0 0 0 0 04 0 0 1 1 1 0 0 05 0 0 1 0 0 1 0 16 0 0 1 1 0 0 0 07 1 0 0 1 0 0 0 08 1 1 0 0 1 0 0 09 1 1 0 0 1 0 1 010 0 0 1 0 0 0 1 1


Table 5. The AGVs that can service di�erent shops.

�nm m 1 2 3 4n

1 0 1 0 12 1 0 1 03 1 0 0 14 0 1 1 1

Table 6. The decision variables' values.

'km m 1 2 3 4k

1 0 0 0 02 0 0 1 03 0 0 0 04 0 0 0 05 1 0 0 06 0 0 0 07 0 0 0 08 0 1 0 09 0 0 0 010 0 0 0 1�n 1 2 3 4- 1 1 0 0

objective functions' value and the decision variables'values are reported in Table 6.

The results show the machines to service theshops, and the chosen AGVs to service the shops aim-ing at maximizing the total reliability of machines inshops in the whole jobshop system, and maximizing thetotal reliability of the AGVs. The strategic viewpointof such computations is to enable the management tocontrol the failures of AGVs and machines to satisfy theoptimization purposes. This way, the proposed multi-objective mathematical model was capable of function-ing as a Markovian reliability assessment model.

6. Conclusions

We proposed a Markovian model for Flexible Manu-facturing Systems (FMSs). The model considered twofeatures of automated exible manufacturing systemsequipped with Automated Guided Vehicle (AGV),namely the reliability of machines and the reliabilityof AGVs in a multiple AGV jobshop manufacturingsystem. We made use of current state transitionmatrix for the failure of the machines and AGVs indi�erent states. Therefore, a Markovian model wasproposed for reliability assessment. Also, for steadystate probability computations, the limiting theoremwas compared with adapted backpropagation neural

network, showing neural network's e�ectiveness. Usingthe reliabilities, we worked out an optimization math-ematical model. The optimization objectives in theproposed model were maximizing the total reliabilityof machines in shops, in the whole jobshop system,and maximizing the total reliability of the AGVs. Thecomputational results illustrated the applicability ofour proposed model. A strategic viewpoint of suchcomputations was to enable the management to controlthe failures of AGVs and machines to satisfy theoptimization purposes.

References

1. Miettinen, K., Eskelinen, P., Ruiz, F. and Luque,M. \NAUTILUS method: An interactive technique inmultiobjective optimization based on the nadir point",European Journal of Operational Research, 206(2), pp.426-434 (2010).

2. Kumar, N.S. and Sridharan, R. \Simulation modellingand analysis of part and tool ow control decisionsin a exible manufacturing system", Robotics andComputer-Integrated Manufacturing, 25(4-5), pp. 829-838 (2009).

3. Das, K., Baki, M.F. and Li, X. \Optimization ofoperation and changeover time for production planningand scheduling in a exible manufacturing system",Computers & Industrial Engineering, 56(1), pp. 283-293 (2009).

4. Browne, J., Dubois, D., Rathmill, K., Sethi, S.P. andStecke, K.E. \Classi�cation of exible manufacturingsystems", FMS Mag., 2(2), pp. 114-117 (1984).

5. Paraschidis, K., Fahantidis, N., Petridis, V., Doulgeri,Z., Petrou, L. and Hasapis, G. \A robotic systemfor handling textile and nonrigid at materials", In:Proceedings of the Computer Integrated Manufacturingand Industrial Automation (CIMIA), Athens, pp. 98-105 (1994).

6. Beamon, B.M. \Performance, reliability, and per-formability of material handling systems", Interna-tional Journal of Production Research, 36(2), pp. 337-393 (1998).

7. Tompkins, J.A., White, J.A., Bozer, Y.A., Frazelle,E.H., Tanchoco, J.M.A. and Trevino, J., FacilitiesPlanning, John Wiley and Sons Inc., NewYork (2002).

8. Terkaj, W., Tolio, T. and Valente, A., \Designing man-ufacturing exibility in dynamic production contexts",In: Design of Flexible Production Systems, Tolio, T.,Ed., Springer, pp. 1-18. (2009).

9. Grieco, A., Semeraro, Q. and Tolio, T. \A review ofdi�erent approaches to the FMS loading problem",International Journal of Flexible Manufacturing Sys-tems, 13(4), pp. 361-384 (2001).

10. Terkaj, W. and Tolio, T. \A stochastic approach to theFMS loading problem", CIRP Journal of Manufactur-ing Systems, 35(5), pp. 481-490 (2006).


11. Egbehi, P.J. \Positioning of automated guided vehiclesin a loop layout to improve response time", EuropeanJournal of Operational Research, 71, pp. 32-44 (1993).

12. Kim, K.H. \Positioning of automated guided vehiclesin a loop layout to minimize the mean vehicle responsetime", Int. J. of Production Economics, 39, pp. 201-214 (1995).

13. Stoop, P.P.M. and Wiers, V.C.S. \The complexityof scheduling in practice", International Journal ofOperations and Production Management, 16(10), pp.37-53 (1996).

14. Sanchez, A.M. \FMS in Spanish industry - lessons fromexperience", Integrated Manufacturing Systems, 5(2),pp. 28-36 (1994).

15. Hilderbrant, R.R. \Scheduling exible machining sys-tems when machines are prone to failure", Ph.D.Thesis, Massachusetts Institute of Technology, USA(1980).

16. Kimemia, J.G. \Hierarchical control of production in

exible manufacturing systems", Ph.D. Thesis, Mas-sachusetts Institute of Technology, USA (1982).

17. Liberopulos, G. \Flow control of failure prone man-ufacturing systems: Controller design theory andapplications", Ph.D. Thesis, Boston University, USA(1993).

18. Viswanadham, N. and Narahari, Y., PerformanceModeling of Automated Manufacturing Systems,Prentice-Hall, Englewood Cli�s, NJ (1992).

19. Vinod, B. \Queuing models for exible manufacturingsystems subject to resource failure", Ph.D. Thesis,Purdue University, US (1983).

20. Vinod, B. and Solberg, J.J. \Performance models forunreliable exible manufacturing systems", Interna-tional Journal of Management Science, 12(3), pp. 299-308 (1984).

21. Choi, S.H. and Lee, J.S.L. \A heuristic approach tomachine loading problem in non-preemptive exiblemanufacturing systems", International Journal of In-dustrial Engineering, 5(2), pp. 105-116 (1998).

22. Reibman, A.L. \Modeling the e�ect of reliabilityon performance", IEEE Transactions on Reliability,39(3), pp. 314-320 (1990).

23. Kan�cev, D. and �Cepin, M. \Evaluation of risk and costusing an age-dependent unavailability modelling of testand maintenance for standby components", Journal ofLoss Prevention in the Process Industries, 24(2), pp.146-155 (2011).

24. Marquez, A.C. and Heguedas, A.S. \Models for main-tenance optimization: A study for repairable systemsand �nite time periods", Reliability Engineering &System Safety, 75(3), pp. 367-377 (2002).

25. Wang, Z-M. and Yang, J-G. \Numerical method forWeibull generalized renewal process and its applica-tions in reliability analysis of NC machine tools",Computers & Industrial Engineering, 63(4), pp. 1128-1134 (2012).

26. Ke, J-C., Hsu, Y-L., Liu, T-H. and George Zhang,Z. \Computational analysis of machine repair problemwith unreliable multi-repairmen", Computers & Oper-ations Research, 40(3), pp. 848-855 (2013).

27. Gupta, V. and Dharmaraja, S. \Semi-Markov model-ing of dependability of VoIP network in the presence ofresource degradation and security attacks", ReliabilityEngineering & System Safety, 96(12), pp. 1627-1636(2011).

28. Zhou, Y., Zhang, Z., Lin, T.R. and Ma, L. \Main-tenance optimization of a multi-state series-parallelsystem considering economic dependence and state-dependent inspection intervals", Reliability Engineer-ing & System Safety, 111, pp. 248-259 (March 2013).

29. Ye, Z-S., Xie, M. and Tang, L-C. \Reliability eval-uation of hard disk drive failures based on countingprocesses", Reliability Engineering & System Safety109, pp. 110-118 (2013).

30. Barron, Y., Frostig, E. and Levikson, B. \Analysis of Rout of N systems with several repairmen, exponentiallife times and phase type repair times: An algorithmicapproach", European Journal of Operational Research,169(1), pp. 202-225 (2006).

31. Ball, M.O. and Lin, F.L. \A reliability model appliedto emergency service vehicle location", OperationsResearch, 41(1), pp. 18-36 (1993).

32. Stewart, W. and Golden, B. \Stochastic vehicle rout-ing: A comprehensive approach", European Journal ofOperational Research, 14(4), pp. 371-385 (1983).

33. Carbone, R. \Public facilities location under stochasticdemand", INFOR, 12(3), pp. 261-270 (1974).

34. Shiode, S. and Drezner, Z. \A competitive facilitylocation problem on a tree network with stochasticweights", European Journal of Operational Research,149(1), pp. 47-52 (2003).

35. Gallinari, P., Thiria, S., Badran, F. and Fogelman-Soulie, F. \On the relations between discriminant anal-ysis and multilayer perceptrons", Neural Networks,4(3), pp. 349-360 (1991).

36. Cheng, B. and Titterington, D.M. \Neural networks:A review from a statistical perspective", StatisticalScience, 9(1), pp. 2-30 (1994).

37. Ripley, B.D. \Neural networks and related methods forclassi�cation", Journal of the Royal Statistical Society,Series B (Methodological), 56(3), pp. 409-456 (1994).

38. Warner, B. and Misra, M. \Understanding neural net-works as statistical tools", The American Statistician,50(4), pp. 284-293 (1996).

39. Vach, W., Robner, R. and Schumacher, M. \Neuralnetworks and logistic regression: Part II", Computa-tional Statistics and Data Analysis, 21(6), pp. 683-701(1996).


40. Zhengrong, G., Lam, L.H. and Dhurjati, P.S. \Featurecorrelation method for enhancing fermentation devel-opment: A comparison of quadratic regression witharti�cial neural networks", Computers and ChemicalEngineering, 20, Supp1, S407-S412 (1996).

41. Smith, A.E. and Mason, A.K. \Cost estimation predic-tive modeling: regression versus neural network", TheEngineering Economist, 42(2), pp. 137-161 (1997).

Biographies

Hamed Fazlollahtabar received his MSc degree inIndustrial Engineering from Mazandaran Universityof Science and Technology, Babol, Iran. He wasawarded a doctorate in Quantitative Approaches inElectronic Systems from the Gulf University of Scienceand Technology. Currently, he is a PhD student inIndustrial Engineering in Iran University of Scienceand Technology, Tehran, Iran. He is the member ofeditorial board of WASET (World Academy of Sci-

ence Engineering Technology), Scienti�c and TechnicalCommittee on Natural and Applied Sciences, reviewercommittee of International Conference on Industrialand Computer Engineering (CIE), and member of theInternational Institute of Informatics and Systemics(IIIS). He is also a member of Iran's National ElitesFoundation. His research interests are mathematicalmodeling, optimisation in knowledge-based systemsand manufacturing systems. He has published over 130research papers in international book chapters, journalsand conferences.

Seyed Gholamreza Jalali Naini is the AssistantProfessor of Industrial Engineering in Iran University ofScience and Technology, Tehran, Iran. He received hisPhD from Nottingham University, UK, 1983. He is alsoa member of the editorial board of several journals. Hehas published over 100 research papers. His researchinterests include risk assessment, stochastic process,health care system and reliability modelling.

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