Scientia Iranica E (2017) 24(1), 355{363

Sharif University of TechnologyScientia Iranica

Transactions E: Industrial Engineeringwww.scientiairanica.com

A multi-objective multi-state degraded system tooptimize maintenance/repair costs and systemavailability

A. Salmasniaa;�, E. Amerib, A. Ghorbanianb;1 and H. Mokhtaric

a. Department of Industrial Engineering, Faculty of Engineering and Technology, University of Qom, Qom, Iran.b. Department of Industrial Engineering, Faculty of Engineering, Tarbiat Modares University, Tehran, Iran.c. Department of Industrial Engineering, Faculty of Engineering, University of Kashan, Kashan, Iran.

Received 17 May 2015; received in revised form 12 December 2015; accepted 9 April 2016

KEYWORDSMulti-state system;Markov process;Multi-objectiveoptimization;Preventivemaintenance;Minimal repair;Desirability function.

Abstract. In this study, a multi-state degraded system is studied, where status of systemis continuously degrading over time. As time progresses, system may either deteriorategradually and go to lower performance state, or it may fail suddenly. If the systemfails, some repairs are carried out to restore the system to the previous state. Whenthe inspections reveal that the system has reached its last acceptable state, a PM is carriedout to restore the system to the higher performance states. The goal is to �nd the optimalPM level, so that the mean availability of the system is maximized and the total cost ofthe system is minimized. In this regard, Markov process is employed to represent di�erentstates of system. An integrated optimization approach is also suggested based on thedesirability function of statistical approach. The suggested aggregation method is robustto the potential dependency between the total cost and the mean availability. It also ensuresthat both objective functions fall in decision-maker's acceptable region. In order to showthe e�ciency of the proposed approach, a numerical example is presented and analyzed.© 2017 Sharif University of Technology. All rights reserved.

1. Introduction

A degraded system is an operating one whose status isdegrading over time, and this degradation may a�ectits performance [1,2]. This phenomenon occurs in manyreal-world applications like transmission line networks,chemical processes, manufacturing industries, etc. Inrecent years, system availability has received extensiveinterest in many real-world systems such as com-puter [3] and communication systems [4], transporta-

1. Present address: Department of Industrial Engineering,Esfarayen University of Technology, North Khorasan, Iran.

*. Corresponding author. Tel.: 02532103585;E-mail addresses: a.salmasnia@qom.ac.ir (A. Salmasnia);ameriehsan@yahoo.com (E. Ameri);ali.ghorbanian83@gmail.com (A. Ghorbanian);mokhtari ie@kashanu.ac.ir (H. Mokhtari)

tion systems [5], oil/gas production systems [6], etc.Most of studies considered two states for each system,so that the system was either working, or it com-pletely failed. This is called a binary-state system [7].However, in many real-world cases, this binary-stateassumption may not be su�cient [8]. System mayhave more than two levels of performance varying fromperfect functioning to complete failure. In other words,system may perform at various intermediate statesbetween working perfectly and total failure [9,10]. Thepresence of degradation is a common situation in whicha system should be considered to be a Multi-State Sys-tem (MSS). The gradual deterioration of system maybe due to several reasons including the operating hours,the environmental conditions of the equipment, thefailures of non-essential components, and the number ofrandom shocks on the system. Basic concepts of multi-

356 A. Salmasnia et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 355{363

stage systems have been introduced in [11-14]. Also,two pieces in literature review on multi-state systemscan be found in [15,16]. A variety of approaches used inmulti-state systems follow di�erent perspectives whichinclude:

1. The structure function approach in which Booleanmodels have been used for the multi-valued cases(e.g., in [12-14]);

2. The MonteCarlo simulation technique(e.g., in [17]);3. The Markov process approach (e.g., in [18,19]);4. The method of Universal Moment Generating Func-

tion (UMGF) (e.g., see [20,21]).

One of the possible ways to improve a multi-statedegraded system and to increase its availability is theuse of Preventive Maintenance (PM). This will dra-matically reduce the costs of stopping the system [9].Perfect PM aims to make the system as good as beforestarting to work, while imperfect PM may bring thesystem back to an intermediate state between thecurrent state and the perfect functioning state [22].In some papers like [23,24], a multi-state system withthe state-dependent cost was investigated. In [25], adeteriorating repairable multi-state system with animperfect policy was o�ered based on the number ofsystem failures. In [26], a monotone process mainte-nance model for a multi-state system was developed.An analytical approach based on the failure numberof the system was used to determine the optimalreplacement policy. Soro et al. [27] suggested that ifthe system reaches the last acceptable degraded state,it is brought back to one of the states with higher ef-�ciency employing preventive maintenance. Nourelfathet al. [28] formulated a joint redundancy and imperfectpreventive maintenance planning optimization modelfor series-parallel multi-state degraded systems.

The majority of solution approaches used in theliterature are single-objective approaches. In otherwords, most of the existing approaches concentratedonly on maximizing system availability, and assumed

that there is no cost limitation [29-37], or consid-ered the cost limitation only as a constraint [38-40].However, it is usually impossible for a single-objectiveapproach to represent a practical problem truly. More-over, we can �nd several reliability/availability ap-proaches to address di�erent problems [41-46].

In this paper, a multi-state degraded system isconsidered. Even if the system deteriorates continu-ously, it is assumed that this process is done in a �nitenumber of discrete states, and Markov chain is usedto show the di�erent states. It is also assumed thatpreventive maintenance can be performed in severaldi�erent levels, varying from minor maintenance tomajor maintenance. A minor PM restores the systemto the previous degraded state, while a major PMrestores it to the \as good as new" state. The goal is to�nd the optimal PM level, so that the mean availabilityof the system is maximized, and the total cost of thesystem is minimized during the useful lifetime of thesystem. The suggested integration approach includesthe following features:

1. It is robust to the potential dependences betweenthe total cost and the mean availability;

2. It aims to identify the settings of the decision vari-ables to maximize the degree of overall satisfactionwith respect to total cost and mean availability.

The remainder of the paper is organized as follows. InSection 2, the assumptions are presented and the multi-state degraded system under study is described. InSection 3, the availability and cost functions are repre-sented and a \minimum" operator for aggregating themis employed. In Section 4, an illustrative example ispresented. Conclusions and some remarks are providedin Section 5.

2. Model description

Initially, there is a system which is in its perfectfunctioning state (state 1). Over time as shown inFigure 1, two things may happen: the system may

Figure 1. System state transition diagram.

A. Salmasnia et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 355{363 357

either deteriorate gradually and go to the �rst degradedstate, or it may suddenly and randomly fail accordingto a Poisson process and go to a failed state. If thesystem fails, minimal repairs are carried out on it thatrestore the system to the previous state. When thesystem reaches the �rst degraded state, it may eithergo to the second degraded state upon degradation, ormay go to a failed state from which a minimal repairis performed. The same process will continue for allacceptable degraded states. When the system reachesan unacceptable state, it cannot satisfy the customers'demand in a required performance level and must betreated as a failure. If the inspection �nds the systemin its last acceptable state (state d), a PM is carried outto restore the system to one of the higher performancestates. The described process is considered as a Markovprocess; the system-state transition diagram is shownin Figure 1.

2.1. AssumptionsThe considered assumptions in the proposed model areas follows:

1. System can have di�erent levels of degradationcorresponding to discrete performance rates, whichvary from perfect state to completely failed state;

2. System can randomly fail from any operationalstate, and then a minimal repair is done on it;

3. The repair time is negligible compared to thesystem's useful life. Therefore, it does not enterinto the calculations;

4. All transition rates are constant with the exponen-tial distribution;

5. The current state of the system can be observed,and the time required for inspection is negligible.In other words, the inspection is carried out imme-diately.

2.2. System descriptionSystem-state transition can be described by the follow-ing notations and is illustrated in Figure 1:

: State (i), i = 1; � � � ; n and n = 2d+m

State (1): Perfect functioning;

State (2j � 1), j = 2; � � � ; d: The system satis�es thecustomer demand with an acceptable performancelevel;

State (2d+1): The system cannot satisfy the requireddemand after a degradation process and is in anunacceptable performance level;

State (2j), j = 1; � � � ; d: The system encountersfailure from an operational state and minimal repairmust be carried out on it.

�j : Failure rate or the transition ratefrom state (2j � 1) to the state (2j);j = 1; � � � ; d;

�j : Minimal repair rate or transition ratefrom state (2j) to state (2j � 1);j = 1; � � � ; d;

�j : Degradation rate or transition ratefrom state (2j � 1) to state (2j + 1);j = 1; � � � ; d;

�j : The preventive maintenance transitionrate from state (2d � 1) to state(2j � 1); j = 1; � � � ; d� 1.

3. The formulation of the problem

As mentioned before, the proposed method aims tosimultaneously optimize the total cost and the meanavailability of the system throughout its useful lifetime.Next, the method of calculating the total cost functionand the mean availability function are described.

3.1. The total cost functionThe total cost is the sum of the costs of preventivemaintenance and repair during the system life cycle.The variable j is de�ned as a binary variable, sothat it is 1 if preventive maintenance is carried outfrom state (2d � 1) to state (2j � 1; j = 1; � � � ; d � 1);otherwise, it is zero. CPMj is the cost per unit timeincurred if a PM is performed from state (2d � 1) to(2j� 1; j = 1; � � � ; d� 1). pi is the probability of beingin the state i. It is assumed that the study period isgiven by the system life cycle T . Considering that T islarge enough, there exists a steady-state distribution ofstate probabilities. In this case, the expected PM costof system is a function of the decision variable, j , andit is written as:

E(CPM) = T:p2d�1

d�1Xj=1

(CPMj � j): (1)

Since in state (2d � 1), a PM is certainly performed.Also, we have

Pd�1j=1 j = 1. In addition, CRj is the

cost per unit time incurred to repair system when itis in failed state (2j); j = 1; � � � ; d. Therefore, theexpected repair cost can be de�ned as follows:

E(CR) = TdXj=1

CRj :p2j : (2)

Consequently, over the time interval [0; T ], the totalcost of PM and repair is:

TC=T

24p2d�1

d�1Xj=1

(CPMj � j)+dXj=1

CRj :p2j

35 : (3)

358 A. Salmasnia et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 355{363

3.2. The availability functionIn Figure 1, each state (2j � 1; j = 1; � � � ; d � 1)is characterized by a performance rate (or capacity)explained by G2j�1, ranging from the best performancerate G1 to the lowest one G2d�1 (G1 � G2 � � � � �G2d�1). It should be noted that the performance rate ofthe failed states is zero (i.e., G2j = 0, j = 1; 2; � � � ; d).Finally, the performance rate of the unacceptable state(2d + 1) is less than customer demand. Since theperformance rate at any time is a random variable,for the time interval [0; T ], the performance rate is astochastic process. The probabilities associated withthe various states at any time t � 0 are given by theset P (t) = fp1(t); � � � ; p2d+1(t)g, where:

pi(t) = Pr(G(t) = Gi); i = 1; � � � ; 2d+ 1: (4)

The system state acceptability depends on the relationbetween the system performance and the customerdemand. This demand W (t) is usually a randomvariable that changes over time and can take a discretevalue of the set W = fW1; � � � ;W2d+1g. It is assumedthat demand is constant over time (W (t) = W ). Asthe system performance should exceed demand W ,the acceptability function is de�ned as F (G(t);W ) =G(t)�W [46]. According to our notations, it is assumedthat G2d+1 < W � G2d�1. Instantaneous availability,A(t), is the probability that the system performancelevel at the moment t is greater than the customers'demand. In other word, system availability is theprobability that the system is in one of acceptablestates at time t.

A(t) = Pr(G(t) �W ) =dXj=1

p2j�1(t): (5)

Since every system enters the absorbing state as tbecomes extremely large, the �nal state probabilitiesare as follows:

limt!1 p2d+1 = 1; (6)

limt!1 pj = 0 j = 1; � � � ; 2d: (7)

According to the proposed Markov model in Figure 1,Chapman-Kolmogorov equations [27] are used to cal-culate pi(t). These equations are as follows:

dP1(t)dt

=�(�1+�1)P1(t)+�1P2(t)+�1P2d�1(t) 1;(8.1)

dP2j�1(t)dt

=� (�j + �j)P2j�1(t) + �jP2j(t)

+ �j�1P2j�3(t) + �jP2d�1(t) j

for j = 2; � � � ; d� 1; (8.2)

dP2d�1(t)dt

=� (�j j + �d + �d)P2d�1(t)

+ �dP2d(t) + �d�1P2d�3(t)

for j corresponding to j = 1; (8.3)

dP2j(t)dt

= ��jP2j(t) + �jP2j�1(t)

j = 1; 2; � � � ; d; (8.4)

dP2d+1(t)dt

= �dP2d�1(t); (8.5)

with the following initial values:p1(t) = 1; p2(t) = p3(t) = � � � = p2d+1(t) = 0: (9)

Also, for each t, 0 � t � T , we have:2d+1Xi=1

pi(t) = 1: (10)

Thus, Eqs. (8)-(10) are the constraints of optimizationproblem for a particular time of t. The objectivefunction is maximizing the mean availability over theuseful life of the system. The mean availability functionof the system during the useful lifetime of the systemis written as follows:

�A =

TRt=0

A(t)

T: (11)

3.3. The integration methodIn order to do simultaneous optimization of the sys-tem cost and availability, an integrated optimizationapproach is suggested based on the desirability func-tion. The desirability function approach systematicallytransforms the objective function into a scale-free valuecalled desirability. It assigns values between 0 and 1to the possible values of objective function, in whichdesirability function gets 1 as the objective functiongets its target value, and it will be zero when objectivefunction lies outside its corresponding acceptable levels.Desirability function for system availability is shown inEq. (12):

d� �A�

=

8><>:�A�LAUA�LA LA � A � UA0; A � LA

(12)

where �A is the obtained value for the system availabilityfrom Eq. (11), and UA can be set at the extreme valueof the system availability:UA = max �A;

S.t. (Eqs. (8) to (10)): (13)

UA, determined by Eq. (13), represents the maximumpossible value of system availability within a feasible

A. Salmasnia et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 355{363 359

region. LA is the lowest acceptable limit of systemavailability which can be determined on the basis ofthe decision-maker's subjective judgment. In otherwords, the degree of satisfaction of a decision-makerwith respect to system availability is maximized when�A equals its target value (UA) and decreases as �A movesaway from UA. Finally, it will be zero when systemavailability is less than LA.

Similarly, the desirability function of system costis obtained from Eq. (14):

d(TC) =

8><>:TC�UTCLTC�UTC LTC � TC � UTC0 TC � UTC

(14)

where TC is the value obtained for the total costfunction from Eq. (3), LTC is acquired by minimizingthe total cost function in a feasible region, and UTC isasked from the decision-maker.

If a minimum operator is employed for aggre-gating the desirability functions of system cost andavailability, an optimization problem can be stated as:

Max �: (15.1)

Subject to:

d� �A� � �; (15.2)

d(TC) � �; (15.3)

dP1(t)dt

=�(�1+�1)P1(t)+�1P2(t)+�1P2d�1(t) 1;(15.4)

dP2j�1(t)dt

=� (�j + �j)P2j�1(t) + �jP2j(t)

+ �j�1P2j�3(t) + �jP2d�1(t) j

for j = 2; � � � ; d� 1; (15.5)

dP2d�1(t)dt

=� (�j j + �d + �d)P2d�1(t)

+ �dP2d(t) + �d�1P2d�3(t)

for j corresponding to j = 1 (15.6)

dP2j(t)dt

= ��jP2j(t) + �jP2j�1(t)

j = 1; 2; � � � ; d; (15.7)

dP2d+1(t)dt

= �dP2d�1(t); (15.8)

A(t) = Pr(G(t) � w) =dXj=1

p2j�1(t); (15.9)

2d+1Xi=1

pi(t) = 1; (15.10)

pi(t) � 0: (15.11)

This formulation aims to maximize the minimum sat-isfaction level of the decision-maker with respect tothe system cost and the availability. It has severaladvantages over the existing methods.

Firstly, the majority of solution approaches inthe literature assumed that there is no cost limitationor considered the cost limitation only as a constraint,whereas the proposed method optimizes total cost andsystem availability simultaneously.

Secondly, in contrast to other existing multi-objective approaches in literature, the suggested max-imin approach is robust to the potential dependencybetween the total cost and availability of the system.In other words, the correlation between the objectivefunctions will not a�ect the optimization process. Asa result, existing aggregation approaches used can leadto misleading results for this problem.

Thirdly, the suggested maximin approach createsa better balance between the objective functions incomparison with other existing aggregation methods.In other words, many aggregation methods may leadto a decision variable vector that the value of someof the objective functions may not be incredible inthe decision-makers' view. In the proposed approach,�rst, the acceptable range for each objective functionis determined separately, and then it maximizes theoverall satisfaction level within the ranges, such thatthe share of each objective function is properly re ectedin the optimization process.

4. Numerical example

Consider a production system that has seven perfor-mance levels measured by hourly production rate, Gi.These production rates are presented in Table 1. Thecustomer demand, W , to be satis�ed is 500 parts perhour. The Markov model of this multi-state system isillustrated in Figure 2.

According to the Markov model, there are twopossible preventive maintenance actions. It is assumedthat the transition rate of perfect PM (�1) is equalto 0.08, and the transition rate of imperfect PM (�2)is 0.02. The other used parameters are presented inTable 2.

The solution space of this problem has two caseswhich are:

Table 1. Hourly production rate for each state.

State i 1 2 3 4 5 6 7Gi 1000 0 750 0 500 0 0

360 A. Salmasnia et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 355{363

Figure 2. System state transition diagram in a numericalexample.

Table 2. Transition rates for the numerical example.

�1 �2 �3 �1 �2 �3 �1 �2 �3

0.03 0.05 0.07 0.005 0.008 0.01 0.01 0.02 0.04

Table 3. The cost per hour incurred to repair system.

j 1 2 3CRj 5.3 3.8 4.5

Table 4. Preventive maintenance cost incurred per hour.

j 1 2CPMj 2.3 1.4

� 1 = 1, 2 = 0,

� 1 = 0, 2 = 1.

In order to evaluate the system availability, �rst,Chapman-Kolmogorov equations are solved for thementioned Markov model to obtain the probabilitiesPi(t); (i = 1; � � � ; 7) separately in both cases. Then,the system availability at time t is calculated usingA(t) = P1(t) + P3(t) + P5(t). Finally, assuming thatthe useful life of the system is determined 80 hours,system availability ( �A) and its corresponding desir-ability (d( �A)) can be obtained by Eqs. (11) and (12),respectively.

The repair costs and the preventive maintenancecosts are presented in Tables 3 and 4, respectively.

The expected repair and PM costs and the totalcost will be calculated using Eqs. (1), (2), and (3),respectively. Then, the desirability function of thetotal cost and system availability will be achieved byassuming that the lower bound of acceptability forthe system availability (LA) and the upper bound ofacceptability for the total cost (UTC) are 0.5 and 80,respectively. After this step, the �nal model can bewritten as follows:

Max �;

Subject to : d(A) � �; d(TC) � �;dP1(t)dt

= �(�1 + �1)P1(t) + �1P2(t) + �1P5(t) 1;

dP3(t)dt

=� (�2 + �2)P3(t) + �2P4(t) + �1P1(t)

+ �2P5(t) 2;

dP5(t)dt

=� (�j j + �3 + �3)P5(t) + �3P6(t)

+ �2P5(t)

for j corresponding to j = 1;

dP2(t)dt

= ��1P2(t) + �1P1(t);

dP4(t)dt

= ��2P4(t) + �2P3(t);

dP6(t)dt

= ��3P6(t) + �3P5(t);

dP7(t)dt

= �3P5(t);

A(t) = P1(t) + P3(t) + P5(t);

7Xi=1

pi(t) = 1;

pi(t) � 0:

After solving the above mathematical programming us-ing MATLAB software package, the optimum solutionis achieved as 1 = 1, 2 = 0.

Subsequently, this 1 = 0; 2 = 0 means thatduring its useful life, whenever this system reaches thestate 3, the perfect preventive maintenance is carriedout; the system is restored to the perfect functioningstate in this way.

5. Conclusion

This paper considered a multi-state degraded systemin which status of system is considered to degradeover time, and this degradation may a�ect systemperformance. In other words, as time progresses,system may either deteriorate gradually and go to lowerperformance state, or it may suddenly and randomlyfail according to a Poisson process called Poissonfailure. If the system fails, minimal repairs are carriedout to restore the system to its previous state. Whenthe inspections showed that the system has reached itslast acceptable state, a PM is carried out to restore thesystem to one of the higher performance states, varyingfrom minor maintenance to major maintenance. Aminor PM restores the system to the previous degradedstate, while a major PM restores it to the \as good asnew" state. To �nd the optimal PM level, such thatthe mean availability of the system is maximized and

A. Salmasnia et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 355{363 361

the total cost of the system is minimized, an integratedoptimization approach is suggested based on the desir-ability function approach. The suggested aggregationmethod is robust to the potential dependency betweenthe total cost and the mean availability. It also ensuresthat both objective functions fall in decision-maker'sacceptable region.

As a direction for future research, the proposedmodel can be extended for a joint redundancy andimperfect preventive maintenance planning optimiza-tion problem for series-parallel multi-state degradedsystems.

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Biographies

Ali Salmasnia is currently an Assistant Professor inUniversity of Qom, Qom, Iran. His research interestsinclude quality engineering, multi-response optimiza-tion, applied multivariate statistics, and multi-criteriondecision making. He is the author or co-author ofvarious papers published in Applied Soft Computing,Neurocomputing, IEEE Transactions on EngineeringManagement, International Journal of Advanced Man-ufacturing Technology, Applied Mathematical Mod-elling, Expert Systems with Applications, InternationalJournal of Information Technology and Decision Mak-ing, and TOP.

Ehsan Ameri received his BS degree from the IslamicAzad University and his MS degree in IndustrialEngineering from the Tarbiat Modares University inIran. His current research interests include reliability,healthcare, and multi-criteria decision making.

Ali Ghorbanian is a Lecturer in the department ofIndustrial Engineering in the Esfarayen University ofTechnology, North Khorasan, Iran. He holds master'sdegree of Industrial Engineering from the TarbiatModares University (TMU). His research interests aredecision support systems, simulation, RSM, knowledgemanagement, and process management.

Hadi Mokhtari is currently an Assistant Professor ofIndustrial Engineering in University of Kashan, Iran.His current research interests include the applicationsof operations research and arti�cial intelligence tech-niques to the areas of project scheduling, production

A. Salmasnia et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 355{363 363

scheduling, and supply chain management. He alsopublished several papers in international journals suchas Computers and Operations Research, InternationalJournal of Production Research, Applied Soft Com-

puting, Neurocomputing, International Journal of Ad-vanced Manufacturing Technology, IEEE Transactionson Engineering Management, and Expert Systems withApplications.