Estimation and Optimization for System Availability Under Preventive Maintenance

Engineers design systems to be reliable and work to fulfil their missions without failure for a specific period. However, the system components deteriorate with time and lead to its failures. A frequent system failure increases the management costs, hence posing a challenge to decision-makers. Therefore, for the avoidance of frequent system failures, preventive maintenance is necessary. The objective of any manufacturing firm is to maximize profit and minimize costs. The interval for preventive maintenance can be optimized if the system’s availability is maximized and its cost function minimized. This study evaluates the availability and cost function for a continuous operating series-parallel system under a fixed time environment. A multiobjective model is formulated to maximize the availability and minimize the cost function of the system. The study illustrated a numerical example and solved using goal programming (GP), fuzzy goal programming (FGP), genetic algorithm (GA), and particle swarm optimization (PSO) techniques. The results are compared using a robust statistical test and, the PSO proves to be better. A simulation study was carried out further to evaluate the availability and cost function using R and MATLAB packages.

or its components, are critical to improve the reliability of 23 such systems at a higher level. Formulating the appropriate 24 mathematical programming model of such scenario has been 25 useful in the system reliability determination. Thus the objec- 26 tive of such problems is to increase the system's availability 27 with some constraints like time, weight, cost, etc. Therefore, 28 The associate editor coordinating the review of this manuscript and approving it for publication was Geng-Ming Jiang . several studies have been carried out in this area. Cox [1] 29 has explained the conditions for the finite optimum solution 30 and [2] suggested the best solution to minimize the expected 31 cost. Wang [3] focused on the series-parallel system and [4] 32 dealt with personal computer design in reliability optimiza-33 tion. Klutke et al. [5] derived the availability by exogenous 34 random environment for an inspected system. Also, they 35 defined a relationship between remaining life, deterioration 36 and repair by the Markovian method. One of the most con-37 tributing factors to workout availability is the failure rate of 38 component. According to [6] availability of extensive data 39 partly led to the creation of predictive maintenance. In the 40 literature of availability, there are two types of the failure 41 rates of components which are considered in general. One 42 is the constant failure rate and other is the time-dependent 43 failure rate; these are mainly based on lifetime distributions 44 like exponential distribution, Weibull distribution, etc. When 45 predictive maintenance strategies for multi-state manufactur-102 ing systems [39]. The authors applied the proposed model 103 in serial manufacturing system and concluded that the sys-104 tems can simultaneously complete production tasks with high 105 quality product, and reduce the maintenance cost in the pro-106 duction cycle. A novel evaluation methodology combining 107 Markov model and dynamic Bayesian networks has been 108 developed to assess systems' resilience under various fixed 109 external disasters [40]. Nourelfath et al. [41] proposed a com-110 bined method based on Markov processes, GA and universal 111 moment generating function to calculate the availability of 112 the multistate system. Reference [42] discussed a methodol-113 ogy to solve the multiobjective reliability optimization model. 114 In their study, the parameters of model are considered impre-115 cise in triangular interval data. They converted the uncertain 116 multiobjective optimization model to a deterministic form 117 and used PSO and GA to solve these problems. Garg [43] pro-118 poses PSO-GA as a hybrid technique for solving constrained 119 non-linear optimization problem. Adhikary et al. [44] used 120 a multiobjective GA to solve a series-parallel system with 121 a preventive maintenance (PM) scheduling model that does 122 not provide PM with an off-working time. Wang et al. [45] 123 uses a numerical algorithm and the PSO to derive an optimal 124 imperfect PM interval considering maximal two-dimensional 125 warranty product availabilty. 126 Many techniques are available to solve optimization prob-127 lems but PSO technique gives the appropriate, convergent 128 and feasible solutions in comparison to other techniques. It is 129 computationally inexpensive in terms of time, memory and 130 speed. Due to its flexibility, the PSO technique is used instead 131 of other available techniques. PSO is considered as a potential 132 competitor to other promising techniques like GA and GP 133 techniques, etc. This paper proposed availability and cost 134 model for series-parallel systems with components which are 135 periodically inspected and managed subject to some main-136 tenance strategy. The objective is to optimize the mainte-137 nance policy for each part of a program and maximizing the 138 availability limit's cost function. The solution procedures are 139 explained by PSO technique. A comparative study is also 140 included by considering some other optimization techniques. 141 Series and parallel systems structures are the most widely and 142 fundamentally used in representing systems structures in a 143 classical reliability theory [46]. Next subsections discusses 144 series, parallel, and series-parallel systems briefly as an over 145 view because of their usefulness in the theory of reliability. 146

147
According to [47], a series system is one of the most impor-148 tant and common systems in reliability theory and applica-149 tions. In a series system, the components or subsystems are 150 arranged and connected in series, and all must work before the 151 system function. In other words, if a component fails, then the 152 system fails as well. In a series system, the total failure rates 153 of its components equal the system's failure rate. Similarly, 154 the series system's lifetime equal to a minor lifetime of its 155 components. It is the simplest form of a system.  The function of a series system is given as: (2) 169 VOLUME 10, 2022 This system consists of m disjoint subsystems (modules) connected in series and each module i (subsystem) has compo-172 nents of n i , connected in parallel. Subsystems can be identical 173 and independently distributed with same size or otherwise.

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The availability function is giving in Eqn. (3).

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This section presents and discusses the optimization tech-234 niques used in solving the availability problems in this study. 235 They include PSO, GA, GP and FGP. It also discusses some 236 lifetime distributions for which some availability functions 237 data follows in real-life situations. The method of estimating 238 parameters of the distribution are also presented. System availability is the probability that a system can per-242 form its mission within a specific time frame without fail-243 ure [28]. In other words, it is a probability that such a sys-244 tem will not fail in a given time. In a real-life engineering 245 problem such as design and preventive maintenance, data 246 related to the failure rate function, mean time to system failure 247 (MTSF), the median time to system failure (MdTSF), mean 248 time between failures (MTBF), mean time to repair (MTTR), 249 etc. follows a particular lifetime distribution. The failure 250 rate can be constant or time-dependent, and they usually 251 are estimated based on the distribution they follow through 252 simulation studies. Before a particular distribution is selected, 253 the data can be fit to observe which probability distribution 254 best fit the data. For so doing, Akaike's information criterion 255 (AIC) and Bayesian information criterion (BIC) techniques 256 help identify the best-fitted model for the failure data set. The 257 On the other hand, the BIC is used for best statistical model

276
The model(s) that has a minimum value(s) of the AIC and 277 BIC is selected as the best-fitted data.
284 where, the baseline distribution function is denoted as G(·) 285 and the unknown parameter with θ > 0. Several well-known 286 distributions covered by this distributions family.

287
While the generalized life distribution function is given by 288 probability model as:  The MLE and UMVUEs for the system availability are given 307 in Eqn. (9) and (10). Optimization, in simple terms, is finding the best possible 315 desired result(s) out of many available feasible solutions. 316 In an optimization problem, the objective could be single 317 or multiple. A multiobjective problem has more than one 318 objective or goal desired to be achieved in some kind. It can be 319 linear or nonlinear function(s) with some constraints or lim-320 itations, which can also be linear or nonlinear. For instance, 321 cost minimization and benefit, profit, or performance max-322 imization. It could be a mixture of both minimization and 323 maximization.

324
In engineering design, an engineer may wish to maximize 325 system availability and, in addition, minimize cost functions, 326 volume or weight of the system. These objectives might be 327 conflicting, and a single optimal value cannot satisfy all the 328 objectives. Here, the designer faces the problem of optimiz-329 ing all the objectives simultaneously. In a single objective 330 optimization, an optimal solution is possible depending on 331 tion problem (MOOP), it is impossible to obtain an optimal 333 solution to all the objectives since they could be conflict- j objectives functions be given as:  A typical GP model is given in Eqn. (12).

374
Here, x j is the j th decision variable and a ij its coefficient, In goal programming formulation, the objective function 384 does not contain decision variables; instead, it includes the 385 deviational variables (δ + i , &δ − i ), representing each type of 386 goal or sub-goal. A deviational variable is typically repre-387 sented in the objective function as a combination of over-388 achieving and underachieving the current goal [see Eqn. 389 (12)]. Deviational variables show the possible deviation 390 below and above the target values. The negative deviation 391 is the deviation for a given goal by which it is less than 392 the aspiration level. The positive deviation is the amount 393 of deviation fo a particular goal by which it exceeds the 394 aspiration level. As previously discussed, some real-life decision-making pro-397 cess involves imprecision. The decision-makers goal value 398 may have some incomplete information or vagueness, and a 399 decision must be taken in such a scenario. Fuzzy sets deal 400 with such goals' parameters or values that are imprecise. 401 The FGP concept is applied to the theory of fuzzy set. This 402 real-life modelling concept is traceable from the Zadeh's 403 work [33]. The first application of fuzzy programming in 404 solving MOOP appeared in [32]. An FGP function is repre-405 sented generally as: Here, G k , represents goals vectors, b i , represent m resources 412 vector, A represent decision variables, coefficient. The sym-413 bol represent fuzzy-maximization objective-type, rep-414 resent fuzzy-minimization objective-type and represent 415 fuzzy-equality constraint-type. Z k denotes the k th objective 416 and X represent n-dimensional vector of decision variables. 417 The fuzzy-minimization-type membership is given as The fuzzy-maximization-type membership is given as The fuzzy-equality-type linear-membership is given by

423
where U k is the upper limit and L k the lower limit, and G k is 424 the aspirational levels given by the DM for the k th goal.

441
Step 3: Step 2, define the payoff matrix utilizing the ideal

449
Step 5: Then using max-min operator, to obtain Step 1: Create a random population, including n chromo-481 some or initial solution.

482
Step 2: Establish in the population the fitness role of each 483 chromosome.

484
Step 3: Building a new population-based on the selection 485 of parent chromosomes by selective 486 methods such as roulette wheel, match, random, competi-487 tive, etc.

488
Mentioning an absolute value for the likelihood of a 489 crossover operator and then conducting a combination pro-490 cedure on parents to create offspring and assuming a specific 491 value for the mutation operator's. Using this procedure into 492 establish a new chromosome shift one or more genes from 493 parent's chromosome.

494
Step 4: Replacing new offspring in the new population.

495
The pseudo code for GA generation is given in Table 4.  (19). Aircraft, computer networks, and other large-scale, com-545 plex systems all significantly impact society. In maintaining 546 these systems, maintainability theory is crucial. An excel-547 lent maintenance strategy development entails mathematical 548 maintenance rules with a focus on preventive maintenance 549 that has primarily been established in the research field 550 of operations research. Designing a maintenance strategy 551 with two maintenance options, preventive replacement and 552 corrective replacement, is the most significant challenge in 553 mathematical maintenance strategies. When a system or unit 554 is changed as part of preventative maintenance, it is done 555 before it breaks down. On the other hand, with the corrective 556 replacement, it is the failed unit that is replaced.

557
Practically important preventive maintenance optimization 558 models that involve age replacement and block replacement 559 are reviewed in the well-known renewal reward argument 560 framework. Some extensions to these basic models and the 561 corresponding discrete-time models are also introduced with 562 the aim of applying the theory to practice.

564
This section discusses system cost function and availability. 565 It present the model formulation of system cost and availabil-566 ity functions for a series-parallel system as a multiobjective 567 optimization. It further uses the concept to demonstrate the 568 solution approaches discussed in Section III for numerical 569 illustratio. Next, cost function is discussed. where c 3 is downtime cost from a failure to its recognition.

597
By Eqns. (20) and (21) the expected cost rate is The Eqn. (22) is used for age replacement. In this case, the  The n subsystems are considered that are connected in series, 631 for i = 1, 2, . . . , n and each i subsystem has m i parallel con-632 nected components. The parallel subsystem works by using 633 the standard series-parallel configuration when at least one of 634 its components are operated and the whole system operates if 635 and only if all subsystems operate. The components are inde-636 pendent in each subsystem i (i = 1, 2, . . . , n) and distributed 637 identically and independently for failure and repair rate. The 638 X failure rate is considered to be independent and has an equal 639 F(t) distribution with finite mean and the Y repair rate is also 640 independent and has an equal finite mean distribution of G(t). 641 Let A(T ) be the availability of the subsystem at time T . Then, 642 .

643
Many authors are worked on optimum PM such as [2] 644 and [37] and others. This model is similar to [37], which 645 describes the assumptions of system repair and failure prob-646 lem. If a device fails, it undergoes under maintenance imme-647 diately and is restored to the operational state after repaired. 648 Repair time is divided into two parts; one is before time 649 T having Y 1 distribution. The repair time is independent 650 and has a finite mean G 1 (t). If a unit's operating time is 651 already known and its failure rate rises over time, it might 652 be prudent to maintain it at the time T preventively until its 653 operating time failure. Other repair time is after time T having 654 distribution of Y 2 . Time distribution of Y 2 to PM completion 655 time distribution is G 2 (t) with a finite mean, which could be 656 lower than the repair time of Y 1 . A new unit begins to work at 657 t = 0. We describe one process from the beginning of service 658 until PM or repair is complete. Then the loss of one cycle is 659 given by: where X k and Y k (k = 1, 2, . . .) as referring to uptime and 662 downtime and A(T ) is the likelihood that the device will work 663 at the time of T , respectively. Hence the availability of the 664 state is Now, this model is defined as a series-parallel system. Let 667 A ij be the availability of the component j (j = 1, 2, . . . , m i ) 668 in subsystem i (i = 1, 2, . . . , n) and let A i be the availability 669 of the subsystem i. That is A ij and A i can be expressed for 670 series-parallel system, such as The system consists of n subsystems connected in series 673 and each i subsystem has components of m i , connected in par-674 allel to i = 1, 2, . . . , n. Subsystems are related in sequence, 675 then the system availability is period. It is thus, formulated as follows: interval from its failure to time T.

722
The next section present the case study.

724
It is a requirement for a military system to complete a series 725 of flight missions in a war. However, it is impossible with-726 out an inevitable break between missions to maintain some 727 components at period T that might fail during the operation. 728 However, not every component may require such mainte-729 nance action. Therefore, to enhance the system availabil-730 ity, let us assume the system is composed of n subsystems 731 in series, each having m components connected in parallel. 732 The subsystems' components assumed further to be identical 733 and independently distributed, and supplied from the same 734 manufacturer. Then, the MTBF and MTTR for every failed 735 component are independent of one another for the mission 736 period T . Let A ij be the j th component availability in the 737 i th system, given by Eqn. (3). Furthermore, if data for the 738 MTBF and MTTR are available, then the procedure discussed 739 in Section III will be used to identify the best-fitted distri-740 bution using Eqn. (5) and (6), respectively. Moreover, the 741 identified best distribution parameters will be estimated using 742 the likelihood function either by MLEs or UMVUEs models 743 proposed by [28]. The MTBF and MTTR data are assumed 744 to follow a lifetime distribution with exponential properties 745 to demonstrate this case study.

746
For illustrating the above case study numerically, the data 747 are simulated by using exponential distribution using R soft-748 ware and information regarding the parameters have been 749 summarized in Table 6  Here, the general MOOP for series-parallel system formu-752 lated in Section V-D is presented using the concept of GP 753 discussed in Section III-C. For all the p functions, the problem 754 can be reported separately as: 755 Firstly, Eqn. (26), as shown at the bottom of the next page, 756 is solved for objective function Z 1 ignoring the Z 2 subject to 757 set of feasible constraints. The optimal individual solution for 758 objective function Z 1 is obtained as Z * 1 . It is reffered to as goal 759 value for Z 1 . Now, an additional constarint with a deviational 760 variable can be defined as Similarly, for the second objective function Z 2 , the additional 763 constarint can be defined as: Finally, the GP model is defined as: The value The FGP procedure discussed in section III-D is used to solve 776 the first objective as follows: The individual solution is obtained as 785 n 1 = 2, n 2 = 2, n 3 = 5, n 4 = 2, n 5 = 1, Z 2 = 159.978.

786
Similarly, the second objective (27)  Now the payoff matrix is given in Table 7.
subject to A s (T ) ≥ 0.85  In order to maximize the above problem, with subject to 808 constraints as described:   (i) Weighted sum approach by assigning weights to each 832 feature to turn the multiobjective into a single objective prob-833 lem.

834
(ii) Optimize goal function with some constraints. The 835 objective functions are transformed to a single objective func-836 tion as follows.

837
Using the values of Table 6, model (29), as shown at the 838 bottom of the next page, is solved using PSO technique to 839 find out optimum solution. Statistical analysis play a vital role in testing validity of 844 arguments. There are different statistical tools for analysing 845 a comparison. Since this study compares four different tech-846 niques, it is necessary to analyse the comparison statistically. 847 Since the population considered is small (less than 30), a t-test 848 can be used to analyse the comparison of methods employed 849 in this study. The t-test conducted on the GA, FGP and GP 850 with PSO technique at α = 0.05 significance level that 851 there is no difference in their population means, and pop-852 ulations have equal variances. Using the pooled t-test for a 853 null hypothesis, the result are presented in Tables 9 and 10, 854 respectively.

856
The four techniques discussed in methodology Section III 857 have been all used to solve the multiobjective optimiza-858 tion problem of a series-parallel system. Table 8 sum-859 marises the results of the solution methods. The aim is 860 to maximize the system's availability and minimize its 861 costs to optimize the interval of preventive maintenance 862 of the components of the subsystems. It can be observed 863 from Table 8 that on comparing these four results, goal 864 particle swarm optimization result. However, it is not the best 866 because it violets the cost constraints, which must not exceed  Furthermore, a statistical test conducted to compare the 883 four techniques using the pooled t-test under two cases.

884
In case one, it assumed that two samples have equal vari-885 ances for the system cost function. In case two, two samples 886 assumed to have equal variances for the system availability.

887
The objective is to test the null hypothesis that no differences  sions to correctly estimate the effects of component failures 902 and optimize the system availability and costs. It brought 903 the most suitable valuable technique to achieving the desired 904 compromise solution when confronted with multiple objec-905 tive goals, seeking to generate a possible and plausible result 906 in an engineering or operational system. In deteriorating 907 systems, component failure is inevitable, but maintenance 908 action prevents unnecessary faults. However, frequent sys-909 tem maintenance increases costs and is not desirable; hence 910 maximizing the system's availability and minimizing its cost 911 function will optimize the interval for the preventive system 912 maintenance. Therefore, this is precisely the present study's 913 features, and it is the most desired goal of decision-makers 914 and field operators in several systems, including military-915 based operations. 916 Additionally, researchers and practitioners will use the 917 present study as a directional guide for adjusting and further-918 ing its applicability in industries where systems deteriorate 919 min Z (A s , C s ) = 0.5