Preventive maintenance and replacement optimization ... - Springer Link

Int J Adv Manuf Technol (2015) 76:2131–2146 DOI 10.1007/s00170-014-6391-5

ORIGINAL ARTICLE

Preventive maintenance and replacement optimization on CNC machine using multiobjective evolutionary algorithms Kamran S. Moghaddam

Received: 5 December 2013 / Accepted: 11 September 2014 / Published online: 7 October 2014 # Springer-Verlag London 2014

Abstract In this paper, a multiobjective optimization model is formulated to determine the optimal preventive maintenance plans for a repairable multicomponent manufacturing system with increasing rate of occurrence of failure. The operational planning horizon is segmented into discrete and equally sized periods, and in each period, three possible maintenance actions (repair, replacement, or do nothing) have been considered for each component. The developed multiobjective nonlinear mixed-integer programming model is then utilized to find the Pareto-optimal preventive maintenance schedules for a computer numerical control (CNC) machine. In order to obtain the Pareto-optimal solutions, five multiobjective evolutionary algorithms are considered, and computational performance of the algorithms is evaluated using different metrics. It is found that three of these algorithms outperform the others in obtaining more and high-quality nondominated preventive maintenance and replacement schedules along with maintaining high level of diversity among generated solutions. Such a modeling approach and the comparison of algorithms would be useful for maintenance planners tasked with the problem of developing optimal maintenance plans for multicomponent manufacturing machines.

Keywords Preventive maintenance and replacement optimization . CNC machine . Multiobjective evolutionary algorithms . Comparison of algorithms

K. S. Moghaddam (*) Department of Information Systems and Decision Sciences, Craig School of Business, California State University-Fresno, Fresno, CA 93740, USA e-mail: [email protected]

1 Introduction Reliability engineering attempts to characterize, measure, and analyze failure and repair behavior of systems in order to improve their operational level by increasing their design life and eliminating or reducing the likelihood of failures and safety risks and downtimes. In this context, preventive maintenance is defined as those activities performed at certain time intervals over a planning horizon to extend useful life of a system and to keep the system in a good condition to be productive and responsive. Preventive maintenance is a broad area of operations that encompasses set of activities in order to improve the overall reliability and availability of systems [1]. All types of complex systems, ranging from manufacturing machines to computer network systems, have prescribed maintenance schedules set forth by the manufacturer aimed at reducing the risk of unexpected failures during their useful life. Preventive maintenance activities generally consist of inspection, cleaning, lubrication, adjustment, alignment, and/or even replacement of components in a system subject to wear out or degradation. These activities reduce the “effective age” of the components and thus improve the rate of occurrence of failure. Regardless of the specific type of system, preventive maintenance activities can be categorized in one of two ways, component repair or component replacement. Note that these activities change the aging characteristics of the components and, if done correctly, ultimately decrease the rate of occurrence of unexpected failures of the system. In addition, preventive maintenance strategies involve a basic trade-off between the minimization of the total operational costs of maintenance and replacement and maximization of the overall reliability of the system.

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Designers of preventive maintenance decisions must balance these expenses along with the cost of unexpected failures in order to minimize the overall operational costs. The population search structure of evolutionary algorithms (EAs) makes them suitable search procedures to find the Pareto-optimal solutions of a multiobjective problem in a single simulation run [2]. With the existence of different types of multiobjective evolutionary algorithms (MOEAs), it is also possible to evaluate their computational performance on different test problems. A multiobjective evolutionary algorithm will be an effective solution method if it can successfully perform three following conflicting tasks [3]: 1. The obtained nondominated solutions should be close enough to the true Pareto front. Ideally, the solutions should be a subset of the Pareto-optimal set. 2. The obtained nondominated solutions should be uniformly distributed over of the Pareto front in order to provide the decision maker a true insight of trade-offs. 3. The obtained nondominated solutions should capture the whole spectrum of the Pareto front. This requires investigating nondominated solutions at the extreme ends of the objective function space. In the following subsections, we briefly address the related literature to preventive maintenance scheduling of manufacturing machines as well as evaluation and comparison of multiobjective evolutionary algorithms. 1.1 Preventive maintenance scheduling of manufacturing machines The effectiveness of preventive maintenance scheduling under different conditions such as shop load, job sequencing rule, maintenance capacity, and strategy was studied in several earlier studies [4–6]. These studies tested the effectiveness of simple preventive maintenance policies using discrete-event simulation, rather than optimizing them along with production scheduling decisions. Integrated preventive maintenance and job shop scheduling problem for a single machine have been also tackled in previous studies [7–11]. In these studies, minimization of total weighted expected completion time is considered as the objective function, while preventive maintenance decision variables are considered in the functional constraints. Chareonsuk et al. [12] developed a multicriteria approach to find optimal preventive maintenance intervals of workstations of a production line at a paper factory with total expected costs and reliability as the objective functions. Leng et al. [13] presented an integrated preventive maintenance scheduling and production planning of a single machine as a

Int J Adv Manuf Technol (2015) 76:2131–2146

multiobjective optimization model. Verma and Ramesh [14] integrated systems and subsystems of a large manufacturing facility into higher modular assemblies and applied a multiobjective preventive maintenance scheduling approach. They modeled this problem as a constrained nonlinear multiobjective mathematical program with reliability, cost, and nonconcurrence of maintenance periods and maintenance start time into the objective functions and used a genetic algorithm to optimize the mathematical model. A multiobjective genetic algorithm in order to solve preventive maintenance scheduling problems is developed by Quan et al. [15]. They considered minimization of workforce idle time and minimization of maintenance time and showed an existing trade-off between the objective functions. Yulan et al. [16] considered five objective functions to minimize maintenance cost, makespan, total weighted completion time of jobs, and total weighted tardiness and to maximize machine availability in a multiobjective integrated production and maintenance planning problem and solved the mathematical model by multiobjective genetic algorithm (MOGA). Berrichi et al. [17] developed a multiobjective integrated production and maintenance scheduling model to determine the Pareto-optimal front of the assignment of production tasks to machines along with preventive maintenance activities in a production system. Berrichi et al. [18] later proposed a heuristic algorithm based on multiobjective ant colony optimization (MOACO) and compared its performance with two MOGAs. Moradi et al. [19] presented a multiobjective optimization model integrating flexible job shop problem with preventive maintenance scheduling by considering minimization of the makespan for the production part and minimization of the system unavailability for the maintenance part. Wang and Pham [20] presented a new multiobjective maintenance optimization model embedded within the imperfect preventive maintenance for a single-unit system. Certa et al. [21] developed a multiobjective maintenance optimization model to determine the Pareto-optimal front while minimizing the total maintenance and downtime costs along with a constraint on the required reliability. Moghaddam [22] recently developed a multiobjective optimization model to determine Pareto-optimal preventive maintenance and replacement schedules for a repairable multiworkstation manufacturing system using hybrid Monte Carlo simulation and goal programming method. 1.2 Comparison of multiobjective evolutionary algorithms Many studies have been conducted with the aim of comparing different types of multiobjective evolutionary algorithms. Zitzler et al. [3] compared eight multiobjective metaheuristic algorithms on six experimental problems. According to their computational study, all multiobjective meta-heuristic


algorithms have a better performance than a basic random search method, while an elitist algorithm, strength Pareto evolutionary algorithm (SPEA), shows superiority over all algorithms on all tested problems. The researchers also reported the observed difficulties due to nonconvexity, discreteness, and multimodality of the objective functions in solving the problems and obtaining converge and diverse set of solutions. Van Veldhuizen [23] conducted a research to compare multiobjective messy genetic algorithm (MOMGA) with MOGA, nondominated sorting genetic algorithm (NSGA), and niched Pareto genetic algorithm (NPGA) on seven test problems by comparing performance metrics of convergence and diversity. It was reported that MOMGA outperforms MOGA, NSGA, and NPGA since it is the only one equipped with elite-preserving feature. In another comparative study, the computational effectiveness of 13 algorithms on six experimental problems is measured [24]. Deb et al. [25] compared three elite-preserving MOEAs on nine test problems and made a conclusion on superiority of NSGA-II over other algorithms on seven problems based on the generational distance values as a convergence metric. Their developed NSGA-II also outperformed all algorithms in terms of diversity of the nondominated solutions measured by spread metric values. Several other comparisons can be found in previous studies [26–29]. A comparative study presented a review of multiobjective evolutionary algorithms for problems with multiple objectives with a specific application to reliability optimization problems [30]. 1.3 Contribution of this research It is found that much of the reviewed literature deals with development of solution methods and algorithms that are only applicable to specific systems such as those found in power plants and production lines. Hence, the main contribution of this research is to extend the previous work of Usher et al. [31] and Quan et al. [15] by defining a general configuration for multicomponent manufacturing systems, in which three different maintenance actions can be taken, and then to develop a mathematical formulation to determine optimal preventive maintenance and replacement schedules. This modeling approach captures the realistic dependency between the machine’s components and enables to develop time-based patterns of repair and replacement actions on a computer numerical control (CNC) machine. The second contribution of this research is to determine the best solution approach from available multiobjective evolutionary algorithms. In order to find the nondominated schedules, five multiobjective evolutionary algorithms are employed, and their computational effectiveness is evaluated using different performance metrics. These performance metrics provide critical benchmarks to select an appropriate solution method for the developed model. By reviewing the existing literature, it is found that there is

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no similar effort in determining an effective solution method to solve preventive maintenance and replacement planning problem for CNC machines. Observing this lack provides an opportunity for this research to fill the existing gap in the current literature. The paper is organized as follows: Sect. 2 describes the formulation procedure of the multiobjective optimization model including the decision variables, objective functions, and the functional constraints. In Sect. 3, a practical application of the proposed optimization model on a CNC machine is demonstrated, and implementation aspects of the algorithms are discussed. Computational results and evaluation of the algorithms to determine the most efficient solution method are provided in Sect. 4. Finally, Sect. 5 concludes the paper and gives venues for future research and extension.

2 Problem formulation 2.1 System specifications Let us consider a new system containing series of repairable and replaceable N components to be operated over a planning horizon. It is important to note that other component configurations (parallel, series–parallel, parallel– series, k-out-of-n, complex, etc.) can be modeled just by modifying the reliability function reflecting the specific structure of the system. It is also assumed that each component i in the system is subject to deterioration and has an increasing rate of occurrence of failure, ρi(t), where t denotes chronological time, (t>0). In this research, because of increasing failure rate and maintainability of the system under study, it is assumed that components failure correspond to the well-known nonhomogeneous Poisson process (NHPP) expressed by ρi ðt Þ ¼ λi β i t βi −1

for i ¼ 1; …; N

ð1Þ

where λi and βi are the scale and the shape parameters of component i, respectively. The nonhomogeneous Poisson process is similar to the homogeneous Poisson process (HPP) with the exception that the rate of occurrence of failure is not constant but is time-dependent. For more details on this well-known stochastic process, see the studies of Elsayed [1] and Ebeling [32]. It would be desirable for the maintenance planner to find a schedule of future repair and replacement actions for each component over the planning horizon [0, L]. The interval [0, L] is segmented into T discrete periods, each of length L/T. At the end of period t, the components may be either repaired, replaced, or no action is to be performed. In most systems, maintenance and repair operations reduce the effective age of the components and subsequently “failure rate” of the system. This kind of maintenance activities are

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known as minimal repairs in the literature [33] since they do not change the failure characteristic of the system. To account for the instantaneous changes in component age and the failure rate, first, the initial age for each new component is set to zero. Let EASi,t denote the effective age of component i at the start of period t, and EAEi,t denotes the effective age of component i at the end of period t. The aging of the component i over each period can be shown as EAEi;t ¼ EAS i;t þ ðL=T Þ

rate of occurrence of failure for component i as formulated in Eq. (5). EAS i;tþ1 ¼ EAEi;t for i ¼ 1; …; N ; t ¼ 1; …; T −1

When a future schedule of maintenance operations for a repairable system is planned, the inevitable costs caused by unexpected component failures must be taken into account. However, it is not possible to exactly determine when such failures occur, but it is possible to anticipate that if the system carries a high rate of occurrence of failure through a period, then the system is at risk of experiencing an upcoming failure resulting to a significant amount of unexpected failure cost. On the other hand, a low rate of occurrence of failure in period t should yield a low cost of failure in that period. In order to consider these unplanned failure costs, the calculation of expected number of failures for each component in each period is proposed. The unexpected failure cost for each component i is defined as FCosti (in units of $/failure event); then, the cost of failures attributable to a component i in period t can be determined as follows (see Elsayed [1] and Ebeling [32] for more discussion and derivation of the expected number of failures of a nonhomogeneous Poisson process (NHPP)).

ð2Þ

for i ¼ 1; …; N ; t ¼ 1; …; T &

Maintenance/minimal repair actions Consider the case in which a component i is minimally repaired at the end of period t. The maintenance operation effectively reduces the effective age of the component at the start of the next period as follows: EAS i;tþ1 ¼ ð1−ImpFactori Þ ⋅ EAEi;t for i ¼ 1; …; N ; t ¼ 1; …; T −1 ð0 ≤ ImpFactori ≤ 1Þ ð3Þ The “Improvement Factor” parameter in the above equation allows for a variable effect of maintenance or repair operation on an aging component. When improvement factor=1, the effect of maintenance is to restore the component to a state of “good-as-new,” and when improvement factor=0, maintenance has no effect and the component remains in a state of “bad-as-old.” For more information about the improvement factor refer to Nakagawa [33], Malik [34], and Jayabalan and Chaudhuri[35]. Note that the maintenance and repair actions at the end of period t partially reduce the rate of occurrence of failure of component i. Furthermore, a maintenance cost MCosti is incurred at the end of that period.

&

Replacement actions If component i is replaced at the end of period t with a new identical component, then it is obvious that the effective age of the component at the start of the next period drops to zero as in Eq. (4) and the component failure behavior is returned to a state of good-as-new in which the rate of occurrence of failure of component i drops from ρi(EAEi,t) to ρi(0). In addition, there is a replacement cost equal to the initial acquisition cost of the component i, denoted as RCosti. EAS i;tþ1 ¼ 0 for i ¼ 1; …; N ; t ¼ 1; …; T −1

&

ð4Þ

Do nothing If no action is planned to be taken in period t, then a continuous increase will be expected on the age and the

ð5Þ

Expected failures costs in EAS i;t ; EAE i;t ¼ FCost i Expected number of failures in EAS i;t ; EAE i;t ! Z EAEi;t β β ¼ FCost i ρi ðt Þdt ¼ FCost i ⋅λi EAE i;t i − EAS i;t i EAS i;t

for i ¼ 1; …; N ; t ¼ 1; …; T

ð6Þ &

Shutdown cost In a multicomponent system with failure, repair, and replacement costs, the preventive maintenance scheduling problem might be considered as a simple problem of finding the optimal sequence of maintenance/repair, replacement, or do-nothing actions for each component, independent of all other components over the planning horizon. As a result, one could simply find the best sequence of operations for component 1 regardless of the operations taken to component 2 and so on. This would result to N-independent scheduling problems. In that case, a system of N components over T time periods has N×3T possible maintenance schedules (i.e., N=10 and T=12 have 5,314,410 possible schedules). Such a modeling approach seems unrealistic, as there should be some large penalty cost whenever an action is performed on any component in the system. It would make sense to group repair and replacement operations to be executed at the


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same time. For instance, while the system is down to replace one component, it is advantageous to perform repair or replacement actions on some other components, even if they are not at their individual optimum point where repair or replacement would ordinarily be performed. Under this scenario, the optimal time to perform these actions on an individual component is completely dependent upon the decisions made for other components. As such, consideration of a penalty cost of shutting down the system is proposed to be charged in period t if any component (one or more) is repaired or replaced in that period. Consideration of this shutdown cost makes the problem much more interesting and very difficult to solve as the optimal sequence of dependent maintenance actions must be determined simultaneously for all components from 3N×T possible combinations (i.e., N=10 and T=12 have 1.79×1057 possible schedules). Equation (7) calculates the total shutdown costs charged whenever any component in each period is repaired or replaced, and it is obvious that if more than one component in a period is undertaken to be repaired or replaced, the function results to a single charge. Total Shutdown Costs " !# T N X ¼ SCost 1−∏ 1− mi;t þ ri;t

ð7Þ

&

System reliability In order to compute the reliability of a series system of components, the reliability function for a repairable component i in period t is defined in Eq. (9), and it can be extended to the reliability function of the entire system as presented in Eq. (10); see the study of Elsayed [1] for more details on reliability of repairable systems having nonstationary number of failures represented by a nonhomogeneous Poisson process (NHPP). " Z Ri;t ¼ exp −

EAS i;t

In the above cost function, mi,t and ri,t are binary decision variables of maintenance and replacement actions for component i in period t. The condition of mi,t + ri,t ≤1 should be held for all components over the intervals of the planning horizon stating that both maintenance and replacement actions should not be carried out on a component simultaneously.

f 2 ¼ System Reliability N

¼

N

T

i¼1 t¼1

Total operational costs From the definitions of each type of cost, the total operational costs of the system can be determined over the T periods of the planning horizon. Therefore, the objective function of the total operational costs can be expressed as f 1 ¼ Total Operational Costs ¼ Expected failures costs þ Maintenance costs þ Replacement costs þ Shutdown costs

t¼1

t¼1

"

β i i

i¼1 t¼1

2.3 Multiobjective optimization model The multiobjective nonlinear mixed-integer optimization model with the total operational costs and system reliability is presented as the following: Min f 1 ¼

N X T h X i¼1 T X

þ

N SCost 1−∏ 1− mi;t þ ri;t i¼1

FCosti ⋅λi

EAEi;t

i β i β − EAS i;t i þ MCost i ⋅mi;t þ RCost i ⋅ri;t

t¼1

2

0 13 N 4SCost@1− ∏ 1− mi;t þ ri;t A5 i¼1

T

∏ ∏ i¼1 t¼1

h β β i exp −λi EAEi;t i − EAS i;t i

subject to : i ¼ 1; …; N EAS i;1 ¼ 0 EAS i;t ¼ 1−mi;t−1 1−ri;t−1 EAEi;t−1 þ mi;t−1 ð1−ImpFactori Þ⋅EAEi;t−1 i ¼ 1; …; N ; t ¼ 2; …; T EAEi;t ¼ EAS i;t þ ðL=T Þ

i ¼ 1; …; N ; t ¼ 1; …; T

mi;t þ ri;t ≤1

i ¼ 1; …; N ; t ¼ 1; …; T

mi;t ; ri;t ¼ 0 or 1

i ¼ 1; …; N ; t ¼ 1; …; T

EAS i;t ; EAEi;t ≥0

i ¼ 1; …; N ; t ¼ 1; …; T

ð11Þ

N X T h i X β β FCost i ⋅λi EAE i;t i − EAS i;t i þ MCost i ⋅mi;t þ RCost i ⋅ri;t

T X

ð10Þ

N

þ

h

T

∏ ∏ Ri;t ¼ ∏ ∏ exp −λi EAEi;t βi − EAS i;t

Max f 2 ¼

2.2 Objective functions

i¼1

ð9Þ

for i ¼ 1; …; N ; t ¼ 1; …; T

t¼1

f1 ¼

# h β β i ρi ðtÞdt ¼ exp −λi EAEi;t i − EAS i;t i

i¼1

t¼1

&

EAEi;t

!#

ð8Þ

In the above optimization model, the first set of constraints indicates that the initial age for each component is equal to zero (all components are assumed to be brand new). The second set mentions that if a component is maintained/

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repaired in the previous period then ri,t−1 =0, mi,t−1 =1 so EASi,t =(1−ImpFactori)⋅EAEi,t−1. If a component is replaced in the previous period, then ri,t−1 =1, mi,t−1 =0, so EASi,t =0, and finally, if no action was taken, ri,t−1 =0, mi,t−1 =0 and EASi,t =EAEi,t−1. The third constraint reflects the aging characteristic of the components over the planning horizon as denoted in Eq. (2). As mentioned in the previous subsection, the condition of mi,t +ri,t ≤1 should be held for all components over the intervals of the planning horizon forcing the model to prevent simultaneously performing both minimal repair and replacement actions on a component in a given period.

were estimated from the historical data. The unexpected failure costs are generally considered higher than the simple replacement of unfailed but degraded components. The shutdown cost of the machine is estimated to be $1,000 whenever a machine is unavailable for a repair or replacement action. The decision-making problem is considered as a tactical problem in which the range of the planning horizon is limited to 12 months. Under this setting, the proposed multiobjective model has 470 variables, 240 of which are binary, 360 functional constraints, 110 of which are nonlinear along with two nonlinear objective functions, total operational costs, and reliability of the system.

3 Practical application

3.2 Solution method

3.1 Data setting

The traditional methods to solve multiobjective optimization problems are based on preference-based approach with predetermined parameters obtained from the decision maker. All the classic methods employ a point-by-point deterministic optimization approach by finding a single optimal solution. Since multiobjective optimization problems have equally important Pareto-optimal solutions (also known as “trade-off optimal solutions”), the ideal approach would be finding multiple Paretooptimal solutions in a single run and let the decision maker choose the desired solution [2]. Evolutionary algorithms (EAs) mimic nature’s evolutionary principles to derive an intelligent search toward an optimal solution using a population of solutions in each generation (i.e., iteration). Since a population of solutions is tracked and evaluated in each generation, the outcome of such an algorithm will be a population of solutions. This feature of genetic algorithms to generate, track, and evaluate multiple solutions in one single run enables this class of algorithms to find multiple optimal solutions and makes the algorithms unique solution methods in solving multiobjective optimization problems [28]. In addition, genetic algorithms have been proved to be able to search simultaneously different regions of a solution space which makes it possible to find a diverse set of solutions for complex problems with nonlinear and

In order to apply the optimization model to a real system, a representative data set is developed for a CNC metalworking machine as shown in Table 1. The reliability characteristics of the components were determined from the historical components’ failures, repairs, and replacements. In a CNC machine, a failure (also known as crash) occurs when the machine moves in such a way that is harmful to the machine, tools, or parts being manufactured, sometimes resulting in bending or breakage of cutting tools, accessory clamps, vices, and fixtures or causing damage to the machine itself by bending guide rails, breaking drive screws, or causing structural components to crack or deform under strain. A minor crash may not damage the machine or tools but may damage the part being machined so that it must be scrapped. Since the CNC machines are repairable systems, the scale and shape parameters of the nonhomogeneous Poisson process (NHPP) representing the distribution of failures of the components were obtained by fitting NHPP function to the observed failure times. In addition to the failure characteristics of components, costs of possible preventive maintenance and replacement operations along with unexpected failure costs

Table 1 Reliability and operational characteristics of the CNC metalworking machine Component

Scale parameter

Shape parameter

Improvement factor

Failure cost ($)

Maintenance cost ($)

Replacement cost ($)

1 2 3

0.0022 0.0035 0.0038

2.20 2.00 2.05

0.38 0.42 0.45

250 240 270

35 32 65

200 210 245

4 5 6 7 8 9 10

0.0034 0.0032 0.0028 0.0015 0.0012 0.0025 0.0020

1.90 1.75 2.10 2.25 1.80 1.85 2.15

0.50 0.52 0.35 0.25 0.32 0.48 0.33

210 220 280 200 225 215 255

42 50 38 45 30 48 55

180 205 235 175 215 210 250

Single parameter (N); well tested; efficient Yes Yes Ranking based on nondomination sorting NSGA-II

Crowding distance

No No No No Fitness sharing by niching Niche count as tiebreaker in tournament selection Ranking based on nondomination sorting No fitness assignment, tournament selection

MOGA

NSGA NPGA

Fitness sharing by niching

No

No

First MOGA; straightforward implementation Simple extension of single objective GA Fast convergence Very simple selection process with tournament selection No

Each subpopulation is evaluated with respect to a different objective Pareto ranking

Diversity mechanism

VEGA

&

Representation of solutions The first step in any genetic algorithm implementation is to develop an encoding of the solution to be represented as unique chromosome. In order to represent the solution of preventive maintenance and replacement scheduling problem with do nothing, maintenance, and replacement actions, an array with length of N×T for N components and T periods is defined. Each cell in this array contains 0, 1, or 2 corresponds to three different actions. Crossover procedures The crossover procedures create a new solution as an offspring from a pair of the selected solutions (parent solutions). The offspring should inherit some useful

Fitness assignment

&

Table 2 Characteristics of the selected multiobjective evolutionary algorithms

MATLAB programming environment is utilized to construct the model and the multiobjective evolutionary algorithms to be all executed on a laptop computer (Intel/Core 2, 1.7 GHz and 2 GB RAM). In metaheuristic algorithms, the quality of solutions is largely affected by the parameter values [2]. Thus, in order to find appropriate values for the parameters of the genetic algorithms, a set of candidate values for each parameter is selected and tested in intensive computational experiments by varying one parameter at each step using a trial and error approach. The final selection of the parameters to be considered for all algorithms is presented in Table 3.

Elitism

3.3 Implementation aspects of the genetic algorithms

No

External population

Multiobjective evolutionary algorithms differ based on their fitness assignment procedure, elite-preserving mechanism, or diversification approaches. Table 2 highlights the well-known MOEAs with their advantages and disadvantages.

No

Advantages

Vector evaluated genetic algorithm (VEGA) [36] Multiobjective genetic algorithm (MOGA) [37] Nondominated sorting genetic algorithm (NSGA) [38] Niched Pareto genetic algorithm (NPGA) [39] Elitist nondominated sorting genetic algorithm (NSGA-II) [40]

Algorithm

& & & & &

Disadvantages

nonconvex functions, discrete variables, and multimodal solution spaces [30]. Since genetic algorithms need a fitness function to evaluate a candidate solution, the main task in modifying a single objective genetic algorithm to tackle multiobjective optimization problems is to find a single metric from a number of objective functions to identify nondominated solutions close enough to the true Pareto-optimal front [2]. Furthermore, the obtained nondominated solutions must be as diverse as possible to be uniformly and sparsely distributed in the Pareto-optimal region reflecting the existing trade-off among different objective functions [2]. In this research, the following algorithms are considered to be used to solve the developed model:

Adopted from Konak et al. [30]VEGA vector evaluated genetic algorithm, MOGA multiobjective genetic algorithm, NSGA nondominated sorting genetic algorithm, NPGA niched-Pareto genetic algorithm, NSGA-II elitist nondominated sorting genetic algorithm

2137 Tends converge to the extreme of each objective Usually slow convergence; problems related to niche size parameter Problems related to niche size parameter Problems related to niche size parameter; extra parameter for tournament selection Crowding distance works in objective space only


2138 Table 3 GA parameters setting


Number of generations

500

Population size Probability of selection Probability of crossover Probability of mutation

500 0.30 0.60 0.10

&

properties of both parents in order to facilitate its propagation throughout the population. In this research, several common crossover procedures (i.e., one-point crossover, two-point crossover, inverse crossover, etc.) were employed and tested in terms of their capability in generating diverse solutions, but it is found that they produce poor solutions resulting to premature convergence of solutions over the generations of the genetic algorithms. Therefore, based on the special structure of the problem, two new crossover procedures were designed to overcome this ineffectiveness of the tested crossovers as follows: (a) Two-point inverse crossover: This type of crossover first generates two random numbers between 1 and N×T and then makes an offspring from selected parents in which all elements outside the position of those random numbers are copied from the first parent but in an inverse order, and inner elements are copied from the second parents in a direct order. If the chosen parents are identical, this type of crossover makes a different offspring, which is not the same to its parents. (b) NT-Point Crossover: In this type of crossover, the even genes are copied from the first parent, and odd genes are copied from the second parent. It then is decided that if two selected solutions are equal to each other, then the algorithm uses twopoint inverse crossover, and if the selected solutions Table 4 A Pareto-optimal schedule obtained by VEGA (cost=$9,413.55, reliability= 0.3086)

are not the same, the algorithm uses NT-point crossover to produce new solutions. Mutation procedure The mutation procedure is applied to the generated offspring solutions. It makes changes into the solutions’ encoding string by randomly modifying some of the string elements. Again, based on the special structure of the problem in which if even one maintenance or replacement action takes place in a period, the whole system encounters large shutdown cost, and special type of mutation procedure is designed to be carried out throughout the execution of the genetic algorithms. In this type of mutation, first, a random number is generated between 1 and N×T to have more randomness and variability over the entire solution genes. Then, the corresponding gene will be changed to 1 or 2 if it is equal to 0, and it will be changed to 0 if it is equal to 1 or 2. The same procedure in the same period for other components will be performed accordingly. This mutation procedure eventually produces scheduled maintenance, and replacement activities tend to be in the same periods across many components.

4 Computational results 4.1 Pareto-optimal preventive maintenance and replacement schedules Tables 4, 5, 6, 7, and 8 depict Pareto-optimal schedules obtained by the tested algorithms. Note that in the presented Pareto-optimal schedules, all maintenance and replacement actions tend to occur in the same period which reflects the effect of the large shutdown cost. Such Pareto-optimal schedules demonstrate the effectiveness of the modeling and implementation approach to determine the nondominated solutions for other types of CNC machines even on a longer planning horizon.

Component/month

1

2

3

4

5

6

7

8

9

10

11

12

1 2 3 4 5 6 7 8 9 10

– – – – – – – – – –

– – – – – – – – – –

M M R R M R M M M M

– – – – – – – – – –

R R M M R M R R M R

– – – – – – – – – –

– – – – – – – – – –

– – – – – – – – – –

R R M R R R R R M R

– – – – – – – – – –

M M R M M M M M R M

– – – – – – – – – –

Int J Adv Manuf Technol (2015) 76:2131–2146 Table 5 A Pareto-optimal schedule obtained by MOGA (cost=$18,645.24, reliability= 0.5576)

Table 6 A Pareto-optimal schedule obtained by NSGA (cost=$14,118.72, reliability= 0.4755)

Table 7 A Pareto-optimal schedule obtained by NPGA (cost=$15,561.98, reliability= 0.5308)

Table 8 A Pareto-optimal schedule obtained by NSGA-II (cost=$14,759.73, reliability= 0.4871)

2139

Component/month

1

2

3

4

5

6

7

8

9

10

11

12

1 2 3 4 5 6 7 8 9 10

– – – – – – – – – –

R R R R M R R M M R

– – – – – – – – – –

R R R R R R R R R R

– – – – – – – – – –

R M R M R R R R M R

– – – – – – – – – –

R R R R R R R M M R

M M M – M M M – – M

R R R R R R R R R R

– – – – – – – – – –

– – – – – – – – – –

Component/month

1

2

3

4

5

6

7

8

9

10

11

12

1 2 3 4 5 6 7 8 9 10

– – – – – – – – – –

– – – – – – – – – –

R R R R R R R R R R

– – – – – – – – – –

R R R R R R R R R R

– – – – – – – – – –

M M M R R M R M M M

R R R R R R R R R R

– – – – – – – – – –

R R R R R R R R R R

– – – – – – – – – –

– – – – – – – – – –

Component/month

1

2

3

4

5

6

7

8

9

10

11

12

1 2 3 4 5 6 7 8 9 10

– – – – – – – – – –

R R R M M R R M R R

– – – – – – – – – –

M R R R R R R R M M

– – – – – – – – – –

R R M M R R R R R R

– – – – – – – – – –

R M R R M M R M R R

– – – – – – – – – –

R R R R R R R R R R

– – – – – – – – – –

– – – – – – – – – –

Component/month

1

2

3

4

5

6

7

8

9

10

11

12

1 2 3 4 5 6 7 8 9 10

– – – – – – – – – –

R R M M R R R M R M

R R M R M R R R M R

– – – – – – – – – –

R R R R R R R R R R

– – – – – – – – – –

– – – – – – – – – –

R R R R R R R R R R

– – – – – – – – – –

R R R M M R R M M R

– – – – – – – – – –

– – – – – – – – – –

2140


Table 9 Summary of computational performance Algorithm Average number of nondominated solutions

Standard deviation of number of nondominated solutions

Average of computational Standard deviation of times (s) computational times (s)

VEGA MOGA NSGA NPGA

32.02 78.95 120.85 70.39

3.30 10.58 6.89 10.70

150 205 387 354

17 29 37 58

NSGA-II

128.56

5.79

401

33

VEGA vector evaluated genetic algorithm, MOGA multiobjective genetic algorithm, NSGA nondominated sorting genetic algorithm, NPGA nichedPareto genetic algorithm, NSGA-II elitist nondominated sorting genetic algorithm

4.2 Computational performance of the MOEAs In order to achieve statistically valid results, 50 independent runs of each algorithm were performed to solve the optimization model. The computational efficiency of the algorithms in terms of average and standard deviation of number of obtained nondominated solutions and also average and variability in run time are examined, and it is shown in Table 9 and Figs. 1 and 2. As can be observed, VEGA is the fastest algorithm with an average run time of 2 min and 30 s; however, comparing to other algorithms, VEGA generates fewer nondominated solutions at the end of the computational experiment. It is useful to reiterate that VEGA is the only algorithm in our study without any mechanism of maintaining diversity among nondominated solutions. On the other side of the spectrum, NSGA-II takes on average 6 min and 41 s to generate 128 nondominated solutions. Again, note that NSGA-II is the only algorithm in our study with elite-preserving mechanism to maintain and pass good nondominated solutions from one generation to another which eventually results to the longest computational time. In largescale optimization problems with more than two objective functions and with thousands of decision variables and functional constraints, the computational time will definitely be an issue that needs to be considered before choosing the most suitable algorithm to solve the problem. As in most applications, the trade-off between desired number of nondominated solutions and computational time must be taken into account for every problem.

Figures 3, 4, 5, 6, 7, and 8 illustrate the obtained nondominated solutions in objective function space by employing the multiobjective genetic algorithms. The most extreme nondominated solutions are identified to be (cost=$37,574.83, reliability=0.731104) generated by VEGA and (cost=$961.82, reliability=0.018988) generated by all algorithms. By examining Figs. 3, 4, 5, 6, 7, and 8, it is subjectively clear that nondominated solutions generated by NSGA and NSGA-II might be more likely closer to the true Pareto-optimal front and also they cover more area in the objective function space to reflect the existing trade-off between the total operational cost and the overall reliability. As stated before, the first task in solving a multiobjective optimization problem is to identify nondominated solutions as close as possible to the unknown Pareto-optimal front. Another necessary feature to be carried is that the obtained nondominated solutions must be uniformly distributed in the Pareto-optimal region reflecting the existing trade-off among different objective functions. Finally, in order to capture the whole spectrum of the Pareto front, extreme solutions at the objective function space should be identified. In order to quantitatively compare the performance of the algorithms in solving the model, four types of performance metrics are chosen to determine the relative domination of the nondominated solutions by each algorithm and also to measure the diversity of those solutions. 480 420

140 120.85

120 100 78.95

80

70.39

60 40

401

387

128.56

354

360 Time (Second)

Non-dominated Soluon

160

300 240 180

205 150

120

32.02

60

20

0

0 VEGA

MOGA NSGA NPGA Average number of non-dominated soluons

NSGA-II

Fig. 1 Average and standard deviation of number of nondominated solutions by the tested algorithms

VEGA

MOGA

NSGA

NPGA

NSGA-II

Average computaonal me

Fig. 2 Average and standard deviation of computational times of the tested algorithms

2141

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5 Reliability

Reliability


0.4 0.3

0.3

0.2

0.2

0.1

0.1

0

0

0.5

1

1.5

2

2.5

3

3.5

0

4

0.7

0.7

0.6

0.6

0.5

0.5 Reliability

0.8

0.4

0.3

0.2

0.2

0.1

0.1

1

1.5 2 Total Cost

2.5

3

3.5 4

x 10

Fig. 4 Nondominated solutions obtained by MOGA after 50 replications

1.5 2 Total Cost

2.5

3

3.5 4

x 10

0.4

0.3

0.5

1

Fig. 6 Nondominated solutions obtained by NPGA after 50 replications

0.8

0

0.5

x 10

Fig. 3 Nondominated solutions obtained by VEGA after 50 replications

0

0

4

Total Cost

Reliability

0.4

0

0

0.5

1

1.5 2 Total Cost

2.5

3

3.5 4

x 10

Fig. 7 Nondominated solutions obtained by NSGA-II after 50 replications

0.8 0.7 0.6

Reliability

0.5 0.4 0.3 0.2 0.1 0

0

0.5

1

1.5 2 Total Cost

2.5

3

3.5 4

x 10

Fig. 5 Nondominated solutions obtained by NSGA after 50 replications

Fig. 8 All nondominated solutions obtained by MOEAs after 50 replications

2142


and VEGA solutions are dominated by the solutions of other algorithms. Besides NSGA, MOGA also performs well in generating nondominated solutions in which only 20 % of their solutions are being dominated by other algorithms. Similar conclusion can be drawn by comparing the average percentage of dominate solutions calculated in the last column. Therefore, NSGA and MOGA satisfactorily demonstrate the capability of solving the multiobjective scheduling problem by identifying nondominated solutions as close as possible to the unknown Pareto-optimal front. It was initially expected that NSGA-II would outperform the other algorithms because it is the only one equipped with elite-preserving mechanism to maintain and pass good nondominated solutions from one generation to another. However, NSGA-II computational performance in generating nondominated solutions is still in an acceptable level since only, on average, 37 % of its solutions are dominated by others’ solutions.

4.3 Metric evaluating closeness to the true Pareto-optimal front Since the true Pareto-optimal solutions, P * , of the multiobjective optimization model are unknown, comparing obtained nondominated solutions with the members of P* is impossible. Therefore, the set coverage metric is used to evaluate the relative domination of nondominated solutions generated by each algorithm. &

Set coverage metric The set coverage metric C(A, B) suggested in the study of Zitzler [41] calculates the proportion of solutions generated by algorithm B which are dominated by solutions generated by algorithm A as follows:

C ðA; BÞ ¼

n o b∈B∃a∈A : a≺b

ð12Þ

jBj

where a≺b means that solution b is dominated by solution a. Hence, if the metric value C(A, B)=1, it reveals that all members of B are dominated by all members of A, and if C(A, B)=0, it shows that no solution of B is dominated by any solution of A. Therefore, the smaller value of the metric is preferred with respect to algorithm B. It is also necessary to clarify that domination operator is not a symmetric logical operator (C(A, B)≠1−C(B, A)), so both C(A, B) and C(B, A) should be calculated. Table 10 provides the set coverage metric values from a pair-wise comparison of the tested algorithms. Each column in the above table shows the percentage of solutions obtained by each algorithm which are dominated by the solutions of the other algorithms. The last row also demonstrates the average percentage of dominated solutions of each algorithm. By carefully reviewing the numbers, it can be stated that only 9.5 % of the solutions generated by NSGA are dominated by the solutions from other algorithms and NSGA performs excellent in producing high quality of nondominated solutions over all other algorithms. On the other hand, about 48 % of the NPGA

4.4 Metric evaluating diversity and spread of the nondominated solutions &

Spacing The spacing metric developed in the study of Schott [42] measures the standard deviation of distance between any solution and all other solutions, and it reveals how well the nondominated solutions are distributed in the objective function space by the following equation: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u jQj u 1 X 2 S¼t ð13Þ d i −d jQj i¼1 where di is the minimum rectilinear distance between each nondominated solution and all other nondominated solutions in set Q in objective function space, whereas d is the average of minimum rectilinear distances of all the solutions in that space. d i ¼ min ¼ k∈Q∧k≠i

M X ðiÞ ðk Þ f −f m

m

ð14Þ

m¼1

Table 10 Evaluating closeness using set coverage metric C(A,B) Algorithm

VEGA

MOGA

NSGA

NPGA

NSGA-II

Average percentage of dominate solutions

VEGA MOGA

– 0.531250

0.012821 –

0.100000 0.175000

0.114286 0.500000

0.171875 0.398438

0.099745 0.401172

NSGA NPGA NSGA-II Average percentage of dominated solutions

0.625000 0.375000 0.375000 0.476563

0.500000 0.102564 0.205128 0.205128

– 0.008333 0.100000 0.095833

0.871429 – 0.428571 0.478571

0.640625 0.296875 – 0.376953

0.659263 0.195693 0.277175



d¼

jQj

X di i¼1

&

2143

jQj

Maximum spread The maximum spread metric first introduced in the study of Zitzler [41] measures the length of the diagonal of a hypercube formed by the extreme obtained nondominated solutions in the normalized objective function space by the following equation: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 0 u jQj u jQj i i u X fm − min fm C u 1 M B max i¼1 C B i¼1 ð16Þ D¯ ¼ u @ F max − F min A tM m¼1

NSGA-II perform very well in generating spread and well-distributed nondominated solutions making these algorithms trustable in achieving the second and third tasks of solving the multiobjective scheduling model.

ð15Þ

m

m

min where F max m and F m are the maximum and minimum values of the objective function m. This metric provides a measurement on how far the extreme nondominated solutions are located. Therefore, D ¼ 1 represents the fact that a widely spread set of nondominated solutions is generated in the last generation, so basically, metric values close to one are preferred. Table 11 demonstrates the spacing and maximum spread metric values calculated based on nondominated solutions. It is revealed that maximum spread of VEGA to be found, in the normalized objective function space, as 0.999989 and the algorithm generates a near-perfect extreme nondominated solutions. However, these nondominated solutions are not uniformly distributed in the objective function space, see Fig. 3, as the spacing metric of VEGA is about 3.4 %. On the other hand, NSGA-II generates excellent uniformly distributed solutions in the objective function space as shown in Fig. 7. However, the algorithm cannot compete with others in generating most extreme nondominated solutions that result to a moderate maximum spread metric value of 0.895663. In conclusion, MOGA, NSGA, NPGA, and

Table 11 Evaluating diversity and spread of the nondominated solutions Algorithm

Spacing (S)

Maximum spread D

VEGA MOGA NSGA NPGA NSGA-II

0.033843 0.016117 0.009580 0.014829 0.006964

0.999989 0.954516 0.958275 0.939300 0.895663

4.5 Metric evaluating closeness, diversity, and spread of the nondominated solutions &

Hyper volume To better evaluate the quality of the obtained nondominated solutions, another metric from the literature is adopted. The hyper volume metric is proposed in the study of Van Veldhuizen [23] that calculates an approximation of the volume of the hypercube formed by the nondominated solutions in the objective function space. This metric can simultaneously measure all three tasks of solving a multiobjective optimization problem, closeness to the true Pareto-optimal front, diversity, and spread of nondominated solutions. Literally, a volume, v i , is constructed by each nondominated solution and a reference point as diagonal corners of the hypercube. ! HV ¼ volume

∪ jQj

vi

ð17Þ

i¼1

In order to calculate the hyper volume metric, the nadir solution (the worst obtained values for each objective function) is selected as a reference point fnadir (C=$37,574.83, R=0.018988). Because of different orders of magnitude of total operational costs and system reliability, the normalized values of the nondominated solutions along with normalized nadir point fnormalized (C=1, R=0) are used to calculate the nadir HVnormalized (with maximum possible value of one). Then, the volume formed by the reference point and all nondominated solutions is to be calculated. To prevent overlapping of the volumes, the nondominated solutions are sorted so the f1,1 is the first value of the first objective function (cost) and f2,1 is the first value of the second objective function (reliability). The rest of the volumes are calculated with respect to the adjacent sorted nondominated solutions using Eq. (18). f normalized HV normalized ¼ f normalized −C normalized −Rnormalized 1;1 nadir 2;1 nadir jQj X normalized normalized þ − f 1;i−1 −Rnormalized f normalized f 1;i 2;i nadir i¼2

VEGA vector evaluated genetic algorithm, MOGA multiobjective genetic algorithm, NSGA nondominated sorting genetic algorithm, NPGA niched-Pareto genetic algorithm, NSGA-II elitist nondominated sorting genetic algorithm

ð18Þ Table 12 presents the normalized hyper volumes formed by the nondominated solutions. It is

2144


Table 12 Evaluating closeness, diversity, and spread Algorithm

Hyper volume (HV)

VEGA MOGA NSGA NPGA NSGA-II

0.645600 0.666090 0.675579 0.652809 0.663407

5 Conclusions and future research In this research, a multiobjective nonlinear mixed-integer optimization model is developed to determine the Paretooptimal preventive maintenance and replacement schedules in multicomponent manufacturing systems with total operational costs and overall system reliability. The developed model was used to find the optimal maintenance schedules of a CNC machine. In order to solve this model, five multiobjective genetic algorithms are employed, and their computational performance is evaluated by different metrics. In addition to this evaluation and based on the special structure of the scheduling problem, two new crossover procedures and a mutation mechanism are also introduced. After computational analyses, the algorithms are ranked based on different performance measures, and it is found that the NSGA, MOGA, and NSGA-II outperform VEGA and NPGA in obtaining more and high-quality nondominated schedules along with maintaining high level of diversity among generated solutions. The presented model in this paper can be directly applied or indirectly integrated in designing maintenance plans in a wide variety of applications such as production assembly lines, manufacturing machines, and material handling systems. The Pareto-optimal solutions identify trade-offs between two objectives, enabling the maintenance planner to consider three types of decisions at the same time. The model and the tested MOEAs can be used to quickly generate new preventive maintenance and replacement plans in complex systems even after occurring unexpected failures. In such a situation, the original schedule should be updated, and a new optimal schedule can be used over the remaining of the planning horizon. This multiobjective model can also be used in condition-based simulation models as a real-time optimization procedure to refine and update maintenance plans during the simulation run. Future work in this area is needed to investi-

VEGA vector evaluated genetic algorithm, MOGA multiobjective genetic algorithm, NSGA nondominated sorting genetic algorithm, NPGA niched-Pareto genetic algorithm, NSGA-II elitist nondominated sorting genetic algorithm

observed that NSGA’s nondominated solutions with HV = 0.675579 form a larger hypercube than the other algorithms and it outperforms in generating diverse and well-distributed solutions while close enough to the true Pareto-optimal front measured by this metric. The overall performance of each algorithm in solving the proposed multiobjective preventive maintenance and replacement optimization model is ranked as in Table 13. It can be seen that NSGA, MOGA, and NSGA-II outperform VEGA and NPGA in most performance measures. However, selection of the most appropriate algorithm depends on the decision maker’s preferences on a specific task such as generating large number of nondominated solutions regardless of how long it takes to identify as many candidate solutions as possible (i.e., NSGA and NSGA-II) or just obtaining a few but extreme nondominated solutions to provide an informative insight about the existing trade-off among conflicting objective functions (i.e., VEGA).

Table 13 Rank of the algorithms with respect to performance metrics (1—best; 5—worst)

Algorithm Number of nondominated solutions Computational time Set coverage (C) Spacing (S) Maximum spread D Hyper volume (HV) VEGA MOGA NSGA

5 3 2

1 2 4

4 2 1

5 4 2

1 3 2

5 2 1

NPGA NSGA-II

4 1

3 5

5 3

3 1

4 5

4 3



gate the effects of other constraints such as limited maintenance resources and system availability requirements by considering other strategies for maintenance activities. It is planned to expand this research by taking other criteria into account like the total tardiness for the manufacturing predetermined schedules. It is also interesting to include constraints related to production capacity. From point of view of solution methods, we are interested to study the effect of hybridization of the genetic algorithms with other methods to be used in an expansion of our study. Acknowledgments The author is grateful to the referees and the editor whose valuable comments and suggestions significantly improved the overall content and presentation clarity of the paper.

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