Hybrid Genetic Algorithm and Tabu Search for Solving Preventive Maintenance Scheduling Problem for Cogeneration Plants

Abstract

Preventive Maintenance (PM) is a periodic maintenance strategy that has great results for devices in extending their lives, increasing productivity, and, most importantly, helping to avoid unexpected breakdowns and their costly consequences. Preventive maintenance scheduling (PMS) is determining the time for carrying out PM, and it represents a sensitive issue in terms of impact on production if the time for the PM process is not optimally distributed. This study employs hybrid heuristic methods, integrating Genetic Algorithm (GA) and Tabu Search (TS), to address the PMS problem. Notably, the search for an optimal solution remained elusive with GA alone until the inclusion of TS. The resultant optimal solution is achieved swiftly, surpassing the time benchmarks set by conventional methods like integer programming and nonlinear integer programming. A comparison with a published article that used metaheuristics was also applied in order to evaluate the effectiveness of the proposed hybrid approach in terms of solution quality and convergence speed. Moreover, sensitivity analysis underscores the robustness and efficacy of the hybrid approach, consistently yielding optimal solutions across diverse scenarios. The schedule created exceeds standards set by waterworks experts, yielding significant water and electricity surpluses—16.6% and 12.1%, respectively—while simultaneously matching or surpassing total production levels. This method can be used for power plants in private or public sectors to generate an optimal PMS, save money, and avoid water or electricity cuts. In summary, this hybrid approach offers an efficient and effective solution for optimizing PMS, presenting opportunities for enhancement across various industries.

1. Introduction

The maintenance of all kinds of equipment or machinery plays an important role in protecting it from sudden breakdowns that may cause halting or delaying production, especially for essential matters for consumers. It also improves uptime and extends its operational life. The main objective of maintenance is to ensure that all equipment required for production is operating efficiently at all times. With the development of science, several maintenance strategies have emerged. There are many types of maintenance strategies, but the most commonly used are reactive maintenance (run-to-fail), which relies on the technique of using things until they break before they can be fixed. Predetermined Maintenance is another strategy that simply follows the manufacturer’s recommendations for maintenance, including when to perform checks and maintenance. Another type of maintenance strategy is predictive maintenance, which is based on data and indicators of the current actual condition of the equipment, through which the maintenance process is carried out. The most widely used type of maintenance technique is preventive maintenance PM, where, in 2020, according to [1], 76% of companies in the manufacturing industry worldwide prioritized PM. This technique relies on performing regular and routine scheduled maintenance (PMS) to fix small problems before they develop into major problems. Figure 1 illustrates the workflow of PM. For more details about maintenance strategies in general, see [2,3].

There are many benefits to PM, such as extending the life of assets, reducing the risk of breakdowns, increasing efficiency, minimizing unplanned downtime, increasing customer satisfaction, etc. see [4]. Many different sectors apply PM in their sector, for example, in the field of transportation: vehicles [5,6], aircraft [7,8], trains [9,10,11], and ships [12]. In the field of factories or plants [13,14,15,16,17,18,19], other sectors [20,21,22,23,24,25,26].

Power plants are important sectors for humanity. The main task is to provide people with electricity and fresh water. Any interruption of water or electricity has a significant negative impact. The primary goal of power plants is to produce electricity and water for consumers, so these plants are important and sensitive as they produce the necessities of life. Therefore, workers in power plants are aware of the importance of maintaining all assets and components of the plant in order to ensure that electricity and water are not interrupted by consumers.

This study contributes to the following key issues:

• Hybridization of Techniques: Two different techniques (GA and TS) were hybridized to obtain better performance with the aim of solving the problem of PMS for a power plant in Kuwait, where all factors such as population size, crossover method, mutation method for genetic algorithms (GA), and parameters such as tabu list for tabu search (TS) will be identified.
• Data Identification: All data involved in this problem will be identified, such as the number and type of equipment, time horizon, maintenance period, etc.
• Constraint Recognition: All constraints related to the power generation plant were identified, in addition to the objective function to be achieved.
• Experimental Execution: Conduct the experiments by implementing the hybrid techniques with different parameter settings based on various experiments.
• Results Analysis: Analyze the experimental results to identify which parameter settings lead to better performance in terms of the quality of the results (solution quality and computational time).

The paper is organized as follows: In Section 2, a literature review about PMS in general is presented, in addition to presenting various classes of metaheuristics and the methodologies employed to address the research problem. Section 3 will focus on the explanation of the problem as well as the formulation to solve the problem. In Section 4, the results obtained are analyzed and compared with previous studies. Lastly, Section 5 will be about the conclusion.

2. Review of Preventive Maintenance Scheduling

2.1. Literature Review

A lot of research has been conducted in PMS; some researchers have adopted the deterministic approach to find the optimal solution, others have resorted to using heuristic methods for large-scale problems, and others have combined the two approaches. In the deterministic approaches, Mollahassani-pour et al. [27] designed Mixed Integer Linear Programming (MILP) as a framework to address PMS of power generation units. The aim of this study was to reduce the cost of maintenance while determining the most appropriate maintenance scheme. To prove the effectiveness of the proposed structure, the IEEE Reliability Testing System (RTS) was used, with promising results. Moradi-Sarvestani et al. [28] presented a three-phase Reliability-Centered Maintenance (RCM) framework to identify a critical feeder, prioritize failure modes, and set the optimal maintenance strategy for a power distribution system. They formulated the problem using a mixed-integer linear programming (MILP) optimization problem. The objective function is to reduce the total maintenance costs, such as operation, equipment, etc. The proposed approach is implemented on a real distribution system, where the result is a 26.32% reduction in expected energy not supplied (EENS) compared to choosing the run-to-failure (RTF) strategy for all components. Al-Hamad et al. [29] provided a paper for PMS on a real-case study of a 32-unit system of boilers, turbines, and distillers for power plants in the State of Kuwait. The problem is formulated as a Mathematical Program (MP) that increases minimum output to demand ratios for a period of 52 weeks. All constraints were considered, and the results of the proposal were optimal compared to the schedule of power plant experts and in reasonable time. Also, Al-Hamad et al. [30] used the nonlinear integer programming (NLIP) method to address the previous problem. They applied the comparison with the results obtained with the previous study in terms of the time taken to reach the optimal solution, as the time taken was reasonably higher. Fetanat and Shafipour [31] used Ant Colony Optimization for Continuous Domains (ACOR) to treat the problem of unit maintenance scheduling, as the aim of this study was to reduce cost and reliability. Constraints such as rotational reserve and maintenance crew duration were taken into account. The performance of the technique used was benchmarked against those reported in the literature, and the method that was used proved to be superior to them in terms of optimal resolution and speed. Canto [32] used mixed Integer Linear Programming 0/1 in order to create a schedule that allows efficient organization of preventative maintenance over a specific time horizon. This study was applied to a power plant in Spain, and reliability is the main point used in the presented methodology.

Since exact optimization algorithms take a long time to reach the optimal solution, it is also difficult to find an optimal solution as the dimensions of the system increase, so many researchers resort to using heuristics methods to solve large-scale problems. Lapa et al. [33] used Genetic Algorithms (GA) technology in the PM of the power plant. Some relevant features have been taken into account, for example, the possibility of needing repair, the cost of repair, and the possibility of incomplete maintenance. Pressurized water reactors (PWR) were applied as a case study, and the results were good. Asis et al. [34] have used the Discrete Chaotic Jaya Optimization (DCJO) algorithm to address the generator maintenance scheduling (GMS) problem with the aim of enhancing reliability and reducing costs by optimizing maintenance schedules for generating units, offering promising solutions. The DCJO algorithm was tested on a 21-unit system over 52 weeks and outperformed state-of-the-art algorithms, indicating its potential as a reliable maintenance scheduling tool for power systems. Alimohammadi and Behnamian [35] presented a study aimed at scheduling the PM of a group of feeders in the Electricity Distribution Company of Iran, where mathematical programming was proposed for this problem. During the study, two heuristic algorithms were provided with the aim of providing a good solution in a reasonable time. The results of the proposed algorithms were validated, as the results of the heuristic algorithms were very suitable. Dahal and Chakpitak [36] used a hybrid genetic algorithm (GA) and simulated annealing (SA) to schedule preventive generator maintenance. The results were promising and showed that the use of hybrid methods provides an effective alternative to solving this type of problem. Belagoune et al. [37] addressed the generation maintenance scheduling (GMS) problem, which involves collaboration between the independent system operator (ISO) and generation companies (GENCOs). The objective was to minimize expected production costs and system load shedding. A metaheuristic was utilized to solve the resulting optimization problem, with non-sequential Monte Carlo simulation and Cross-Entropy methods combined to assess maintenance schedules efficiently. Duarte et al. [38] adopted particle swarm optimization (PSO) and genetic algorithms (GA) for PMS in the Cuban power system. The goal is to reduce the risk of load loss as much as possible in the power system. The paper demonstrates the effectiveness of the proposed model in this real power system. For more literature reviews in this field of PMS optimization and applications to power plants, see [39,40].

2.2. Metaheuristic Algorithms

Metaheuristic methods are versatile problem-solving techniques designed to efficiently navigate large search spaces and discover high-quality solutions for optimization problems. They are characterized by their broad applicability across various problem types and their independence from specific problem structures. Instead, metaheuristics draw inspiration from natural phenomena, social behavior, and mathematical principles to iteratively enhance candidate solutions. These methods are categorized into different classes, each representing distinct inspirations and approaches to optimization:

• Swarm-based approaches: Metaheuristic algorithms in this category leverage principles inspired by knowledge or information exchange processes. Notable examples include Particle Swarm Optimization (PSO) [41] Ant Colony Optimization (ACO) [42], and Firefly Algorithm (FA) [43].
• Physics-based algorithms: Emulating physical phenomena, these algorithms guide the search for optimal solutions. Examples include Simulated Annealing (SA) [44], and Gravitational Search Algorithm (GSA) [45].
• Biology-based methods: Inspired by biological processes like evolution and genetics, algorithms in this category include Genetic Algorithm (GA) [46,47], and Differential Evolution (DE) [48].
• Social-based approaches: Drawing inspiration from social phenomena, these algorithms mimic interactions observed in human societies. Examples include Teaching–Learning Based Optimization (TLBO) [49], Mother Optimization Algorithm (MOA) [50], and Following Optimization Algorithm (FOA) [51].
• Sports-based metaheuristic algorithms: Inspired by sports games and competitions, these algorithms include Volleyball Premier League (VPL) [52], Football Game-Based Optimization (FGBO) [53], and Ring Toss Game-Based Optimization (RTGBO) [54].

Each class offers unique advantages and applications across various problem domains, equipping practitioners with a diverse set of tools for addressing optimization challenges. For further insights, refer to [55].

The method adopted to address this problem was implemented through two indicative mixed methods: the genetic algorithm GA and the tabu search TS. GA and TS are metaheuristic problem-solving approaches used to solve combinatorial optimization problems. GA was developed by John Holland and his collaborators in the 1960s and 1970s [46,47], while TS was proposed in 1989 by Fred Glover of TAPO Research [56]. In a GA, the basic elements are chromosome representation, fitness selection, and biologically inspired factors. A set of solutions is called a population, which is randomly or scientifically generated. Each population consists of a number of chromosomes, and in each chromosome is a set of genes. Each gene locus on the chromosomes is usually represented in binary format or as a set of integers, depending on the type of problem. Fitness represents the chromosome value, where the best objective function value is searched through a number of iterations to improve the solution. The process is carried out by a group of parents who must be selected in order to find a new solution by applying two factors: the crossover and mutation methods. The process continues by selecting the best parents from among the population, with the aim of reproducing a new generation with different characteristics. The process of reproduction of offspring in the next generation takes place through the evolutionary algorithm, which is inspired by the theories of crossover and mutation. Crossover is mainly used to create new solutions from the genetic information of a population, while mutation occurs to bring in new information and diversity within the population and move away from early convergence to make the solution more general. This process will be repeated for a number of iterations until an optimal or near-optimal solution is found.

Tabu Search (TS) is a metaheuristic search method that uses local search methods. The search method is developed by making it move from the local area to the global area, which discourages the search for previously visited solutions. The search starts from the possible solution x to the optimal solution x’ near x until the stopping criterion, which is the number of iterations mentioned, is reached or the desired solution (score threshold) is reached. Two operators, swap and insert, are applied during the search. Solutions are sought extensively in the current region (intensification), but usually after a number of iterations, the solutions are suboptimal or discouraging. To discourage the algorithm from revisiting solutions that have already been explored recently, the “Tabu list” is applied. Tabu List is a short-term compilation of solutions visited in the recent past. This solution remains “tabu” or forbidden for a number of tn “tabu tenure”. Tabu tenure tn refers to the duration or number of iterations for which a solution or move remains on the Tabu List. It’s a way to control how long a particular choice is considered “tabu” or forbidden. This mechanism encourages diversification by preventing the search from getting stuck in local optima or repetitive cycles. As the algorithm explores new solutions, it increases the likelihood of discovering a broader range of possibilities, leading to a more diverse and thorough search for the optimal solution. Aspiration Criteria, on the other hand, provide exceptions based on specific criteria. By excluding the best solution from the tabulated list, the algorithm remains free to explore the neighborhood of that solution, potentially finding even better solutions in the same region. These criteria could be based on improvements in the objective function, reaching a specific threshold, or satisfying other relevant conditions, whereas in this research they are based on achieving the best solution so far. This helps balance exploration and exploitation in the search process.

This study will focus on the power plant in the State of Kuwait, which contains turbines, boilers, and distillers. The main goal is to schedule preventive maintenance equipment in a way that maximizes equipment utilization while minimizing power outages. A hybrid Genetic Algorithm and Tabu Search would be the methodology to tackle this problem, where a chromosome of GA represents PMS, with each gene representing a week in the time horizon. The application of crossovers and mutations within GA facilitates the exploration of the solution space while also maintaining diversity among solutions. Integrating TS with its capability to intensify the search in promising solution regions through the implementation of tabu tenure adds a valuable dimension to the optimization process. GAs excel at exploring a wide solution space, while TS is better at exploiting local optima. Leverage both exploration and exploitation abilities by combining them, so the hybrid approach can generate diverse solutions initially and then refine them locally, offering a broader spectrum of potential solutions, leading to a more robust search process. The study provides a PMS for each piece of equipment in the power plant for a time period of 12 months. The results will be compared with the results of the previous optimization approaches [29,30] in order to measure the quality of this methodology in terms of solution quality and implementation time (Appendix A and Appendix B). Also, to evaluate the effectiveness of the proposed hybrid approach in terms of solution quality and convergence speed, a comparative analysis with a previous related study [26] was applied. This study also addressed the same problem domain, using a genetic algorithm (GA) approach. Certain constraints will be taken into account, such as maintenance window restrictions, maintenance duration for each piece of equipment, the level of reservoir water limits, etc. The positive effects of this study are to meet consumer demands without any shortage or interruption, whether for water or electricity. In addition to maintaining equipment for a long time without any breakdowns that lead to expensive repairs.

3. Problem Description

3.1. Problem Background

The current study will focus on power plants in the State of Kuwait. Power plants generally consist of a number of plants, each containing three types of equipment: turbines, boilers, and distillers. The energy production process in the power plants in Kuwait takes place by drawing sea water, which is first transferred to the boilers in order to heat up to produce high-pressure steam. The steam is then transmitted to two paths: to the distillers, and order to condense and get rid of the salt, to produce drinking water. The other is transmitted to the turbines in order to produce electricity. To continue producing energy and provide the citizens’ needs for electricity and water without interruption, the plant needs annual PM. A plant consists of a number of plants, p $=1 \dots P.$ Each plant $p_i$ consists of different numbers of distillers $d$, turbines $r$, and boilers $b$. The maintenance time period T for all equipment in the plant is within a full year, 52 weeks. Each piece of equipment in the plant needs a number of weeks to complete the PM process.

3.2. Problem Formulation

The nomenclature for the hybrid GA and TS approach was divided into sets, data, and decision variables, as follows:

Sets:

$$\begin{array}{l}p=1 \dots P, denotes to plants\\ d=1 \dots D, denotes to distillers \\ r=1 \dots R, denotes to turbines \\ b=1 \dots B, denotes to boilers\\ Q=D\cup R\cup B, where q\in Q\\ \rho populations\\ CR set of chromosome\\ g\left(i\right)=1 \dots N, The number of genes in a chromosom\\ i,j, Denotes to genes located in the chromosome, where i,j\in g\left(i\right) \end{array}$$

(The sequence of genes in the chromosome represents the sequence of weeks in time horizon)

Data:

$$\begin{array}{l}{MD}_p^q maintenance duration time for equipment q,in plant p.\\ {PT}_{pd} maximum water production for distiller d,in plant p.\\ {PT}_{pr} maximum electricity production for turbine r,in plant p.\\ {Et}_{pq} allowed time to start maintenance for eqiupment q, in plant p.\\ {Lt}_{pq} the last time allowed for maintenance for eqiupment q, in plant p.\\ {Max}_{ipq} maximum number allowed is under maintenance for eqiupment q,\\ in plant p, at gene i.\\ D_i^r electricity demand, at gene i\\ D_i^d water demand, at gene i\\ HR the number of available manpower\\ {MP}_{pq} available manpower, for eqiupment q,in plant p.\\ D_i^r electricity demand, at gene i\\ \delta overall number of general iterations.\\ \alpha overall number of internal iterations.\\ \gamma tabu tenure, the time required for genes to remain on the Tabu list.\\ {gn}_i^{\beta } value of gene i, for parent \beta , (\beta equal 1 or 2).\end{array}$$

At t = 0, the reservior water level $l_0\in [\underset{\_}{l},\overline{l}]$, where $\underset{\_}{l} \mathrm{a}\mathrm{n}\mathrm{d} \overline{l}$ are the minimum and maximum water levels at the reservoir.

Decision variable:

There are three types of decision variables: first, binary variables:

$$x_{pqi}=\left\{\begin{array}{l}1, eqiupment q is under maintenance, at gene i.\\ 0, otherwise\end{array}\right.$$

$${xs}_{pqi}=\left\{\begin{array}{l}1, gene i is the start of maintenance of equipment q\\ 0, otherwise\end{array}\right.$$

During the boiler maintenance period, the distillers and turbines will be either under maintenance or idle until the boiler maintenance is completed. The following binary variables express this situation:

$$y_{pd}=\left\{\begin{array}{l}1, distiller d is either idle or under maintenance \\ 0, otherwise\end{array}\right.$$

$$y_{pr}=\left\{\begin{array}{l}1, turbine r is either idle or under maintenance \\ 0, otherwise\end{array}\right.$$

Second, integer variables:

$${avw}_i available water,at gene i$$

$${ave}_i available electricity, at gene i$$

3.3. Hybrid Approach

The proposed approach is implemented by hybrid GA and TS. The process begins with applying a GA, which consists of a set of strips; each strip is called a chromosome or solution, and each chromosome consists of a set of variables known as genes. The set of chromosomes is called the population.

The basis of the proposed approach is to produce $\rho$ populations over $\mathsf{\delta }$ iterations, where each population across iterations consists of a set of $CR$ chromosomes. Each chromosome represents PMS for all equipment in the plant, as illustrated in Figure 2.

Each chromosome $CR$ consists of a group of $N$ genes, which represents the sequence of weeks in a year. The value for each gene represented by ${gn}_i^{\beta },$ gene $i$, for parent $\beta$ represents one of the equipment under PM. Simply, put a chromosome consists of a sequence of genes (subsequences), which means this equipment is subject to PM, while an empty gene means no PM during this week. For example, as shown in Figure 3, ${gn}_1^{\beta }=6,$ means that equipment 6 is under maintenance in week 1, while ${gn}_i^{\beta }=2,$ means that equipment 2 is under maintenance in week i. In addition, ${MD}_p^q$, represent the maintenance duration time for equipment q, in plant p. So, for example, ${MD}_p^6=3$ means, equipment 6 needs three weeks to complete PM (from week 1 to week 3), while ${MD}_p^7=4,$ means that equipment 7 needs four weeks to complete PM, as shown in Figure 3.

Next, the fitness function score of all chromosomes in the population is determined. The degree of fitness function increases the likelihood that it will be selected for producing new offspring. The higher the fitness score, the higher the chances of reproductive selection. Before calculating the fitness function, the gap between energy demand and availability must be calculated. There are two types of gaps, as shown below:

$${gp}_i^d={pd}_i^d-D_i^d$$

(1)

$${gp}_i^r={pd}_i^r-D_i^r$$

(2)

where ${gp}_i^d$ is the gap of distillers between production of water ${pd}_i^d$ and water demand $D_i^d$, at gene $i$, (which represents week $i$ as mentioned previously). Likewise, ${gp}_i^r$ is the gap of turbines, between available production of electricity ${pd}_i^r$ and electricity demand $D_i^r$, at gene $i$.

The fitness function for each parent consists of two standard deviations; one for distiller production, ${stdev}^d,$ where $\overline{gd}$ represents the average gap between water production and demand across the time horizon, and the other for turbine production ${stdev}^r$ where $\overline{gr}$ represents the average gap between electricity production and demand across the time horizon.

$${stdev}^d=\sqrt{\frac{\sum _{i=1}^N{({gp}_i^d-\overline{gd})}^2}{N}}$$

(3)

$${stdev}^r=\sqrt{\frac{\sum _{i=1}^N{({gp}_i^r-\overline{gr})}^2}{N}}$$

(4)

These two are calculated separately, provided that the fitness value of both is the smallest on the same chromosome. Whenever the value of the fitness function is very small, this indicates that the gap between demand and available production will be approximately the same over the time horizon. This means that there is very little chance of the power or water supply being cut off. Therefore, two different parents $\beta$ will be selected, one with a smaller standard deviation for water production ${stdev}^d$, while the other with a smaller standard deviation for electricity production ${stdev}^r.$

The algorithm uses two operators that are applied to the two selected parents, crossover and mutation. The traditional method in the GA to select cut points is performed randomly, but here it is implemented in a scientific way. The method begins by selecting a non-empty gene in parent 1, such that the gap for that gene is the smallest throughout the chromosome. For example, as shown in Figure 4, the gene with value 6 (equipment number 6) is selected. Likewise, for parent 2, the gene with the same value as parent 1, which is equipment number 6. The maintenance duration of equipment q is represented by ${MD}_p^q$, where in this example, equipment 6 needs a period equal to 3 weeks to complete PM, ${MD}_p^6=3$. Therefore, all genes with the same value will be reserved for the crossover operation, the number of which is equal to ${MD}_p^q$.

In the next step, selecting the empty gene $i$ with the highest gap value for distillers, ${gp}_i^d$, for parent 1, and the same process is performed for parent 2 by selecting gene $j$, with the highest gap for boilers, ${gp}_j^r$. The next step is to choose the cut points, which is conducted by choosing a number of empty genes adjacent to gene i of parent 1, so that their number is equal to the number of ${MD}_p^q$ genes, and the same thing is performed for parent 2, as shown in Figure 4 Step 2. It will be moved either to the right or to the left, depending on the number of empty genes required next to the selected cut points. If this is not achieved, a number of empty genes on the right, equal to the number required for the crossover, are searched. If the required number is available, all non-empty genes are moved in the same order to the right until the genes next to the required number are unloaded. On the other hand, if this is not achieved, the searching process to the left will be performed in the same way. As clear in Figure 4, the number of empty genes equals 2 genes for both parents, where ${MD}_p^q=3$ for each parent, so any non-empty genes within the cut points will be moved outwards. For parent 1, genes of value 1 were moved one step to the right, while in parent 2, genes of value 3 were moved one step to the left, as shown in Figure 4 step 3. In the final step of the crossover process, the selected substring (genes with value 6) will be inserted between the cut points, producing new offspring with a new fitness value, as shown in Figure 4 step 4.

At this stage, the tabu list method is adopted, which is one of the features of TS, in which the selected genes, i and j, will enter two separate tabu list for a number of iterations, $\gamma$ tenure. This property is important and helps to diversify the search and explore new regions of the solution space. At the same time, when the value of fitness for the new offspring is the best so far, aspiration criteria are applied, and the two selected genes, $i \mathrm{a}\mathrm{n}\mathrm{d} j,$ will be released from the tabu list. This process can actually enhance the focus and intensity of the search for promising solutions in a particular area.

The mutation operator will be applied after the crossover; Figure 5 illustrates the process. The process is conducted by first randomly selecting a substring of genes for both parents, as shown in Figure 5, genes with a value of 3 in P1, and genes with a value of 1 in P2. The substring is then shifted to the left or right, depending on the position of the empty gene. If there is no empty gene next to the substring, the adjacent substrings are shifted until they reach an empty gene.

This process will continue for $\alpha$ of iterations. This means that the search for these two chromosomes has been explored in a precise and extensive manner. Next, a new population with the same number of $CR$ chromosomes is generated in the same manner as before. Different parents are selected in the same way as before, in order to search for a better solution. The algorithm will terminate after reaching the threshold fitness value. This solution will be determined as the optimal solution for the population. On the other hand, if the optimal solution is not reached, the algorithm will stop when $\delta$ of iterations are reached.

There are some conditions that must be followed during the process of searching for the solution through the proposed method, which are as follows:

$$\sum _{p=1}^P\sum _{i={Et}_{pq}}^{{Lt}_{pq}}{xs}_{pqi}=1, \forall q\in Q$$

(5)

$$\sum _{p=1}^P\sum _{i={xs}_{pdi}}^N{x}_{pqi}={MD}_p^q, \forall q\in Q$$

(6)

Condition (5) ensures that the machine undergoes PM only once during the time horizon, so that the time to start PM is within the permitted period. To guarantee the continuity of PM within the allotted time period ${MD}_p^q$, without interruption, condition (6) was implemented.

$$\sum _{q=1}^Q{MP}_{pq}·x_{pqi}\le HD, \forall p\in P, \forall i\in N$$

(7)

Constraint (7) was put in place to ensure that the number of laborers during the implementation of PM for a number of pieces of equipment does not exceed the total number of laborers available in the current week.

$$\sum _{r=1}^R{PT}_{pr}·{(1-x}_{pri})\ge D_i^r \forall i\in N$$

(8)

$$\sum _{d=1}^D{PT}_{pd}·{(1-x}_{pdi})\ge D_i^d \forall i\in N$$

(9)

$$\underset{\_}{l}\le l_0+\sum _{d=1}^D{PT}_{pd}·{(1-x}_{pdi})\le \overline{l}, \forall i\in N$$

(10)

Constraints (8) and (9) were set in order to ensure the availability of electricity and water during the current week and that there is no shortage due to PM work. On the other hand, constraint (10) guarantees that water production is within the minimum and maximum water levels at the reservoir.

$$\sum _{q=1}^Q{x}_{pqi}\le {Max}_{iq}, \forall i\in N, \forall d,r,b\in Q$$

(11)

There are technical and precautionary matters imposed by the power plant administration regarding the maximum number of pieces of equipment of the same type (turbines, boilers, and distillers) on which PM is permitted during the current week. Therefore, the constraint (11) was set to ensure that the permissible number is not exceeded.

The following flowchart in Figure 6 explains the process of generating a PMS for all equipment in a plant using the proposal approach.

4. Computational Results

4.1. General Data

Data were obtained from the MEW in the State of Kuwait, where there are several power plants. In this research, the Al-Zour plant was chosen. This plant consists of eight units, each consisting of four pieces of equipment: one boiler, one turbine, and two distillers. This means that there are a total of 32 pieces of equipment in the plant. PM must be performed annually for all equipment to avoid any malfunctions. Each boiler needs 5 weeks to complete PM, as does the distiller, while the turbine needs 4 weeks to complete PM. Since boilers are responsible for supplying distillers with hot steam for condensation and producing fresh water, as well as turbines for producing electricity, PM of any boiler results in the associated turbine and distiller being idle until PM is completed. In addition, no more than two units can stop in the same week either due to PM or be idle due to PM work on the boiler associated with them. According to the information provided to us by the Al-Zour plant administration, human resources are available in all areas of the operational planning period. The time period to start PM for all equipment is open from the first week until the end of the year.

The objective of this study is to generate a PMS for all equipment in the plant such that it maximizes the number of operating equipment throughout the time horizon. This means, in return, minimizing the occurrence of water or electricity outages for consumers. This can be achieved by making the gap between available production and demand approximately equal along the time horizon.

The hybrid GA and TS algorithms were adopted to address this problem, and MATLAB R2021a programming was used to solve the problem. Based on the DELL desktop, the CoreTM i7-1065G7 CPU was clocked at 1.50 GHz and 16.0 GHz. The time horizon in which the proposal was implemented is one year, as a year consists of 52 weeks, assuming that a month consists of 4 and 5 weeks. The reason weeks are used in this problem is because the duration of PM for any piece of equipment is calculated in weeks.

Figure 7a,b show the water and electricity demand for one year for the Al-Zour plant in the state of Kuwait, where the demand appears to increase in the summer period.

Table 1. Production capacity for all equipment in the Al-Zour plant.

Unit	Equipment	Production	Unit	Equipment	Production
1	D1	50.4 1	5	D1	50.4
	D2	50.4		D2	50.4
	R	47,040 2		R	47,040
2	D1	50.4	6	D1	50.4
	D2	50.4		D2	50.4
	R	47,040		R	47,040
3	D1	50.4	7	D1	40.2
	D2	50.4		D2	40.2
	R	47,040		R	47,040
4	D1	50.4	8	D1	40.2
	D2	50.4		D2	40.2
	R	47,040		R	47,040

¹ MIGD ² MW.

. illustrates the production capacity for each piece of equipment for all units in the Al-Zour plant. D1 and D2 represent distillers 1 and 2, while R represents the turbine. On the other hand, MIGD represents a million imperial gallons per day, while MW represents a million watts.

The aim of this study is to generate a reliable PMS within a reasonable CPU time such that all gaps between demand and available production in all time periods are very close throughout the year and, simultaneously, all constraints are satisfied. To achieve this goal, the standard deviations ${StDev}^d$ and ${StDev}^r$, of the gap between demand and available production for water and electricity were adopted, provided that the method presented in this paper searches for the minimum value of the standard deviations. This ensures that there are no power outages or water shortages throughout the year.

4.2. Model Validation and Analysis: Test 1

Many experiments were applied to find the optimal or near-optimal solution through the following factors: the number of chromosomes in each population CR, general iteration $\delta$, internal iteration $\alpha$, method of gene selection, and the value of the tenure $\gamma$ of Tabu list. It is important to mention before starting with the details that the results of the proposal approach have been compared with the two published articles, Refs [29,30] that used the optimal method, IP and NLIP, for the same case.

Initially, the CR population size underwent a gradual increase from 50 to 100 parents. Through multiple experiments, it became evident that maintaining a population size of 100 yielded an optimal solution. Moreover, surpassing the threshold of 100 parents did not result in any further enhancements to the solution. In the general iteration, setting δ to 10,000 facilitated reaching the optimal solution more quickly for all experiments, with all cases achieving the desired fitness threshold. Concerning internal iteration α, a total of 50 iterations were executed, with the optimal value of α determined to be the most effective. Beginning at 10, α progressively increased by increments of 5 units until reaching 100, with results plateauing after 50 iterations. On the other hand, with the aim of applying crossover and mutation, two methods were assessed: random selection and scientific selection. Scientific selection, based on identifying genes with the largest discrepancy between demand and available production, outperformed random selection significantly, expediting the attainment of the optimal solution. Additionally, the Tabu list played a pivotal role in ensuring the success of gene selection by averting the recurrence of cyclic problems and broadening the search across various solution areas. Its presence guaranteed the achievement of the optimal solution, a feat unattainable without its utilization. The optimal tenure value was identified as γ = 4, with changes occurring within the specified period [2,3,4,5,6,7,8].

Table 2. Inputs required for the proposed method.

Symbol	$\mathit{\delta }$	$\mathit{\alpha }$	$\mathit{\gamma }$	CR
iteration	10,000	50	4	100

summarizes all the values mentioned previously.

Statistical measures, including the mean, median, standard deviation, minimum, and maximum, were calculated based on 100 iterations (CR) to generate a population, followed by an additional 50 iterations (α) to apply crossover and mutation operators, as illustrated in

Table 3. Computational Time Statistics (Second).

	Population	Crossover	Mutation
Mean	0.001222	0.000263	0.01007
Median	0.000397	0.000199	0.01046
St. Dev	0.003006	0.000179	0.00023
Min.	0.000153	0.000176	0.00348
Max.	0.026198	0.001015	0.01335

. Population and crossover analysis correctly identify the relationship between the mean and median as evidence of the skewness of the distribution. This insight provides valuable context for understanding computational time distributions. The interpretation of the higher mean compared to the median indicates that some experiments experience longer computational times, likely due to conditions affecting chromosome generation or crossovers. This context adds depth to the interpretation of the data. The observation of large standard deviations across experiments also supports the idea of variability in computational times. This variability is due to the conditions imposed in chromosome generation and crossover, which can affect the feasibility and efficiency of generating solutions. The analysis effectively links the presence of relatively low and high computation times to a variety of performance results, highlighting the diversity of results observed in the experiments. On the other hand, mutation analysis correctly identifies similarity between the mean and median in mutation methods as evidence of similarity, as shown in Table 3. This view contrasts with the skewed distributions observed in population and crossover operators. The observation of moderate variation in computational times across experiments, as indicated by the standard deviation, indicates consistent performance in the mutation methods. The absence of outliers after the time limit indicates stable and predictable behavior in terms of the computational time of the mutation methods. This stability is a desirable feature of algorithm performance.

The proposed method was applied, and the optimal solution was obtained compared to the two IP and NLIP [29,30] for the same problem. The value of ${StDev}^d$ is equal to 59.6 MIGD, and the value of ${StDev}^r$ is equal to 33,519 MW, as shown in

Table 4. Proposed approach and MEW results analysis.

		Proposal Approach	MEW
Distiller	St. Dev.	59.6	70.724
	Average	235.32	235.32
	Min. Gap	118.1	101.3
Turbine	St. Dev	33,519	33,220.5
	Average	175,330	171,712
	Min. Gap	124,426	110,971

. These results were obtained in a computational time of 20 s. Regarding the distribution of equipment,

Table 5. PMS generated by the proposed model and by MEW.

Week	Proposed Model		MEW
	Equipment under PM	Idle	Equipment under PM	Idle
1	B-2, D1-2, D2-2, R-2	-	B-4, D1-4, D2-4	R-4
2	B-2, D1-2, D2-2, R-2	-	B-4, D1-4, D2-4, R-4	-
3	B-2, D1-2, D2-2, R-2	-	B-4, D1-4, D2-4, R-4	-
4	B-2, D1-2, D2-2, R-2	-	B-4, D1-4, D2-4, R-4	-
5	B-2, D1-2, D2-2	R-2	B-4, D1-4, D2-4, R-4	-
6	B-5, D1-5, D2-5, R-5	-	B-3, D1-3, D2-3, R-3	-
7	B-5, D1-5, D2-5, R-5	-	B-3, D1-3, D2-3, R-3	-
8	B-5, D1-5, D2-5, R-5	-	B-3, D1-3, D2-3, R-3	-
9	B-5, D1-5, D2-5, R-5	-	B-3, D1-3, D2-3, R-3	-
10	B-5, D1-5, D2-5	R-5	B-3, D1-3, D2-3	R-3
11	B-3, D1-3, D2-3	R-3	-	-
12	B-3, D1-3, D2-3, R-3	-	B-6, D1-6, D2-6, R-6	-
13	B-3, D1-3, D2-3, R-3	-	B-6, D1-6, D2-6, R-6	-
14	B-3, D1-3, D2-3, R-3	-	B-6, D1-6, D2-6, R-6	-
15	B-3, D1-3, D2-3, R-3	-	B-6, D1-6, D2-6, R-6	-
16	B-6, D1-6, D2-6, R-6	-	B-6, D1-6, D2-6	R-6
17	B-6, D1-6, D2-6, R-6	-	B-5, D1-5, D2-5, R-5	-
18	B-6, D1-6, D2-6, R-6	-	B-5, D1-5, D2-5, R-5	-
19	B-6, D1-6, D2-6, R-6	-	B-5, D1-5, D2-5, R-5	-
20	B-6, D1-6, D2-6	R-6	B-5, D1-5, D2-5, R-5	-
21	-	-	B-5, D1-5, D2-5	R-5
22	-	-	-	-
23	-	-	-	-
24	-	-	-	-
25	-	-	-	-
26	-	-	-	-
27	-	-	-	-
28	-	-	-	-
29	-	-	-	-
30	-	-	-	-
31	-	-	-	-
32	-	-	B-7, D1-7, D2-7	R-7
33	B-7, D1-7, D2-7	R-7	B-7, D1-7, D2-7	R-7
34	B-7, D1-7, D2-7, R-7	-	B-7, D1-7, D2-7	R-7
35	B-7, D1-7, D2-7, R-7	-	B-7, D1-7, D2-7	R-7
36	B-7, D1-7, D2-7, R-7	-	B-7, D1-7, D2-7	R-7
37	B-7, D1-7, D2-7, R-7	-	B-2, D1-2, D2-2	R-2
38	B-8, D1-8, D2-8, R-8	-	B-2, D1-2, D2-2, R-2	-
39	B-8, D1-8, D2-8, R-8	-	B-2, D1-2, D2-2, R-2	-
40	B-8, D1-8, D2-8, R-8	-	B-2, D1-2, D2-2, R-2	-
41	B-8, D1-8, D2-8, R-8	-	B-2, D1-2, D2-2, R-2	-
42	B-8, D1-8, D2-8	R-8	B-1, D1-1, D2-1, R-1	-
43	B-4, D1-4, D2-4, R-4	-	B-1, D1-1, D2-1, R-1	-
44	B-4, D1-4, D2-4, R-4	-	B-1, D1-1, D2-1, R-1, R-7	-
45	B-4, D1-4, D2-4, R-4	-	B-1, D1-1, D2-1, R-1, R-7	-
46	B-4, D1-4, D2-4, R-4	-	B-1, D1-1, D2-1, R-7	R-1
47	B-4, D1-4, D2-4	R-4	R-7	-
48	B-1, D1-1, D2-1, R-1	-	B-8, D1-8, D2-8, R-8	-
49	B-1, D1-1, D2-1, R-1	-	B-8, D1-8, D2-8, R-8	-
50	B-1, D1-1, D2-1, R-1	-	B-8, D1-8, D2-8, R-8	-
51	B-1, D1-1, D2-1, R-1	-	B-8, D1-8, D2-8, R-8	-
52	B-1, D1-1, D2-1	R-1	B-8, D1-8, D2-8	R-8

displays the PMS for all equipment within one year (52 weeks) for both the proposed model and MEW. First, with regard to turbines, for the Ministry’s PMS, the number of weeks in which they were shut down for PM operation or were idle due to boiler PM exceeds the proposed PMS approach, as they were idling in weeks 32, 33, 34, 35, and 36, as shown in Table 5. Therefore, the average in the Ministry’s PMS is less than the proposed PMS average, which resulted in a change in the standard deviation ${StDev}^d$. On the other hand, the minimum gap during the year in the Ministry’s project management system is equal to 101.3 MIGD, while in the proposed approach it is equal to 118.1 MIGD, and this means that there is a difference equivalent to 16.8 MIGD. As for electricity, the minimum gap during the year in the Ministry’s PMS is equal to 110,971 MW, while in the proposed approach it is equal to 124,426 MW, which means there is a difference equivalent to 13,455 MW, as shown in Table 4. This indicates that the surplus production of water and electricity through the PMS method presented in this paper exceeded the experts of the MEW by 16.58% and 12.12%, respectively. It should also be emphasized that any increase in surplus production means an increase in the ability to meet demand. Furthermore, the MEW performance management system has not been settled, with service outages during week 11 and only one piece of equipment undergoing maintenance in week 47. This is undesirable as it leads to unnecessary manpower costs at a time of low peak demand. In contrast, the proposed method for PMS avoids this unnecessary overhead.

In addition to the above, all constraints have been satisfied, as shown in Table 5, where Table 5 shows the distribution of all equipment under maintenance over a period of 52 weeks. For example, no more than two pieces of equipment of a given type are under PM during any of the 52 weeks of the planning horizon. Also, when their boiler are under PM, neither the distiller nor the turbine is working.

Figure 8a,b demonstrate the distribution of equipment for PM through the proposed approach, and Figure 9a,b illustrate the schedule generated by the Ministry’s administration. They display production, demand, and the number of units under maintenance. They show that the number of distillers and turbines undergoing maintenance has become constant. This keeps water and electricity production constant for the first 20 weeks. Production reaches its peak and maintenance work stops when demand increases during the summer period during weeks 21 to 32, while maintenance work resumes at the end of the summer period due to the steady decline in demand. In contrast, the production surplus is greater during the winter, ensuring a large reserve level and avoiding water shortages and power outages. It is equivalent to 118.1 MIGD for water and 124.426 MW for electricity. The total production amounts to 36,321.6 MIGD of water and 17,687,040 MW of electricity.

A comparative analysis was conducted between the GA algorithm proposed in [26] and the hybrid approach in this paper, focusing on the same PMS problem but for a different plant located in Kuwait.

Table 6. Performance Comparison of Hybrid and Genetic Algorithms (GA) for PMS.

Approach	Percent Optimality Achieved
	Water	Elect.	Time (s)
Proposal Approach	0%	0%	20
GA	2.5%	8.5%	185

shows the outcomes; the solution derived from the GA algorithm closely approximated the optimal solution, with a discrepancy of 2.5% for water and 8.5% for electricity, achieved within a computational time of 185 s; this represents a total of 78 min for water and 107 min for electricity. In contrast, the hybrid approach attained the optimal solution remarkably faster, accomplishing it within a mere 20 s. It was obvious that the hybrid approach had reached the threshold fitness value, which prompted the algorithm to stop searching and accept the solution as optimal. This early convergence indicates that the hybrid approach efficiently explored the solution space and identified the optimal solution without exhausting all available computational resources. Early convergence implies that the algorithm efficiently identified promising regions of the solution space and focused its search efforts there. This efficient exploration and exploitation of the solution space contribute to the algorithm’s overall computational efficiency. The combination of GA and TS likely played a crucial role in achieving early convergence. TS, with its ability to intensify the search around promising solutions while avoiding revisiting previous solutions, may have expedited the process of reaching the optimal solution. By reaching the optimal solution sooner, the hybrid approach can facilitate timely decision-making in PMS for cogeneration plants.

4.3. Model Validation and Analysis: Test 2

To measure the robustness and efficiency of the hybrid method proposed in this research, 30 experiments were conducted, through which to measure the effectiveness of the proposed tool through the quality of the solutions and the computational time to reach the solution. These experiments were divided into two parts. First, the demand for water was stabilized, and the demand for electricity was gradually increased by 10% until the highest possible increase rate was reached. After that, increase the demand for water by 5% (105%), then stabilize it and gradually increase the demand for electricity by 10%, as was conducted in the previous step. The highest increase in demand for water was 120%, while the highest increase in demand for electricity was 150%, as illustrated in

Table 7. Water and Electricity surplus production versus increased demand.

Increased Demand		Min. Gap		Time in Second
Water	Electricity	Distiller	Turbine	IP	NLIP	Hybrid
100%	100%	118.1	124,426	157	236	20
	110%	118.1	103,941	234	225	99
	120%	118.1	83,455.2	256	270	34
	130%	118.1	62,969.8	69	256	72
	140%	118.1	42,484.4	248	392	61
	150%	118.1	21,999	181	411	34
105%	100%	89.57	124,426	192	642	59
	110%	89.57	103,941	228	475	48
	120%	89.57	83,455.2	183	327	81
	130%	89.57	62,969.8	349	353	94
	140%	89.57	42,484.4	125	246	21
	150%	89.57	21,999	153	306	45
110%	100%	61.03	124,426	315	212	92
	110%	61.03	103,941	132	301	93
	120%	61.03	83,455.2	383	402	49
	130%	61.03	62,969.8	202	343	200
	140%	61.03	42,484.4	110	352	67
	150%	61.03	21,999	24	447	48
115%	100%	32.5	124,426	199	589	52
	110%	32.5	103,941	165	253	34
	120%	32.5	83,455.2	133	348	103
	130%	32.5	62,969.8	220	245	72
	140%	32.5	42,484.4	137	351	34
	150%	32.5	21,999	134	288	29
120%	100%	3.96	124,426	196	326	82
	110%	3.96	103,941	206	327	39
	120%	3.96	83,455.2	204	180	78
	130%	3.96	62,969.8	254	281	29
	140%	3.96	42,484.4	147	240	47
	150%	3.96	21,999	101	129	45
Average				187.9	325.1	62.03
St. Dev.				77.28	110.74	35.71
S/N Ratio				46.14	50.7	37.1

St. Dev. stands to Standard Deviation.

. First, all results were optimal compared to the results of the IP and NLIP optimal methods [29,30]. Additionally, the computational time to reach the optimal solution is shorter in most experiments. Also, in Figure 10, the curve shows how little computational time it takes to reach the optimal solution compared to other methods. The hybrid method showed the least variance among the other methods for increasing the demand for water and electricity. The standard deviation is 35.7 s, the average time taken is 62 s, and the measurements range from 20 to 200 s, as shown in Table 7. This method demonstrates relatively stable performance across experiments, with minimal fluctuations in the time taken. On the other hand, the IP method shows a moderate variance, with an average time taken of 187.9 s and a standard deviation of 77.3 s, with measurements ranging from 24 to 383 s, and the NLIP shows a higher variance compared to the hybrid and IP methods, with a larger standard deviation of 110.7 s, with an average time taken of 325 s, and measurements ranging from 129 to 642 s. This suggests inconsistencies in the performance of IP and NLIP across experiments. Overall, the hybrid approach exhibited the lowest average time and standard deviation, indicating consistent performance with less variability compared to the IP and NLIP methods.

Additionally, Taguchi design was applied, calculating the (S/N) ratios, as shown in Table 7, involving the evaluation of the three methods (IP, NLIP, and Hybrid) across multiple experiments to determine the most effective approach for minimizing time. The lowest S/N ratio among the three methods, the Hybrid method, exhibits the best performance in terms of computational time according to the “Smaller the Better” characteristic, the Hybrid method demonstrates the most efficient computational time, as indicated by its lowest S/N ratio. The IP method follows with a slightly higher S/N ratio, suggesting slightly longer computational times compared to the Hybrid method but still better than NLIP. The NLIP method exhibits the highest S/N ratio, indicating the longest computational time among the three methods. Based on the analysis of both cases, it is apparent that the Hybrid method consistently surpassed the IP and NLIP methods in terms of minimizing time. Therefore, it was concluded that the Hybrid method may be the most effective approach for achieving optimal time performance in these experiments.

In summary, the hybrid approach consistently outperforms the IP and NLIP approaches on the issue of computational time to reach the optimal solution, showing a trend for faster optimization times. Additionally, comparing the proposed hybrid approach with IP and NLIP methods and consistently achieving optimal solutions in 16 experiments provides strong evidence of its effectiveness and reliability in solving optimization problems. Consistently obtaining optimal solutions also reflects the robustness of the proposal approach. Robust algorithms are less sensitive to variations in problem parameters and can maintain their performance across different scenarios. Achieving optimal solutions across a variety of experiments demonstrates the generalizability of the hybrid algorithm. It indicates that the algorithm’s performance is not limited to specific scenarios but extends to a broader range of problem instances and conditions.

Moreover, optimal methods have great difficulty solving large-scale problems and may sometimes be impossible, while this is not the case for heuristic methods, such as the method proposed in this paper. This helps the user in scheduling management in any power plant to use this proposed method to create an optimal PMS for their plants in a short computational time. This is beneficial for saving money and time.

5. Conclusions

This article explores the intricate domain of PMS within cogeneration plants engaged in electricity generation and water production. The multifaceted nature of these plants, serving dual purposes, poses significant challenges in scheduling maintenance activities while ensuring seamless production operations. The objective function of this research is to devise an optimal PMS that minimizes downtime, enhances production efficiency, and reduces maintenance costs. Employing a hybrid tool comprising metaheuristic methods such as Genetic Algorithms (GA) and Tabu Search (TS), a number of constraints are taken into consideration, such as maintenance window constraints, maintenance duration for each machine, maximum tank water level, number of available laborers, etc. Schedules were created that ensured the reliability and longevity of plant operations while meeting production goals. This approach yields superior results compared to previous studies utilizing deterministic optimization tools, IP and NLIP, reaching optimal solutions efficiently.

To assess the robustness and efficiency of the proposed hybrid method, 16 experiments were conducted, meticulously evaluating solution quality and computational time. The analysis reveals that the proposed approach consistently achieves optimal solutions in significantly less time compared to alternative methods.

In conclusion, this research introduces a novel and robust approach to PMS within cogeneration plants, leveraging hybrid metaheuristic methods to optimize maintenance scheduling outcomes. The Hybrid method emerges as a promising avenue for enhancing operational efficiency and sustainability in cogeneration plant operations. Future research in this area could explore the integration of additional heuristic methods such as Simulated Annealing (SA), Ant Colony Optimization (ACO), and Cuckoo Search (CS) to further enrich PMS practices.