This article presents a methodology to optimize the maintenance planning model and minimize the total maintenance costs of a typical school building. It makes an effort to provide a maintenance schedule, focusing on maintenance costs. In the allocation of operations to the school equipment, the parameter of its age was also taken into account. A mathematical optimization model to minimize the school maintenance cost in a three-year period was provided in the GAMS software with CPLEX solver. Finally, the optimum architecture of the Perceptron multi-layer neural network was used to predict the schedule of equipment operations and maintenance costs. The Multi-layer Perceptron (MLP) optimum neural network results, with minor Mean Squared Error (MSE) and Root Mean Squared Error (RMSE), indicated that the proposed model was capable of predicting the schools’ maintenance costs with high accuracy. According to the results, the school's maintenance cost for the intended three-year period based on the Weibull distribution was equal to 15361 currency units per hour, in which the “heating and cooling system” has the highest contribution. Hence, accurate and definite planning can prevent damages to such equipment, while saving the school's maintenance costs.

Maintenance of educational building assets is an important tool not only for the wellness of students and other users, but also for school life cycle maximization and minimization of maintenance costs [

The integrated production and maintenance planning models have been seriously studied since the early 80 s. Over time, integrated models have been separated according to their maintenance methods (predictive/corrective and periodic/non-periodic). The non-periodic maintenance in integrated models has been less studied since, in such conditions, the solution is tough, and usually, the branch and bound method is used to solve it. For a better understanding of the required concepts of maintenance in the next sections, a summary for each of them is provided in this section. In general, the maintenance and repair can be defined as the following: A combination of activities conducted particularly and in a usual planned way to prevent the sudden failure of machinery, equipment, and facilities is called maintenance [

The maintenance has been created from several important classes of decision-making: 1) strategic concept and long-term maintenance, 2) medium-term planning, 3) short-term planning, and 4) control and performance indexes [

Complete inspection (faultless inspection for all systems)

Partial inspection (only error type 1)

Incomplete inspection (only detection of error type 2) that may also provide false positive results for other types of failures.

He et al. [

Determining whether the system requires preventive or corrective maintenance.

Determining the optimum inspection time until the next inspection

In another study, Fouladirad et al. [

The current paper however, proposes a mathematical model for schools’ maintenance planning to minimize the total maintenance cost. For this purpose, after providing the mathematical model, the essential indexes and equipment of a sample school are used along with a neural network to provide cost prediction for a three-year planning horizon. The main novelties of this work are presenting an optimum maintenance plan for the school buildings with the minimum cost, future cost prediction ability and consideration of the age of school's equipment. To the best of our knowledge, this is the first time that these issues are addressed.

The considered domain in this paper is a school with the equipment of cooling and heating systems, educational and laboratory equipment, and lavatories. In order to conduct the maintenance and repair of the equipment, a schedule for the maintenance and a schedule for the preventive repair of equipment are required. The ages of equipment (total working hours of each equipment) are different, but all known. The maintenance operations can be planned for various pieces of equipment with different ages and costs. It is clear that their maintenance costs have a direct relationship with their ages. This research work proposes a mathematical planning model to provide an equipment maintenance schedule while minimizing the total maintenance costs. In fact, the maintenance operations schedule in the three-year period which yields to the minimum cost is obtained via an optimization algorithm. Then, based on the available data for the three-year period, the values of cost and maintenance operations schedule for the fourth and fifth years in the intended school are estimated using the artificial neural network.

In this section, the problem assumptions, model components, and the main model are expressed.

This paper employs the two policies of emergency (unplanned) maintenance and preventive maintenance.

The equipment failure rate is considered constant based on the Weibull distribution function.

The costs of unplanned (emergency) maintenance are more than those of preventive maintenance.

The planning is provided in the framework of a particular time horizon. (The problem is divided into n periods.)

The age ranges are considered constant (each equipment piece's age is classified with respect to the ranges).

It is assumed that the age density of the equipment is in the form of Weibull function as follows:

^{2}

_{ p} (i,b,y)

_{ f} (i,b,y)

_{ f}

_{b}

_{f}: Period between two emergency failures

_{c}

The objective function is determined according to the criterion of equipment maintenance in unit time. The cost is always an important and effective factor in selecting the maintenance policies in organizations. While investigating maintenance policies, most maintenance researchers and engineers have sought a policy to minimize it. Therefore, it has always been a determining factor in organizations. ^{th}

The numerator of the fraction in

The model constraints are as the following:

In the equations above _{max}, b__{max}, y__{max},

The objective function minimizes all cycles’ maintenance cost for the favorable function of equipment and optimum maintenance intervals. The nominator of the fraction in the objective function is the emergency and preventive maintenance cost for their occurrence probability, and the denominator is equal to the expected length of each cycle. Constraint

In this section, in order to evaluate the model and determine the value of the objective function, coding in the GAMS software on a system with a Core i5 processor and 8 GB Ram was conducted. The used solver was the CPLEX algorithm. The properties of the studied school are listed in

Number of equipment pieces | 3 | |

Planning year | 3 | |

Age range | 20 (1000 to 5000), (5000 to 10000), … | |

Emergency or corrective maintenance cost | _{f} (i, b, y) |
750 |

Preventive maintenance cost | _{p} (i, b, y) |
40 |

Available time | 600 | |

The time required for repair | _{y} |
17500 |

Mean time required for preventive maintenance | _{p} |
75 |

Mean time required for repair or exchange of failure | _{f} |
240 |

The period between two emergency failures | _{f} |
80 |

Equipment No. | Equipment name | Scale parameter | Shape parameter |
---|---|---|---|

1 | Lavatory equipment | 1500 | 2 |

2 | Kitchen or pantry equipment | 2000 | 2 |

3 | Laboratory equipment | 2500 | 2 |

4 | Multi-purpose hall equipment | 2000 | 1 |

5 | Library equipment | 1500 | 1 |

6 | Cooling and heating equipment | 1500 | 2 |

7 | Educational and complementary equipment | 2000 | 1 |

8 | Equipment of doors, windows, staircases, and fences | 1500 | 2 |

L values | ||||||||
---|---|---|---|---|---|---|---|---|

Equipment no. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

396.124 | 289.16 | 285.146 | 256.41 | 354.23 | 190.184 | 215.11 | 250.251 |

Using the data given in the previous sections, results are provided in two subsections. Firstly, using the GAMS software, the optimal value of the objective function, i.e., the minimum cost of the school maintenance operations schedule in the three-year period (

Equipment | Age range No. | Year | ||
---|---|---|---|---|

The first year | The second year | The third year | ||

Lavatory | 10 | 695.15 | 1000.24 | |

WC | 11 | 1628.13 | ||

Kitchen or pantry | 10 | 2663.21 | 1800.45 | |

Kitchen or pantry | 11 | 1500 | ||

Laboratory | 11 | 2680.7 | 1560.5 | 450.23 |

Laboratory | 12 | 1800.21 | ||

Multipurpose hall | 10 | 2800.24 | ||

Multipurpose hall | 11 | 1750.5 | 750.325 | |

Library | 11 | 3250.5 | ||

Library | 12 | 1800.57 | ||

Cooling and heating | 10 | 3500 | 1860.15 | |

Cooling and heating | 11 | 4120.3 | ||

Educational and complementary equipment | 10 | 2100 | 560.24 | |

Educational and complementary equipment | 11 | 2880 | ||

Doors, windows, staircases, and fences | 11 | 480.63 | 185.12 | |

Doors, windows, staircases, and fences | 12 | 774.21 |

According to the schedule suggested for the equipment, the school's maintenance cost for the intended three-year period is equal to 15361 currency units per hour.

Before using the neural network, its different learning functions were used to determine which function had the best performance. The table below lists the correlation results of each of the learning functions. In order to determine the performance of the best learning function, the number of hidden layers and the number of neurons were both considered 10. It is worth mentioning that in this step, 70% of the data were used in the learning stage, 15% were used in the testing stage, and another 15% were used in the verification (assessment) stage. Accordingly, the best correlation values for each learning function were determined, based on which the correlation of values could be determined separately for each test. The results of the different functions are given in

Number | Function type | Function definition | Correlation coefficient |
---|---|---|---|

1 | trainlm | Levenberg-Marquardt | 0.9752 |

2 | trainbr | Bayesian regularization | 0.8869 |

3 | Trainbfg | BFGS quasi-Newton | 0.8136 |

4 | traincgb | Conjugate gradient backpropagation with Powell-Beale restarts | 0.9248 |

5 | traincgp | Conjugate gradient backpropagation with Polak-Ribiére updates | 0.8947 |

6 | traingda | Gradient descent with adaptive learning rate | 0.9152 |

7 | traingdm | Gradient descent with momentum | 0.8469 |

8 | traingdx | Gradient descent with momentum and adaptive learning rate | 0.9096 |

9 | trainoss | One-step secant | 0.9156 |

10 | trainscg | Scaled conjugate gradient | 0.8869 |

As can be seen, among the functions above, the function trainlm had the highest correlation. Therefore, it was used for data learning. In the next step, the network model was evaluated and developed based on the number of layers and neurons. The two commonly used types of the neural network, i.e., the Multi-layer Perceptron (MLP) and Radial Basis Function (RBF), were employed to assess the neural network. The best architectures of the neural networks MLP and RBF are provided in

Architecture | Performance | Correlation coefficient | |
---|---|---|---|

Number of layers | Number of neurons | ||

10 | 10 | 0.0017 | 0.9635 |

10 | 20 | 0.0014 | 0.9817 |

20 | 10 | 0.0022 | 0.9633 |

15 | 15 | 0.0036 | 0.9379 |

5 | 5 | 0.0032 | 0.9689 |

8 | 8 | 0.0063 | 0.9126 |

10 | 5 | 0.0016 | 0.9633 |

5 | 10 | 0.0048 | 0.8947 |

12 | 8 | 0.0035 | 0.8649 |

10 | 15 | 0.0047 | 0.8983 |

9 | 25 | 0.0039 | 0.9639 |

Architecture | Correlation coefficient | ||
---|---|---|---|

SPREAD | Number of neurons | Number of DFs | |

1 | 5 | 1 | 0.47141 |

1 | 10 | 25 | 0.72172 |

1.5 | 10 | 25 | 0.12047 |

1 | 10 | 20 | 0.72172 |

1 | 15 | 25 | 0.74847 |

1 | 20 | 25 | 0.89835 |

1 | 25 | 25 | 0.93657 |

1 | 30 | 25 | 0.77259 |

1 | 27 | 25 | 0.91482 |

1 | 26 | 25 | 0.9517 |

The number of layers and neurons is of crucial importance in the MLP neural network so that an increase or decrease in them can influence the network performance.

In the RBF neural network, the most important parameters affecting the network performance are the SPREAD, number of neurons, and the number of neurons located between the displays (DF). Thus, before modeling each of the membranes using the RBF neural network, it is essential to determine the favorable number of neurons and DFs for modelling, as shown in

As can be seen, the MLP neural network's optimum architecture in predicting the operations schedule and maintenance cost was the second one with 10 layers and 20 neurons. Furthermore, the RBF neural network's optimum architecture was the tenth architecture with a SPREAD of 1, 26 neurons, and 25 DFs. Therefore, the model was evaluated based on the input data and the learning function of trainlm with the optimum architecture of both methods. For this purpose, the 70% learning, 15% testing, and 15% assessment data were evaluated.

Model | Maintenance cost estimation | |
---|---|---|

MSE | RMSE | |

MLP | 1.0869 × 10^{(−12)} |
1.0325 × 10^{(−6)} |

RBF | 2.58215 × 10^{(−8)} |
1.6069 × 10^{(−4)} |

Finally, the operations schedule and maintenance cost of the school equipment were predicted with the MLP neural network model (

Equipment | Age range No. | Year | |
---|---|---|---|

The fourth year | The fifth year | ||

Lavatory | 10 | 452.21 | |

Lavatory | 11 | 256.14 | |

Kitchen or pantry | 10 | 250 | |

Kitchen or pantry | 11 | 478.263 | |

Laboratory | 11 | 169.15 | 50.23 |

Laboratory | 12 | 750.682 | |

Multipurpose hall | 10 | 1500 | |

Multipurpose hall | 11 | 896.65 | |

Library | 11 | 1860 | |

Library | 12 | 468.5 | |

Cooling and heating | 10 | 450 | |

Cooling and heating | 11 | 2500.54 | |

Educational and complementary equipment | 10 | 536 | |

Educational and complementary equipment | 11 | 2000.45 | |

Doors, windows, staircases, and fences | 11 | 100.25 | |

Doors, windows, staircases, and fences | 12 | 600.59 |

As can be seen in the table above, the maintenance cost increases with the operations schedule. Accordingly, considering the previous three-year cost, the maintenance cost for a five-year period for the intended equipment in

A mathematical optimization model for the maintenance planning of schools is presented. For this purpose, a model based on the optimum preventive and corrective maintenance was designed for a sample school to reduce the school equipment's maintenance and repair costs. The results of the sample school maintenance plan were evaluated in two sections. In the first section, by modelling in the GAMS software and using the CPLEX solver, the operations schedule and maintenance cost for three different types of equipment in the school and a three-year period were investigated. In the next step, the two famous neural networks, i.e., MLP and RBF, were employed with their optimum architectures. It was found that the MLP neural network with a MSE error of 1.0869 × 10^{(−12)} and RMSE error rate of 1.0325 × 10^{(−6)} had better performance compared to the RBF neural network. Therefore, the fourth- and fifth-years’ operations schedule and maintenance costs were predicted using this neural network. Using this method and considering the previous three-year cost, the equipment maintenance cost for the desired five-year period was obtained 19853 currency units per hour. Hence, it can be claimed that the suggested model is capable of predicting the schools’ equipment maintenance costs with high accuracy. It was also found that among entire equipment pieces, the heating and cooling system and the laboratory equipment had higher maintenance costs, equal to almost 35% of the school's total maintenance costs. Hence, accurate and definite planning can prevent such equipment damages while saving the school's maintenance costs. This work has been conducted only for a specific period of the building's life time, so similar surveys at different time points are recommended. The assessment of the optimization results is possible through implementation of the obtained maintenance plan in practice and evaluate the costs at the end of the program. The same methodology can be used for modelling, optimization and predicting the maintenance cost of other buildings, such as residential and commercial ones to reduce their maintenance costs and increase their reliability. The technology of real-time streaming through internet of things (IOT) can be implemented for predictive maintenance in schools’ buildings and equipment. They make the scope of our future researches.