Opportunity-Maintenance-Based Scheduling Optimization for Ship-Loading Operation Systems in Coal Export Terminals

Abstract

As important nodes of the global coal supply chain, coal export terminals bear the tasks of coal storage, processing, and handling, whose efficiency and stability are of great importance with the growing coal shipping market in recent years. However, poor working conditions of the handling equipment in the coal export terminal, together with its relatively fixed layout and poor flexibility, allow frequent equipment failures to seriously affect the ship-loading operations. To solve the problem, this paper constructs a scheduling optimization model for ship-loading operation systems considering equipment maintenance and proposes an opportunity-maintenance-based two-layer algorithm to solve the model. The upper layer aims to optimize the scheduling scheme of the ship-loading operation system under a certain maintenance plan. The lower layer of the algorithm, an opportunity-maintenance-based “equipment-level–flow-level” maintenance optimization method, determines the best equipment maintenance plan. A coal export terminal in China is employed as the case study to verify the effectiveness of the proposed method. The results show that the proposed method can reduce the average dwell time of ships at the terminal by 15.8% and save total scheduling and maintenance costs by 10.3%. This paper shows how to make full use of equipment failure historical data and integrate equipment maintenance schemes into the scheduling problem of the ship-loading operation system, which can effectively reduce the impact of equipment failures on ship-loading operations and provide decision support for coal export terminal management.

1. Introduction

With the continuous increase in demand for coal and iron ore, the global dry bulk shipping market is developing strongly. According to the market research forecast, the global dry bulk shipping market will grow at a compound annual rate of 4% from 2022 to 2030, which will reach 547.971 billion U.S. dollars by 2030. As an important energy and strategic material, coal is mostly transported by sea. Coal ports, which are an important node of the coal supply chain, bear the important tasks of coal storage, processing, and handling. Therefore, ensuring the handling efficiency of coal ports is crucial to the stability of the coal supply chain.

At present, the working conditions of coal terminals are relatively poor, and the handling equipment often runs continuously under high load and high power, leading to frequent failures. According to statistics, at China’s main coal export port Huanghua Port, equipment downtime due to equipment failures and other reasons accounts for 20.5% of the total handling operations duration. Meanwhile, compared with other types of terminals (e.g., container terminals and general cargo terminals), horizontal transportation of coal terminals is completed by belt conveyors and other continuous operating equipment, whose layout is relatively fixed, with a strong correlation between handling equipment and poor flexibility. The failure of the handling equipment will lead to the stagnation of the terminal’s entire handling operation.

Furthermore, compared with the coal import terminals, the upstream industry of coal export terminals is usually more concentrated, and the goods are often loaded at several specific terminals and then transported to the destination port by water. As a result, the coal export terminals have a higher degree of specialization and larger transportation volume, lending the handling stagnation a greater impact on the entire coal supply chain. Therefore, putting forward a reasonable maintenance method of handling equipment to reduce the ship waiting time caused by the maintenance of handling equipment is of great significance to improve the handling efficiency of coal export terminals and the stability of the coal supply chain.

This paper aims to solve the scheduling optimization problem of ship-loading operation systems in coal terminals considering handling equipment maintenance, and a two-layer solution algorithm is proposed to support managers in making reasonable scheduling and maintenance decisions. The main contributions are as follows:

First of all, to the best of our knowledge, this is the first work to consider the impact of equipment failure and maintenance when addressing coal terminal handling operations. The interaction between scheduling and maintenance of handling equipment is further elucidated by considering the constraints of complex decision-making caused by equipment maintenance.

Secondly, an opportunity maintenance (OM)-based algorithm is developed to optimize the ship operation sequence, handling operation sequence, and equipment maintenance plan at the same time. Specifically, the lower layer of the algorithm makes full use of equipment failure historical data and introduces opportunity maintenance to obtain an equipment-level and flow-level optimal maintenance plan step by step, considering the series-parallel relationship between equipment. The upper layer optimizes the scheduling scheme of the ship-loading operation system under a certain maintenance plan obtained from the lower layer.

Finally, the validity and practicability of the proposed method are verified by taking the ship-loading operation system in a coal terminal in north China as an example.

The remainder of this paper is organized as follows. Section 2 reviews the related studies. Section 3 describes the studied system and problem, builds the mathematical model, and describes the solution algorithm. Section 4 conducts numerical experiments, followed by Section 5 which summarizes the whole paper and puts forward the future research direction.

2. Literature Review

The existing studies related to this paper can be divided into three parts: dry bulk terminal operating system scheduling, port infrastructure operation and maintenance, and opportunity maintenance.

2.1. Dry Bulk Terminal Operating System Scheduling

At present, the research of operating system scheduling in dry bulk cargo terminals can be divided into three parts: yard scheduling, berth allocation, and multi-system joint scheduling.

In the study of storage yard scheduling, some scholars devote themselves to the development of storage plans. Ouhaman et al. studied a real-world storage space allocation problem at an export bulk terminal [1]. Xin et al. proposed a modeling and control methodology for allocating materials in a dry bulk terminal [2]. Savelsbergh and Smith proposed a tree search algorithm for managing the stockyard at a coal export terminal considering coal assembly planning [3]. Burdett et al. proposed a new method to characterize the stockpiles in dry bulk terminals called bench block models [4]. Angelelli et al. and Kalinowski et al. provided constant-factor approximation algorithms and exact branch-and-bound algorithms to solve the scheduling problem of reclaimers in the stockyard of a coal export terminal [5,6]. Belassiria et al. developed a branch-and-bound method to solve the stacker and reclaimer scheduling problem in the coal storage area [7]. van Vianen et al. proposed a simulation-based method to schedule the stacker–reclaimer operation in a dry bulk yard [8]. Hu et al. proposed a genetic algorithm for stacker–reclaimer scheduling in an iron ore terminal [9]. Unsal et al. studied a complex parallel scheduling problem with non-crossing constraints, sequence-dependent setup times, and eligibility restrictions [10].

In the study of berth allocation, Ernst et al. proposed a two-phase method for solving the berth allocation problem in dry bulk terminals [11]. Cheimanoff et al. proposed a reduced VNS-based approach to solving the dynamic continuous berth allocation problem in bulk terminals considering tidal constraints [12]. de Leon et al. developed a machine-learning-based system for berth scheduling at bulk terminals [13]. Hu et al. established the mathematical model of berth allocation strategies considering the quayside’s limitation and the loading capacity of ships [14]. Guo et al. developed an integrated scheduling model managing the scheduling process of vessel traffic and deballasting operations [15].

Some researchers also study the integrated scheduling problem of the yard operating system and dock operating system. Burdett et al. presented an improved scheduling approach for optimizing the activities of a coal export terminal (CET) [16]. Belov et al. developed a metaheuristic logistics planning system integrating train scheduling, stockpile management, and vessel scheduling [17]. Burdett et al. modeled CET as a flexible job shop with operators and used an advanced meta-heuristic algorithm to solve the model [18]. de Andrade et al. proposed a mixed-integer linear programming formulation to deal with the integration problem in dry bulk export port terminals [19]. Unsal and Oguz proposed a logic-based Benders decomposition algorithm to solve the problem consisting of berth allocation, reclaimer scheduling, and stockyard allocation [20]. Menezes et al. presented a hierarchical approach to solve the production planning and scheduling problem in bulk cargo terminals [21]. Jiang et al. studied an optimization problem of integrated scheduling of restricted channels, berths, and yards in bulk cargo ports considering carbon emissions [22].

The mathematical model and solution algorithms proposed in these studies are important references for this study. However, these studies did not consider the impact of handling equipment failures on schedule, making them unsuitable to be used in this study.

2.2. Port Infrastructure Operation and Maintenance

In terms of port infrastructure operation and maintenance, some scholars studied the impact of infrastructure disruption caused by extreme weather (hurricanes, etc.) on ports [23], while few studies have studied the impact of equipment failure on port operation. Lewandowski and Scholz-Reiter proposed a framework for the systematic design and operation of condition-based maintenance systems [24]. Widarto and Handani developed the maintenance scheduling for the diesel generator support system of the container crane [25]. Mhalla et al. proposed a selective maintenance scheme combining failure rate and age for container port handling equipment [26]. Szpytko and Duarte developed a conceptual model of maintenance decisions for container terminal cranes based on digital twins [27]. Lin et al. built an age-reduction maintenance model to describe the deterioration situation of the port facility [28]. Tchotang et al. combined the Pareto method and trend test to analyze the reliability of a reach stacker [29]. Zheng et al. studied an integrated berth allocation, quay crane assignment, and specific quay crane assignment problem where quay crane maintenance is involved [30]. Krimi et al. proposed a mixed-integer programming model and investigated a set of heuristics algorithms based on general variable neighborhood search [31]. Sepehri et al. analyzed historical automatic identification system data to identify the interaction between maintenance dredging and seagoing vessels and quantify the hindrance periods for the case study in the Port of Rotterdam [32].

It can be seen that most of these studies focus on handling equipment in container ports, such as quay cranes, stackers, etc. However, since the handling operation conditions of coal terminals are different from those of container terminals, these studies cannot be used in this study.

2.3. Opportunity Maintenance

Opportunity maintenance (also called opportunistic maintenance), which reduces maintenance costs or time by combining the maintenance of multiple components, has been widely used in complex systems (e.g., wind turbines, PV power generation systems, CNC gear grinding machines) and has been attracting wide attention from researchers in recent years [33]. Huang et al. proposed a condition-based opportunistic maintenance policy for series systems with dependent competing failure processes [34]. Li et al. developed a maintenance optimization model designed for a system comprising two distinct components that are arranged in series and susceptible to common-cause failure [35]. Su et al. presented a condition-based opportunistic maintenance strategy for multi-component wind turbines by using stochastic differential equations [36]. To solve the problem of the high maintenance cost of PV power generation systems due to unreasonable opportunity maintenance grouping, Chen et al. proposed an opportunity maintenance strategy for PV power generation systems considering the structural relevance [37]. Wang et al. presented a novel optimal opportunistic maintenance strategy for wind turbines considering wind speed as a stochastic process [38]. Chen et al. studied an opportunistic maintenance optimization problem of continuous process manufacturing systems considering imperfect maintenance with epistemic uncertainty [39]. Dinh et al. developed an opportunistic predictive maintenance framework for a system comprising one continuous-operation component and an intermittent-operation component [40]. Dinh et al. discussed how disassembly actions can affect the system reliability and opportunistic maintenance optimization of disassembled components in a multicomponent system subjected to structural dependence [41]. Li et al. studied an opportunistic maintenance strategy optimization problem for a CNC gear grinding machine considering imperfect maintenance under a hybrid unit-level maintenance strategy [42]. Wei et al. proposed an optimal opportunistic maintenance planning for the critical structure of circulating pumps integrating discrete- and continuous-state information [43]. Chen et al. developed an opportunistic maintenance strategy for PV power generation systems under the action of random shocks [44]. Li et al. studied the influence mechanism of the interaction between operation and maintenance activities within the whole life cycle of electric multiple-unit components on the maintenance strategy, based on the concept of lean maintenance [45]. Yuan et al. presented an opportunistic maintenance model of multi-component complex systems based on a time threshold, with the consideration of the disassembly sequence and hybrid evolution factors influencing maintenance and learning–forgetting effects [46].

Most of the studies above focus on series systems instead of series-parallel systems, and few of them consider the interaction between maintenance and scheduling of the system, making them unsuitable to be used to solve the maintenance-based scheduling optimization problem for a ship-loading operation system, which is a series-parallel system.

3. Research Methods

In this section, the problem discussed in this paper is first presented in Section 3.1. The method of building the optimization model is then expressed in Section 3.2, followed by the proposed OM-based two-layer algorithm in Section 3.3.

3.1. Problem Description

This paper attempts to solve the scheduling optimization problem of the ship-loading operation system, considering equipment maintenance; the complexity of the problem is detailed in the following subsection.

3.1.1. Scheduling of Ship-Loading Operation System

As shown in Figure 1, after a ship arrives at the berth, the corresponding reclaimers (R) are invoked to unload the coal at the storage yard according to the operation plan. Then, the coal is transported to the front of the terminal via a series of belt conveyors including BQ, BC, and BM. Finally, the corresponding ship loader (SL) is called to complete the dry bulk cargo loading operation.

The complexity of shipping operation system scheduling is mainly reflected in the following aspects: (1) a ship with multiple tasks; (2) mixing operation of reclaimers; (3) combined operation of ship-loaders; and (4) uniqueness and accessibility constraints.

3.1.2. Maintenance of Equipment in the Ship-Loading Operation System

The handling equipment in the ship-loading operation system, including reclaimers, belt conveyors, and ship-loaders, may be in a series or parallel relationship (see the red dashed line in Figure 1). In this study, based on the characteristics of scheduling and maintenance of handling equipment in coal export terminals, its maintenance is treated as follows.

(1)

Repairable equipment. When the handling equipment encounters failure, it is repaired rather than replaced.

(2)

Predictive maintenance of handling equipment. After the handling equipment of the ship-loading operation system is put into use for some time, its failure rate will increase. In this case, predictive maintenance (PdM, i.e., maintenance according to the status of the handling equipment) should be carried out to restore the failure rate of the handling equipment to a lower layer. The normal working duration between two PdMs is called the predictive maintenance interval (PdMI), and the time of each PdMI is recalculated from 0 and denoted as (0, T_ij), where j is the number of PdMs. T_ij is denoted as the predictive maintenance interval length (PdMIL).

(3)

Minimal repair of handling equipment. For the handling equipment faults occurring between two PdMs, the concept of minimal repair is introduced. The duration of minimal repair is shorter than that of PdM, and will not change the failure rate of the handling equipment, but only restore the handling equipment to the state before its failure.

3.2. Model Establishment

The basic assumptions, notations, objective function, and constraints of the optimization model are introduced as follows.

3.2.1. Basic Assumptions and Notations

The basic assumptions of the optimization model are shown as follows:

(1)

The time of the ship arriving at the terminal and the arranged berth are known.

(2)

The ship’s time at the terminal counts from the ship arriving at the berth to the ship leaving the berth.

(3)

The pile to reclaim, amount of reclaiming, and operation duration of ship-loading task are known.

(4)

Before a ship-loading task is finished, the operation line occupied cannot be used for other tasks.

(5)

The parameters in the failure rate evolution function have been obtained by fitting historical data. The age and status of the handling operation equipment are known as well. The notations used in the model are shown in

Table 1. Notations and descriptions.

Notation	Description
Sets and indices
S	Set of ships arriving at the terminal, $s\in \left\{1,\dots ,\left|S\right|\right\}$
U	Set of reclaiming operation line, $u\in \left\{1,\dots ,\left|U\right|\right\}$
V	Set of horizontal operation line, $v\in \left\{1,\dots ,\left|V\right|\right\}$
W	Set of ship-loading operation line, $w\in \left\{1,\dots ,\left|W\right|\right\}$
M	Set of ship-loading task, $m\in \left\{1,\dots ,\left|M\right|\right\}$
$I_m$	Set of handling equipment used by task m, $i\in \left\{1,\dots ,\left|I_m\right|\right\}$
$N_{im}$	Set of PdM for equipment i during task m, $j\in \left\{1,\dots ,\left|N_{im}\right|\right\}$
K	Set of berth, $k\in \left\{1,\dots ,\left|K\right|\right\}$
T	Set of time, $t\in \left\{0,\dots ,\left|T\right|\right\}$
P	Set of pile, $p\in \left\{1,\dots ,\left|P\right|\right\}$
Parameters
$C_{total}$	Total cost of maintenance and scheduling for ship-loading operation system in coal export terminal
$C_{time}$	Dwell time cost of ships at the terminal
$C_{energy}$	Energy consumption cost of operation lines
$C_{maintenance}$	Handling equipment maintenance cost
$t_s^{arr},\forall s\in S$	Time of ship arriving at the terminal
$t_s^{dep},\forall s\in S$	Time of ship leaving the terminal
$E_s^{sh},\forall s\in S$	Dwell time cost of ship s at port per unit time
$E_u^{bq},\forall u\in U$	Energy cost of transporting unit coal for reclaiming operation line u
$E_v^{bc},\forall v\in V$	Energy cost of transporting unit coal for horizontal operation line v
$E_w^{bm},\forall w\in W$	Energy cost of transporting unit coal for ship-loading operation line w
$d_m,\forall m\in M$	Coal required by task m
$A_u^{bq},\forall u\in U$	Start-up energy cost each time for reclaiming operation line u
$A_v^{bc},\forall v\in V$	Start-up energy cost each time for horizontal operation line v
$A_w^{bm},\forall w\in W$	Start-up energy cost each time for ship-loading operation line w
$a_{m,u}^{bq},\forall m\in M,\forall u\in U$	Number of start-ups for reclaiming operation line u to finish task m
$a_{m,v}^{bc},\forall m\in M,\forall v\in V$	Number of start-ups for horizontal operation line v to finish task m
$a_{m,w}^{bm},\forall m\in M,\forall w\in W$	Number of start-ups for ship-loading operation line w to finish task m
$E_i^{sm},\forall i\in I_m$	Maintenance cost of each PdM for equipment i
${\mathcal{N}}_{imj},\forall i\in I_m,\forall m\in M,\forall j\in N_{im}$	Number of minimal repairs for equipment i in its jth PdMI during ship-loading task m
$E_{imj}^{um},\forall i\in I_m,\forall m\in M,\forall j\in N_{im}$	Maintenance cost of each minimal repair for equipment i in its jth PdMI during ship-loading task m
$\mathcal{M}$	Large positive constant
$t_m^s$	Start time of task m
${\tau }_m^{ta}$	Operation duration of task m
${\tau }_m^{down}$	Maintenance duration of task m
${\tau }_s^{un}$	Unmoor duration of ship s
${\tau }_s^{aux}$	Auxiliary operation duration of ship s
${\phi }_{m,m^{\prime }}$	Operation time interval between task m and m’
${\pi }_{k,k^{\prime }}$	Duration of the ship-loader moving from berth k to berth k’
$\eta$	Operation time interval of two tasks that belong to the same berth and occupy two ship-loaders next to the berth respectively
Variables
Decision variables
$y_{m,u}^{bq}\in \left\{0,1\right\},\forall m\in M,\forall u\in U$	1 if reclaiming operation line u is occupied by task m, 0 otherwise
$y_{m,v}^{bc}\in \left\{0,1\right\},\forall m\in M,\forall v\in V$	1 if horizontal operation line v is occupied by task m, 0 otherwise
$y_{m,w}^{bm}\in \left\{0,1\right\},\forall m\in M,\forall w\in W$	1 if ship-loading operation line w is occupied by task m, 0 otherwise
$z_{s,s^{\prime }}^{sh}\in \left\{0,1\right\},\forall s,s^{\prime }\in S$	1 if priority of ship s is higher than that of ship s’, 0 otherwise
$z_{m,m^{\prime }}^{ta}\in \left\{0,1\right\},\forall m,m^{\prime }\in M$	1 if priority of task m is higher than that of task m’, 0 otherwise
$T_{imj},\forall i\in I_m,\forall m\in M,\forall j\in N_{im}$	PdMIL for equipment i in its jth PdMI during task m
Auxiliary variables
${\beta }_{s,k}\in \left\{0,1\right\},\forall s\in S,\forall k\in K$	1 if ship s is arranged to berth k, 0 otherwise
${\theta }_{s,m}\in \left\{0,1\right\},\forall s\in S,\forall m\in M$	1 if task m belongs to ship s, 0 otherwise
${\gamma }_{m,p}\in \left\{0,1\right\},\forall m\in M,\forall p\in P$	1 if pile p is selected to unload coal by task m, 0 otherwise
${\epsilon }_{m,m^{\prime }}\in \left\{0,1\right\},\forall m,m^{\prime }\in M$	1 if task m and m’ belong to the ship berthing at the same berth and occupy two ship-loaders next to the berth respectively, 0 otherwise
${\delta }_{m,k}\in \left\{0,1\right\},\forall m\in M,\forall k\in K$	1 if task m belongs to the ship berthing at berth k, 0 otherwise
$x_{m,t}^{ta}\in \left\{0,1\right\},\forall m\in M,\forall t\in T$	1 if task m is in process at time t, 0 otherwise

.

3.2.2. Objective Function

$$Min C_{total}=C_{time}+C_{energy}+C_{maintenance}$$

(1)

$$C_{time}=\sum _{s\in S}\left(t_s^{dep}-t_s^{arr}\right)\times E_s^{sh}$$

(2)

$$\begin{array}{cc}\hfill C_{energy}=& {\displaystyle \sum _{m\in M}}[\left({\displaystyle \sum _{u\in U}}{E}_u^{bq}\times y_{m,u}^{bq}+{\displaystyle \sum _{v\in V}}{E}_v^{bc}\times y_{m,v}^{bc}+{\displaystyle \sum _{w\in W}}{E}_w^{bm}\times y_{m,w}^{bm}\right)\times d_m\hfill \\ & +{\displaystyle \sum _{u\in U}}{A}_u^{bq}\times a_{m,u}^{bq}\times y_{m,u}^{bq}+{\displaystyle \sum _{v\in V}}{A}_v^{bc}\times a_{m,v}^{bc}\times y_{m,v}^{bc}\hfill \\ & +{\displaystyle \sum _{w\in W}}{A}_w^{bm}\times a_{m,w}^{bm}\times y_{m,w}^{bm}]\hfill \end{array}$$

(3)

$$C_{maintenance}=\sum _{m\in M}\sum _{i\in I_m}\left(N_{im}\times E_i^{sm}+\sum _{j\in N_{im}}{\mathcal{N}}_{imj}\times E_{imj}^{um}\right)$$

(4)

The objective function is the total cost of maintenance and scheduling for ship-loading operation system $C_{total}$, consisting of dwell time cost of ships $C_{time}$, energy consumption cost of operation lines $C_{energy}$, and handling equipment maintenance cost $C_{maintenance}$. $C_{time}$ mainly considers the easing cost of ships, which is determined by the well time of ship s at port $\left(t_s^{dep}-t_s^{arr}\right)$, and cost per unit time $E_s^{sh}$. $C_{energy}$ consists of transportation energy cost and start-up energy cost. Transportation energy cost is calculated by unit cost of reclaiming operation line $\sum _{u\in U}{E}_u^{bq}\times y_{m,u}^{bq}$, unit cost of horizontal operation $\sum _{v\in V}{E}_v^{bc}\times y_{m,v}^{bc}$, unit cost of ship-loading operation $\sum _{w\in W}{E}_w^{bm}\times y_{m,w}^{bm}$, and coal required by ship-loading task m, $d_m$. Start-up energy cost considers cost of idling during the start-up phase of three operating lines ${\sum }_{u=1}^U{A}_u^{bq}\times a_{m,u}^{bq}\times y_{m,u}^{bq},{\sum }_{v=1}^V{A}_v^{bc}\times a_{m,v}^{bc}\times y_{m,v}^{bc}$, and ${\sum }_{w=1}^W{A}_w^{bm}\times a_{m,w}^{bm}\times y_{m,w}^{bm}$. $C_{maintenance}$ considers PdM cost $N_{im}\times E_i^{sm}$ and minimal repair cost ${\sum }_{j=1}^{N_{im}}{\mathcal{N}}_{imj}\times E_{imj}^{um}$.

3.2.3. Constraints

(1)

Constraints connecting ships mooring and leaving the berths

$$z_{s,s^{\prime }}^{sh}+z_{s^{\prime },s}^{sh}=\sum _{k\in K}{\beta }_{s,k}\times {\beta }_{s^{\prime },k},\forall s,s^{\prime }\in S,s\ne s^{\prime }$$

(5)

$$t_{s^{\prime }}^{arr}\ge t_s^{dep}-\mathcal{M}\times \left(1-z_{s,s^{\prime }}^{sh}\right),\forall s,s^{\prime }\in S,s\ne s^{\prime }$$

(6)

$$t_s^{dep}=\underset{\forall m}{\mathrm{m}\mathrm{a}\mathrm{x}}\left[\left(t_m^s+{\tau }_m^{ta}+{\tau }_m^{down}\right)\times {\theta }_{s,m}\right]+{\tau }_s^{un},\forall s\in S$$

(7)

$$t_m^s+\mathcal{M}\times \left(1-{\theta }_{s,m}\right)\ge t_s^{arr}+{\tau }_s^{aux},\forall s\in S,\forall m\in M$$

(8)

Equation (5) indicates that ships with higher priority occupy the berth preferentially. In the equation, $z_{s,s^{\prime }}^{sh}$ is a 0–1 variable, whose value is 1 if the priority of ship s is higher than that of ship $s^{\prime }$, 0 otherwise. k is the serial number of berth $k=1,2,\dots ,K$, while $K$ is the total number of berths in the terminal. ${\beta }_{s,k}$ is the 0–1 variable, whose value is 1 if berth k is occupied by ship s, 0 otherwise. Equation (6) indicates that the mooring time of any ship should be later than the departure time of the previous ship at corresponding berth, where $\mathcal{M}$ is a sufficiently large positive real number. Equation (7) indicates that the ship can leave the berth after completing all of its ship-loading tasks, where $t_m^s$, ${\tau }_m^{ta}$, and ${\tau }_m^{down}$ are the start time, operating duration, and maintenance duration of task m, respectively. ${\theta }_{s,m}$ is a 0–1 variable, whose value is 1 if task m belongs to ship s, 0 otherwise. ${\tau }_s^{un}$ is the unmoor duration of ship s. Equation (8) indicates that the ship can only start handling operations after completing auxiliary operations, whose duration is ${\tau }_s^{aux}$.

(2)

Constraints connecting ship-loading tasks

$$\begin{gathered} z_{m,m^{\prime }}^{ta}+z_{m^{\prime },m}^{ta}=\underset{\forall p,u,v,w}{\mathrm{m}\mathrm{a}\mathrm{x}}\left[\left({\gamma }_{m,p},{\gamma }_{m^{\prime },p}\right),\left(y_{m,u}^{bq},y_{m^{\prime },u}^{bq}\right),\left(y_{m,v}^{bc},y_{m^{\prime },v}^{bc}\right),\left(y_{m,w}^{bm},y_{m^{\prime },w}^{bm}\right),{\epsilon }_{m,m^{\prime }}\right], \\ \forall m,m^{\prime }\in M,m\ne m^{\prime } \end{gathered}$$

(9)

$$\begin{gathered} {\epsilon }_{m,m^{\prime }}=\left(\sum _{k\in K}{\delta }_{m,k}\times {\delta }_{m^{\prime },k}\right)\times \left[{\sum }_{w=2}^{\left|W\right|-1}\left(y_{m,w-1}^{bm}\times y_{m^{\prime },w+1}^{bm}+y_{m^{\prime },w-1}^{bm}\times y_{m,w+1}^{bm}\right)\right], \\ \forall m,m^{\prime }\in M,m\ne m^{\prime } \end{gathered}$$

(10)

$$\begin{gathered} t_{m^{\prime }}^s\ge t_m^s+{\tau }_m^{ta}+{\tau }_m^{down}-\mathcal{M}\times \left(1-z_{m,m^{\prime }}^{ta}\right)+{\phi }_{m,m^{\prime }}, \\ \forall m,m^{\prime }\in M,m\ne m^{\prime } \end{gathered}$$

(11)

$${\phi }_{m,m^{\prime }}=\left(\sum _{w\in W}{y}_{m,w}^{bm}\times y_{m^{\prime },w}^{bm}\right)\times \left(\sum _{k\in K}\sum _{k^{\prime }\in K}{\delta }_{m,k}\times {\delta }_{m^{\prime },k^{\prime }}\times {\pi }_{k,k^{\prime }}\right)+{\epsilon }_{m,m^{\prime }}\times \eta ,\forall m,m^{\prime }\in M,m\ne m^{\prime }$$

(12)

$$t+\mathcal{M}\times \left(1-x_{m,t}^{ta}\right)\ge t_m^s,\forall m\in M,\forall t\in T$$

(13)

$$t-\mathcal{M}\times \left(1-x_{m,t}^{ta}\right)<t_m^s+{\tau }_m^{ta}+{\tau }_m^{down},\forall m\in M,\forall t\in T$$

(14)

$$\sum _{t\in T}{x}_{m,t}^{ta}={\tau }_m^{ta}+{\tau }_m^{down},\forall m\in M$$

(15)

Equation (9) represents the priority that two loading tasks m and $m^{\prime }$ should meet, which occupy the same pile, the operation lines, or two ship-loaders on either side of the same berth, where $z_{m,m^{\prime }}^{ta}$ is a 0–1 variable, whose value is 1 when priority of task m is higher than task $m^{\prime }$, 0 otherwise. p is the serial number of pile $p=1,2,\dots ,P$. ${\gamma }_{m,p}$ is a 0–1 variable, whose value is 1 if task m unloads coal from pile p, 0 otherwise. ${\epsilon }_{m,m^{\prime }}$ is a 0–1 variable representing the relationship between two tasks occupying two ship-loaders on either side of the same berth, 1 if tasks m and $m^{\prime }$ meet the condition, 0 otherwise. ${\epsilon }_{m,m^{\prime }}$ can be calculated by Equation (10), where ${\delta }_{m,k}$ is 0–1, whose value is 1 if task m is finished by the ship mooring at berth k, 0 otherwise. Equation (11) represents the start time constraints of two ship-loading tasks with priority constraint, where ${\phi }_{m,m^{\prime }}$ is the operation time interval between task m and $m^{\prime }$, considering the possible operating time of the ship-loader shifting, as well as the operating time interval of two tasks occupying two ship-loaders on either side of the same berth (see Equation (12)). ${\pi }_{k,k^{\prime }}$ is the duration to move the ship loader from berth k to $k^{\prime }$, while $\eta$ means operating time interval. Equations (13)–(15) indicate that the same task cannot be interrupted, except for predictive maintenance and minimal repairs during the operation. Note that the time is discretized in this paper, thus $t=1,2,\dots T$, and T is the optimization period length. $x_{m,t}^{ta}$ is a 0–1 variable, whose value is 1 if task m is in progress at time t, 0 otherwise.

(3)

Constraints connecting the handling equipment selection

$$\left(\sum _{u\in U}{y}_{m,u}^{bq}\right)\times \left(\sum _{v\in V}{y}_{m,v}^{bc}\right)\times \left(\sum _{w\in W}{y}_{m,w}^{bm}\right)=1,\forall m\in M$$

(16)

$$y_{m,u}^{bq}\le \sum _{p\in P}{\sum }_{p^{\prime }=0}^{p-1}{\gamma }_{m,p}\times {\gamma }_{m,p^{\prime }}\times {\alpha }_{p,p^{\prime },u}^{bq},\forall m\in M,\forall u\in U$$

(17)

$$y_{m,v}^{bc}\le \sum _{u\in U}\sum _{w\in W}{y}_{m,u}^{bq}\times y_{m,w}^{bm}\times {\alpha }_{u,v,w}^{bc},\forall m\in M,\forall v\in V$$

(18)

$$y_{m,w}^{bm}\le \sum _{k\in K}{\delta }_{m,k}\times {\alpha }_{w,k}^{bm},\forall m\in M,\forall w\in W$$

(19)

$$\begin{array}{c}{\displaystyle \sum _{m\in M}}{\displaystyle \sum _{k\in K}}{x}_{m,t}^{ta}\times y_{m,w}^{bm}\times {\delta }_{m,k}\times k\le {\displaystyle \sum _{m\in M}}{\displaystyle \sum _{k\in K}}{x}_{m,t}^{ta}\times y_{m,w^{\prime }}^{bm}\times {\delta }_{m,k}\times k+\\ M\times \left(1-{\displaystyle \sum _{m\in M}}{x}_{m,t}^{ta}\times y_{m,w^{\prime }}^{bm}\right),\forall t\in T,w,w^{\prime }\in W,w<w^{\prime }\end{array}$$

(20)

Equation (16) indicates that a ship-loading task requires only one reclaiming operation line, horizontal operation line, and ship-loading operation line, respectively. Equations (17)–(19) indicate that there are accessibility constraints among the pile, reclaiming operation line, horizontal operation line, ship-loading operation line, and berth occupied by task, where ${\alpha }_{p,p^{\prime },u}^{bq}$ is a 0–1 variable, whose value is 1 if reclaiming operation line u can access pile p and $p^{\prime }$, 0 otherwise. Note that $p^{\prime }$ is the serial number of the pile that is used for mixing operation with pile p. When $p^{\prime }=0$, pile p is used for reclaiming operation only. ${\alpha }_{u,v,w}^{bc}$ is a 0–1 variable, whose value is 1 if horizontal operation line v can access reclaiming operation line u and ship-loading operation line w, 0 otherwise. Similarly, ${\alpha }_{w,k}^{bm}$ is a 0–1 variable, whose value is 1 if ship-loading operation line w can access berth k, 0 otherwise. Equation (20) indicates that the ship-loaders cannot cross each other.

(4)

Constraints connecting the PdM and minimal repair

$$\begin{array}{c}2\times {\displaystyle \sum _{u\in U}}{a}_{m,u}^{bq}\times y_{m,u}^{bq}+{\displaystyle \sum _{v\in V}}{a}_{m,v}^{bc}\times y_{m,v}^{bc}+2\times {\displaystyle \sum _{w\in W}}{a}_{m,w}^{bm}\times y_{m,w}^{bm}=\\ {\displaystyle \sum _{i\in I_m}}\left(N_{im}+{\displaystyle \sum _{j\in N_{im}}}{\mathcal{N}}_{imj}+1\right),\forall m\in M\end{array}$$

(21)

$${\sum }_{j=1}^{\left|N_{im}\right|-1}{T}_{imj}<{\tau }_m^{ta}\le {\sum }_{j=1}^{\left|N_{im}\right|}{T}_{imj}$$

(22)

$$\sum _{i\in I_p}\sum _{j\in N_{im}}{T}_{imj}-T_{iP}-\underset{i\in I_p}{\mathrm{m}\mathrm{i}\mathrm{n}}\left(T_{imj}\right)<{\tau }_m^{ta}$$

(23)

$${\tau }_m^{ta}\le \sum _{i\in I_p}\sum _{j\in N_{im}}{T}_{imj}-T_{iP}$$

(24)

Equation (21) indicates that start-up times of the occupied operation line m should correspond to the number of PdMs and minimal repairs included in the operation line. Equation (22) indicates the relationship between the number of PdMs of handling equipment i during ship-loading task m (denoted as $N_{im}$) and the PdMIL of equipment i in that period (denoted as $T_{imj}$) when equipment i is not paralleled with other equipment. Similarly, Equations (23) and (24) indicate the relationship between $N_{im}$ and $T_{imj}$ when equipment i is paralleled with other equipment, where $I_p$ is the set of equipment in the parallel part of equipment i, $i\in I_p$. $T_{iP}$ is the duration when equipment i and its paralleled equipment work simultaneously.

3.3. OM-Based Two-Layer Algorithm

To solve the scheduling optimization model for the ship-loading operation system considering equipment maintenance, this paper proposes an OM-based two-layer algorithm.

3.3.1. Algorithm Framework

As shown in Figure 2, the upper layer of the algorithm is used to determine the optimal scheduling plan of the system, while the lower layer optimizes the maintenance plan and feedback to the upper layer.

3.3.2. Upper Layer

The upper layer is a genetic algorithm (GA), whose objective is to minimize $C_{total}$. It is used to determine the scheduling plan of the ship-loading operating system under a specific equipment maintenance plan.

(1)

Chromosome coding

The chromosome coding of GA is shown in

Table 2. The chromosome coding structure.

Sequence	Length	Berth 1	Berth 2	…	Berth m
Berthing priority of ship	m	$b_1$	$b_2$	…	$b_m$
Priority of ship-loading task	n	$c_1,c_2,\dots c_{n_1}$	$c_{n_1+1},c_{n_1+2},\dots c_{n_2}$	…	$c_{n_{m-1}+1},c_{n_{m-1}+2},\dots c_n$
Reclaiming operation line	n	$e_1,e_2,\dots e_{n_1}$	$e_{n_1+1},e_{n_1+2},\dots e_{n_2}$	…	$e_{n_{m-1}+1},e_{n_{m-1}+2},\dots e_n$
Horizontal operation line	n	$f_1,f_2,\dots f_{n_1}$	$f_{n_1+1},f_{n_1+2},\dots f_{n_2}$	…	$f_{n_{m-1}+1},f_{n_{m-1}+2},\dots f_n$
Ship-loading operation line	n	$g_1,g_2,\dots g_{n_1}$	$g_{n_1+1},g_{n_1+2},\dots g_{n_2}$	…	$g_{n_{m-1}+1},g_{n_{m-1}+2},\dots g_n$

. Where $b_i$ is the berthing priority of ship i at the corresponding berth, the smaller the value of $b_i$, the higher the berthing priority. Similarly, $c_j$ represents the operation priority of the ship-loading task j. The smaller the value, the higher the operation priority. $e_j$, $f_j$, and $g_j$ are the serial numbers of the reclaiming operation line, horizontal operation line, and ship-loading operation line occupied by task j, respectively.

(2)

Population initialization

The rules for generating the initial population are as follows.

(a)

The berthing priority of the ship’s sequence is randomly generated according to the ship’s assigned berth.

(b)

After the priority sequence of the ship-loading task is randomly generated, it is adjusted according to the berthing priority of the ship to which the task belongs. The higher the berthing priority, the higher the priority of the ship-loading task.

(c)

The sequence of reclaiming operation line and ship-loading operation line sequence is randomly selected when the accessibility requirements of berth and stack are met.

(d)

The horizontal operation line sequence is randomly selected to meet the accessibility requirements of the reclaiming operation line and ship-loading operation line.

(3)

Fitness function selection

$C_{total}$ is selected as the fitness function.

(4)

Genetic operator

(a)

Genetic operator selection: the roulette method is used first to select a certain number of chromosomes from the parents for crossover and mutation operations, to produce offspring individuals. After that, the optimal strategy is adopted to select the fitter individuals in the parents and offspring to carry out the next generation of genetic operation, to retain all the excellent individuals in the evolution process.

(b)

Crossover strategy: The overall single-point crossover is adopted as the crossover strategy, the five sequences are crossed at the same time, and the following three possible new chromosome situations that do not meet the constraint conditions are corrected:

①

In the berthing priority-of-ships sequence, if the berthing priorities of two ships in the same berth are equal, one of the repeated priorities is randomly selected and replaced with the missing priority in the sequence.

②

In the priority of ship-loading task sequence, if the priority of two tasks is equal, the amendment principle is the same as ①.

③

Between the priority of ship-loading task sequence and the berthing priority of ship sequence, if the priority of the task of the ship with low priority is higher than that of the ship with high priority at the same berth, it will be adjusted.

(c)

Mutation strategy: Three mutation strategies are adopted, and the adoption probabilities of the three mutation strategies are 0.4, 0.3, and 0.3, respectively.

①

The gene locations of two ships berthing at the same berth are randomly selected and their berthing priorities are exchanged. If there is a contradiction between the priority of ship-loading task sequence and berthing priority-of-ship sequence, it will be adjusted according to the berthing priority of the corresponding ship-loading task. The higher the berthing priority, the higher the priority of the ship-loading task.

②

In the ship-loading task priority sequence, two genes are randomly selected for exchange, whose correction principle is the same as ①.

③

The gene location of the ship-loading operation line sequence is randomly selected and replaced by another ship-loading operation line occupied by the same task. If the accessibility constraint between the new ship-loading operation line and the original horizontal operation line is not satisfied, a new horizontal operation line is randomly selected to meet the accessibility constraint for correction.

3.3.3. Lower Layer

The lower layer is an OM-based equipment maintenance method, which is used to obtain optimal equipment maintenance plans under a specific scheduling plan. The objective function of the lower layer is to minimize ${\tau }_m^{down}$, which can also be expressed as Equation (25) since ${\tau }_m^{ta}$ is fixed.

$${ETA}_m={\displaystyle \frac{{\tau }_m^{ta}}{{\tau }_m^{ta}+{\tau }_m^{down}}}$$

(25)

The proposed OM-based “equipment-level–flow-level” maintenance optimization method is shown in Figure 3.

(1)

Failure rate evolution function modeling

Considering the impact of the increased failure rate caused by the external environment, the increased failure rate caused by wear and tear, and the imperfect maintenance of handling equipment, the failure rate evolution function of equipment i can be modeled by Equation (26).

$$f_{i\left(j+1\right)}\left(t\right)={\gamma }_i{\beta }_i{f}_{ij}\left(t+a_i{T}_{ij}\right),t\in \left(0,T_{i\left(j+1\right)}\right),j=1,2,\dots ,N_i$$

(26)

where ${\gamma }_i$ is the environmental factor, ${\beta }_i$ is the failure rate factor, $a_i$ is the recursively decreasing factor, and ${\gamma }_i>1$, ${\beta }_i>1$, $0<a_i<1$. $N_i$ is the number of PdMIs for equipment i in its life cycle. In this paper, ${\displaystyle \frac{T_{ij}}{T_{i1}}}$ is used as the parameter to judge whether to replace equipment i. When ${\displaystyle \frac{T_{ij}}{T_{i1}}}\le C,C\in (0,1)$ is satisfied for the first time, equipment i will be replaced and the corresponding j is recorded as $N_i$, and the replacement of equipment i is treated as its $N_i$th PdM. $T_{ij}$ is jth PdMIL of equipment i, $T_{ij}=t_{ij}^{sms}-t_{i\left(j-1\right)}^{sme}$, where $t_{ij}^{sms}$ is the start time of jth PdM of equipment i, $t_{i\left(j-1\right)}^{sme}$ is the end time of the (j − 1)th PdM of equipment i, and $t_{i0}^{sme}$ is the time when equipment i is put to use. Thus, the cumulative failure rate function of equipment i at time t in its jth PdMI, $F_{ij}\left(t\right)$, can be expressed as follows.

$$F_{ij}\left(t\right)={\int }_0^t{f}_{ij}\left(x\right)dx$$

(27)

For repairable equipment i, the concept of failure rate threshold is introduced to ensure its stable working state. The failure rate threshold of equipment i in its jth PdMI, $H_{i0}$, is defined. Once the cumulative failure-rate function of equipment i in its jth PdMI reaches $H_{i0}$, PdM should be performed.

$$H_{i0}={\int }_0^{T_{ij}}{f}_{ij}\left(t\right)dt=F_{ij}\left(T_{ij}\right)$$

(28)

(2)

Failure rate evolution function parameter fitting

According to the historical data, the failure rate evolution function of the equipment is fitted to a three-parameter Weber distribution, shown in Equation (29).

$${\lambda }_{ij}\left(t\right)={\displaystyle \frac{{\eta }_{ij}}{{\theta }_{ij}}}{\left({\displaystyle \frac{t-t_{ij}^0}{{\theta }_{ij}}}\right)}^{{\eta }_{ij}-1},t\in \left(0,T_{ij}\right),j=1,2,\dots ,N_i$$

(29)

Then, Equation (26) can be expressed as follows.

$$f_{ij}\left(t\right)=\left\{\begin{array}{c}{\displaystyle \frac{{\eta }_{ij}}{{\theta }_{ij}}}{\left({\displaystyle \frac{t-t_{ij}^0}{{\theta }_{ij}}}\right)}^{{\eta }_{ij}-1},t\in \left(0,T_{ij}\right),j=1\\ {\gamma }_i{\beta }_i{f}_{i\left(j-1\right)}\left(t+a_i{T}_{i\left(j-1\right)}\right),t\in \left(0,T_{ij}\right),j=2,3,\dots ,N_i\end{array}\right.$$

(30)

When j = 1, $f_{i1}\left(t\right)$ is the first term of the recursion formula of $f_{ij}\left(t\right)$, thus $f_{i1}\left(t\right)={\lambda }_{i1}\left(t\right)$. When $j>1$, $f_{ij}\left(t\right)$ meet Equations (26) and (29) at the same time, thus $f_{ij}\left(t\right)={\gamma }_i{\beta }_i{f}_{i(j-1)}\left(t+a_i{T}_{i(j-1)}\right)={\lambda }_{ij}\left(t\right)$. Furthermore, let j = 2 and 3, Equations (31) and (32) can be obtained.

$${\gamma }_i{\beta }_i{\displaystyle \frac{{\eta }_{i1}}{{\theta }_{i1}}}{\left({\displaystyle \frac{t-t_{i1}^0+a_i{T}_{i1}}{{\theta }_{i1}}}\right)}^{{\eta }_{i1}-1}={\displaystyle \frac{{\eta }_{i2}}{{\theta }_{i2}}}{\left({\displaystyle \frac{t-t_{i2}^0}{{\theta }_{i2}}}\right)}^{{\eta }_{i2}-1}$$

(31)

$${\gamma }_i{\beta }_i{\displaystyle \frac{{\eta }_{i2}}{{\theta }_{i2}}}{\left({\displaystyle \frac{t-t_{i2}^0+a_i{T}_{i2}}{{\theta }_{i2}}}\right)}^{{\eta }_{i2}-1}={\displaystyle \frac{{\eta }_{i3}}{{\theta }_{i3}}}{\left({\displaystyle \frac{t-t_{i3}^0}{{\theta }_{i3}}}\right)}^{{\eta }_{i3}-1}$$

(32)

Let t = 0 in Equations (31) and (32),

$${\gamma }_i{\beta }_i{\displaystyle \frac{{\eta }_{i1}}{{\theta }_{i1}}}{\left({\displaystyle \frac{-t_{i1}^0+a_i{T}_{i1}}{{\theta }_{i1}}}\right)}^{{\eta }_{i1}-1}={\displaystyle \frac{{\eta }_{i2}}{{\theta }_{i2}}}{\left({\displaystyle \frac{-t_{i2}^0}{{\theta }_{i2}}}\right)}^{{\eta }_{i2}-1}$$

(33)

$${\gamma }_i{\beta }_i{\displaystyle \frac{{\eta }_{i2}}{{\theta }_{i2}}}{\left({\displaystyle \frac{-t_{i2}^0+a_i{T}_{i2}}{{\theta }_{i2}}}\right)}^{{\eta }_{i2}-1}={\displaystyle \frac{{\eta }_{i3}}{{\theta }_{i3}}}{\left({\displaystyle \frac{-t_{i3}^0}{{\theta }_{i3}}}\right)}^{{\eta }_{i3}-1}$$

(34)

After the former translation, Equations (33) and (34) can be expressed as follows.

$${\displaystyle \frac{{{\theta }_{i1}}^{{\eta }_{i1}}{{\theta }_{i3}}^{{\eta }_{i3}}}{{{\theta }_{i2}}^{2{\eta }_{i2}}}}{\displaystyle \frac{{\eta }_{i2}^2}{{\eta }_{i1}{\eta }_{i3}}}{\displaystyle \frac{{\left(-t_{i2}^0\right)}^{{\eta }_{i2}-1}}{{\left(-t_{i3}^0\right)}^{{\eta }_{i3}-1}}}={\displaystyle \frac{{\left(a_i{T}_{i1}-t_{i1}^0\right)}^{{\eta }_{i1}-1}}{{\left(a_i{T}_{i2}-t_{i2}^0\right)}^{{\eta }_{i2}-1}}}$$

(35)

Let ${\displaystyle \frac{{{\theta }_{i1}}^{{\eta }_{i1}}{{\theta }_{i3}}^{{\eta }_{i3}}}{{{\theta }_{i2}}^{2{\eta }_{i2}}}}{\displaystyle \frac{{{\eta }_{i2}}^2}{{\eta }_{i1}{\eta }_{i3}}}{\displaystyle \frac{{\left(-t_{i2}^0\right)}^{{\eta }_{i2}-1}}{{\left(-t_{i3}^0\right)}^{{\eta }_{i3}-1}}}=A$, and take the natural logarithm of both sides of Equation (35):

$$\mathrm{l}\mathrm{n}A=\left({\eta }_{i1}-1\right)\mathrm{l}\mathrm{n}\left(a_i{T}_{i1}-t_{i1}^0\right)-\left({\eta }_{i2}-1\right)\mathrm{l}\mathrm{n}\left(a_i{T}_{i2}-t_{i2}^0\right)$$

(36)

In Equation (36), except unknown parameter $a_i$, all parameters can be obtained from the three-parameter Weber distribution of Equation (29), thus $a_i$ can be determined by numerical solution algorithm and ${\gamma }_i{\beta }_i$ can be calculated by Equation (33).

(3)

Impact of minimal repair

Assume that the probability of minimal repair occurring is related to the cumulative failure rate function of the equipment $F_{ij}\left(t\right)$. The random variable representing the occurrence of minimal repairs at time t, denoted as $X\left(t\right)$, follows a 0–1 distribution and can be calculated by Equation (37).

$$P\left\{X\left(t\right)=x\right\}={\left[{\epsilon }_i\times F_{ij}\left(t\right)\right]}^x\times {\left[1-{\epsilon }_i\times F_{ij}\left(t\right)\right]}^{1-x},x=0,1$$

(37)

When minimal repair happens, $x=1$, 0 otherwise. ${\epsilon }_i$ is the minimal repair correction factor.

Denote the start time of the kth minimal repair in jth PdMI of equipment i as ${\tau }_{ijk}^{um}$ and its duration as ${\tau }_{ijk}^{um}$, $k=1,2,\dots ,{\mathcal{N}}_{ij}$, where ${\mathcal{N}}_{ij}$ is the total number of minimal repairs in that PdMI. Assuming that no other minimal repairs will occur in the course of a minimal repair, i.e., $t_{ij\left(k+1\right)}^{ums}>t_{ijk}^{ums}+{\tau }_{ijk}^{um}$, then the jth PdMIL of equipment i considering minimal repair is $T_{ij}+{\sum }_{k=1}^{{\mathcal{N}}_{ij}}{\tau }_{ijk}^{um}$, whose cumulative failure rate function $G_{ij}\left(t\right),t\in \left(0,T_{ij}+{\sum }_{k=1}^{{\mathcal{N}}_{ij}}{\tau }_{ijk}^{um}\right)$ can be calculated by Equation (38).

$$G_{ij}\left(t\right)=\left\{\begin{array}{c}{F}_{ij}\left(t_{ijk}^{ums}\right),t\in M_{ij}\\ F_{ij}\left(t-{\sum }_{k=0}^n{\tau }_{ijk}^{um}\right),t\notin M_{ij}\end{array}\right.$$

(38)

where $n$ is the number of minimal repairs that have occurred in the PdMI. ${\tau }_{ij0}^{um}=0$. $M_{ij}$ is the duration of minimal repair, which can be calculated by Equation (39).

$$M_{ij}=\left(t_{ijk}^{ums},t_{ijk}^{ums}+{\tau }_{ijk}^{um}\right),k=1,2,\dots ,{\mathcal{N}}_{ij}$$

(39)

Thus, when minimal repair occurs, the curve of $G_{ij}\left(t\right)$ will extend in the horizontal direction the minor repair duration ${\tau }_{ijk}^{um}$, and then continue to increase with the trend of $F_{ij}\left(t\right)$.

Since minimal repairs do not change the trend of $G_{ij}\left(t\right)$, when quantifying the relationship between $H_{i0}$ and $T_{ij}$, the effect of minimal repair is ignored and the equation $F_{ij}\left(t\right)$ is used directly.

Moreover, when $t\in \left(t_{ijk}^{ums},t_{ij\left(k+1\right)}^{ums}\right),k=0,1,\dots ,{\mathcal{N}}_{ij}$, the total time of occurred minimal repair at time t in the jth PdMI of equipment i, $T_{ijk}^{um}(t)$, can be calculated by Equation (40).

$$T_{ijk}^{um}\left(t\right)={\sum }_{q=0}^k{\tau }_{ijq}^{um}$$

(40)

where ${\tau }_{ijk}^{um}$ is the duration of the kth minimal repair in the jth PdMI of equipment i, and ${\tau }_{ij0}^{um}=0$.

(4)

Maximizing availability of single equipment

The availability ${ETA}_i\left(H_{i0}\right)$ of handling equipment i during its entire life cycle using $H_{i0}$ is set as the objective function, which is shown in Equation (41).

$$Max ET{A}_i\left(H_{i0}\right)={\displaystyle \frac{T_i^{up}\left(H_{i0}\right)}{T_i^{up}\left(H_{i0}\right)+T_i^{down}\left(H_{i0}\right)}}={\displaystyle \frac{1}{1+\frac{T_i^{down}\left(H_{i0}\right)}{T_i^{up}\left(H_{i0}\right)}}}$$

(41)

where $T_i^{up}\left(H_{i0}\right)$ is operation duration of equipment i during its entire life cycle.

$$T_i^{up}\left(H_{i0}\right)={\sum }_{j=1}^{N_i}{T}_{ij}\left(H_{i0}\right)$$

(42)

where $N_i$ is the number of PdMIs of equipment i and $T_{ij}\left(H_{i0}\right)$ can be determined as follows.

$$T_{ij}\left(H_{i0}\right)=\left\{\begin{array}{c}{\left[{\theta }_{i1}^{{\eta }_{i1}}\times H_{i0}+{\left(-t_{0i}\right)}^{{\eta }_{i1}}\right]}^{{\displaystyle \frac{1}{{\eta }_{i1}}}}+t_{0i},j=1\\ \begin{array}{c}{\left[{\left({\gamma }_i{\beta }_i\right)}^{-\left(j-1\right)}\times {\theta }_{i1}^{{\eta }_{i1}}\times H_{i0}+{\left(a_i\times {\sum }_{k=1}^{j-1}{T}_{ik}-t_{0i}\right)}^{{\eta }_{i1}}\right]}^{{\displaystyle \frac{1}{{\eta }_{i1}}}}\\ -a_i\times {\sum }_{k=1}^{j-1}{T}_{ik}+t_{0i},j=2,3,\dots ,N_i\end{array}\end{array}\right.$$

(43)

In Equation (41), $T_i^{down}\left(H_{i0}\right)$ is maintenance duration of equipment i during its entire life cycle considering PdM and minimal repair, which can be calculated by Equation (44).

$$T_i^{down}\left(H_{i0}\right)={\sum }_{j=1}^{N_i}{T}_{ij}^{sm}+{\sum }_{j=1}^{N_i}{T}_{ij}^{um}\left(T_{ij}\right)$$

(44)

where $T_{ij}^{sm}$ is the duration of the jth PdM for equipment i, determined by historical data. $T_{ij}^{um}\left(T_{ij}\right)$ is the total time of minimal repair in the jth PdMI of equipment i, calculated by Equation (40).

(5)

PdMIL threshold of equipment

Denote the Pth PdMIL of equipment i after adjustment as $T_{iP}$, whose corresponding failure rate threshold is $H_{iP}$, while the other failure rate threshold remains $H_{i0}$, then the relationship between $H_{iP}$ and $T_{iP}$ can be determined by Equation (45).

$$H_{iP}=\left\{\begin{array}{c}{\displaystyle \frac{1}{{{\theta }_{i1}}^{{\eta }_{i1}}}}\times \left[{\left(T_{iP}-t_{0i}\right)}^{{\eta }_{i1}}-{\left(-t_{0i}\right)}^{{\eta }_{i1}}\right],P=1\\ \begin{array}{c}{\left({\gamma }_i{\beta }_i\right)}^{\left(P-1\right)}\times {\displaystyle \frac{1}{{{\theta }_{i1}}^{{\eta }_{i1}}}}\times \\ \left[{\left(a_i\times {\sum }_{j=1}^{P-1}{T}_{ij}\left(H_{i0}\right)+T_{iP}-t_{0i}\right)}^{{\eta }_{i1}}-{\left(a_i\times {\sum }_{j=1}^{P-1}{T}_{ij}\left(H_{i0}\right)-t_{0i}\right)}^{{\eta }_{i1}}\right]\\ P=2,\dots ,N_i\end{array}\end{array}\right.$$

(45)

where $T_{ij}\left(H_{i0}\right)$ can calculated by Equation (43). Then ${ETA}_i\left(H_{iP}\right)$ is calculated by Equations (46)–(48).

$$ET{A}_i\left(H_{iP}\right)={\displaystyle \frac{1}{1+\frac{T_i^{down}\left(H_{iP}\right)}{T_i^{up}\left(H_{iP}\right)}}}$$

(46)

$$T_i^{up}\left(H_{iP}\right)={\sum }_{j=1}^{P-1}{T}_{ij}\left(H_{i0}\right)+T_{iP}+{\sum }_{j=P+1}^{N_i}{T}_{ij}\left(H_{i0}\right)$$

(47)

$$T_i^{down}\left(H_{iP}\right)=N_i\times T_i^{sm}+{\sum }_{j=1}^{P-1}{T}_{ij}^{um}\left(H_{i0}\right)+T_{iP}^{um}\left(H_{iP}\right)+{\sum }_{j=P+1}^{N_i}{T}_{ij}^{um}\left(H_{i0}\right)$$

(48)

Let $∆{ETA}_i={ETA}_i\left(H_{iP}\right)-{ETA}_i\left(H_{i0}\right)$, then the PdMIL threshold of equipment i, $T_{iP}\in (T_{iP}^{lead},T_{iP}^{lag})$ can be obtained by Equations (49)–(51).

$$\Delta ET{A}_i\ge 0$$

(49)

$$\Delta ET{A}_i\ge 0$$

(50)

$$H_{iP}\le 1$$

(51)

where Equation (49) is the availability increment non-negative constraint, Equation (50) is the PdMIL non-negative constraint, and Equation (51) is the failure rate threshold constraint.

(6)

OM-based flow-level maintenance optimization

After determining the PdMIL threshold for each piece of equipment, the concept of OM is introduced for flow-level maintenance optimization, which takes advantage of PdM opportunities for high-priority equipment in the ship-loading operation flow to perform PdM on low-priority equipment in series. The objective function of flow-level maintenance optimization is shown in Equation (52).

$$Max {ETA}_F=1-{\displaystyle \frac{T_F^{down}}{T_F}}$$

(52)

where $T_F$ is the optimization period length, which is constant. $T_F^{down}$ is the maintenance duration in the optimization period considering PdM and minimal repair, which can be determined by Equation (53).

$$T_F^{down}={\sum }_{i=1}^{I_S}{N}_{Fi}\times T_i^{sm}+{\sum }_{i=1}^{I_S}{\sum }_{j=1}^{N_{Fi}}{T}_{ij}^{um}-{\sum }_{i=2}^{I_S}{T}_i^{om}$$

(53)

where $I_S$ is the number of pieces of equipment in the serial part. $T_i^{sm}$ is the mean value of PdM duration of equipment i. $N_{Fi}$ is the number of PdMs for equipment i during the optimization period. $T_{ij}^{um}$ is the total time of minimal repair in the jth PdM of equipment i. $T_i^{om}$ is saved total time by using OM strategy considering PdM and minimal repair. The OM procedure is shown in

Table 3. OM procedure for ship-loading operation flow.

Step 1. Calculate each PdMIL threshold for equipment i in the flow, $\left(T_{iP}^{lead},T_{iP}^{lag}\right), i\in \left\{1,2,\dots ,I\right\}, P\in \left\{1,2,\dots ,N_i\right\}$, where $I$ is the total number of pieces of equipment in the flow and $N_i$ is the total number of PdMIs in life cycle of equipment i.

Step 2. Judge whether equipment i is in parallel with other handling equipment. If so, go to Step 4; otherwise, rank the priority of equipment in parallel part according to the duration of PdM $T_i^{sm}$. The larger the $T_i^{sm}$, the higher the priority (the smaller the priority number). Step 3. When $i<N_{pi}$, where $N_{pi}$ is the total number of pieces of equipment in parallel part, repeat the step as follows, otherwise go to Step 4.

Step 3.1. Determine equipment i’s shutdown interval caused by PdM $\left(t_{ij}^{sms},t_{ij}^{smf}\right)$ and minimal repair $\left(t_{ijk}^{ums},t_{ijk}^{ums}+{\tau }_{ijk}^{um}\right)$, to determine its actual shutdown interval $\left(t_i^{downs},t_i^{downf}\right)$.

Step 3.2. Determine the initial shutdown interval of equipment $m=i+1$ caused by PdM $\left(t_{mn}^{sms},t_{mn}^{smf}\right)$ and minimal repair $\left(t_{mnp}^{ums},t_{mnp}^{ums}+to\right)$, to determine its actual shutdown interval $\left(t_m^{downs},t_m^{downf}\right)$.

Step 3.3. When no matter how adjusted $T_{mn}\in \left(T_{mn}^{lead},T_{mn}^{lag}\right)$, $\left(t_i^{downs},t_i^{downf}\right)\cap \left(t_m^{downs},t_m^{downf}\right)$ cannot be an empty set, treat equipment $i$ and m as equipment $i^{\prime }$ with actual shutdown interval $\left(t_{i^{\prime }}^{downs},t_{i^{\prime }}^{downf}\right)$ = $min\left(\left(t_i^{downs},t_i^{downf}\right)\cap \left(t_m^{downs},t_m^{downf}\right)\right)$, set priority of equipment $i^{\prime }$ as lowest, and consider its effect on availability of the ship-loading operation flow. Otherwise, treat equipment $i$ and m as equipment $i^{\prime }$ which is always working, and it is unnecessary to consider the series-parallel relationship with other handling equipment.

Step 3.4. Consider the maintenance opportunities provided by equipment i and equipment m, and determine the next equipment.

Step 4. Rank the priority of equipment in serial part according to the duration of PdM $T_i^{sm}$. The larger the $T_i^{sm}$, the higher the priority (the smaller the priority number). Step 5. When $i<N_{si}$, where $N_{si}$ is the total number of pieces of equipment in serial parts, repeat the step as follows, otherwise go to Step 6. Step 5.1. Determine equipment i’s shutdown interval caused by PdM $\left(t_{ij}^{sms},t_{ij}^{smf}\right)$ and minimal repair $\left(t_{ijk}^{ums},t_{ijk}^{ums}+{\tau }_{ijk}^{um}\right)$, to determine its actual shutdown interval $\left(t_i^{downs},t_i^{downf}\right)$. Step 5.2. Determine the initial shutdown interval of equipment $m=i+1$ caused by PdM $\left(t_{mn}^{sms},t_{mn}^{smf}\right)$ and minimal repair $\left(t_{mnp}^{ums},t_{mnp}^{ums}+{\tau }_{mnp}^{um}\right)$, to determine its actual shutdown interval $\left(t_m^{downs},t_m^{downf}\right)$. Step 5.3. Maximize $\left(t_{mn}^{sms},t_{mn}^{smf}\right)\cap \left(t_{ij}^{sms},t_{ij}^{smf}\right)$ by adjusting $T_{mn}\in \left(T_{mn}^{lead},T_{mn}^{lag}\right)$ according to $\left(t_{ij}^{sms},t_{ij}^{smf}\right)$. Step 5.4. Consider the maintenance opportunities provided by equipment i and equipment m, and determine the next equipment.

Step 6. Consider the effect of parallel parts, and calculate the availability of the whole ship-loading operation flow ${ETA}_F$ by Equations (52) and (53).

.

4. Case Study

In this section, the input parameters of the case study are first introduced in Section 4.1, followed by results and analysis in Section 4.2

4.1. Input Parameters

The terminal in the case study is located in northern China. There are three berths (201#–203#), equipped with three ship-loaders (SL4–SL6) and three belt conveyors (BM4–BM6) to connect them, forming the ship-loading operation line. There are 40 piles (P01–P40) in the storage yard, equipped with five reclaimers (R5–R9) and three belt conveyors (BQ3–BQ5), forming the reclaiming operation line. Another two belt conveyors (BC3 and BC4) are used as horizontal operation lines. The layout for the terminal is shown in Figure 4, the relationship between operating lines, equipment, and piles is shown in

Table 4. Relationship between operating line, handling equipment, and pile.

Operation Line	No.	Equipment 1	Equipment 2	Pile
Reclaiming operation line u	1	BQ3	R5	P01–P08
	2	BQ4	R6(R7)	P09–P24
	3	BQ5	R8(R9)	P25–P40
Horizontal operation line v	1	BC3	-
	2	BC4	-
Ship-loading operation line w	1	BM4	SL4
	2	BM5	SL5
	3	BM6	SL6

, and accessibility between them is shown in

Table 5. Accessibility of operating line.

Berth	Reclaiming Operation Line u	Horizontal Operation Line v	Ship-Loading Operation Line w
201#	1	-	1
	2	1	1
	3	1	1
202#	2	1	2
	3	1	2
	2	2	2
	3	2	2
203#	1	-	3
	2	2	3
	3	2	3

.

Furthermore, the ship-loading operation lines allow for combined operation of adjacent berths. For example, w1 ship-loading operation line, which is in charge of 201# berth, can move to 202# berth. The optimization period is January 2017, and its length is 744 h. During this time, 63 ships arrived at the terminal, including 368 ship-loading tasks. The information on ship arrivals is listed in

Table 6. Basic information on ship arrivals.

Ship’s No.	Arrival Date	Arrival Time	Berth	Task No.	Unit Dwell Time Cost of Ship (CNY/h)
1	2 January 2017	9:49	203#	1–4	1416
2	2 January 2017	5:40	202#	5–8	1416
3	2 January 2017	5:38	201#	9–10	1416
4	2 January 2017	22:50	202#	11–14	1416
5	5 January 2017	2:56	202#	15–19	1416
6	5 January 2017	1:44	201#	20–22	1416
7	5 January 2017	14:23	203#	23–24	1416
8	9 January 2017	23:33	203#	25–28	1416
9	6 January 2017	9:07	201#	29–34	1416
10	7 January 2017	3:56	203#	35–39	1416
11	8 January 2017	14:02	203#	40–45	1416
12	7 January 2017	8:43	201#	46–50	1416
13	8 January 2017	19:46	202#	51–58	1416
14	11 January 2017	7:10	203#	59–61	1416
15	10 January 2017	0:00	201#	62–69	1416
16	7 January 2017	15:58	202#	70–72	1416
17	6 January 2017	13:04	202#	73–78	1416
18	8 January 2017	15:11	201#	79–84	1416
19	11 January 2017	13:52	202#	85–89	1416
20	12 January 2017	15:15	203#	90–101	1592
21	12 January 2017	5:36	201#	102–106	1416
22	14 January 2017	7:46	203#	107–114	1416
23	10 January 2017	6:50	202#	115–120	1416
24	13 January 2017	21:43	201#	121–125	1416
25	11 January 2017	1:26	201#	126–130	1416
26	12 January 2017	16:42	202#	131–136	1416
27	15 January 2017	2:36	201#	137–142	1416
28	14 January 2017	0:39	202#	143–145	1416
29	15 January 2017	9:11	202#	146–151	1416
30	17 January 2017	4:34	203#	152–159	1416
31	17 January 2017	17:43	202#	160–169	1592
32	15 January 2017	19:40	203#	170–175	1416
33	16 January 2017	18:16	202#	176–183	1592
34	19 January 2017	6:37	201#	184–189	1416
35	16 January 2017	4:09	201#	190–193	1416
36	23 January 2017	0:41	202#	194–199	1416
37	21 January 2017	1:17	201#	200–203	1416
38	20 January 2017	15:51	203#	204–210	1416
39	18 January 2017	8:27	201#	211–213	1416
40	18 January 2017	12:50	203#	214–219	1416
41	17 January 2017	2:02	201#	220–225	1416
42	19 January 2017	1:15	202#	226–231	1416
43	22 January 2017	15:32	203#	232–239	1416
44	25 January 2017	19:29	202#	240–249	1592
45	24 January 2017	9:38	203#	250–253	1416
46	21 January 2017	4:48	202#	254–262	1592
47	24 January 2017	4:03	202#	263–270	1416
48	24 January 2017	17:26	201#	271–271	1416
49	22 January 2017	19:28	201#	272–279	1416
50	25 January 2017	20:36	203#	280–284	1416
51	27 January 2017	8:40	202#	285–293	1592
52	29 January 2017	8:28	203#	294–299	1416
53	28 January 2017	13:01	202#	300–311	1592
54	25 January 2017	13:28	201#	312–315	1416
55	27 January 2017	16:59	203#	316–318	1416
56	26 January 2017	21:43	201#	319–325	1416
57	31 January 2017	14:27	201#	326–328	1416
58	31 January 2017	15:44	203#	329–336	1416
59	30 January 2017	13:27	203#	337–346	1416
60	31 January 2017	2:15	202#	347–352	1416
61	28 January 2017	0:52	201#	353–357	1416
62	28 January 2017	19:34	201#	358–363	1416
63	30 January 2017	17:11	201#	364–368	1592

. Unit cost of maintenance and energy consumption are shown in

Table 7. Unit cost of maintenance and unit energy consumption cost.

Operation Line	Unit Cost of PdM (Thousand CNY)	Unit Cost of Minimal Repair (Thousand CNY)	Transportation Cost per Ton (CNY)	Start-Up Cost Each Time (CNY)
u1	78	18	0.49	102
u2	72	19	0.63	130
u3	85	16	0.63	130
v1	52	10	0.08	5.2
v2	55	9	0.08	5.2
w1	124	20	0.39	61
w2	117	23	0.39	91
w3	130	26	0.39	123

.

4.2. Results and Analysis

In this section, the initial maintenance optimization scheme and scheduling optimization results considering maintenance are presented in Section 4.2.1 and Section 4.2.2, respectively.

4.2.1. Initial Maintenance Optimization Scheme

According to historical data, values of ${\gamma }_i{\beta }_i$ and $a_i$ in the failure rate evolution function are first fitted, and the duration of PdM and minimal repair are determined, whose results are shown in

Table 8. Fitting results of failure rate evolution function parameters, PdM duration, and minimal repair duration of equipment.

Serial No. i	Equipment	${\mathit{\gamma }}_{\mathit{i}}{\mathit{\beta }}_{\mathit{i}}$	${\mathit{a}}_{\mathit{i}}$	${\mathit{T}}_{\mathit{i}}^{\mathit{s}\mathit{m}}$ (h)	${\mathit{\tau }}_{\mathit{i}\mathit{j}}^{\mathit{u}\mathit{m}}$ (h)
1	BQ3	1.03	0.011	17	3
2	BQ4	1.06	0.010	18	3.5
3	BQ5	1.02	0.025	16	3
4	R5	1.03	0.089	23	3.5
5	R6	1.04	0.018	25	4
6	R7	1.02	0.060	24	3.5
7	R8	1.03	0.018	24	3.5
8	R9	1.04	0.088	23	3
9	BC3	1.04	0.126	18	3
10	BC4	1.02	0.090	19	3
11	BM4	1.06	0.019	18	3.5
12	BM5	1.03	0.018	17	3
13	BM6	1.04	0.020	19	3.5
14	SL4	1.02	0.069	27	4
15	SL5	1.03	0.070	26	3.5
16	SL6	1.04	0.066	28	4

. After that, the $H_{i0}$, $N_i$, and $T_{ij}$ for equipment i can be determined, as shown in

Table 9. $H_{i0}$, $N_i$, and $T_{ij}$ for equipment i.

i	Equipment	${\mathit{H}}_{\mathit{i}0}$	${\mathit{N}}_{\mathit{i}}$	${\mathit{T}}_{\mathit{i}1}$	${\mathit{T}}_{\mathit{i}2}$	${\mathit{T}}_{\mathit{i}3}$	${\mathit{T}}_{\mathit{i}4}$	${\mathit{T}}_{\mathit{i}5}$	${\mathit{T}}_{\mathit{i}6}$	${\mathit{T}}_{\mathit{i}7}$	${\mathit{T}}_{\mathit{i}8}$	${\mathit{T}}_{\mathit{i}9}$	${\mathit{T}}_{\mathit{i}10}$	${\mathit{T}}_{\mathit{i}11}$	${\mathit{T}}_{\mathit{i}12}$
1	BQ3	0.87	26	626	609	592	575	559	544	529	514	500	486	473	460
2	BQ4	0.72	16	547	523	499	477	455	435	415	396	378	361	344	328
3	BQ5	0.86	24	600	579	559	540	523	506	490	474	460	446	432	419
4	R5	0.86	11	570	515	470	432	399	371	346	324	305	287	271	-
5	R6	0.86	19	542	522	502	482	464	446	429	413	397	382	368	354
6	R7	0.85	16	588	549	515	486	459	436	415	395	377	361	346	332
7	R8	0.86	23	576	556	538	520	503	486	470	455	440	426	412	399
8	R9	0.95	10	585	525	476	434	399	368	341	317	295	276	-	-
9	BC3	0.72	8	541	468	413	369	334	304	278	256	-	-	-	-
10	BC4	0.85	12	608	552	506	468	436	408	383	362	342	325	309	295
11	BM4	0.70	15	515	491	467	445	423	403	383	365	347	330	313	298
12	BM5	0.70	23	526	509	492	477	461	447	432	418	405	392	380	368
13	BM6	0.69	19	537	517	497	478	460	442	425	409	393	378	363	349
14	SL4	0.85	13	627	579	537	500	467	438	412	389	367	348	330	314
15	SL5	0.84	11	610	560	515	476	441	410	383	358	335	315	296	-
16	SL6	0.85	10	650	597	548	505	467	432	401	372	347	324	-	-

and

Table 10. $H_{i0}$, $N_i$, and $T_{ij}$ for equipment i.

i	${\mathit{T}}_{\mathit{i}13}$	${\mathit{T}}_{\mathit{i}14}$	${\mathit{T}}_{\mathit{i}15}$	${\mathit{T}}_{\mathit{i}16}$	${\mathit{T}}_{\mathit{i}17}$	${\mathit{T}}_{\mathit{i}18}$	${\mathit{T}}_{\mathit{i}19}$	${\mathit{T}}_{\mathit{i}20}$	${\mathit{T}}_{\mathit{i}21}$	${\mathit{T}}_{\mathit{i}22}$	${\mathit{T}}_{\mathit{i}23}$	${\mathit{T}}_{\mathit{i}24}$	${\mathit{T}}_{\mathit{i}25}$	${\mathit{T}}_{\mathit{i}26}$
1	447	435	423	411	400	389	378	368	357	348	338	329	320	311
2	313	299	285	271	-	-	-	-	-	-	-	-	-	-
3	407	395	384	373	363	352	343	333	324	316	307	299	-	-
4	-	-	-	-	-	-	-	-	-	-	-	-	-	-
5	340	327	315	303	291	280	270	-	-	-	-	-	-	-
6	318	306	295	284	-	-	-	-	-	-	-	-	-	-
7	386	374	362	350	339	329	318	308	299	289	280	-	-	-
8	-	-	-	-	-	-	-	-	-	-	-	-	-	-
9	-	-	-	-	-	-	-	-	-	-	-	-	-	-
10	-	-	-	-	-	-	-	-	-	-	-	-	-	-
11	283	269	256	-	-	-	-	-	-	-	-	-	-	-
12	356	345	334	323	313	303	294	284	275	267	258	-	-	-
13	336	323	310	298	287	275	265	-	-	-	-	-	-	-
14	300	-	-	-	-	-	-	-	-	-	-	-	-	-
15	-	-	-	-	-	-	-	-	-	-	-	-	-	-
16	-	-	-	-	-	-	-	-	-	-	-	-	-	-

. $(T_{iP}^{lead},T_{iP}^{lag})$ can thus be obtained. Next, the equipment-level and initial flow-level maintenance optimization scheme for operation lines can be obtained by our proposed method.

The equipment-level maintenance timetables of reclaiming operation line u1 (BQ3–R5) and u2 (R6(R7)–BQ4) are shown in Figure 5a,b, where the blue section indicates normal operation of handling equipment, the red section indicates minimal repair, and the purple section indicates PdM. Since the adjustment of the lower-priority BQ3 within its PdMI threshold does not enable it to take advantage of the maintenance opportunities provided by the higher-priority R5, the maintenance interval of u1 is the union of the maintenance intervals for BQ3 and R5, whose initial flow-level maintenance timetable is shown in Figure 6a. Since R6 and R7 are in parallel, while the intersection of their actual shutdown interval can be an empty set by adjusting the PdMIL of lower-priority equipment R7, they can be equivalent to a piece of handling equipment running all the time, whose impact on the handling equipment BQ4 can be neglected. Therefore, the initial flow-level maintenance timetable of u2 is the same as the equipment-level maintenance timetable of BQ4 (shown in Figure 6b). Similarly, the initial flow-level maintenance timetables of u3, w1, w2, w3, v1, and v2 are shown in Figure 6c–h.

4.2.2. Scheduling Optimization Considering Maintenance

The basic parameters of the genetic algorithm are shown in

Table 11. Parameter setting of genetic algorithm.

Genetic Algebra	Population Number	Crossover Probability	Mutation Probability
200	200	0.9	0.2

. The two-layer algorithm is used to solve the scheduling optimization problem considering maintenance, and the ship scheduling scheme of the optimal solution is shown in Figure 7a–d. It can be seen from the calculation process diagram of the genetic algorithm (see Figure 8) that the algorithm has good convergence.

By comparing the optimal solution with the actual schedule of the terminal (see

Table 12. Comparison of the optimal solution and actual schedule of the terminal.

Item	Actual Schedule of Terminal (CNY)	Optimal Solution (CNY)	Cost Saving (%)
$C_{time}$	2,111,353	1,794,197	15.0
$C_{energy}$	3,353,103	3,342,973	0.3
$C_{maintenance}$	340,000	77,000	77.4
$C_{total}$	5,804,456	5,204,170	10.3

), it can be seen that the proposed method in this study can effectively reduce $C_{time}$ by 15.0% and $C_{maintenance}$ by 77.4%. On the one hand, the “equipment-level–flow-level” maintenance method reduces unnecessary equipment maintenance. On the other hand, the use of OM strategy, together with suitable scheduling of operation lines, decreases the effect of equipment maintenance on ship-loading operation, and reduces the average dwell time of a ship at terminal by 15.8% from 23.4 h to 19.7 h. For $C_{energy}$, since the start-up energy cost is relatively small compared with the transportation energy cost, while the transportation energy cost is almost fixed once the ship-loading operation task is determined, $C_{energy}$ can only be saved by 0.3%. Overall, the total cost of maintenance and scheduling for the ship-loading operation system in the coal export terminal $C_{total}$ can be saved by 10.3%, which verifies the effectiveness of the proposed method.

Furthermore, we employed the method in Gan et al. [47] and Abdollahzadeh-Sangroudi et al. [48] to solve the scheduling optimization problem of the ship-loading operation system considering equipment maintenance, and the comparison of optimal solutions by different methods is listed in

Table 13. Comparison of the optimal solution by using different methods.

Item	Gan et al. [47]	Abdollahzadeh-Sangroudi et al. [48]	This Paper
$C_{time}$	1,865,295	2,015,783	1,794,197
$C_{energy}$	3,342,973	3,342,973	3,342,973
$C_{maintenance}$	104,000	77,000	77,000
$C_{total}$	5,312,268	5,435,756	5,204,170

.

From the table, we can see that although Gan et al. [47] also used GA, the OM-based “equipment-level–flow-level” maintenance optimization in this paper shows its superiority, obtaining lower $C_{time}$ and $C_{maintenance}$ due to a better equipment maintenance scheme. As for the method in Abdollahzadeh-Sangroudi et al. [48], it can obtain the same optimal equipment maintenance scheme. However, compared with the Cuckoo Optimization Algorithm, GA with specific population initialization and genetic operator in this paper are more suitable for a ship-loading operation system, as it can obtain lower $C_{time}$.

Furthermore, we can find that better equipment maintenance schemes not only reduce unnecessary equipment maintenance time and cost but also effectively save the waiting time of ships at terminal caused by equipment failure and maintenance, leading to higher handling operation efficiency and lower operation cost of the coal export terminal.

5. Conclusions and Future Work

As important nodes of the global coal supply chain, coal export terminals meet with challenges including poor working conditions of the handling equipment and relatively fixed layout, such that frequent equipment failures seriously affect the efficiency and stability of ship-loading operations. To solve the problem, in this study, a scheduling optimization model for the ship-loading operation system in the coal export terminal considering equipment maintenance is established. To deal with the complicated interaction between scheduling and maintenance of the system, the concept of opportunity maintenance is introduced and a two-layer algorithm is proposed. The actual operation and maintenance data of a coal export terminal in China are taken as an example to verify the effectiveness of the proposed method, and the results show that the average dwell time reduction and total cost saving reach 15.8% and 10.3%, respectively. A comparison of the proposed method and the opportunity-maintenance-based scheduling method in existing studies is also presented.

To our knowledge, this is the first work that considers the effect of equipment maintenance on handling operation scheduling in coal export terminals. This study expresses how to deal with equipment maintenance in scheduling problems of port systems through actual operation and maintenance data. The proposed method can effectively reduce unnecessary equipment maintenance and decrease the waiting time of ships at terminal caused by equipment maintenance, which can provide ship-loading operation system scheduling plans for port managers, as well as equipment maintenance plans for equipment maintenance workers.

One of the main limitations of this paper lies in that the arranged berth of the ship arriving at the terminal is assumed to be known. In the future, the joint optimization of berth allocation and ship-loading operation system scheduling with the consideration of equipment maintenance may be another topic worth exploring.