# Optimization of Fleet’s Level Repair Plan for Diverse Missions

Authors: Tiejun Jiang; Chun Yu
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

Various missions significantly impact the fleet’s level repair plan. The mission requirement model is established based on the “regular and dual control” repair mode. Considering the fleet’s overall operation, capability decay characteristics, and planner’s expectations and preferences, the fleet’s capability evaluation model is constructed. Furthermore, aiming at the maximum capability value and the minimum number of the mission ships and comprehensively considering constraints such as sailing rate, mission period, repair period, repair cost, capability restoration, and mission capability, the optimization model of the fleet’s level repair plan for diverse missions is established. The particle swarm optimization algorithm based on the hierarchical sequence method is used to solve the model, and the fleet’s level repair plan and mission configuration plan are obtained, which solves the coordination problem of the two types of plans. The results show that, compared with the traditional planning method, this method can fully consider the actual requirements of diverse missions and has better coordination of the relationship between the ship’s use and repair. This method can provide strong technical support for the scientific preparation of level repair plans and the effective completion of combat readiness training missions.


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## Body

## 1. Introduction

Due to the long repair period, large cost requirements, high resource consumption, and significant impact on use, how to make a fleet’s repair plan scientifically is a critical concern in the field of ship repair. Especially with the expansion of naval missions, the increase in the ship’s use intensity, and the improvement of capability requirements, how to ensure that the ship’s capabilities can effectively meet the mission requirements and can scientifically coordinate the relationship between the ship’s use and repair is a critical issue in this field, which is a technical problem that needs to be solved urgently.Repair strategies for ships usually include preventive repair and postfailure repair. In this article, the ship’s level repair refers to a kind of preventive repair, and a ship’s level repair plan refers to the arrangement of different levels of preventive repair. The primary method of compiling a ship’s level repair plan is to arrange specific repair activities under a budget with a clear repair structure in relevant regulations. The repair structure includes the repair level, the repair interval time range, and the repair period range. Most of the related research works are carried out under this framework.In related research, the ship’s level repair plan optimization is usually regarded as a multiobjective decision-making problem under multicondition constraints. The ship’s level repair plan optimization is modeled as a constraint satisfaction problem, and an optimization algorithm is used to find the optimal solution that meets the constraints [1, 2].Considering various optimization objectives and constraints, different repair plan decision models are established accordingly. From the perspective of a single ship, Gu et al. proposed a state-based ship repair interval model [3]. He et al. established the ship’s LORA optimization model based on LINGO [4]. Considering mission factors, Liu et al. established a preventive repair planning model [5]. Cullum et al. proposed a risk-based repair scheduling method for naval vessels [6]. Ayik established the set partition model to determine the repair structure [7]. Ayik and Hall studied the impact of aircraft carrier repair schedules on deployment plans [8, 9]. Zhu et al. optimized the repair plan from the fleet’s perspective [10–12]. Li et al. used simulation analysis to realize the optimal arrangement of the repair structure [13]. Yardley et al. analyzed the repair cycles of an aircraft carrier and its effects [14–16]. Kamel et al. established a multilevel integrated preventive repair scheduling model [17]. Du et al. proposed a delay-time model for repair schedules [18]. Paralikar optimized the repair intervention-based machine learning methods [19–21]. According to the characteristics of problems, other methods, such as fuzzy logic, stochastic programming, and greedy heuristic local search algorithm, were also used to determine repair intervals [22, 23].According to the current research results, there are still some gaps in solving practical problems, which are mainly reflected in the following points:(1)
The existing works of literature seldom solve the problem from the perspective of the equipment system but simply optimize the repair plan from the perspective of sailing rate and integrity, without considering the effectiveness of the equipment system and the correlation between the use and repair of different types of ships. In fact, ships usually work in clusters during exercises, training, and operations. In order to give full play to the ship’s effectiveness, they are usually used in the fleet. Correspondingly, the ship’s level repair plan should also be arranged from the overall perspective of the fleet.(2)
Some existing works of literature optimize the repair plan from the perspective of mission availability or mission time requirements and rarely consider the impact of mission capability requirements. In fact, the ship’s fundamental purpose is to meet the requirements of a wide variety of missions, such as training, exercises, escort, combat, and other operations. In essence, the requirements on ships for various missions are requirements on their capabilities, and different missions need different types and strengths of ship capabilities. The ship’s level repair is an important means to ensure the timely and effective restoration of the ship’s capability. Therefore, the capability requirements of the mission should be fully considered when the ship’s level repair plan is optimized.(3)
Most of the existing works of literature do not follow the typical repair mode actually implemented at present, and there is a certain inadaptability in the application. The ship’s repair mode is the basis for optimizing the ship’s level repair plan. The current ship’s level repair arrangement is based on regular repairs with a fixed interval period, called as the “regular and dual control” repair mode. In this mode, the repair level, repair start time, and repair period are determined according to the ship’s repair structure in relevant regulations, the so-called control plan. The repair scope and depth are set according to the budget, the so-called control cost. The “regular and dual control” repair mode is a typical mode for naval ship repair. Therefore, the optimization of the ship’s level repair plan should be considered in this mode.Based on the current research status, this article analyzes the changes in the ship’s technical state, the relevant regulations on the ship repair structure, the repair cost, the sailing rate, the fleet operation, and the impacts of the diverse mission requirements on the ship’s level repair. The fleet’s level repair plan for various missions has been optimized.The primary motivation of this study is to propose the optimization method of the fleet’s level repair plan for diverse missions in the “regular and dual control” repair mode. The rest of this study is organized as follows. Section2 describes the mission fleet, the mission requirements model, and the evaluation model of the mission fleet’s capability. The optimization model of the fleet’s level repair plan is established in Section 3. Section 4 describes the model-solving process based on a hierarchical sequential particle swarm optimization algorithm. For illustration and verification purposes, a realistic problem scenario is used to test the effectiveness of the proposed methodology, and the corresponding results are reported in Section 5. Finally, some concluding remarks are drawn in Section 6.


## 2. Problem Analysis

The current problem is that in a periodT1,T2, for n ships of k types, how to make the fleet’s level repair plan in the case of v missions. In addition to the constraints of repair cost, repair time, and sailing rate, the fleet’s level repair plan also needs to meet the mission capability requirements and mission time constraints of various types of ships. Based on these conditions, planners decide on the ships to be repaired, the ship’s repair time, and the ships to be used for each mission. The goal is to maximize the capability of mission ships and minimize the number of mission ships in the planning period.


### 2.1. Mission Fleet

The set of all ships can be expressed as(1)S=S1,S2,⋯,Si,⋯,Sn,where S is the set of all ships, Si is the i-th ship i=1,2⋯,n, type is the ship type type=1,2,⋯,k, ntype is the number of various ships, and n=∑type=1kntype.The mission fleet is a subset of all ships. The mission fleet can be expressed as(2)SRj=SRj1,SRj2,⋯,SRji,⋯,SRjnRj,where Rj is the j-th mission j=1,2,⋯,v, SRj is the fleet set for the mission Rj, and nRj is the number of ships for the mission Rj.SRj is a subset of SSRj⊆S. There is a one-to-one correspondence between SRji in SRj and Si in S, as shown in Figure 1.Figure 1
Correspondence betweenS and SRj.


### 2.2. Mission Requirements Model

Mission requirements mainly include availability requirements and capability requirements. The availability requirement means that all the ships in the fleet are not in the repair state and remain available during the mission. The capability requirement implies that the fleet’s various subcapabilities meet the mission requirements [24–26].The fleet’s subcapability integrates the subcapability of a single ship, which can be generally divided into the following five primary types: command-and-control capability, maneuver capability, defense capability, offensive capability, and support capability [27].The mission is expressed as(3)R=R1,R2,⋯,Rj,⋯,Rvj=1,2,⋯,v,where R is the mission set faced by n ships in a period T1,T2.For the mission’s availability requirement, the mission ship cannot perform level repairs in the mission period, which means(4)WhenSi∈SRj,Rjs,Rje∩Tis,Tie=Φ,where Rjs,Rje is the execution period of the mission Rj, Tis,Tie is the repair period of the ship Si, and Φ is the empty set.For the mission’s capability requirement, the fleet’s subcapability should be greater than the minimum requirement value of the corresponding mission capability, which is expressed as(5)CXRjRjs≤t≤Rje≥EXminRj,CMRjRjs≤t≤Rje≥EMminRj,CFRjRjs≤t≤Rje≥EFminRj,CJRjRjs≤t≤Rje≥EJminRj,CLRjRjs≤t≤Rje≥ELminRj,where CXRj,CMRj,CFRj,CJRj,andCLRj , respectively, are the mission fleet’s subcapability values, which are command-and-control capability, maneuver capability, defense capability, offensive capability, and support capability; Rjs and Rje are the start time and end time of the mission Rj; and EXminRj,EMminRj,EFminRj,EJminRj,andELminRj are the minimum requirement values of the mission fleet’s subcapability.For the missionRj, k types of ships are used. The number of each type of ship is ntypeRjtype=1,2,⋯,k, and the total number of mission ships is(6)nRj=∑type=1kntypeRj.It is worth noting that the number of ships to be used for each mission is not directly given by the mission but is determined by combining the constraints of mission capability requirements and the comprehensive optimization of the ship’s level repair plan.


### 2.3. Evaluation Model of Mission Fleet’s Capability

Many practices show that as a ship performs various missions, its capability will decay [28, 29], which is shown as an S-shaped decreasing law with time. Considering the connotation and characteristics of capability decay and the advantage of the logistic function used to describe the growth process, the inverse form of the logistic function is used to describe capability decay [30].Generally, for the fleet’s subcapability, taking the command-and-control capability as an example (others are similar), it is expressed as(7)CXRjt=∑i=1nRjwRji∗XRjiekRji∗t−aRji+1,where CXRjt is the command-and-control capability value of the mission fleet, wRji is the weight of the i-th ship SRji, XRji is the initial command-and-control capability of SRji, kRji is the capability decay rate, aRji is the capability decay length, and t is the time.It should be noted that whent takes a particular time as the origin (such as the beginning of a particular year), the capability decay length should be aRji−bRji, where bRji is the elapsed time since the ship entered service or finished the last level of repair.Furthermore, considering the differences in capability expectations and risk preferences of different decision-makers [31, 32], the prospect theory is adopted to establish the capability value function. Taking the command-and-control capability of the mission fleet as an example, its value function is expressed as(8)UXRjt=CXRjt−EXRjα,CXRjt≥EXRj,−EXRj−CXRjtβ,CXRjt<EXRj,where UXRjt is the command-and-control capability value function of the fleet for the mission Rj, CXRjt is the actual value of the fleet’s subcapability, EXRj is the expected value of the fleet’s subcapability, and α and β are the risk aversion coefficients 0<α<1<β.Therefore, the fleet’s capability value function for the missionRj is(9)URjt=wc1UXRjt+wc2UMRjt+wc3UFRjt+wc4UJRjt+wc5ULRjt,where URjt is the mission fleet’s capability value function, wc1∼wc5 are the contribution weights of the subcapabilities to the overall capability, and UXRjt,UMRjt,UFRjt,UJRjt,andULRjt are the mission fleet’s subcapability value functions.


## 2.1. Mission Fleet

The set of all ships can be expressed as(1)S=S1,S2,⋯,Si,⋯,Sn,where S is the set of all ships, Si is the i-th ship i=1,2⋯,n, type is the ship type type=1,2,⋯,k, ntype is the number of various ships, and n=∑type=1kntype.The mission fleet is a subset of all ships. The mission fleet can be expressed as(2)SRj=SRj1,SRj2,⋯,SRji,⋯,SRjnRj,where Rj is the j-th mission j=1,2,⋯,v, SRj is the fleet set for the mission Rj, and nRj is the number of ships for the mission Rj.SRj is a subset of SSRj⊆S. There is a one-to-one correspondence between SRji in SRj and Si in S, as shown in Figure 1.Figure 1
Correspondence betweenS and SRj.


## 2.2. Mission Requirements Model

Mission requirements mainly include availability requirements and capability requirements. The availability requirement means that all the ships in the fleet are not in the repair state and remain available during the mission. The capability requirement implies that the fleet’s various subcapabilities meet the mission requirements [24–26].The fleet’s subcapability integrates the subcapability of a single ship, which can be generally divided into the following five primary types: command-and-control capability, maneuver capability, defense capability, offensive capability, and support capability [27].The mission is expressed as(3)R=R1,R2,⋯,Rj,⋯,Rvj=1,2,⋯,v,where R is the mission set faced by n ships in a period T1,T2.For the mission’s availability requirement, the mission ship cannot perform level repairs in the mission period, which means(4)WhenSi∈SRj,Rjs,Rje∩Tis,Tie=Φ,where Rjs,Rje is the execution period of the mission Rj, Tis,Tie is the repair period of the ship Si, and Φ is the empty set.For the mission’s capability requirement, the fleet’s subcapability should be greater than the minimum requirement value of the corresponding mission capability, which is expressed as(5)CXRjRjs≤t≤Rje≥EXminRj,CMRjRjs≤t≤Rje≥EMminRj,CFRjRjs≤t≤Rje≥EFminRj,CJRjRjs≤t≤Rje≥EJminRj,CLRjRjs≤t≤Rje≥ELminRj,where CXRj,CMRj,CFRj,CJRj,andCLRj , respectively, are the mission fleet’s subcapability values, which are command-and-control capability, maneuver capability, defense capability, offensive capability, and support capability; Rjs and Rje are the start time and end time of the mission Rj; and EXminRj,EMminRj,EFminRj,EJminRj,andELminRj are the minimum requirement values of the mission fleet’s subcapability.For the missionRj, k types of ships are used. The number of each type of ship is ntypeRjtype=1,2,⋯,k, and the total number of mission ships is(6)nRj=∑type=1kntypeRj.It is worth noting that the number of ships to be used for each mission is not directly given by the mission but is determined by combining the constraints of mission capability requirements and the comprehensive optimization of the ship’s level repair plan.


## 2.3. Evaluation Model of Mission Fleet’s Capability

Many practices show that as a ship performs various missions, its capability will decay [28, 29], which is shown as an S-shaped decreasing law with time. Considering the connotation and characteristics of capability decay and the advantage of the logistic function used to describe the growth process, the inverse form of the logistic function is used to describe capability decay [30].Generally, for the fleet’s subcapability, taking the command-and-control capability as an example (others are similar), it is expressed as(7)CXRjt=∑i=1nRjwRji∗XRjiekRji∗t−aRji+1,where CXRjt is the command-and-control capability value of the mission fleet, wRji is the weight of the i-th ship SRji, XRji is the initial command-and-control capability of SRji, kRji is the capability decay rate, aRji is the capability decay length, and t is the time.It should be noted that whent takes a particular time as the origin (such as the beginning of a particular year), the capability decay length should be aRji−bRji, where bRji is the elapsed time since the ship entered service or finished the last level of repair.Furthermore, considering the differences in capability expectations and risk preferences of different decision-makers [31, 32], the prospect theory is adopted to establish the capability value function. Taking the command-and-control capability of the mission fleet as an example, its value function is expressed as(8)UXRjt=CXRjt−EXRjα,CXRjt≥EXRj,−EXRj−CXRjtβ,CXRjt<EXRj,where UXRjt is the command-and-control capability value function of the fleet for the mission Rj, CXRjt is the actual value of the fleet’s subcapability, EXRj is the expected value of the fleet’s subcapability, and α and β are the risk aversion coefficients 0<α<1<β.Therefore, the fleet’s capability value function for the missionRj is(9)URjt=wc1UXRjt+wc2UMRjt+wc3UFRjt+wc4UJRjt+wc5ULRjt,where URjt is the mission fleet’s capability value function, wc1∼wc5 are the contribution weights of the subcapabilities to the overall capability, and UXRjt,UMRjt,UFRjt,UJRjt,andULRjt are the mission fleet’s subcapability value functions.


## 3. Optimization Model of Fleet’s Level Repair Plan

### 3.1. Optimization Goals

Ships exist for missions. For the fleet’s level repair plan for diverse missions, its goal is to maximize the mission fleet’s capability value in a certain period, and it is expressed as(10)maxfU=∑j=1v∫RjsRjeURjtdt,where fU is the sum of the capability value of the fleet for v missions in the planning period T1,T2 and URjt is the capability value function of the fleet for the mission Rj at a time t.In (10), when the mission has already started before the planning period, i.e., Rjs<T1, then let Rjs=T1; when the mission has not ended after the planning period, i.e., Rje>T2, then let Rje=T2.If the goal is only to maximize the mission fleet’s capability value, the optimization result will be to send as many ships as possible to perform the mission, which is not in line with reality and will cause a significant waste of resources. Therefore, based on meeting the mission’s capability requirements, a smaller number of ships should be dispatched to carry out the mission as much as possible to preserve the maximum combat readiness force and save resources. This is expressed by the following equation:(11)minnR=∑j=1vnRj=∑j=1v∑type=1kntypeRj,where nR is the sum of the number of ships for v missions, nRj is the number of ships for the mission Rj, ntypeRj are the number of ships of each type for the mission Rj, and k is the number of ship types.


### 3.2. Constraints

#### 3.2.1. Overall Sailing Rate

The ship’s sailing rate refers to the percentage of the number of ships that can generally sail to the total number. The overall sailing rate constraint can be expressed as(12)n−Nr−Nun≥r,where n is the total number of ships, Nr is the number of ships under repair, Nu is the number of ships under disrepair, and r is the specified minimum sailing rate requirement.


#### 3.2.2. Sailing Rate for Different Types of Ships

In the relevant regulations, different requirements are also put forward for the sailing rate of different types of ships. The sailing rate constraints of different types of ships are expressed as(13)ntype−Nrtype−Nutypentype≥rtype,where type is the ship type type=1,2,⋯,k, Nrtype is the number of ships under repair, Nutype is the number of ships under disrepair, and rtype is the minimum sailing rate requirement.


#### 3.2.3. Mission Requirements

The mission requirement constraints include mission availability constraints and mission capability constraints, and the fleet’s mission requirement model established in Section2.2 is directly used to establish the mission requirement constraints.


#### 3.2.4. Mission Time Overlap

When multiple missions overlap in a certain periodT11,T12, as shown by Ri and Rj in Figure 2, the same ship cannot appear in the two missions to avoid the time conflict, and this is expressed as(14)WhenRis,Rie∩Rjs,Rje≠Φ,Si∉SRii∩SRji.Figure 2
Mission time distribution.


#### 3.2.5. Mission Period

When a particular mission enters the next periodT21,T22 in a certain period T11,T12, as shown by Rk in Figure 2, the mission fleet’s capability is only calculated in a certain period T11,T12.


#### 3.2.6. Repair Period

The repair period is determined for the relevant regulations:(15)Tie−Tis=ti,where Tie is the level repair end time of the ship Si, Tis is the level repair start time of the ship Si, and ti is the prescribed level repair period of the ship Si by regulations.


#### 3.2.7. Repair Cost

The total budget price of the ships to be repaired will be lower than that of the total budget. The total cost constraint is expressed as(16)∑i=1nxici≤C.Whenxi is 0, the ship Si is not repaired; when xi is 1, the ship Si needs to be repaired; ci is the budget price of the ship Si; and C is the total budget.


#### 3.2.8. Intertemporal Repair Cost

When there are ships under repair across the planning period, the budget for this period is the total budget price multiplied by the ratio of the repair time in this period to the total repair time, that is,(17)ci=Ci⋅ti′ti,where ci is the budget for the intertemporal repair of the ship Si in this period, Ci is the level repair budget price of the ship Si, ti′ is the repair time of the ship Si in the planning period, and ti is the level repair period of the ship Si.


#### 3.2.9. Capability Restoration

In general, the repair will only allow a limited restoration of the ship’s capability and will not reach the initial capability at the time when the ship entered service or finished the last level of repair; thus, we get(18)Pio′+ΔPi≤Pio,where Pio′ is the capability of the ship Si before the level repair; ∆Pi is the capability of the ship Si that can be restored after the level repair; ∆Pi=fci,ti is the relationship among the budget price ci, the repair period ti, and the restored capability ∆Pi; the positive correlation function ∆Pi can be regarded as a fixed value in the “regular and dual control” repair mode; and Pio is the initial effectiveness of the ship Si, that is, the effectiveness at the starting time of decision.From another point of view,Pio′ should be lower than Pio−∆Pi, that is, before the ship’s capability drops to ∆Pi, the ship is still in a relatively good state, and the repair is usually not considered at this time.


## 3.1. Optimization Goals

Ships exist for missions. For the fleet’s level repair plan for diverse missions, its goal is to maximize the mission fleet’s capability value in a certain period, and it is expressed as(10)maxfU=∑j=1v∫RjsRjeURjtdt,where fU is the sum of the capability value of the fleet for v missions in the planning period T1,T2 and URjt is the capability value function of the fleet for the mission Rj at a time t.In (10), when the mission has already started before the planning period, i.e., Rjs<T1, then let Rjs=T1; when the mission has not ended after the planning period, i.e., Rje>T2, then let Rje=T2.If the goal is only to maximize the mission fleet’s capability value, the optimization result will be to send as many ships as possible to perform the mission, which is not in line with reality and will cause a significant waste of resources. Therefore, based on meeting the mission’s capability requirements, a smaller number of ships should be dispatched to carry out the mission as much as possible to preserve the maximum combat readiness force and save resources. This is expressed by the following equation:(11)minnR=∑j=1vnRj=∑j=1v∑type=1kntypeRj,where nR is the sum of the number of ships for v missions, nRj is the number of ships for the mission Rj, ntypeRj are the number of ships of each type for the mission Rj, and k is the number of ship types.


## 3.2. Constraints

### 3.2.1. Overall Sailing Rate

The ship’s sailing rate refers to the percentage of the number of ships that can generally sail to the total number. The overall sailing rate constraint can be expressed as(12)n−Nr−Nun≥r,where n is the total number of ships, Nr is the number of ships under repair, Nu is the number of ships under disrepair, and r is the specified minimum sailing rate requirement.


### 3.2.2. Sailing Rate for Different Types of Ships

In the relevant regulations, different requirements are also put forward for the sailing rate of different types of ships. The sailing rate constraints of different types of ships are expressed as(13)ntype−Nrtype−Nutypentype≥rtype,where type is the ship type type=1,2,⋯,k, Nrtype is the number of ships under repair, Nutype is the number of ships under disrepair, and rtype is the minimum sailing rate requirement.


### 3.2.3. Mission Requirements

The mission requirement constraints include mission availability constraints and mission capability constraints, and the fleet’s mission requirement model established in Section2.2 is directly used to establish the mission requirement constraints.


### 3.2.4. Mission Time Overlap

When multiple missions overlap in a certain periodT11,T12, as shown by Ri and Rj in Figure 2, the same ship cannot appear in the two missions to avoid the time conflict, and this is expressed as(14)WhenRis,Rie∩Rjs,Rje≠Φ,Si∉SRii∩SRji.Figure 2
Mission time distribution.


### 3.2.5. Mission Period

When a particular mission enters the next periodT21,T22 in a certain period T11,T12, as shown by Rk in Figure 2, the mission fleet’s capability is only calculated in a certain period T11,T12.


### 3.2.6. Repair Period

The repair period is determined for the relevant regulations:(15)Tie−Tis=ti,where Tie is the level repair end time of the ship Si, Tis is the level repair start time of the ship Si, and ti is the prescribed level repair period of the ship Si by regulations.


### 3.2.7. Repair Cost

The total budget price of the ships to be repaired will be lower than that of the total budget. The total cost constraint is expressed as(16)∑i=1nxici≤C.Whenxi is 0, the ship Si is not repaired; when xi is 1, the ship Si needs to be repaired; ci is the budget price of the ship Si; and C is the total budget.


### 3.2.8. Intertemporal Repair Cost

When there are ships under repair across the planning period, the budget for this period is the total budget price multiplied by the ratio of the repair time in this period to the total repair time, that is,(17)ci=Ci⋅ti′ti,where ci is the budget for the intertemporal repair of the ship Si in this period, Ci is the level repair budget price of the ship Si, ti′ is the repair time of the ship Si in the planning period, and ti is the level repair period of the ship Si.


### 3.2.9. Capability Restoration

In general, the repair will only allow a limited restoration of the ship’s capability and will not reach the initial capability at the time when the ship entered service or finished the last level of repair; thus, we get(18)Pio′+ΔPi≤Pio,where Pio′ is the capability of the ship Si before the level repair; ∆Pi is the capability of the ship Si that can be restored after the level repair; ∆Pi=fci,ti is the relationship among the budget price ci, the repair period ti, and the restored capability ∆Pi; the positive correlation function ∆Pi can be regarded as a fixed value in the “regular and dual control” repair mode; and Pio is the initial effectiveness of the ship Si, that is, the effectiveness at the starting time of decision.From another point of view,Pio′ should be lower than Pio−∆Pi, that is, before the ship’s capability drops to ∆Pi, the ship is still in a relatively good state, and the repair is usually not considered at this time.


## 3.2.1. Overall Sailing Rate

The ship’s sailing rate refers to the percentage of the number of ships that can generally sail to the total number. The overall sailing rate constraint can be expressed as(12)n−Nr−Nun≥r,where n is the total number of ships, Nr is the number of ships under repair, Nu is the number of ships under disrepair, and r is the specified minimum sailing rate requirement.


## 3.2.2. Sailing Rate for Different Types of Ships

In the relevant regulations, different requirements are also put forward for the sailing rate of different types of ships. The sailing rate constraints of different types of ships are expressed as(13)ntype−Nrtype−Nutypentype≥rtype,where type is the ship type type=1,2,⋯,k, Nrtype is the number of ships under repair, Nutype is the number of ships under disrepair, and rtype is the minimum sailing rate requirement.


## 3.2.3. Mission Requirements

The mission requirement constraints include mission availability constraints and mission capability constraints, and the fleet’s mission requirement model established in Section2.2 is directly used to establish the mission requirement constraints.


## 3.2.4. Mission Time Overlap

When multiple missions overlap in a certain periodT11,T12, as shown by Ri and Rj in Figure 2, the same ship cannot appear in the two missions to avoid the time conflict, and this is expressed as(14)WhenRis,Rie∩Rjs,Rje≠Φ,Si∉SRii∩SRji.Figure 2
Mission time distribution.


## 3.2.5. Mission Period

When a particular mission enters the next periodT21,T22 in a certain period T11,T12, as shown by Rk in Figure 2, the mission fleet’s capability is only calculated in a certain period T11,T12.


## 3.2.6. Repair Period

The repair period is determined for the relevant regulations:(15)Tie−Tis=ti,where Tie is the level repair end time of the ship Si, Tis is the level repair start time of the ship Si, and ti is the prescribed level repair period of the ship Si by regulations.


## 3.2.7. Repair Cost

The total budget price of the ships to be repaired will be lower than that of the total budget. The total cost constraint is expressed as(16)∑i=1nxici≤C.Whenxi is 0, the ship Si is not repaired; when xi is 1, the ship Si needs to be repaired; ci is the budget price of the ship Si; and C is the total budget.


## 3.2.8. Intertemporal Repair Cost

When there are ships under repair across the planning period, the budget for this period is the total budget price multiplied by the ratio of the repair time in this period to the total repair time, that is,(17)ci=Ci⋅ti′ti,where ci is the budget for the intertemporal repair of the ship Si in this period, Ci is the level repair budget price of the ship Si, ti′ is the repair time of the ship Si in the planning period, and ti is the level repair period of the ship Si.


## 3.2.9. Capability Restoration

In general, the repair will only allow a limited restoration of the ship’s capability and will not reach the initial capability at the time when the ship entered service or finished the last level of repair; thus, we get(18)Pio′+ΔPi≤Pio,where Pio′ is the capability of the ship Si before the level repair; ∆Pi is the capability of the ship Si that can be restored after the level repair; ∆Pi=fci,ti is the relationship among the budget price ci, the repair period ti, and the restored capability ∆Pi; the positive correlation function ∆Pi can be regarded as a fixed value in the “regular and dual control” repair mode; and Pio is the initial effectiveness of the ship Si, that is, the effectiveness at the starting time of decision.From another point of view,Pio′ should be lower than Pio−∆Pi, that is, before the ship’s capability drops to ∆Pi, the ship is still in a relatively good state, and the repair is usually not considered at this time.


## 4. Model Solving Based on the Hierarchical Sequential Particle Swarm Optimization Algorithm

There are two optimization objectives in the model, and they conflict with each other. To ensure combat readiness and prevent the resource waste caused by too many ships to perform the mission, the minimum number of mission ships is identified as the primary objective. The minimum number of ships to be used for each mission is obtained when the constraints are met. Therefore, a rough mission plan is made. With the maximum capability value of the mission fleet, the optimization solution can obtain the ships to be repaired, the ship’s repair time, and the ships to be used for each mission.Considering the model’s characteristics of multiobjective hierarchical optimization, the hierarchical sequence method in multiobjective optimization is suitable to solve the model. The hierarchical sequence method refers to sorting the importance of multiple objectives required by the problem, first obtaining the optimal solution for the objective with the first importance and then obtaining the next objective based on the optimal solution of the previous objective. The optimal solution is obtained until the last goal [33].The implementation steps of the model-solving based on the hierarchical sequence method are as follows:(1)
Regardless of the maximum capability value of the mission fleet, the objective is to minimize the number of mission ships, and the optimal results are obtained asnRj∗j=1,2,⋯,v and ntypeRj∗type=1,2,⋯,k.(19)minnR=∑j=1vnRj=∑j=1v∑type=1kntypeRj,s.t.I∑i=1nxici≤C,Tie−Tis=ti,Pio′+∆Pi≤Pio,n−Nr−Nun≥r,ntype−Nrtype−Nutypentype≥rtype,type=1,2,⋯,k,CXRjRjs≤t≤Rje≥EXRj,CMRjRjs≤t≤Rje≥EMRj,CFRjRjs≤t≤Rje≥EFRj,CJRjRjs≤t≤Rje≥EJRj,CLRjRjs≤t≤Rje≥ELRj,whenSi∈SRji,Rjs,Rje∩Tis,Tie=Φ,whenRis,Rie∩Rjs,Rje≠Φ,Si∉SRii∩SRji.(2)
The number and type of ship to be used for each mission through the first step are used as additional constraints for solving the second objective. At this time, the optimized model is(20)maxfU=∑j=1v∫RjsRje∑i=1nURjtdt,s.t.IIs.t.I,ntypeRj=ntypeRj∗,j=1,2,⋯,v,type=1,2,⋯,k.The decision variables of the model mainly include whether the ship is repaired, the ship’s repair start time, and the number of ships to be used for each mission.The particle swarm optimization algorithm is a parallel, efficient, and global search method, and it can use binary coding or continuous coding. Its application effect is better in multiobjective, nonlinear integer, and mixed-integer constrained optimization problems [34, 35]. This article uses the particle swarm optimization algorithm to solve the model.


### 4.1. Coding of Particles

In the decision variables of the model, there are both discrete and continuous variables, both 0-1 and integer variables. A hybrid coding method is used to express the decision variables, as shown in Figure3. The decision variables are specifically expressed as x1,x2,⋯,xn,T1s,T2s,⋯,Tns,SR11,SR12,⋯,SR1nR1,⋯,SRv1,SRv2,⋯,SRvnRv. x1,x2,⋯,xn adopt discrete binary coding and take 0 or 1 to indicate whether the ship is repaired. T1s,T2s,⋯,Tns adopt continuous coding and take a number between T1 and T2 to indicate the ship’s repair start time. SR11,SR12,⋯,SR1nR1,⋯,SRv1,SRv2,⋯,SRvnRv adopt integer coding and take 1,2,⋯,n to indicate the number of the ship to be used for missions R1 to Rv.Figure 3
Particle coding for diverse missions.For a population composed ofN particles, the current position Qm of the particle mm=1,2,⋯,N is xm1,xm2,⋯,xmn,Tm1s,Tm2s,⋯,Tmns,SmR11,SmR12,⋯,SmR1nR1,⋯,SmRv1,SmRv2,⋯,SmRvnRv.xm1,xm2,⋯,xmn are a set of solutions corresponding to the decision variables x1,x2,⋯,xn of the m-th particle, indicating whether the ship is repaired. Tm1s,Tm2s,⋯,Tmns are a set of solutions corresponding to the decision variables T1s,T2s,⋯,Tns of the m-th particle, indicating the ship’s start repair time. SmR11 and SmR12,⋯,SmR1nR1,⋯,SmRv1,SmRv2,⋯,SmRvnRv are a set of solutions of the m-th particle corresponding to the decision variables SR11,SR12,⋯,SR1nR1,⋯,SRv1,SRv2,⋯,SRvnRv, indicating the number of the ship to be used for missions R1 to Rv.The current flight speedVm is Vxm1,Vxm2,⋯,Vxmn,VTm1s,VTm2s,⋯,VTmns,VSmR11,VSmR12,⋯,VSmR1nR1,⋯,VSmRv1,VSmRv2,⋯,VSmRvnRv. The particle position is a multidimensional variable, and the flight speed is different for each dimension. Therefore, the flight speed is also multidimensional. Vm represents the flight speed of the corresponding particle position Xm in different dimensions. Vxm1,Vxm2,⋯,Vxmn represent the flight speed of xm1,xm2,⋯,xmn. VTm1s,VTm2s,⋯,VTmns represent the flight speed of Tm1s,Tm2s,⋯,Tmns. VSmR11,VSmR12,⋯,VSmR1nR1,⋯,VSmRv1,VSmRv2,⋯,VSmRvnRv represent the flight speed of SmR11,SmR12,⋯,SmR1nR1,⋯,SmRv1,SmRv2,⋯,SmRvnRv.The number of swarm particlesN, the number of iterations Kmax, the initial position Qm1, the initial velocity Vm1, the local optimal position Pm, the best fitness of individual particles pbest, the global optimal position g, and the best fitness of all particles gbest are initialized. Parameters such as learning factors c1 and c2, inertia weight w, maximum position Xmax, minimum position Xmin, maximum speed Vmax, and minimum speed Vmin are set.


### 4.2. Calculation of Fitness

The second objective functionfU in the optimization model is directly used as the fitness function in the particle swarm optimization algorithm, and the fitness of each particle is calculated. When the iterative calculation reaches the u-th generation, the position of the particle m is Qmu and the velocity is Vmu.Pm is the position of the particle m which has the best fitness in the previous u−1 generations, called the individual optimal position. If the fitness corresponding to Qmu is better than the fitness corresponding to Pm then Pm=Qmu; otherwise, Pm remains unchanged. Pg is the particle’s position which has the best fitness in the previous u−1 generation, called the global optimal position. If the fitness corresponding to a particle’s position is better than the fitness corresponding to Pg, then Pg is updated to the particle’s position; otherwise, Pg remains unchanged.


### 4.3. Update of Speed and Position

The flight speed and position of each particle are updated synchronously.


#### 4.3.1. Update of Flight Speed

The flight speed is expressed as(21)Vmu+1=wu∗Vmu+c1∗rand∗Pm−Qmu+c2∗rand∗Pg−Qmu,where Vmu+1 is the speed of the u+1-th generation particle m, Vmu is the speed of the u-th generation particle m, c1 is the preset learning factor relative to the local optimal position Pm of the particle m, c2 is the preset learning factor relative to the global optimal position Pg of the particle m, rand is a random number in the range of (0, 1), wu is the inertia weight used to control the search range, and Qmu represents the position of the u-th generation particle m.When the particle’s speed is too large, it may jump out of the optimization range. So, its speed in each dimension is limited withinVmin,Vmax. For the particle exceeding the speed range, its speed is reinitialized. When the flight speed is updated, for the inertia weight wu, a linear decreasing weight strategy is adopted [36, 37] which is expressed as(22)wu=wmax−wmax−wminTmax∗u,where wmax is the maximum inertia weight, wmin is the minimum inertia weight, k is the current iteration number, and Tmax is the maximum iteration number.


#### 4.3.2. Update of Particle Position

Different update methods are adopted for different coding methods [38, 39].For discrete binary coding, the update expression is(23)Qmu+1=1rand<sVmu+1,0rand≥sVmu+1,sVmu+1=11+e−Vmu+1.For continuous coding, the update expression is(24)Qmu+1=Qmu+Vmu+1.For integer coding, the update expression is(25)Qmu+1=roundQmu+Vmu+1,where Qmu+1 is the position of the u+1-th generation particle m, Qmu is the position of the u-th generation particle m, s is the transformation function of particle flight speed, and round is the rounding function according to the rounding criterion.Since the position with integer coding describes the ship’s number (take1,2,⋯,n), those positions that exceed the range will be initialized, which is expressed by the following equation: (26)WhenQmu+1<1orQmu+1>n,Qmu+1=roundrand∗n.


### 4.4. Iterative Optimization

The iterative calculation is carried out according to the abovementioned steps. The optimal moderate value of the particle population and the corresponding particle position in the previous iterations are recorded and updated. The flight speed and position of the particles are updated synchronously.The particle position with the optimal fitness value is obtained when the maximum iteration numberTmax is reached. The particle position corresponds to the optimal decision variables in the fleet’s level repair plan model, namely, the ships to be repaired, the ship’s repair start time, and the ships to be used for each mission. According to the optimization calculation results, the fleet’s level repair plan can be obtained.


## 4.1. Coding of Particles

In the decision variables of the model, there are both discrete and continuous variables, both 0-1 and integer variables. A hybrid coding method is used to express the decision variables, as shown in Figure3. The decision variables are specifically expressed as x1,x2,⋯,xn,T1s,T2s,⋯,Tns,SR11,SR12,⋯,SR1nR1,⋯,SRv1,SRv2,⋯,SRvnRv. x1,x2,⋯,xn adopt discrete binary coding and take 0 or 1 to indicate whether the ship is repaired. T1s,T2s,⋯,Tns adopt continuous coding and take a number between T1 and T2 to indicate the ship’s repair start time. SR11,SR12,⋯,SR1nR1,⋯,SRv1,SRv2,⋯,SRvnRv adopt integer coding and take 1,2,⋯,n to indicate the number of the ship to be used for missions R1 to Rv.Figure 3
Particle coding for diverse missions.For a population composed ofN particles, the current position Qm of the particle mm=1,2,⋯,N is xm1,xm2,⋯,xmn,Tm1s,Tm2s,⋯,Tmns,SmR11,SmR12,⋯,SmR1nR1,⋯,SmRv1,SmRv2,⋯,SmRvnRv.xm1,xm2,⋯,xmn are a set of solutions corresponding to the decision variables x1,x2,⋯,xn of the m-th particle, indicating whether the ship is repaired. Tm1s,Tm2s,⋯,Tmns are a set of solutions corresponding to the decision variables T1s,T2s,⋯,Tns of the m-th particle, indicating the ship’s start repair time. SmR11 and SmR12,⋯,SmR1nR1,⋯,SmRv1,SmRv2,⋯,SmRvnRv are a set of solutions of the m-th particle corresponding to the decision variables SR11,SR12,⋯,SR1nR1,⋯,SRv1,SRv2,⋯,SRvnRv, indicating the number of the ship to be used for missions R1 to Rv.The current flight speedVm is Vxm1,Vxm2,⋯,Vxmn,VTm1s,VTm2s,⋯,VTmns,VSmR11,VSmR12,⋯,VSmR1nR1,⋯,VSmRv1,VSmRv2,⋯,VSmRvnRv. The particle position is a multidimensional variable, and the flight speed is different for each dimension. Therefore, the flight speed is also multidimensional. Vm represents the flight speed of the corresponding particle position Xm in different dimensions. Vxm1,Vxm2,⋯,Vxmn represent the flight speed of xm1,xm2,⋯,xmn. VTm1s,VTm2s,⋯,VTmns represent the flight speed of Tm1s,Tm2s,⋯,Tmns. VSmR11,VSmR12,⋯,VSmR1nR1,⋯,VSmRv1,VSmRv2,⋯,VSmRvnRv represent the flight speed of SmR11,SmR12,⋯,SmR1nR1,⋯,SmRv1,SmRv2,⋯,SmRvnRv.The number of swarm particlesN, the number of iterations Kmax, the initial position Qm1, the initial velocity Vm1, the local optimal position Pm, the best fitness of individual particles pbest, the global optimal position g, and the best fitness of all particles gbest are initialized. Parameters such as learning factors c1 and c2, inertia weight w, maximum position Xmax, minimum position Xmin, maximum speed Vmax, and minimum speed Vmin are set.


## 4.2. Calculation of Fitness

The second objective functionfU in the optimization model is directly used as the fitness function in the particle swarm optimization algorithm, and the fitness of each particle is calculated. When the iterative calculation reaches the u-th generation, the position of the particle m is Qmu and the velocity is Vmu.Pm is the position of the particle m which has the best fitness in the previous u−1 generations, called the individual optimal position. If the fitness corresponding to Qmu is better than the fitness corresponding to Pm then Pm=Qmu; otherwise, Pm remains unchanged. Pg is the particle’s position which has the best fitness in the previous u−1 generation, called the global optimal position. If the fitness corresponding to a particle’s position is better than the fitness corresponding to Pg, then Pg is updated to the particle’s position; otherwise, Pg remains unchanged.


## 4.3. Update of Speed and Position

The flight speed and position of each particle are updated synchronously.


### 4.3.1. Update of Flight Speed

The flight speed is expressed as(21)Vmu+1=wu∗Vmu+c1∗rand∗Pm−Qmu+c2∗rand∗Pg−Qmu,where Vmu+1 is the speed of the u+1-th generation particle m, Vmu is the speed of the u-th generation particle m, c1 is the preset learning factor relative to the local optimal position Pm of the particle m, c2 is the preset learning factor relative to the global optimal position Pg of the particle m, rand is a random number in the range of (0, 1), wu is the inertia weight used to control the search range, and Qmu represents the position of the u-th generation particle m.When the particle’s speed is too large, it may jump out of the optimization range. So, its speed in each dimension is limited withinVmin,Vmax. For the particle exceeding the speed range, its speed is reinitialized. When the flight speed is updated, for the inertia weight wu, a linear decreasing weight strategy is adopted [36, 37] which is expressed as(22)wu=wmax−wmax−wminTmax∗u,where wmax is the maximum inertia weight, wmin is the minimum inertia weight, k is the current iteration number, and Tmax is the maximum iteration number.


### 4.3.2. Update of Particle Position

Different update methods are adopted for different coding methods [38, 39].For discrete binary coding, the update expression is(23)Qmu+1=1rand<sVmu+1,0rand≥sVmu+1,sVmu+1=11+e−Vmu+1.For continuous coding, the update expression is(24)Qmu+1=Qmu+Vmu+1.For integer coding, the update expression is(25)Qmu+1=roundQmu+Vmu+1,where Qmu+1 is the position of the u+1-th generation particle m, Qmu is the position of the u-th generation particle m, s is the transformation function of particle flight speed, and round is the rounding function according to the rounding criterion.Since the position with integer coding describes the ship’s number (take1,2,⋯,n), those positions that exceed the range will be initialized, which is expressed by the following equation: (26)WhenQmu+1<1orQmu+1>n,Qmu+1=roundrand∗n.


## 4.3.1. Update of Flight Speed

The flight speed is expressed as(21)Vmu+1=wu∗Vmu+c1∗rand∗Pm−Qmu+c2∗rand∗Pg−Qmu,where Vmu+1 is the speed of the u+1-th generation particle m, Vmu is the speed of the u-th generation particle m, c1 is the preset learning factor relative to the local optimal position Pm of the particle m, c2 is the preset learning factor relative to the global optimal position Pg of the particle m, rand is a random number in the range of (0, 1), wu is the inertia weight used to control the search range, and Qmu represents the position of the u-th generation particle m.When the particle’s speed is too large, it may jump out of the optimization range. So, its speed in each dimension is limited withinVmin,Vmax. For the particle exceeding the speed range, its speed is reinitialized. When the flight speed is updated, for the inertia weight wu, a linear decreasing weight strategy is adopted [36, 37] which is expressed as(22)wu=wmax−wmax−wminTmax∗u,where wmax is the maximum inertia weight, wmin is the minimum inertia weight, k is the current iteration number, and Tmax is the maximum iteration number.


## 4.3.2. Update of Particle Position

Different update methods are adopted for different coding methods [38, 39].For discrete binary coding, the update expression is(23)Qmu+1=1rand<sVmu+1,0rand≥sVmu+1,sVmu+1=11+e−Vmu+1.For continuous coding, the update expression is(24)Qmu+1=Qmu+Vmu+1.For integer coding, the update expression is(25)Qmu+1=roundQmu+Vmu+1,where Qmu+1 is the position of the u+1-th generation particle m, Qmu is the position of the u-th generation particle m, s is the transformation function of particle flight speed, and round is the rounding function according to the rounding criterion.Since the position with integer coding describes the ship’s number (take1,2,⋯,n), those positions that exceed the range will be initialized, which is expressed by the following equation: (26)WhenQmu+1<1orQmu+1>n,Qmu+1=roundrand∗n.


## 4.4. Iterative Optimization

The iterative calculation is carried out according to the abovementioned steps. The optimal moderate value of the particle population and the corresponding particle position in the previous iterations are recorded and updated. The flight speed and position of the particles are updated synchronously.The particle position with the optimal fitness value is obtained when the maximum iteration numberTmax is reached. The particle position corresponds to the optimal decision variables in the fleet’s level repair plan model, namely, the ships to be repaired, the ship’s repair start time, and the ships to be used for each mission. According to the optimization calculation results, the fleet’s level repair plan can be obtained.


## 5. Application Analysis

### 5.1. Problem Description

The problem scenario is that there are seven ships in a fleet in a particular year, and the repair budget is 70 million yuan. The annual fleet’s level repair plan is made under the “regular and dual control” repair mode. Due to confidentiality, the ship’s repair-related data are adequately processed.(1)
The ship’s initial capabilities are shown in Table1. All ships are divided into the following three types: destroyer, frigate, and supply ships. Experts evaluate the initial capability according to the various performance indicators. The planner shall determine the contribution weight of each ship’s capability to the overall capability by the strategic objectives, missions, relevant regulations, and instructions, and the opinions of the unit, that is, wc1,wc2,wc3,wc4,wc5 = (0.2, 0.15, 0.25, 0.25, 0.15).(2)
The ship’s repair-related data are shown in Table2. Relevant regulations and historical standards determine the repair level, the budget price, and the repair period. The restored capability is evaluated based on the historical data of the same type of ship on capability changes. The capability decay function is obtained by fitting various capabilities’ monitoring and evaluation data at different time points. The used time refers to the interval between the time when the ship entered service or finished the last level of repair and the beginning of the year. The ship’s operating department shall determine the lower limit of the ship’s operational capability based on experiences and historical operational conditions.Ship 1 has been repaired from the previous year to the current year, and the plan has been determined. Specifically, it has been repaired for 2.5 months in the previous year, and it needs to be repaired for 2.5 months this year. The other ship’s level repair plans are to be arranged. For ease of calculation, the unit “month” is counted as 30 days.(3)
The importance and mission assignment determine the weight of each ship in the fleet, that is,w1,w2,w3,w4,w5,w6,w7 = (0.17, 0.17, 0.14, 0.14, 0.14, 0.12, 0.12). The strategic objectives, superior instructions, and troop opinions determine the fleet’s expected capability. To simplify the calculation, the fleet’s expected capabilities for different missions are set as the same, that is, EXRj,EMRj,EFRj,EJRj,ELRj = (25, 40, 25, 25, 30). The planner’s risk preference determines the risk aversion coefficients α and β, which are 0.6 and 1.2, respectively.(4)
The overall minimum sailing rate is determined asr=60% according to the troop type and current regulations. These ships are divided into the following three types: destroyer, frigate, and supply ships. According to the statistical evaluation of the ship’s sailing rate data over the years, the minimum sailing rate of destroyer, frigate, and supply ships is determined as r1=50%, r2=60%, and r3=50%.(5)
The mission requirements are shown in Table3. The mission period is given according to the mission plan. The minimum requirement values of the mission fleet’s subcapability are obtained according to the historical execution of the mission and the experience of the ship’s operation department.Table 1
Ship’s initial capabilities.
Ship numberShip typeCommand-and-control capabilityManeuver capabilityDefense capabilityOffensive capabilitySupport capabilityComprehensive capabilityShip 1Destroyer909090909090Ship 2Destroyer909075757380Ship 3Frigate506080604060Ship 4Frigate356070504052Ship 5Frigate356060453547.5Ship 6Supply106025259037Ship 7Supply106025259037Table 2
Ship’s repair-related data.
Ship numberRepair levelBudget price (RMB 104 yuan)Repair period (month)Restored capabilityCapability decay functionTime used (month)The lower limit of capabilityShip 1Level 1400054590/1+e0.4∗t−2424.530Ship 2Level 315001.52080/1+e0.4∗t−231850Ship 3Level 120002.53060/1+e0.4∗t−231020Ship 4Level 210001.51752/1+e0.5∗t−21.51322Ship 5Level 350011247.5/1+e0.45∗t−19.5625Ship 6Level 29001.51943.75/1+e0.35∗t−2114.514Ship 7Level 34501937/1+e0.35∗t−211316Table 3
Mission requirements.
Mission numberMission periodRequirements for subcapabilityMission 1(April 1st, May 30th)(15, 16, 16, 19, and 22)Mission 2(August 1st, September 30th)(10, 13, 16, 17, and 19)


### 5.2. Parameter Setting and Model Solution

According to relevant research studies and historical experiences [40, 41], the parameters of the particle swarm optimization algorithm are set as N=100, Tmax=200, c1=1.5, c2=1.5, wmax=0.9, wmin=0.4, Vmax=2, and Vmin=−2.Through the first step of the hierarchical sequence method, the calculation result obtained isn1R1=1, n3R1=1, n2R2=2, and n3R1=1; that is, Mission 1 requires one destroyer and one supply ship; Mission 2 requires two frigates and one supply ship.Through the second step of the hierarchical sequence method, the calculation result is the optimal position of the particleg = (1, 0, 1, 0, 1, 1, 2.84, 7.19, 7.16, 3.24, 6.01, 4.76, 1, 6, 3, 5, 7) and the optimal fitness gbest = 2.0275.As Ship 1 crosses into this year from the previous year, its repair time is known. The 1st to 6th digits ofg represent whether Ship 2 to Ship 7 are repaired. The 7th to 12th digits represent the repair start time of Ship 2 to Ship 7.The 13th and 14th digits represent the ship’s number for Mission 1. The 15th to 17th digits represent the ship’s number for Mission 2.gbest represents the mission fleet’s system capability value.The mission plan is shown in Table4. The fleet’s level repair plan is shown in Table 5. The available status of the ship is shown in Figure 4, where 0 means available and 1 means unavailable.Table 4
Mission plan.
Mission numberMission periodCapability requirementShips for the missionMission fleet’s capabilityMission 1(April 1, May 30)(15, 19, 16, 16, and 22)Ship 1, 62.0275Mission 2(August 1st, September 30th)(10, 17, 19, 13, and16)Ship 3, 5, 7Table 5
Ship’s level repair plan for diverse missions.
Ship numberShip typeRepair levelBudget price (RMB 104 yuan)Repair start and end timeShip 1DestroyerLevel 12000January 1–March 15Ship 2DestroyerLevel 31500March 25–May 10Ship 4FrigateLevel 21000August 4–September 19Ship 6Supply shipLevel 2900July 1–August 15Ship 7Supply shipLevel 3450May 22–June 22Figure 4
Ship’s availability for diverse missions.


### 5.3. Analysis of Calculation Results

The calculation results show that in the “regular and dual control” repair mode, the fleet’s level repair plan for diverse missions considers the conventional constraints and focuses on the impact of mission requirements; that is, the mission capability requirement constraints and mission availability constraints are considered.This method can simultaneously obtain the fleet’s mission and level repair plans and can effectively unify the use and repair. It can solve the more common problem of conflict between the use and repair at present.To further analyze the model’s validity, in the current repair mode, under the two conditions of whether the influence of the mission is considered, the arrangement of the fleet’s level repair plan is compared and analyzed. Table6 shows the detailed results.Table 6
Comparison of ship’s level repair plans.
Ship numberRepair levelRepair period by considering diverse missionsRepair period without considering diverse missionsShip 1Level 1January 1–March 15January 1–March 15Ship 2Level 3March 25–May 10March 15–May 1Ship 4Level 2August 4–September 19August 2–September 17Ship 6Level 2July 1–August 15June 25–August 10Ship 7Level 3May 22–June 22May 25–June 25It can be seen from Table6 that the ships to be repaired have the same numbers in both cases, which are Ship 1, Ship 2, Ship 4, Ship 6, and Ship 7. Among them, since Ship 1 entered the current year from the previous year, the repair time shall be implemented by the current regulations, and the repair time of the ship shall remain unchanged in both cases.For Ships 6 and 7, since both ships are supply ships, they are used for different types of missions. Under the traditional planning method, there is a certain overlap risk of repair time, which may lead to the sailing rate constraint of the same type of ships not to be met. Therefore, under the condition of not affecting the mission use requirements, their repair time should be staggered appropriately.For Ships 2 and 4, since these two ships do not perform missions, their repair time is adjusted appropriately according to the repair requirements of other ships in order to ensure that the repair resources can better meet the requirements of the mission ships.In general, compared with the case without considering the mission, the final optimization results are different due to the different optimization objectives, decision variables, and constraints. However, this method considers more comprehensive factors, and the conclusion is also in line with the relevant regulations of current ship repair and meets the requirements for diverse missions. This method is closer to the actual situation of the current ship’s use and repair.


## 5.1. Problem Description

The problem scenario is that there are seven ships in a fleet in a particular year, and the repair budget is 70 million yuan. The annual fleet’s level repair plan is made under the “regular and dual control” repair mode. Due to confidentiality, the ship’s repair-related data are adequately processed.(1)
The ship’s initial capabilities are shown in Table1. All ships are divided into the following three types: destroyer, frigate, and supply ships. Experts evaluate the initial capability according to the various performance indicators. The planner shall determine the contribution weight of each ship’s capability to the overall capability by the strategic objectives, missions, relevant regulations, and instructions, and the opinions of the unit, that is, wc1,wc2,wc3,wc4,wc5 = (0.2, 0.15, 0.25, 0.25, 0.15).(2)
The ship’s repair-related data are shown in Table2. Relevant regulations and historical standards determine the repair level, the budget price, and the repair period. The restored capability is evaluated based on the historical data of the same type of ship on capability changes. The capability decay function is obtained by fitting various capabilities’ monitoring and evaluation data at different time points. The used time refers to the interval between the time when the ship entered service or finished the last level of repair and the beginning of the year. The ship’s operating department shall determine the lower limit of the ship’s operational capability based on experiences and historical operational conditions.Ship 1 has been repaired from the previous year to the current year, and the plan has been determined. Specifically, it has been repaired for 2.5 months in the previous year, and it needs to be repaired for 2.5 months this year. The other ship’s level repair plans are to be arranged. For ease of calculation, the unit “month” is counted as 30 days.(3)
The importance and mission assignment determine the weight of each ship in the fleet, that is,w1,w2,w3,w4,w5,w6,w7 = (0.17, 0.17, 0.14, 0.14, 0.14, 0.12, 0.12). The strategic objectives, superior instructions, and troop opinions determine the fleet’s expected capability. To simplify the calculation, the fleet’s expected capabilities for different missions are set as the same, that is, EXRj,EMRj,EFRj,EJRj,ELRj = (25, 40, 25, 25, 30). The planner’s risk preference determines the risk aversion coefficients α and β, which are 0.6 and 1.2, respectively.(4)
The overall minimum sailing rate is determined asr=60% according to the troop type and current regulations. These ships are divided into the following three types: destroyer, frigate, and supply ships. According to the statistical evaluation of the ship’s sailing rate data over the years, the minimum sailing rate of destroyer, frigate, and supply ships is determined as r1=50%, r2=60%, and r3=50%.(5)
The mission requirements are shown in Table3. The mission period is given according to the mission plan. The minimum requirement values of the mission fleet’s subcapability are obtained according to the historical execution of the mission and the experience of the ship’s operation department.Table 1
Ship’s initial capabilities.
Ship numberShip typeCommand-and-control capabilityManeuver capabilityDefense capabilityOffensive capabilitySupport capabilityComprehensive capabilityShip 1Destroyer909090909090Ship 2Destroyer909075757380Ship 3Frigate506080604060Ship 4Frigate356070504052Ship 5Frigate356060453547.5Ship 6Supply106025259037Ship 7Supply106025259037Table 2
Ship’s repair-related data.
Ship numberRepair levelBudget price (RMB 104 yuan)Repair period (month)Restored capabilityCapability decay functionTime used (month)The lower limit of capabilityShip 1Level 1400054590/1+e0.4∗t−2424.530Ship 2Level 315001.52080/1+e0.4∗t−231850Ship 3Level 120002.53060/1+e0.4∗t−231020Ship 4Level 210001.51752/1+e0.5∗t−21.51322Ship 5Level 350011247.5/1+e0.45∗t−19.5625Ship 6Level 29001.51943.75/1+e0.35∗t−2114.514Ship 7Level 34501937/1+e0.35∗t−211316Table 3
Mission requirements.
Mission numberMission periodRequirements for subcapabilityMission 1(April 1st, May 30th)(15, 16, 16, 19, and 22)Mission 2(August 1st, September 30th)(10, 13, 16, 17, and 19)


## 5.2. Parameter Setting and Model Solution

According to relevant research studies and historical experiences [40, 41], the parameters of the particle swarm optimization algorithm are set as N=100, Tmax=200, c1=1.5, c2=1.5, wmax=0.9, wmin=0.4, Vmax=2, and Vmin=−2.Through the first step of the hierarchical sequence method, the calculation result obtained isn1R1=1, n3R1=1, n2R2=2, and n3R1=1; that is, Mission 1 requires one destroyer and one supply ship; Mission 2 requires two frigates and one supply ship.Through the second step of the hierarchical sequence method, the calculation result is the optimal position of the particleg = (1, 0, 1, 0, 1, 1, 2.84, 7.19, 7.16, 3.24, 6.01, 4.76, 1, 6, 3, 5, 7) and the optimal fitness gbest = 2.0275.As Ship 1 crosses into this year from the previous year, its repair time is known. The 1st to 6th digits ofg represent whether Ship 2 to Ship 7 are repaired. The 7th to 12th digits represent the repair start time of Ship 2 to Ship 7.The 13th and 14th digits represent the ship’s number for Mission 1. The 15th to 17th digits represent the ship’s number for Mission 2.gbest represents the mission fleet’s system capability value.The mission plan is shown in Table4. The fleet’s level repair plan is shown in Table 5. The available status of the ship is shown in Figure 4, where 0 means available and 1 means unavailable.Table 4
Mission plan.
Mission numberMission periodCapability requirementShips for the missionMission fleet’s capabilityMission 1(April 1, May 30)(15, 19, 16, 16, and 22)Ship 1, 62.0275Mission 2(August 1st, September 30th)(10, 17, 19, 13, and16)Ship 3, 5, 7Table 5
Ship’s level repair plan for diverse missions.
Ship numberShip typeRepair levelBudget price (RMB 104 yuan)Repair start and end timeShip 1DestroyerLevel 12000January 1–March 15Ship 2DestroyerLevel 31500March 25–May 10Ship 4FrigateLevel 21000August 4–September 19Ship 6Supply shipLevel 2900July 1–August 15Ship 7Supply shipLevel 3450May 22–June 22Figure 4
Ship’s availability for diverse missions.


## 5.3. Analysis of Calculation Results

The calculation results show that in the “regular and dual control” repair mode, the fleet’s level repair plan for diverse missions considers the conventional constraints and focuses on the impact of mission requirements; that is, the mission capability requirement constraints and mission availability constraints are considered.This method can simultaneously obtain the fleet’s mission and level repair plans and can effectively unify the use and repair. It can solve the more common problem of conflict between the use and repair at present.To further analyze the model’s validity, in the current repair mode, under the two conditions of whether the influence of the mission is considered, the arrangement of the fleet’s level repair plan is compared and analyzed. Table6 shows the detailed results.Table 6
Comparison of ship’s level repair plans.
Ship numberRepair levelRepair period by considering diverse missionsRepair period without considering diverse missionsShip 1Level 1January 1–March 15January 1–March 15Ship 2Level 3March 25–May 10March 15–May 1Ship 4Level 2August 4–September 19August 2–September 17Ship 6Level 2July 1–August 15June 25–August 10Ship 7Level 3May 22–June 22May 25–June 25It can be seen from Table6 that the ships to be repaired have the same numbers in both cases, which are Ship 1, Ship 2, Ship 4, Ship 6, and Ship 7. Among them, since Ship 1 entered the current year from the previous year, the repair time shall be implemented by the current regulations, and the repair time of the ship shall remain unchanged in both cases.For Ships 6 and 7, since both ships are supply ships, they are used for different types of missions. Under the traditional planning method, there is a certain overlap risk of repair time, which may lead to the sailing rate constraint of the same type of ships not to be met. Therefore, under the condition of not affecting the mission use requirements, their repair time should be staggered appropriately.For Ships 2 and 4, since these two ships do not perform missions, their repair time is adjusted appropriately according to the repair requirements of other ships in order to ensure that the repair resources can better meet the requirements of the mission ships.In general, compared with the case without considering the mission, the final optimization results are different due to the different optimization objectives, decision variables, and constraints. However, this method considers more comprehensive factors, and the conclusion is also in line with the relevant regulations of current ship repair and meets the requirements for diverse missions. This method is closer to the actual situation of the current ship’s use and repair.


## 6. Conclusions

Based on the “regular and dual control” repair mode, this article fully considers the ship’s objective requirements for diverse missions, the mission execution effect, and the ship’s resource consumption. It establishes the optimization model of the fleet’s level repair plan for diverse missions. This article proposes the particle swarm optimization algorithm based on the hierarchical sequence method to solve the model and to obtain the fleet’s level repair and mission plan for diverse missions. This method solves the coordination problem of the mission and level repair plans and can provide technical support to facilitate adequate arrangements for the ship’s use and repair. The “regular and dual control” repair mode will change with the fleet’s mission expansion. The following study focuses on the ship’s mission and repair arrangements based on the “regular and conditional” repair mode.



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*Source: 3037495-2023-02-15.xml*