A multi-objective optimization problem research for amphibious operational mission of shipboard helicopters

Wei HAN, Yulin WANG, Xiho SU, Bing WAN, Yujie LIU

a Aeronautical Foundation College, Naval Aviation University, Yantai 264001, China

b Aeronautical Operations College, Naval Aviation University, Yantai 264001, China

c Coastal Defense College, Naval Aviation University, Yantai 264001, China

KEYWORDS

Abstract Airborne landing with shipboard helicopters gradually replaces surface landing to dominate joint amphibious operations.A problem with shipboard helicopter mission planning is conducted in the context of amphibious operations.First,the typical missions of shipborne helicopters in amphibious operations are analyzed.An Amphibious Operational Mission Planning Model for Shipboard Helicopters(AOMPMSH)is established,with the objectives of minimizing the completion time of the amphibious campaign and minimizing troop and helicopter losses, taking the available operational resources and the order of the mission sub-phases into account.Then, a simulationbased amphibious operations effectiveness assessment model is constructed to calculate the optimization objectives of AOMPMSH by simulating the campaign development with an amphibious objective area situation transfer model and simulating the engagement process with a modified Lanchester model.A reference point based multi-objective optimization algorithm is designed to solve the proposed AOMPMSH.The population iteration mechanism employs an initial population generation method and a local search method to solve the problem of vast definition space.The population ranking selection mechanism employs a population distribution based reference point generation method to solve the problem of population irregular distribution.Finally, a simulation case with the background of a battalion-scaled amphibious campaign is presented.The calculation results verify the rationality of the proposed model and the superiority of the designed algorithm and have some reference value for the operational applications of shipboard helicopters in amphibious operations.

1.Introduction

Relying on the tactical characteristics of flexible deployment,concealment and fastness, shipboard helicopters can surpass the obstacles, minefields and reinforced fortresses set up on the inter-water beaches by the anti-landing parties using multipurpose amphibious assault ship (Landing Helicopter Dock,LHD) and other forward bases as landing platforms in Amphibious Operations (AO).The shipboard helicopters can normally deliver Airborne Landing Force (ALF) and carry out airborne fire support coordinated with Surface Landing Force (SLF) in the Amphibious Objective Area (AOA), thus creating local advantages and achieving suddenness and rapidity of AO.However, the shipboard helicopters have hardware defects such as complex support, low sortie intensity, limited carrying capacity, and vulnerability to attack, making their applications more restricted and operationally risky.Consequently, shipboard helicopter missions need to be precisely planned to maximize their advantages while minimizing losses.

Currently,mission planning research on vehicle formations mainly concentrates on UAV cluster operation planning,1which is roughly divided into two parts: mission assignment2and flight path planning.3The UAV mission assignment investigation mainly focuses on mission assignment decisions among different resources and platforms, and the optimal scheduling of mission execution sequences.This kind of problem has been transformed into a classical combinatorial optimization problem in most research and has repeatedly been shown to be an NP-hard problem.There are some wellknown models in this domain, such as Traveling Salesman Problem (TSP) model,4,5Vehicle Routing Problem (VRP)model6–8and Mixed Integer Linear Programming (MILP)model.9However, neither the TSP model, the VRP model,nor their variants rarely involve a target trade-off when planning the order of service for each customer in an ideal environment such as an urban roadway.The Mixed Integer Nonlinear Programming (MINLP) model converts smaller-scale nonlinear problems with explicitly nonlinear objectives and constraints into linear programming problems, which are solved using traditional linear programming methods,making it difficult for problems with larger-scale and more complex objectives and constraints.

Traditional mission planning models are usually optimized to achieve maximum outcomes, minimize losses, complete the mission in the shortest time,and make optimum use of operational resources.Since the Amphibious Operational Mission Planning Model for Shipboard Helicopters (AOMPMSH) in this research focuses on the overall situation development of the amphibious campaign, its objective function not only can account for the indicators of the shipboard helicopter mission itself mentioned above,but also needs to integrate the mission into the amphibious campaign system for the overall effectiveness evaluation and investigate its contribution to the situation development of the amphibious campaign.The operational system effectiveness evaluation methods can roughly be divided into multi-attribute assessment, formula analysis,deduction, and complex network analysis methods.The multi-attribute assessment method10,11is the most widely used military operations analysis method in the past, which evaluates the system’s effectiveness through the weighted synthesis of various qualitative and quantitative indicators.However,the multi-attribute assessment method is susceptible to human subjective influence and is tricky to describe the complex system.The analytical method directly calculates the system effectiveness based on the functional relationship between effectiveness indicators and known conditions, which is the closest to the objective function calculation method commonly used in mission planning models.The most representative analysis method is the Lanchester model, which has been widely used in combat model construction12and troop attrition calculation13since it was proposed.The Lanchester model has evolved with new warfare styles such as information-based warfare,14asymmetric warfare,15and multi-party conflict.16Ref.17 developed Lanchester combat models to investigate how superior intelligence can compensate for an inferior force ratio.Ref.18 analyzed the Crete airborne campaign using the proposed innovative framework that supports decisionmaking at the strategic level and the operational level.Ref.19 examined the effect of troop reinforcements on engagement dynamics using the Lanchester model.Nevertheless, the analysis method also suffers from the difficulty of modeling the confrontation process of complex operation systems.The deduction method20,21constructs a computer simulation model to simulate the operation rules of the system and uses the computer simulation results to evaluate the system effectiveness directly or indirectly.Although it can provide a complete representation of the operation system from function to performance, the model usually has high complexity and consumes enormous computational resources.Representation of operation system from the perspective of complex network theory has been a popular research domain in recent years.The one that has been applied more often is the OODA loop theory.22

In reality, most operational mission planning problems are Multi-objective Optimization Problems (MOPs).Multi-Objective Evolutionary Algorithm (MOEA) has been proven to be the most effective way to solve MOPs at this stage.23According to the solution concept, MOEA can be roughly divided into three categories which are respectively based on the Pareto dominance relationship,24indicators,25and decomposition.26As the research advances, the traditional MOEA encounters many obstacles in solving the Many-objective Optimization Problem (MaOP)27and the irregular Pareto Front problem.28Much research on the irregular Pareto Front problem has been conducted,which can be roughly categorized into adjusted reference vector method,29fixed reference vector plus auxiliary method,30reference point method,31and clustering and partitioning method32according to their implementation approaches.

The amphibious operational mission planning problem of shipboard helicopters investigated in this research can be abstractly interpreted as the landing side (hereafter referred to as the Red side) choosing to commit how many resources,at what time, deploying how many and which type of ALFs against which nodes of the defending side (hereafter referred to as the Blue side) in the AOA, which is guided by the development of the amphibious battlefield situation, optimized by the goal of campaign overall effectiveness, and involving constraints based on equipment performance, military synergy and other aspects.It is a matter of studying Red’s available operation resources allocation and scheduling Blue’s target nodes.Based on the analysis above,there are some main shortcomings of the current research in addressing the operational mission planning of shipboard helicopters: (A) In terms of operational mission optimization,research on aviation mission planning in the context of amphibious operations is still in a gap.Other existing models lack attempts to optimize the selection and execution order of missions uniformly as decision variables in a tremendous adversarial environment.(B) In terms of optimization objectives, the effectiveness assessment of complex systems for amphibious operations with low computational complexity is hard to achieve with a single existing method.(C)In terms of algorithms,the existing algorithms are relatively slow to converge when solving realistic problems with a vast definition space and have a high computational complexity when solving MOPs with irregular Pareto Fronts.

The main contributions of this research are shown as follows:

(1) The AOMPMSH is proposed based on reasonable assumptions about the amphibious operational background, which pioneered the operational mission planning problem within this domain.The presented AOMPMSH is formulated as a mathematical programming model that exceeds most existing cases2in the dimensionality of the decision variables, including the allocation of limited operational resources to different objectives, the order in which missions are performed,and the frequency of mission repetition for the same objective.

(2) Contrary to the mission planning model for vehicle formations in a generic context,1the optimization objectives of AOMPMSH are to minimize the overall casualties of all combat units in the joint landing force as well as to minimize the total elapsed time of the amphibious operation.A combat simulation model based on the modified Lanchester equation is proposed to calculate the above optimization objectives.In contrast to the ideal Lanchester square law model,33this research accounts for the influence of factors such as intelligence information, Command and Control (C2),morale levels, and multi-troop engagements on both belligerents.

(3) A reference point based MOEA is designed.The Reference Point Adaptive Generation based on Population Distribution(RPAG-PD)is presented in terms of population ranking and selection mechanism, enhancing the search capability for irregular populations without increasing computational complexity.New initial population generation, crossover, mutation and local optimization methods are presented in terms of the population update and iteration mechanism, improving the search efficiency for definition space.The proposed algorithm performs better than the existing studies27,30,34–35in optimizing efficiency and maintaining population diversity.

The rest of this paper is organized as follows.Two typical amphibious operational mission models of shipboard helicopters are described and analyzed in Section 2.Based on the above two mission models,the mathematical programming model of AOMPMSH is presented in Section 3,which includes a detailed description of the variable assumptions, constraints and optimization objectives.A reference point based MOEA is presented in Section 4.In Section 5, a simulation example is developed based on a battalion-scaled joint amphibious campaign scenario.The proposed algorithm’s optimization results are first compared with the state-of-the-art multi-objective optimization algorithm in several performance indicators,and then analyzed in the context of amphibious operation theory.In Section 6, the conclusion is drawn.

2.Mission introduction

The shipboard helicopter formations consisting of Utility Helicopters(UHs)and Attack Helicopters(AHs)can perform various missions in AO, including vertical delivery of soldiers,equipment and supplies, close fire support, seizing lowaltitude air domination, strike surface targets, and antisubmarine minesweeping.Considering the operational requirements and current capability of the shipboard helicopters,two typical mission modes of airborne landing and close fire support in the beach landing phase of an amphibious campaign are selected for analysis in this research.

2.1.Airborne landing

In amphibious operations, by taking off from LHDs, landing and delivering airborne combat power on a beachhead near the coastline, the shipboard helicopter formations can seize the target nodes collaborating with the surface landing force to assist in their rapid advance or encircle and annihilate the enemy force.At this stage,shipboard helicopters can only deliver light combat forces on a small scale due to the constraints of LHDs’ sortie frequency and helicopter’s load capability.Therefore, the ALF in a sizeable amphibious campaign is mainly responsible for the coordinated assault mission with the SLF.

According to the above analysis, the Airborne Landing Missions(ALMs)can be abstracted as the distribution of shipboard UHs’ loads (Airborne Landing Units, ALUs) to different enemy targets in the AOA, including the selection and calculation of target nodes, assault timing, landing zone and helicopter formation routes.The primary performers of the ALM are shipboard UHs, which also include the shipboard AHs that provide accompanying escort, fire reconnaissance,and cover.

The coordination among different combat elements during the execution of ALMs can impact their effectiveness,which is briefly examined in this research.The coordination between LHDs and shipboard helicopters mainly concerns the impact of LHDs’support capability on the sortie frequency.The coordination between shipboard helicopters and ALUs mainly contains the effect of UH’s capabilities of load, speed, and flight range on force delivery efficiency.The coordination between shipboard aviation forces and other heterogeneous forces mainly involves the impact of battlefield information control,air control, and the effect of suppression of enemy air defense on the flight safety of helicopter formations, as well as the impact of battlefield situation on the operational effectiveness of aviation combat power.

2.2.Close fire support

With unique flight characteristics,shipboard AHs can fly much lower than fixed-wing aircraft and maneuver much faster than wheeled or tracked equipment.Therefore, during the Close Fire Support Missions (CFSMs), AH formations taking off from the LHDs or standing by in forwarding airspace can maneuver to the target stealthily and quickly by flying at ultra-low altitude,carrying out precise fire assaults on enemy’s tanks, armored vehicles, radar stations, artillery positions,communication nodes, forward posts, fire fortifications, surface ships, and other high-value targets, which makes them irreplaceably important.

Similar to the ALMs, the CFSMs can be abstractly interpreted as the distribution of shipboard AHs’ loads (airborne munitions) to different enemy targets, specifically including the selection and calculation of target nodes, assault timing,helicopter formation routes and other elements.The primary performer of CFSMs is the shipboard AHs, which also include drones that provide battlefield situational awareness.In the mission implementation process, it is also necessary to consider the impact of LHDs’ support capabilities, AHs’capabilities of load, speed and flight range, enemy air defense fire, and battlefield situation on the effectiveness of CFSMs.

3.Mathematical formulation for AOMPMSH

3.1.Model assumptions

As the most complex warfare in history, joint amphibious operations contain too many elements in their model of operational missions,so it is necessary to make appropriate normative assumptions to simplify the model.

(1) The shipboard helicopters mainly undertake ALMs and CFSMs, which are planned only in this research and developed by the pre-war staff plan together with the SLF’s operational missions.

(2) The ALMs are performed by a combination of UHs and AHs.The type and number of ALUs delivered to each node in the AOA and the number of required UHs and AHs are developed in the pre-war staff plan.The CFSTs are performed only by AHs.Additionally, each helicopter formation of CFSM consists of a fixed number of AHs.

(3) The shipboard helicopters will not have their operation missions compromised by malfunctioning or being shot down.The LHD is not subject to attack and can provide sufficient fuel, ammo supply and all kinds of maintenance and service support for shipboard helicopters.Besides, the landing force’s Command, Control, Communication, Computer, Intelligence, Surveillance, and Reconnaissance (C4ISR) system can provide stable and effective command and communications for all operational missions.

(4) The case of Amphibious Ready Group close to the landing beachhead (50 km) is explored in this research, which results in a short mission execution time for the helicopter formations.Therefore, the shipboard helicopters sortie in a two-wave cycle.Besides,if the shipboard helicopters are of the same mission,they will depart in the same or close waves as possible.

3.2.Notations

The relevant notations of the AOMPMSH are provided in Table 1.

Table 1 Relevant notations of AOMPMSH.

3.3.Constraints of AOMPMSH

The constraints of AOMPMSH can be roughly divided into two categories: available operational resource constraints and mission sub-phase time and order constraints.The available operational resource constraints can be further divided into the constraints of ALU number,available shipboard helicopter number,and LHD’s support resource.The mission subphase time and order constraints can be further divided based on the sub-phase execution sequence and the LHD deck scheduling.

3.3.1.Operational resource constraints

(1) ALU number constraints

According to Assumption 4, in order to drop more ALUs simultaneously in a wave, the ALUs should be deployed scattered on different LHDs.Since the Red side is limited to the total number of ALUs that can be committed during an amphibious campaign, the number of ALMs that each LHD can support has to meet the limit on the number of ALUs loaded on that LHD as well.Specifically,should satisfy the following constraints:

Meanwhile,the total number of ALUs planned to be delivered by all ALMs in an amphibious campaign should satisfy the following constraints:

(2) Shipboard helicopter number constraints

According to Assumption 2, the sum of AHs or UHs required for operational missions in adjacent wave cycles on a particular LHD is limited by the total number of AHs or UHs carried on that LHD

(3) LHD support resource constraints

According to Assumption 4,limited by the number of takeoff spots on the flight deck of each LHD, the number of helicopters in formations for the same sortie wave should satisfy

In addition, helicopter formations assigned to the same operational mission should be concentrated in the same or adjacent waves so far as possible, soin Eq.(6) should satisfy the constraint:

where c=1,2, p=1,2,???,P, w=1,2,???,W,W′=min ｛w+nw-1,W｝, nwdenotes the minimum number of waves required to deliver all helicopter formations assigned to the category c operational mission for the p th target node,which is calculated as the value of up-rounding the quotient.The quotient is obtained by dividing the sum of AHs and UHs by the number of all takeoff spots of all Red’s LHDs.

3.3.2.Mission sub-phase time and order constraints

(1) Constraints based on sub-phase execution sequence

A complete operational mission of shipboard helicopters can be divided into nine sub-phases in the order of execution,which consist of deck transfer sub-phase,maintenance and service support sub-phase, takeoff and departure sub-phase,cruising flight sub-phase (from LHD to coastline), ultra-low altitude flight sub-phase (from coastline to target nodes), goal accomplishment sub-phase, ultra-low altitude flight sub-phase(from target nodes to coastline), cruising flight sub-phase(from coastline to LHD), approach and landing sub-phase.Among the above sub-phases, the fourth to eighth subphases can be combined into the preparation phase, while the second and third sub-phases can be combined into the sortie phase.In each individual operational mission of shipboard helicopters, the following constraints are satisfied between the start and end moments of each sub-phase:

Notably, not all operational-mission-assigned helicopter formations will go through all these nine sub-phases.The exception is the deck transfer sub-phase.Suppose the same number of AHs and UHs are required for all missions in the w th and w+2 th waves of a cyclic sortie on a certain LHD.In that case,the helicopters in the w th sortie wave can receive maintenance and service support directly at the takeoff spots after landing and then take off directly for operational missions in the w+2 th wave.Thus(1 ) in Eq.(9) satisfies the w th wave landing,helicopter formation ①,which is ready first, needs to wait for helicopter formation ②to finish the maintenance and service support before taking off together.After both helicopter formations ①and ②are in the air,helicopter formations ③and ④,which have returned in the w-1 th wave, will be able to land on the cleared flight deck.

3.4.Optimization objective of AOMPMSH

Amphibious landing is the most harrowing style of offensive operation in history.Due to the limited size of the force that could be deployed on the beachhead, the Red side could only commit landing force wave by wave.The first wave of landing force must quickly seize and consolidate the beachhead positions to prevent the Blue side from launching counterimpacts at any time and ensure that subsequent forces land safely and quickly.Furthermore, due to the lack of heavy

where w′=w′′+2, w′′? ｛1,2,???,W-2｝, T1represents the assumed deck transfer time.

(2) Constraints based on LHD deck scheduling

Because of the space limitation on the LHD flight deck,the shipboard helicopters usually have to occupy the takeoff spots for their maintenance and service support,which leads to shipboard helicopters formations preparing to approach for landing having to wait for the LHD flight deck to clear (after the next wave of helicopters takes off)before they can land.Therefore, for a shipboard helicopter formation on the same LHD sortieing in two adjacent waves, the takeoff and landing moments must satisfy the following constraints:

It is assumed that all the Red’s helicopter formations departing in the same sortie wave must take off simultaneously from all LHDs.Then, as shown in Fig.1, after helicopters in equipment,the first wave of landing forces is at a disadvantage in terms of firepower and protection against the Blue side which relies on firm fortifications.Therefore, the Red side must rationally plan the usage of landing forces to minimize casualties while ensuring the completion of established operational missions on time.The combat time consumption, force attrition, and helicopter threat from the enemy’s air defense fire in the beach landing phase of an amphibious campaign are selected as the optimization objectives of AOMPMSH based on the above analysis.

(1) Minimize campaign time consumption

The defending side in amphibious campaigns usually organizes mobile forces for counter-impacts while the attacking side’s advantage is still unstable.The ability to quickly seize and effectively consolidate the beachhead positions before the enemy’s counter-impact directly determines the success of a landing campaign.Therefore,one of the optimization objectives of AOMPMSH is chosen as the total time taken by all landing forces on the Red side to complete all scheduled operational missions in the beach landing phase.

where Tfindenotes the end of the beach landing phase,which is the moment when the Red’s landing forces complete the missions required to seize and consolidate the beachhead positions, and T0denotes the start of the beach landing phase,which is the moment when the first wave of Red’s shipboard helicopters starts to take off.

Fig.1 Sortie and preparation phase description of shipboard helicopter formation.

Fig.2 Schematic diagram of battlefield situation in AOA.

(2) Minimize force attrition

Since the Red side can usually hold battlefield air control during amphibious operations, it is difficult for the Blue side to conduct large-scale force movements.Hence, the Red’s attack operations mainly determine the battlefield situation shift in the AOA.As shown in Fig.2, the node state vector Δ= [δ1,δ2,???,δP] is constructed based on the control rights of each node in the AOA,where δp=0 indicates that Blue still controls the p th node and δp=1 indicates that Red has captured the p th node or paralyzed the node’s primary function through a fire strike.It is assumed that the Blue side has constructed forward node p1, command post p2, artillery position p3,air defense missile position p4,traffic node p5,bridge p6and counter-impact force assembly point p7respectively in the AOA.The value transitions of state vector Δ for each node above correspond to different battlefield situation shift processes, which are shown in Table 2.

Table 2 Value transitions of Δ corresponding to battlefield situation shift in AOA.

Red’s landing force attrition includes attrition of both ALFs and SLFs, which can be expressed as

The Lanchester equation is used to calculate the force attrition of the Red and Blue sides in this research.When the two sides use rifles, small-caliber artillery, anti-tank missiles and other aimed fire weapons to engage in in-sight combat,Lanchester’s square law can be used to calculate the attrition of both sides

where r and b are the number of combat units destroyed by each other per unit of time.

When one or both of the two belligerents uses area fire weapons such as howitzers in out-of-sight combat, attrition on the other side can be calculated using Lanchester’s second linear law

where ξ denotes the Red’s combat units killed within the unit area per unit of time by each Blue’s area fire combat unit.

Eqs.(14) and (15) are applicable to the case where both belligerents contain only homogeneous combat units.Suppose both belligerents have a high level of C4ISR, and in that case, they can adjust their fire allocation according to the number of each type of combat unit of the opponent at each moment in real time.For the case where one or both forces in an engagement contain multiple heterogeneous combat units, their casualties for each type of combat unit can be expressed as

where‘‘*”denotes the multiplication of the corresponding elements of the matrix.R and B are non-negative matrices,called the damage factor matrix of both belligerents.

rjidenotes the number of Blue’s category j combat unit destroyed by Red’s category i combat unit per unit of time,and the same to bij.m and n denote the number of Red’s and Blue’s combat units respectively.Both Φ and Ψ are non-negative matrices with column sums not exceeding 1,called the fire distribution matrix of both belligerents.

φjidenotes the proportion of Red’s category i combat unit that attacks Blue’s category j combat unit,and the same to φij.Ref.33 investigated how to assign values to Ψ and Φ to maximize B*Ψ and R*Φ.

However,it is hard for belligerents to ensure the ideal level of C4ISR in the reality of operations.Therefore, Eq.(16) is amended to take into account the inadequate situational awareness of the Red forces and the inability of the forces to fully deploy15,36

where α′denotes the maximum strength vector of forces that the Red side can deploy under the influence of tactics, terrain and other constraints; β0denotes the initial strength vector of the Blue side;ε ?[0, 1]denotes the proportion of Blue’s forces detected by the Red side accurately.The right side of the equation is equivalent to Lanchester’s square law when ε=1, and the right side of the equation is equivalent to Lanchester’s second linear law when ε=0.

Finally,the surrender index μ is introduced,considering the morale factor of both belligerents.Therefore,

where

αp(m )and βp(n )denote the infantry units of the Red and Blue sides respectively, and αp0and βp0are the initial values of αpand βprespectively.

In order to avoid the interference of tactical factors from both belligerents to Ψ and Φ, it is assumed that both belligerents adopt the optimal strategy in the case of a multi-troop engagement, where each column of Ψ and Φ has and only has one element of 1 and rest of 0.33

(3) Minimize helicopter threat from the enemy’s air defense fire

Calculating the impact of the random event of a shipboard helicopter being shot down on mission effectiveness is tricky and often requires the use of Monte Carlo simulations.Therefore, the cumulative threat to the Red’s shipboard helicopters from Blue’s air defense fire during flight is used to quantify the losses.

where kScaledenotes the scale factor;andindicate the time when the helicopter formation assembles after taking off and the time when it begins its approach to the ship after returning from the AOA respectively; E(x,y,t) denotes the sum of the threat values of all ground air defense fire from the Blue side per helicopter on the Red side per unit time at coordinate (x,y ) at time t.The calculation of E(x,y,t) is simplified by accounting only for the air defense fire factors at each node,the distance between the helicopters and the nodes,and the role of terrain in blocking air defense fire on the Blue side37, and E(x,y,t) can be calculated by

where ηSAMis a Boolean variable indicating whether the p th node is blocked by terrain from the coordinate (x,y ).It takes the value of 1 when not blocked and 0 when blocked.denotes the air defense fire factor at the p th node.D(x,y,p)denotes the plane distance between the p th node and coordinate (x,y ).The coordinate position (x,y ) of each helicopter on the Red side at the moment t is obtained by the Dijkstra path planning algorithm38in the decoding operation of Section 4.3.

4.Proposed algorithm

Traditional mathematical programming methods are inefficient in solving problems with nonlinear objective functions and constraints, and are sensitive to weights and optimization order.The MOEAs have become the primary tool for solving the MOPs evolving toward realism and complexity.A reference point based MOEA is proposed in this research, with innovations in population iteration mechanism and ranking mechanism, respectively.

A form of discrete mission list coding similar to VRP was adopted to express the sequential relationship between the operational missions.According to constraint (2) in Section 3.3.1, the number of shipboard helicopters required for all operational missions in the code is determined when the number of LHDs and sortie waves are provided.Thus operational missions of AOMPMSH that require different numbers of helicopters can cause differences in the coding length of each chromosome.Moreover, the chromosome code of AOMPMSH can hardly contain complete information about all the operational missions.Therefore, the genetic operators in traditional evolutionary algorithms have trouble in implementing crossover for different length chromosomes and searching the defining space of the problem with chromosomes that contain limited information.Improvements are made regarding the MOEA’s population update and iteration mechanism.The crossover and mutation operation methods for different length chromosome codes are redesigned.An initial population generation method and a local optimization method are innovatively proposed to improve the optimization efficiency of the proposed algorithm.

The RPAG-PD is designed in terms of population ranking and selection mechanism.The main idea is to generate a fixed number of reference points to guide spatial decomposition and environmental selection according to the density of individual distribution in each generation of the population.In addition,a novel sorting strategy employing fitness values determined via scalarization in the SDEA34is used to rank the niche solutions associated with each reference point.

4.1.Algorithm framework

The main framework of the proposed algorithm is shown in Algorithm 1.The known parameters of the algorithm include the objective space dimension m,the simplex-lattice parameter H,and the maximum iteration number gIter.In step 1,the population size is calculated according to Eq.(24)39(Line 1)

Then the initial population P0is obtained by the Initial Population Generation based on Traversing Class 1 Neighborhoods (IPG-T1N)(Line 2).In step 2, crossover (Lines 7–8),mutation (Line 9) and local optimization (Line 10) operations are performed on the parent population Pgto produce the offspring population.In step 3, the offspring populationis merged with the parent population Pgto obtain(Line 12).Update zIdealand zNadiraccording to the solution setcorresponding to, and further obtain the normalized solution set Z～according to Eq.(25) (Lines 13–14)

Algorithm 1.Algorithm framework.

images/BZ_268_1297_1596_2289_2952.png

Fig.3 Relationship among main components of proposed algorithm.

Fig.4 An example of chromosome coding.

The relationship of several main components of the proposed algorithm is shown in Fig.3.

4.2.Chromosome coding

As shown in Fig.4, all of the operational missions satisfying≥1 are decomposed into a list of individual missions satisfying=1, comprising a discrete two-line chromosome coding.

σidenotes the chromosome-encoded gene,and the meaning of element c in the first row and element p in the second row are the same as those in Table 1.denotes the length of the chromosome code, which means the total number of operational missions in code.

According to Eq.(3),if the total number of AHs or UHs on the LHD is not sufficient to meet the consecutive calls for two adjacent waves of operational missions,limited shipboard helicopter resources must be allocated to one wave, resulting in only missions requiring another type of helicopters or no missions being scheduled in the other wave.Therefore,adding null genes to code ^C is necessary to indicate the idle status of some takeoff spots of a particular LHD in a certain wave.This idle state is denoted byand the number of idle takeoff spots is assumed to be the maximum common factor of the number of helicopters required for all operational missions.According to hypothesis 2 in Section 3.1, the neighborhoods where each gene is located can be classified into two categories.Genes corresponding to ALMs requiring a variable number of helicopters (hereafter collectively referred to as Class 1 missions) are located in Class 1 neighborhoods.Genes corresponding to CFSMs requiring a fixed number of helicopters (hereafter collectively referred to as Class 2 missions)are located in the Class 2 neighborhoods.It is easy to know that the change in chromosome length is caused by the change in the number of Class 1 missions in the chromosome.

4.3.Chromosome decoding

The specific steps of the decoding operation are given in Algorithm 2.The input parameter is the chromosome code ^C.The decoding process is roughly divided as follows.First, the discrete code ^C characterizing the operational mission sequence is converted into a sortie schedule ^F, which characterizes the sortie plan for each shipboard helicopter (Lines 1–15).Then,the start moment list ^Q of the Red’s ALMs and CFSMs to each target node in the AOA is obtained based on ^F (Lines 16–19).Based on the intended operational plan of the SLF,the changes in the belligerents’ strength vectors over time in local battles occurring at each target node in the AOA are calculated through a modified Lanchester model.Based on the results of Lanchester model, each node’s state vector is derived.Finally, the battlefield situation shift model derives the amphibious campaign progress and determines whether the Red side will ultimately achieve the campaign objectives.The objective ^Z of operational mission effectiveness in Section 3.4.is obtained by the way (Lines 20–37).

Fig.5 An example of a shipboard helicopter sortie schedule.

The solution process for ^F and ^C′consists of the following steps.Step 1 calculates the number uiof all ALUs to be delivered of each ALM in ^C and the number hiof all helicopters required for each operational mission (Lines 1–2).In Step 2,the Class 1 missions that satisfy Eqs.(1)–(7) are selected from the code ^C based on uiand hi.The elements in ^F are assigned values according to Eq.(27).The executed missions are added to the code ^C′in the same order as in the original code ^C(Lines 9–11).After all the Class 1 missions satisfying the constraints are inserted into ^F,Step 3 searches the Class 2 missions in ^C in the same way as Step 2, and update ^F and ^C′accordingly (Lines 3–15).

Assume that Red has three LHDs with sufficient helicopters on board and plans to sortie three waves for the beach landing phase.Then the chromosome code ^C in Fig.1 can be transcribed into the sortie schedule ^F as shown in Fig.5, which is a matrix with w rows andcolumns.denotes the mission performed by the shipboard helicopter that took off from the j th takeoff spot of the l th LHD in the w th wave.

The solution process for ^Q consists of two steps.In step 1,the helicopter route archive ^R to each node piwith the least risk is programmed using the Dijkstra algorithm, based on the target piof each operational mission in the helicopter sortie schedule ^F,combined with the known or assumed distribution of enemy’s air defense fire in the AOA(Line 17).Step 2 solves the archive ^Q of all mission’s goal accomplishment sub-phase start moment,which is the moment when ALUs start airborne landing or AHs start the air-to-ground strike.(Line 18).

The solution process for ^Z consists of four steps.Step 1 assumes that the node state Δ(p )=0 for ?p ? ｛1 ,2,???,P｝at the moment T0when the beach landing is initiated (Line 20).Step 2 solves the risk taken by each helicopter formation on the Red side during the flight in the AOA (Lines 23–26).In Step 3,the nodes where the local engagements occur in each simulation moment are found according to the helicopter operational mission list and combined with the assault plan intended for the SLF on the Red side(Line 27).The force attrition and battle elapsed time for both belligerents in the local engagements at each node are calculated using Eqs.(13)–(23)(Line 28).The above result updates the node state vector Δ(Lines 29–33).The value change of both belligerents’ strength vector at each target node is numerically integrated with tstepas the simulation step,until Δ meets Red’s campaign objective or Δ indicates that Red is definitely unable to reach it.Finally,the total consumption time z1for the beach landing phase, the cumulative force attrition z2for the Red’s ALF and SLF,and the risk z3taken by all shipboard helicopters performing operational missions are calculated to form the solution ^Z.

Algorithm 2.Chromosome decoding method.

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4.4.Initial population generation

Since the chromosome codes of AOMPMSH only contain the operational missions performed in their corresponding decision variables, no one code can contain information about all missions.An Initial Population Generation based on Traversing Class 1 Neighborhoods (IPG-T1N) is designed to make the initial population contain more operational mission information satisfying the constraints of Section 3.3.1 by performing an adequate search for the Class 1 neighborhoods.A feasible coding archive containing all different Class 1 mission permutations is generated and taken as the initial population, which guarantees the diversity of the initial population.The pseudo-code is shown in Algorithm 3.

Algorithm 3.Initial Population Generation based on Traversing Class 1 Neighborhoods (IPG-T1N).

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4.5.Crossover

A Double Point Crossover based on Chromosome Length

Algorithm 4.Double Point Crossover based on Chromosome Length Transformation (DPC-CLT).

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4.6.Mutation

A mutation method based on positional gene swapping is proposed in Algorithm 5.The algorithm inputs are the initial chromosome code,the initial solution,and the mutation probability PMut.

Algorithm 5.Mutation method.

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4.7.Local optimization

It is assumed that the helicopter formations performing CFSMs consist of a fixed number of AHs.The fixed number is the maximum common factor of AHs required for each CFSM.The CFSM requiring a large AH formation is expressed in the code as multiple Class 2 missions arranged next to each other.Therefore, the genes corresponding to a particular Class 2 mission may not occur or recur more than once in a single chromosome code, giving Pnassignments for the code in Class 2 neighborhoods of length n.A Class 2 Neighborhoods Local Optimization based on Tabu Search(2NLO-TS) is proposed in Algorithm 6 to solve the above problem.

Algorithm 6.Class 2 Neighborhoods Local Optimization based on Tabu Search (2NLO-TS).

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4.8.Reference point adaptive generation

As the basis for spatial decomposition and environment election,reference points have a decisive influence on the searching performance of MOEAs.The initial population (assuming H=8) generated by IPG-T1N in Section 4.4 is used as an example computational dataset,based on which the limitations of the uniformly distributed reference points generated by the two methods40are compared.Then, the Reference Point Adaptive Generation based on Population Distribution(RPAG-PD) is proposed with a sample calculation result.

The linear hyperplane in Fig.6 is generated based on the Nadir points generated by the minimum Euclidean distance from the coordinate axes.30The intercept of the above linear hyperplane with each coordinate axis of the objective space may be less than 0,which means that the above method cannot guarantee to convert all solutions into the first quadrant.In addition, the normalization of the solution set by the above method is not satisfactory due to the randomness of intercepts between the hyperplane and each coordinate axis of the objective space.

Fig.6 Hyperplane constructed using the nearest point to coordinate axis as Nadir point.

Fig.7 Hyperplane constructed using the point with maximum objective in each coordinate axis as Nadir point.

Fig.8 Results obtained by clustering normalized solutions.

A resolution for the problem in Fig.6 is given in Ref.40,which constructs the hyperplane by taking the maximum objective in each dimension of the objective space as the Nadir points.However, it is susceptible to the influence of outlying points.As shown in Fig.7, the outlying point in the z1direction makes the nadir point in that direction significantly farther away from the solution set.As shown in Fig.8, the intercept of the linear hyperplane in the z1direction is significantly larger than the value of most individuals in the normalized solution set.Therefore, as shown in Fig.8, during the inner angle measure based clustering operation, considerable reference points in the uniformly distributed reference archive on the linear hyperplane have no associated solutions.

According to the above analysis,two ideas can improve the association adequacy between the normalized solutions and the reference points.The first is to remove the outlying points in the objective space,which makes the reference points fit the solution set better.The second is to change the distribution of the reference points on the hyperplane,which makes the reference points converge towards the dense region of the solution set.Since the chromosomes in this research are discretely coded,the first method may lead to the loss of potentially optimal solutions.The idea of the second method has already been applied in the MOEAs based on adjusting the reference vector.41–43However, most of these methods require clustering operations with high computational complexity.

Fig.9 Pseudo linear hyperplane and reference points generated using RPAG-PD.

As shown in Fig.11,due to the non-uniform partitioning of the objective space, the sum objective values of the reference points generated by the proposed algorithm are not the same and are distributed on a nonlinear surface.Meanwhile, the generated reference points are more densely distributed in the regions where the solutions are aggregated.Thus, they have a better mapping effect on solutions in the case of irregular distribution.

Algorithm 7.Pseudocode greedy assignment of customers.

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5.Computational experiments and discussion

5.1.Landing campaign scenario

A typical battalion-scale joint amphibious campaign scenario is presented.The deployment of each target node in the AOA is shown in Fig.12(a).The topography of the AOA is shown in Fig.12(b).The Blue’s air defense fire deployed in the AOA consists of short-range air defense missile systems at F1and F2, and man-portable air defense missiles at each node.As shown in Fig.12(c), the threat of Blue’s air defense fire to the Red’s helicopters flying at an ultra-low altitude of 10 m can be calculated by Eq.(23) according to the terrain cover.

Fig.10 Schematic diagram of polar coordinate space Γ3 being partitioned.

Fig.11 Results obtained by clustering the normalized solutions.

It is assumed that the Red’s amphibious ready group consists of 3 LHDs assigning 12 UHs,12 AHs,and 6 takeoff spots for ALMs and CFSMs.As shown in Table 3, the Red’s landing force is divided into a left flank surface landing cluster(landing force ①), a right flank surface landing cluster (landing force ②), a follow-on assault cluster (landing force ③),and an airborne landing cluster (landing force ④).The airborne landing cluster consists of 5 ALUs.Infantry unit 1 is deployed in LHD 1.Infantry unit 2 and fire support unit 1 are deployed in LHD 2.Infantry unit 3 and reconnaissance unit 1 are deployed in LHD 3.Each infantry and fire support unit requires 6 UHs for transportation, while each reconnaissance unit requires 4 UHs.

As shown in Fig.12(a), the Blue side has three garrison infantry companies A,B and C under the X garrison battalion,a battalion artillery unit, a brigade artillery unit, and two short-range air defense missile systems deployed in the AOA.At the moment T0, the Red side learns the deployment of Blue’s personnel and equipment at each node based on reconnaissance intelligence as shown in Table 4: frontier positions A1-3, B1-3, C1-3, battalion command post Y, company command post A0, B0, C0, air defense positions F1, F2, brigade artillery P1, and battalion artillery P2.In addition, the Red’s intelligence indicates that the Blue’s armored unit will complete its assembly and arrive at the beachhead via Y 3 to 4 hours after the moment T0.

Fig.12 A case of landing campaign scenario.

Table 3 Weapon deployment of Red forces.

The Red’s combat operations will be executed following a predetermined staff plan.In the Red’s main attack direction,the right flank surface landing cluster plans to capture the Blue nodes in the order of C2(node 8),C1(node 7),C0(node 1),and take control of the battalion command post Y (node 2) as the objective of beach landing phase.The airborne landing cluster,including helicopters and soldiers for ALMs and CFSMs, is primarily responsible for assisting the offensive operations of the right flank surface landing cluster, with targets including the Blue’s artillery positions P1(node 5) and P2(node 6), air defense system positions F1(node 3)and F2(node 4),and command posts C0and Y.In order to ensure clearance of the Blue’s anti-aircraft fire threat in the AOA, the Red side must assign ALUs to take control of F1and F2.The expected ALU number assigned by the Red side against each node in the AOA is shown in Table 5.

Table 4 Weapon deployment of Blue forces.

Table 5 Red’s expected number of ALUs assigned to each node on Blue side.

Fig.13 A Gantt chart expressing flight mission and deck preparation phase helicopter formations.

Each LHD on the Red side carries 2 UH formations and 4 AH formations.Each UH formation has 6 UHs.Each AH formation has 3 AHs.It is assumed that the time consumed for the deck transfer, maintenance and service support, takeoff and departure, cruising flight, and approach and landing sub-phases in a complete operational mission process are constant values of 30, 55, 10, 14, and 10 min, respectively.The helicopter formation flies at a constant speed of 210 km/h during the ultra-low altitude flight sub-phase, whose elapsed time is calculated from the flight path length divided by the formation flight speed.The time consumed for the goal accomplishment sub-phase depends on the mission category, with 15 min for ALUs landing and 10 min for air-to-ground fire support.Suppose all helicopters in the same wave take off at the same time.In that case, the sub-phase distribution of operational missions exemplified in Fig.5 over time is shown in Fig.13,where the color blocks mean the same as in Fig.1 and the numbers mean the same as in Fig.5.

According to the stuff plan, the Red’s SLF expects to start minesweeping and barrier-breaking at T0, and start beach landing at T0+1.The Red’s helicopter formations expect to take off and depart at T0, and start ultra-low altitude covert flight after crossing the coastline.The helicopter formation flight paths corresponding to the coding example in Fig.4 are shown in Fig.14, where the chosen landing zone needs to meet three conditions: close to the target node, less terrain undulation, and far from the Blue’s fire range.

Fig.14 Flight paths for helicopter formations.

The Red’s and Blue’s damage factor matrices are constructed based on the damage capability data of each type of equipment in Ref.43.In order to simplify the calculation,the Red’s infantry fighting vehicle and amphibious assault vehicle in Table 3 are combined into vehicled artillery, rocket launcher and mortar are combined into individual artillery,heavy machine gun, light machine gun and assault rifle are combined into light infantry.Similarly,the Blue’s 60 mm mortar and man-portable missile in Table 4 are combined into individual artillery, and heavy machine gun, light machine gun and assault rifle are combined into light infantry.

According to Red’s operation program,the combined combat units and damage factor matrices are brought into the Lanchester equations in Section 3.4.for a solution.The variation of the strength vector taken for each combat unit at the node C0, Y, F1, F2, P1, P2, C1, and C2with time is shown in Fig.15.In the course of beach landing, the Red’s SLF leaves 24 light infantry units in place to defend each node it captures,while the rest launches an attack to the next node.

5.2.Algorithm simulation results analysis

The real Pareto Front(PF)of the AOMPMSH is hard to identify because the problem is investigated in a realistic rather than an ideal situation.Thus, two evaluation indicators22for MOEAs that do not require PF information are selected: the general comprehensive indicator Hyper Volume (HV), which measures the convergence and diversity of the solution set,44and the comprehensive diversity indicator linear distribution(ΔLine), which measures the distributivity and extension of the solution set.45

5.2.1.Population update and iteration mechanism

The innovations proposed in terms of population update and iteration mechanism are first evaluated, including IPG-T1N and 2NLO-TS.We assume that the initial population sizeand the simplex-lattice parameter H = 9.Following the idea of variable control, 50 iterations are performed respectively using the single SDEA,34 the SDEA with IPG and 2NLO both, the SDEA with IPG only, and the SDEA with 2NLO only.

The HV of the normalized solution set iterated 50 generations by the 4 algorithms with (1, 1, 1) as the reference point is shown in Fig.16.For the evaluation indicator HV, the two simulation cases with IPG slightly outperform the other two cases at generation 0, while the two cases with 2NLO significantly outperform the other two cases in convergence efficiency.

ΔLineof the normalized solution set iterated 50 generations by the 4 algorithms is shown in Fig.17.For the evaluation indicator ΔLine, the two simulation cases with the IPG significantly outperform the other two cases and have more potential for optimization.

Combining Fig.16 and Fig.17,it can be demonstrated that the IPG and 2NLO operations proposed in terms of population update and iteration mechanism effectively improve the initial distributivity of the solution set and the convergence efficiency during the iteration.

5.2.2.Population ranking and selection mechanism

The RPAG-PD is proposed in terms of population ranking and selection mechanism.In order to demonstrate the superiority of RPAG-PD, the proposed algorithm is compared with the SDEA based on uniform reference points,34the NSGAIII,35and two state-of-the-art MOEAs: the ASEA+30and the NSGA-II/SDR.27Assuming the initial population sizesimplex-lattice parameter H=9, and 50 iterations of computation, the performance of the above 5 algorithms with IPG and 2NLO on evaluation indicators HV and ΔLineis compared.

The HV of the normalized solution set iterated 50 generations by the 5 algorithms with (1, 1, 1) as the reference point is shown in Fig.18, where the proposed reference point based MOEA has the best performance in optimization results and is the second only to NSGA-III in convergence efficiency.

ΔLineof the normalized solution set iterated 50 generations by the 5 algorithms is shown in Fig.19, where the proposed algorithm is the second only to ASEA+.It is worth noting that both NSGA-III and ASEA+can only perform better than the proposed algorithm on either indicator HV or ΔLine,while performing significantly worse on the other indicator.

The proportion of Pareto dominant solution iterated 50 generations by the 5 algorithms is shown in Fig.20,to explore its effect on the above 2 evaluation indicators.

Fig.15 Value change of Red’s and Blue’s strength vector at each node.

Fig.16 Performance of algorithm with or without IPG and 2NLO on HV.

Fig.18 Performance of five algorithms on HV.

It can be observed from Fig.20 that the Pareto dominant solutions obtained by the NSGA-III quickly fill up the entire population at the initial stage, resulting in the NSGA-III searching for the optimal solution only in the Pareto dominant solution set.In contrast, the proportion of Pareto dominant solutions computed by the ASEA+remains low,reaching only half of the entire population at the highest.Therefore, the superior performance of NSGA-III on HV is achieved at the expense of the diversity of the solution set, which converges rapidly at the beginning and tends to lose the potential for further optimization.In contrast, the superior performance of ASEA+on ΔLineis achieved at the expense of convergence efficiency.The paranoid pursuit of population diversity affects its evolutionary efficiency.

Fig.19 Performance of five algorithms on ΔLine.

Fig.20 Performance of five algorithms on Pareto solution proportion.

Fig.22 Average mission frequency in codes of Pareto dominant solutions.

According to the ‘‘No Free Lunch” principle, no algorithm can simultaneously be optimal in both population diversity and convergence efficiency.The proposed algorithm incorporates the RPAG-PD while retaining the SSM in the original SDEA.As a result, the solutions obtained by the proposed algorithm achieve a certain degree of Pareto optimality, which means maintaining a better diversity while achieving a stronger search capability and a higher convergence efficiency.

5.3.Case simulation results analysis

The multi-objective optimization of the shipboard helicopter missions for 3 waves on the Red side is performed based on the landing campaign scenario presented in Section 5.1.Assuming the population sizethe simplex-lattice parameter H=9, and the iteration number gIter=50.

All the solutions obtained during 50 iterations by the proposed algorithm are shown in Fig.21.The average mission frequency in the corresponding chromosome codes of all the Pareto dominant solutions is shown in Fig.22.Comparing the proportion of the mission frequency in Fig.22, we can see that the Red side most prefers to perform ALMs assaulting the Blue’s battalion artillery position P2and CFSMs assaulting the Blue’s brigade artillery position P1, battalion artillery position P2, battalion command post Y and company command post C0.In addition, operation missions assaulting Blue’s artillery positions make up the majority of the Red’s first sortie wave.

Considering the limited number of helicopters and ALUs,the Red will give priority to assaulting the Blue’s artillery positions and command posts, especially preferring to assault the Blue’s artillery positions in the first wave.In addition, the Red side prefers to deliver ALUs to the Blue’s company command posts and battalion artillery positions,where the ground defenses are weaker than other nodes.

6.Conclusions and future work

The shipboard helicopter mission planning problem in the amphibious beach landing phase is investigated in order to arrange the missions more rationally, with the following main conclusions: (A) The AOMPMSH is constructed and formulated as a multi-objective optimization model for the optimization of target selection and order arrangement.(B) A simulation-based amphibious campaign effectiveness evaluation model is constructed by drawing on the idea of system effectiveness evaluation, by which the optimization objectives are calculated.(C)The proposed reference point based MOEA has an excellent performance in convergence efficiency and population diversity due to the innovation in both population iteration mechanism and sorting mechanism.(D) An operational mission optimization case in the context of a typical battalion-scale joint amphibious campaign is presented, which provides a reference for operational mission planning of shipborne helicopters in actual warfare.

In the future, there is still some profound work that needs to be done: (A) The shipboard helicopter mission planning model should be improved for situations that may occur in the real combat environment, such as helicopter failure and command communication failure.(B)More categories of operational missions and other types of shipboard helicopters and even unmanned and fixed-wing shipboard aircraft should be included in the programming model to make the model fit more application scenarios.(C) The complex network of battlefield situations should be modeled in more details by combining the relevant theories of amphibious operation and equipment performance.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.