Ziehms et al. [10] first proposed the concept of Phased-Mission Systems (PMS) in 1975. It is a system where the mission is composed of multiple non-overlapping consecutive stages, i.e., the system mission can be decomposed into several consecutive stages. In each stage, the system must perform specific missions. Various studies on reliability calculation and solutions for PMS missions exist [11]. The existing reliability analysis methods of PMS missions can be categorized into two categories [12]: analytical and simulation methods. The analytical approaches can be further divided into static model and dynamic model methods [13].

The static model methods mainly include reliability block diagram, fault tree, binary decision diagram (BDD) [14–16], and multi-binary decision diagram (MDD) [17–20]. Esary et al. [21] proposed a micro-component method for PMS reliability analysis, which decomposed each component into a series of statistically independent series units to solve the cross-stage dependency problem of PMS. However, this method is only suitable for small, non-repairable PMS because of the enormous computational burden. Zhang et al. [22] proposed a PMS-BDD method that combined stage algebra and binary decision trees, which effectively solved the problem of component sharing among stages to solve the problem of computational explosion. However, BDD has strict requirements on variable sorting, and it can barely solve the system’s reliability when the system contains components following multiple failure distributions. Bian et al. [23] regarded the convoy escorting group as a PMS. They assumed all the weapon systems contained in the convoy escorting group followed either exponential or Weibull distribution and then employed BDD to analyze and solve the mission reliability of the convoy escorting group in each stage of the mission. Recently, related studies on static models mainly focus on common cause failures [24,25], multi-mode failures [20], stage combination needs [26], and others. However, due to the assumption that component failures are mutually independent [19,27,28] and without considering the issue of component sharing in each stage, the application of this kind of model is limited.

Dynamic model methods consider the state mapping among different components with a certain probability, based on which the system’s change process is described and the mission reliability is obtained. Dynamic model analysis methods include Markov models [29–31], Petri net models [32–35], stochastic processes (CTMC) [36–38], semi-Markov processes [39,40], and others. Bayesian networks (BN) methods [41,42] can use conditional probabilities between nodes to represent the component sharing in and among stages. This method can also model PMS using discrete BN after discretizing mission time. Dynamic methods consider component sharing in and among stages, but with the increase of component count, the system state scale will face combined explosion problems.

Monte Carlo simulation, based on the law of large numbers [12,43], is a flexible tool for solving the reliability of PMS missions. This method has broad applications, but the only disadvantage is that it requires multiple simulations to reach satisfactory accuracy, which is usually time-consuming.

Analytical models enable accurate modeling, and some can handle complex systems, such as components following different distributions with various durations. However, as the number of stages and components increases, the computational complexity rises exponentially, making it an NP-hard problem. Simulation methods typically have great generality and flexibility but are sometimes computationally expensive and only produce approximate results. Hence, it is necessary to fully understand the specific problem to be solved and consider the practical situation of engineering applications to select appropriate methods.

Table 1 shows relative research on the Multi-Stage Mission Reliability of UAVs.

There are various studies on multi-stage missions for UAVs, but two main problems exist: (1) there is little research on the reliability of PMS missions for UAV swarm; (2) the modeling of mission reliability for single-stage, especially for combat-stage missions is not detailed enough, which only represents the performance of UAVs, lack of consideration of the impact of interceptions by the enemy as well as the impact of different allocation plans on mission reliability.

This study uses mission reliability as one factor (another is cost) to find optimal operational test plans. The problem of selecting optimal operational plans considering more than one objective (mission reliability and cost) is a multi-objective integer optimization problem. In recent years, evolutionary algorithms have incorporated biological information into meta-heuristic algorithms, making many breakthroughs in research in optimization algorithms [50]. Some typical multi-objective evolutionary algorithms are multi-objective particle swarm (MOPSO) [50], multi-objective ant colony (MOACO) [51], multi-objective bee colony (MABC) [52], multi-objective simulated annealing (MOSA) [53,54], multi-objective genetic (MOGA) [55], Multi-Objective Discrete Grey Wolf Optimization (MODGWO) [56], and some mixed use of these algorithms [57], to name a few [58,59].

These multi-objective optimization algorithms are used in many fields, such as in scheduling issues [56,60], route planning [61], communications technology [55], weapons target allocation [62,63], structural health monitoring [64,65], and others. For example, Gu et al. [56] established an energy-saving scheduling model to solve the Energy-Saving Job Shop Scheduling Problem (EJSP). The proposed algorithm can help optimize the total energy consumption and the makespan. Bin et al. [61] constructed an improved multi-objective genetic algorithm to solve the problem of rural logistics distribution routing optimization. This algorithm aids in route planning under demand uncertainty and carbon emission constraints. Hu et al. [55] introduced an improved technique based on a genetic algorithm to get optimization solutions in the communication field, such as energy consumption, cost-effective edge user distribution, and efficient scheduling of multi-objective optimization problems. Qiu et al. [62] proposed a multi-objective simplified swarm optimization algorithm to address the dynamic weapon target assignment problem. The full-variable update mechanism and the harmonic step strategy are introduced in the algorithm to make it more efficient. Cha et al. [64] proposed the hybrid multi-objective NS2-IRR GA to detect structural damages. The proposed algorithm integrates the IRR GAs’ strengths as an encoding policy and non-dominated sorting GA-II (NSGA-2) as a selection method. The proposed NS2-IRR GA performs exceptionally well in detecting the exact locations and extents of the induced minor damages in the structure, even though the damaged element lacks measured information. Cha et al. [65] proposed a multi-objective genetic algorithm (MOGA) for optimal placements of control devices and sensors in seismically excited civil structures. The proposed algorithm combines an implicit redundant representation genetic algorithm with a strong Pareto evolutionary Algorithm 2. MOGA is good at developing optimal Pareto front curves for optimal placement of actuators and sensors in seismically excited large buildings and satisfying the performance of dynamic responses.

This study investigates the planning of the UAV swarm’s operational test plans from the perspective of mission reliability to support the planning of the UAV swarm’s operational test. Firstly, the mission reliability in the combat stage is modeled, which considers factors such as the flight height of the UAVs, the area allocation plan, and the enemy threat in the area. Then, the PMS mission reliability model of the UAV swarm is constructed, and the reliability calculated in the combat phase comes from the model at the first step. The PMS mission reliability of the UAV swarm is calculated by considering factors such as flight height, allocation plan, and enemy threat of the UAV swarm. Finally, subjected to both reliability and cost, the optimal plans for the operational test of the UAV swarm are derived. The main merits of the study are as follows: 1)

UAV swarm’s operational test is regarded as a multi-stage mission, and a corresponding multi-stage mission reliability model is proposed. Typically, the mission reliability calculation model for the combat stage differs from the classical calculation model, which only considers UAV performance. Instead, it fully considers UAV allocation, flight altitude, and interceptions by the enemy, which are more realistic in real cases.

The simulation results based on the above model show that the model has good application value and can provide technical support for command and control personnel.

The rest of this paper is arranged as follows: Section 2 comprehensively introduces the modeling of mission reliability of UAV swarm. Section 3 presents the planning method for the operational reconnaissance test and attack UAV swarm. Section 4 describes a case to demonstrate the validation of the proposed method. Section 5 concludes this paper.

Many types of missions can be performed by UAV swarms [66,67], and this study mainly focuses on the reconnaissance and attack missions of UAV swarms. The reconnaissance sub-swarm and attack sub-swarm execute reconnaissance and attack sub-missions, respectively. The attack-type UAV refers specifically to suicide UAVs. The total mission is divided into three stages: the launch, the flight to the combat area, and the combat (including reconnaissance and attack sub-missions). When calculating the reliability of the reconnaissance and attack mission of a UAV swarm, the following considerations are made: 3)

The recovery of the UAVs is not considered because the UAVs are usually of low value, which means the cost-efficiency of recovery is not high, so they are usually not recovered in actual combat.

In this section, the reliability model for the combat stage is developed.

As the combat area is large, it is necessary to divide it into smaller subareas to get better executions of reconnaissance and attack missions. The division mainly considers the reconnaissance coverage of a single reconnaissance UAV because a reconnaissance UAV swarm needs to reconnoiter the whole combat area, while an attack UAV swarm only must attack targets in the combat area. When the detection angle of a single reconnaissance UAV’s sensor is α and the reconnaissance altitude is h, the maximum reconnaissance width is 2htan⁡α. Completing the mission requires that all combat areas be covered, so the combat area A is divided, as shown in Fig. 1.

The mission reliability of the reconnaissance UAV sub-swarm executing the reconnaissance mission in the combat stage can be referred to in the literature [68]. This study introduces the calculation of the mission reliability of the suicide UAV sub-swarm executing the attack mission.

The relevant indicators of the suicide UAVs are listed in Table 2.

Table 2 indicates that: (1)N=∑i=1aniA

The reliability of the attack mission of the suicide sub-swarm includes three aspects: flight reliability, target recognition capability, and target attack capability, as shown in Fig. 2. The calculation of flight reliability and target recognition reliability are the same as that of reconnaissance UAVs [68]. The calculation of target attack ability is related to factors such as attack distance, combat environment, and UAV payload [2]. The target attack capability of the ith suicide UAV can be expressed as depicted in Eq. (2):

(2)ria=kD×kE×DAMwhere kD [3] is the distance coefficient, which is related to the straight-line distance between the suicide UAV and the enemy target, whose calculation method is shown in Eq. (3); kE is the environmental impact coefficient reflecting the impact of visibility, wind, and other combat environments [69–71] on the attack ability of the suicide UAV, whose value is listed in Table 3; DAM is the damage probability [72,73], the possibility that the suicide UAV destroys the target after hitting it, whose calculation method is as Eq. (4):

(3)kD={0.5d≤RiA and d≤Rj0.5+0.2(d−RjRiA−Rj)Rj<d<RiA0RiA<d<Rj0.2max(RiA,Rj)<d<RiAI  where d is the straight-line distance between the launch position of the suicide UAV and the target position, which can be confirmed before the UAV swarm executes the attack mission. In particular, to simplify the calculation, we ignore the difference of d when the suicide UAV is under different flight altitudes. RiA is the damage radius of the ith suicide UAV (i.e., the maximum distance at which it can initiate a suicide attack on the enemy). Rj is the attack radius of the jth enemy target (if the enemy target lacks of attack ability, Rj is set to 0). RiAI is the reconnaissance radius of the ith suicide UAV.

(4)DAM=1−(12)RiA2CEPi2where CEPi is the attack circle’s calculated deviation of the ith suicide UAV, which reflects the attack accuracy of the suicide UAV [73]. The larger the CEP value, the worse the attack accuracy of the suicide UAV.

As shown in Eq. (3), due to the different positions of enemy targets, the reliability of the attack mission for a single suicide UAV must be calculated point-to-point. Eq. (5) represents the actual attack reliability of a single ith suicide UAV at a flight altitude h against g targets: (5)rigA=ri0AI×fiA×Rantiemis×Rantifps×riga(i=1,2,…,a2,g=1,2,…,e) t=dvAwhere rigA is the actual attack reliability of a single ith suicide UAV against g targets. Rantiemis is the anti-jamming capability of a single suicide UAV on the sub-region As, Rantifps is the anti-attack capability of a single suicide UAV on the sub-region As. The calculation method is the same as that of the reconnaissance aircraft. ri0AI is the basic mission reliability of a single ith suicide UAV, fiA is the flight reliability of the ith suicide UAV, and t is the time for a suicide UAV to travel from its launch position to the enemy target position. We can see from Eq. (5) that for the same type of enemy weapon, the final mission reliability of the same single suicide UAV varies when it is located in different sub-regions, that is because the interference in various sub-regions on the same suicide UAV will vary.

In particular, to complete the attack mission, the sub-region containing the enemy targets must be assigned suicide UAVs.

The target attack reliability of the suicide UAV swarm for the gth target can be calculated by Eq. (6): (6)RgA=1−∏i=1a2(1−rigA)lgiwhere RgA is the target attack reliability of the suicide UAV swarm for the gth target. lgi is the number of ith suicide UAVs assigned to attack the gth enemy target.

Then, Eq. (7) can be applied to calculate the target attack reliability of the entire suicide UAV swarm across the entire operational area, represented by RAA: (7)RAA=∏g=1eR gA

BDD and MOQPSO are used to build the PMS mission reliability model of the UAV swarm. The mission reliability of the reconnaissance UAV sub-swarm and the suicide UAV sub-swarm in the combat stage are obtained from the calculation model in Section 4.1.

The cost is another critical factor for the operational test planning of a UAV swarm; thus, the calculation method is provided.

Based on the hypothesis in Section 2.1, the maintenance of launch systems, ground stations, and others is not considered. Thus, when calculating the cost of one plan, the cost of the UAVs is only considered. For UAV swarms taking up reconnaissance and attack missions, the cost of reconnaissance UAV sub-swarms executing the reconnaissance mission in the combat stage can be indicated in the literature [68]. As for the suicide swarm, since recovery is not considered, the cost of the attack mission is calculated as illustrated in Eq. (20): (19)CostA=∑i=1a2Costi

When planning an operational test, it is necessary to compare the calculated mission reliability with the lower bound of the required mission reliability, Assuming the mission reliability of the reconnaissance sub-swarm is RAI whose calculating method can be indicated in the literature [68]. The comparison is as shown in Eq. (21), where RA is the total mission reliability of reconnaissance and attack UAV swarm, and ⌢RA is the required lower limit. (20)RAI⋅RAA=RA≥R^A

Then, the operational test planning model is: (21)RA≥R^AMIN(CostA)MAX(RA)

The problem of selecting optimal operational plans considering more than one objective (mission reliability and cost) is a multi-objective integer optimization problem. This study uses the multi-objective quantum particle swarm optimization algorithm (MOQPSO) to solve the problem. In this section, the algorithm’s basic process is introduced first and then applied to solve an example.

For integer optimization problems with multi-dimensional inputs, Particle Swarm Optimization (PSO) or its variants can be adopted. However, the traditional PSO method has some disadvantages: (1) it requires many preset parameters, making it challenging to find the optimal parameters; (2) the change in particle positions lacks randomness, making it easy to fall into the trap of local optimization [76]. A good solution is to adopt the Quantum Particle Swarm Optimization (QPSO) algorithm, which increases particle position’s randomness by eliminating particle movement’s direction attribute to remove the correlation between location updates and previous motion of particles [77]. QPSO can also make integer particle positions suitable for addressing the integer optimization issue.

For a multi-objective optimization problem, since objectives can constrain each other, indicating that one objective can improve at the expense of another, achieving optimal performance for all objectives is hard. Hence, a multi-objective optimization problem can only get a set of non-inferior solutions, i.e., the Pareto solution set [78]. Therefore, multi-objective optimization algorithms must be employed to solve this problem. The Multi-Objective Particle Swarm Optimization (MOPSO) algorithm proposed by Cortés et al. [79], in 2003, is a multi-objective optimization algorithm originating from PSO that can only be used for single-objective optimization.

Accordingly, compared to PSO, QPSO improves the calculation method of pbest, MOPSO improves the selection method of gbest according to multi-objective problems. Thus, a multi-objective quantum particle swarm optimization algorithm (MOQPSO) is proposed to combine the strengths above. Referring to the algorithm procedure of QPSO and MOQPSO [80], the algorithm of MOQPSO is shown in Fig. 10.

Based on the flowchart, the algorithm of MOQPSO [80,81] is: 1)

Randomly initialize the particle swarm coordinates and some basic parameters, such as population size, archive size, iteration count, weight coefficient, and others.

First, the following assumptions are proposed: 1)

The UAV swarm performs a reconnaissance and attack mission, where the reconnaissance UAV sub-swarm completes the reconnaissance mission first, and then the suicide UAV sub-swarm completes the attack mission.

The time taken to fly to the operational area can be calculated by Eq. (22). The time spent in the operational area is determined by the sum of the time taken by the reconnaissance UAV swarm to perform reconnaissance missions and the longest attack duration among the suicide UAVs where the time taken by the reconnaissance UAV swarm to perform reconnaissance missions can be determine by Eq. (23), and the longest attack duration among the suicide UAV swarm is assumed to be 3 min.

(22)t12=dreachvflywhere dreach is the distance from the launch point of the UAV swarm to the boundary of the operational area, and vfly is the speed of the UAV swarm during the flight phase.

(23)t23_1=Lvreconnaissancewhere L is the length of the long side of the operational area boundary, and vreconnaissance is the reconnaissance speed of the UAV reconnaissance sub-swarm.

The flight speeds of various types of UAVs during different mission stages are listed in Table 7.

The probability of UAV swarm mission failure depends not only on flight reliability but also on the threat level of the operational area, enemy target threats, and the allocation plan of the UAV swarm. Some academic methods have been proposed to solve this problem [82–84], which are not introduced in this study. The calculations of single-stage mission reliabilities are not included in this research to illustrate the effects of the proposed method on operational test planning. The reconnaissance mission reliability of the reconnaissance sub-swarm is assumed to be 0.999, the attack mission reliability of the suicide sub-swarm is 0.998 during the combat stage, and the launch phase of the UAV swarm is 5 min. The flight stage time can be calculated as 45 min, and the operation stage time can be determined as 9 min.

Hence, the simplified reliability of each essential mission variable can be computed as shown in Table 8.

Using the BDD algorithm, the mission reliability of the UAV swarm executing the mission of reconnaissance and attack during the launch phase, flight to the operational area phase, and combat phase can be calculated as 0.9999, 0.9995, and 0.9956, respectively. Based on Eq. (12), the mission reliability of the last stage is equivalent to the overall mission reliability.

Therefore, the reliability of the UAV swarm for reconnaissance and attack missions is 0.9956.

The method introduced in Section 3 will be applied to find optimal plans for the UAV swarm operational test. This study sets the MOQPSO algorithm to iterate 10,000 times and designates a capacity of 50 for the optimal solution set. Then, it gets the total mission reliability and cost for a UAV swarm performing multi-stage reconnaissance and attack missions under different allocation plans and flight altitudes.

Regarding the total cost, the cost of the launch system and ground station are not counted since they can be reused. The cost of the communication link (which refers explicitly to the communication between UAVs) is already included in the cost of UAVs and does not need to be considered separately. Hence, the total cost of the reconnaissance and attack mission for the UAV swarm is the sum of the costs of the reconnaissance sub-swarm and attack sub-swarm.

Two situations can be met in actual military use: enemy target locations are settled or not settled. For both scenarios, the calculation of total mission reliability and cost and the process to get the optimal plans can be conducted. (1)

When Enemy Target’s Locations are Settled

Assuming that the enemy target’s locations are settled, as shown in Table 9.

Fig. 12 shows the final converged Pareto curve (shown by red scatters) of the case using the algorithm we propose. The simulation process requires about 369.77 s. Table 10 lists the first two (ranked by mission reliability) Pareto-optimal plans for the UAV swarm’s reconnaissance and attack mission, along with their total mission reliability and cost.

(2)

When Enemy Target’s Locations are Not Settled

Assuming that the location of each enemy target is uncertain, the ranges of distances between the starting point of the UAV swarm and enemies are shown in Table 11.

The 100 times simulation results of each preliminary optimal plan are drawn in scatter-plots shown in Figs. 13 and 14, where the vertical axis represents mission reliability, and the horizontal axis represents the number of simulation times. Taking the comparison between Plan 1 and 2 as an example, it is evident that Plan 2 has a more stable and overall better reliability value than Plan 1. Therefore, based on this comparison, Plan 2 is selected.

This subsection compares the MOQPSO algorithm with NSGA-II [85]. The NSGA-II mentioned is based on a Genetic Algorithm (GA) and is suitable for solving multi-objective optimization problems. Another difference between NSGA-II and classical GA is that NSGA-II has a more efficient selection method for choosing parent and child generations [85]. This study mainly compares the Pareto curve and computational efficiency of MOQPSO and NSGA-II.

Fig. 15 shows the final converged Pareto curve (shown by red scatters) of the case using NSGA-II. The simulation process needs about 772.77 s.

Figs. 12 and 15 indicate that MOQPSO has a better Pareto curve with more centralized optimal particles. In addition, MOQPSO is about 52% quicker than NSGA-II for a few reasons: (1) when updating, MOPSO can preserve the information of relatively good particles while NSGA-II has no memory and changes particles randomly; (2) MOPSO only shares the information of present optimal particles to iterate while NSGA-II shares the information of all the particles, indicating the iteration of the whole generation of uniform; thus, MOPSO can have the chance of iterating faster; (3) compared to NSGA-II, MOQPSO does not have coding, crossover, or a mutation process; it has an easier way of iterating.

Accordingly, the results prove that the proposed MOQPSO performs better when responding to the proposed model.