A Novel Tent-Levy Fireworks Algorithm for the UAV Task Allocation Problem Under Uncertain Environment

Recently, unmanned aerial vehicle (UAV) task allocation is a hot topic both in the civilian and military, while the research of considering uncertainty and multi-objective is still in its infancy. Firstly, based on the uncertainty theory, a mathematical model of the uncertain multi-objective UAV task allocation problem with uncertain variables in both objective function and constraint conditions is established. The expected value criterion and opportunity constraint are introduced to transform the model into a deterministic optimization model. Furthermore, because traditional fireworks algorithm (FWA) has the shortcomings of low solution accuracy and slow convergence speed in solving the UAV task allocation problem, a novel Tent-Levy FWA (TLFWA) based on discrete update process is designed by introducing integer coding, Tent chaotic mapping and Levy variation. Experimental results show that the mean cost calculated by TLFWA is 8.17% and 13.73% lower than that of FWA and particle swarm optimization algorithm respectively, which proves the effectiveness of TLFWA. This study provides a new way to solve multi-objective and uncertain decision-making problems.

bad weather conditions and other uncertain factors, the fuel 27 The associate editor coordinating the review of this manuscript and approving it for publication was Emre Koyuncu . consumption, flight time and threat in the process of UAV 28 task allocation are unmeasurable. Those uncertain variables 29 will result in the loss of optimality or even infeasibility of the 30 task allocation scheme under certain conditions. Traditional 31 methods to solve uncertain problems include probability the-32 ory, fuzzy set theory and robust optimization theory. How-33 ever, probability theory cannot solve the uncertain problem 34 when probability distribution of variables cannot estimate [6], 35 [7]. Fuzzy set theory is not self-consistent in mathematics 36 [8], and cannot solve uncertain problems in some specific 37 situations [9], [10]. The results of robust optimization theory 38 are relatively conservative [11]. Liu indicated that probabil-39 ity theory, fuzzy set theory and robust optimization theory 40 may lead to counterintuitive results when dealing with these 41 uncertainties [12]. Therefore, Liu [13], [14] proposed a com-42 plete theoretical system with normativity, subadditivity and 43 self-duality-uncertainty theory to solve uncertain problems. 44 The main contributions of this paper are as follows.

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(3) The performance of TLFWA is tested by six benchmark 103 functions. And the effectiveness of the proposed model 104 and method in this paper are verified by UAV task 105 allocation examples. 106 The paper is organized as follows. Section 2 introduces 107 the basic definitions and theorems of uncertainty theory. The 108 uncertain UAV task allocation model is established in Sect. 3. 109 In order to solve the model, a novel improved fireworks 110 algorithm is designed in Sect. 4. In section 5, the effectiveness 111 of the model and algorithm proposed in this paper is verified 112 by experiments. Section 6 concludes this work in this paper. 113

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Definition 1 [13]: Let be a nonempty set, and L is a 115 σ -algebra over . Each element in L is called a measurable 116 set, which is renamed event in uncertainty theory. A set 117 function M from L to [0, 1] is called an uncertain measure 118 if it satisfies axioms 1, 2, 3, and the triplet ( , L, M ) is called 119 an uncertainty space. Axiom 4 defines an uncertain measure 120 in product space.

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• Y k : the kth UAV task allocation sequence (l ≤ 184 m). Y k indicates that the kth UAV attacks task targets 185 y 1 , y 2 , · · · , y l in turn and then returns to the base.

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• T k : T k = T k y 1 , T k y 2 , · · · , T k y l is the time of attacking  The task allocation problem is essentially to allocate a task 200 attack sequence to each UAV. The objective decision variable 201 is denoted as X . If UAV attacks the target j directly after 202 attacking task target i, x ij = 1, otherwise x ij = 0.
The objective function of the UAV task allocation problem 205 includes fuel consumption cost and delay penalty. Due to the 206 influence of flight altitude, speed, atmospheric disturbance 207 and other uncertain factors, both the fuel consumption per 208 unit flight distance ε ij and the flight time t ij in the actual 209 flight process are regarded as uncertain variables. Let ε ij and 210 t ij be a series of independent uncertain variables satisfying 211 normal distribution N (e 1 , σ 1 ) and N (e 2 , σ 2 ), respectively. 212 Therefore, the fuel consumption cost F 1 can be expressed 213 by And the delay penalty F 2 can be expressed by where C k i is the delay penalty of the kth UAV, λ is the 220 coefficient of the delay penalty, T k j is the time of attacking 221 task target j by the kth UAV, Time j is the time window of the 222 task target j. 223 In conclusion, the objective function of the UAV attack task 224 allocation problem is where λ 1 and λ 2 are the weight coefficient of fuel con-228 sumption cost and delay penalty respectively, and satisfy 229 λ 1 + λ 2 = 1. it must be satisfying 240 where A k is the total amount of fuel carried by the kth UAV.
268 Therefore, the expected value model of the UAV attack task 269 allocation problem can be expressed as Each firework represents a feasible task allocation scheme in 284 the UAV attack task allocation problem. Let x is a firework, 285 M is number of fireworks, and D is the number of task targets 286 to ensure that each task target can be attacked only once. The 287 initial solution space is initialized according to the number of 288 fireworks, and the ith firework x i can be calculated by where rand is random number generator.
Explosion is the core step in the FWA. The explosion ampli-293 tude and the spark number depend on the fitness of fireworks. 294 The good fitness of fireworks makes amplitude smaller and 295 generate more sparks. The optimal value can be found more 296 quickly by strengthening local search. In contrast, the global 297 search is strengthened, which enhances the diversity of solu-298 tion space. The fireworks explosion amplitude and spark 299 number are calculated by Calculate the explosion amplitude and the spark number by formula (20) and (20) (6) Generate explosion sparks by formula (22)  (7) end for (8) Generate variation sparks based on Levy variation by formula (32) (9) Map sparks according to mapping rules which is out of bounds (10) Calculate all fireworks and sparks fitness (11) Selected next generation fireworks according to the selection strategy (12) end if  The sparks will shift within amplitude by the explosion, 309 which can be calculated by 311 where x ik is the value of the ith firework in the kth dimension.

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To avoid overwhelming effects of extreme sparks, the number 313 of sparks generated by each firework explosion is corrected 314 as follows.
In order to increase the diversity of fireworks, the Gaussian 318 variation is carried out fireworks, which can be expressed by  In order to ensure that sparks generated by firework explosion 324 and variation are in the feasible region, the modular operation 325 is adopted in this paper. The formula is expressed by

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The fireworks and sparks which produced by exploding and 332 varying are the candidate set K for the next generation of fire-333 works. The convergence speed of the algorithm can be accel-334 erated theoretically by choosing optimal individual directly, 335 whereas it may also cause the algorithm to fall into local opti-336 mal solution. To avoid this problem, roulette wheel strategy 337 is introduced on the basis of elite strategy, which select the 338 best firework by elite strategy. The remaining fireworks will 339 be determined by roulette wheel strategy, together being the 340 next generation of fireworks. This method not only ensures 341 that high fitness individuals can be selected with a high 342 probability, but also ensures that low fitness individuals have 343 chance to be selected, which effectively avoids the algorithm 344 falling into local optimal solution. The selection probability 345 of candidate individual x i can be calculated by where D (x i ) is the sum of distances from the firework x i to 349 all fireworks in the candidate set except itself.  will gather at the peak point, which is not conducive to 393 optimization. However, the probability distribution curve of 394 Tent chaotic mapping is almost a straight line. The initial 395 solution after chaotic mapping will be evenly distributed in 396 the range, which avoids falling into local optimization caused 397 by excessive search in some local areas.

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Therefore, Tent chaotic mapping is adopted in this paper 399 to generate initial task allocation schemes, which can be 400 calculated by

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FWA carries out the explosion operation by Gaussian 405 variation. However, Gaussian variation greatly reduces the 406 variation efficiency due to its small perturbation and the vari-407 ation of all real number range. It is not applicable to the 408 task allocation problem. Fig. 3 shows the probability density 409 function and distribution function of Gaussian distribution, 410 Cauchy distribution and Levy distribution. Compared with 411 Gaussian distribution and Cauchy distribution, Levy distri-412 bution has a higher wave peak, wider tail, and a positive 413 real number range of variation. Therefore, the Levy variation 414 with a large variation range and strong disturbance ability is 415 used in this paper (see Fig. 3), which can increase the search 416 scope and search efficiency. This is more conducive to finding 417 the optimal task allocation scheme. The probability density 418 function f (x, c) and distribution function F (x, c) of Levy 419 distribution are as follows.   The exact expression of the distribution function and its 436 inverse function is the precondition for inverse transformation 437 method. Therefore, rounding method, which is more widely 438 applicable, is adopted in this paper.    Step 1: Generate independent random numbers X ∼ 446 U (x min , x max ) and Y ∼ U (y min , y max ) subject to uniform 447 distribution.

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Step 2: X is a random number subject to Levy distribution 449 if Y ≤ f (X ); Otherwise, repeat step 1.

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The spark variation formula of Levy variation operator is

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where β is a random number subject to the Levy distribution.      and TLFWA are as follows: the number of fireworks is 5, 480 the maximum explosive spark number is 40, the minimum 481 explosive spark number is 2, the explosion radius coefficient 482 is 40, the explosive spark coefficient is 50, the variable spark 483 number is 5, and the maximum iteration number is 1000.

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MATLAB R2018a software is used for simulation in this 485 paper. In order to reduce errors, the optimal fitness obtained 486 by running 50 times is taken as the test result. The fitness 487

Remark 2:
The mean of Sphere and Schwefel_1.2 func-509 tion's fitness obtaining by FWA and TLFWA are less than 510 1E-16, which may be caused by value rounding error or 511 truncation error, so the value is approximated to zero.

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In summary, TLFWA, which introduced integer coding, 514 Tent chaotic mapping and Levy variation operator, improves 515 its convergence speed and the ability to jump out of local 516 optimal solution. TLFWA is superior to the traditional FWA 517 and PSO algorithm.

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In order to verify the effectiveness and feasibility of the 520 proposed model and TLFWA, UAV task allocation scenarios 521 with different scales are designed in this paper. Scenario 1: 522 two UAVs attack seven task targets. Scenario 2: four UAVs 523 attack fourteen task targets. The uncertain variables ε ij and t ij 524 satisfy normal distribution N (1, 2) and N (1, 1), respectively. 525 λ 1 = 0.6, λ 2 = 0.4, α = 0.85. To facilitate the analysis, 526 UAVs and task targets are regarded as particle, and commu-527 nication constraints between UAVs are not considered in the 528 task allocation process. In the Fig. 11, the first and second 529 coordinate of task target are horizontal coordinate and vertical 530 coordinate, which represent the position information of task 531 target. And the third coordinate of task target is the time limit. 532 TLFWA is used to solve the two UAV task allocation 533 scenarios, and the results are as shown in Table 5, Fig. 12-13. 534   UAVs may abandon the relatively closest task targets due 535 to the time limit. As shown in Fig. 12, UAV 2 directly attacks 536 the task target 7 rather than the closest task target 6 after 537 attacking task target 4. This is because the delay penalty 538 is higher than fuel consumption cost if UAV attacks the 539 closest task target 6. Therefore, UAV task allocation results

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Parameter settings of the three algorithms are the same as 547 section 5.1. Each algorithm runs independently for 30 times. 548 The maximum iteration is 200. The task allocation results are 549 shown in Table 6. 550 Table 6 shows that the minimum cost, mean cost and 551 variance of different task allocation scenarios calculated by 552 TLFWA are less than or equal to the traditional FWA and 553 PSO algorithm. Taking scenario 2 for example, the mean 554 cost calculated by TLFWA is 8.17% lower than that of FWA 555 and 13.73% lower than that of PSO algorithm. FWA and 556 PSO algorithm basically fall into the local optimal solution. 557 TLFWA successfully jumps out of local optimal solution and 558 obtains the optimal task allocation scheme. In conclusion, 559 TLFWA can stably obtain feasible task allocation scheme 560 with a small cost, and its stability is superior to FWA and 561 PSO algorithm. Parameter settings of the three algorithms are the same as 564 section 5.1. The three algorithms are iterated 200 times in the 565 same scene, and the convergence curves is shown in Fig. 14. 566 It can be seen from Fig. 14 that the costs obtained by the 567 three algorithms can converge to a stable value. When the 568 problem scale is small (scenario 1), all three algorithms can 569 obtain the optimal task allocation scheme. With the increase 570 of problem scale (scenario 2), TLFWA and FWA can obtain 571 the optimal solution, whereas PSO algorithm falls into local 572 optimal solution. In addition, TLFWA has the fastest con-573 vergence speed. This is the inherent advantage of TLFWA.

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On the one hand, TLFWA uses Tent chaotic mapping, which 575 increases the diversity of initial task allocation schemes.  Experimental results show that the mean cost calculated by 600 TLFWA is 8.17% and 13.73% lower than that of FWA and 601 PSO algorithm respectively, which proves the effectiveness 602 of TLFWA.

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In the future research, the following three aspects would be 604 concerned mainly. From the theoretical perspective, we con-605 sider using the optimistic value and pessimistic value criteria 606 of uncertainty theory to solve the multi-objective problem 607 directly. From the model perspective, the model is refined 608 by considering the time-varying value of mission objectives, 609 the change of battlefield threat, and the battle loss of UAV to 610 make it closer to the battlefield reality. From the algorithm 611 perspective, it is considered to design appropriate coding 612 method and optimization mechanism algorithm to solve the 613 task allocation problem more efficiently.