多目标应用：多目标蜣螂优化算法求解多旅行商问题（Multiple Traveling Salesman Problem, MTSP）
一、多旅行商问题
多旅行商问题（Multiple Traveling Salesman Problem, MTSP）是著名的旅行商问题（Traveling Salesman Problem, TSP）的延伸，多旅行商问题定义为：给定一个座城市的城市集合，指定个推销员，每一位推销员从起点城市出发访问一定数量的城市，最后回到终点城市，要求除起点和终点城市以外，每一座城市都必须至少被一位推销员访问，并且只能访问一次，需要求解出满足上述要求并且代价最小的分配方案，其中的代价通常用总路程长度来代替，当然也可以是时间、费用等。围绕着各推销员的起始点和终止点来划分，多旅行商问题大致可以分为四种，其中单仓库多旅行商问题是其中一种。多旅行商问题
单仓库多旅行商问题（Single-Depot Multiple Travelling Salesman Problem, SD-MTSP）：个推销员从同一座中心城市出发，访问其中一定数量的城市并且每座城市只能被某一个推销员访问一次，最后返回到中心城市，通常这种问题模型被称之为SD-MTSP。
1.1多旅行商多目标问题描述
对于推销员人数为，所需要访问的城市总数为的多旅行商问题而言，记中心城市为0，其双目标的数学模型可以表述为：
min f = [ f 1 , f 2 ] T \min f=\left[f_{1}, f_{2}\right]^{T} minf=[f1,f2]T
f 1 = ∑ k = 1 m ∑ i = 0 n ∑ j = 0 n d i j x i j k f 2 = max 1 ≤ k ≤ m ∑ i = 0 n ∑ j = 0 n d i j x i j k − min 1 ≤ k ≤ m ∑ i = 0 n ∑ j = 0 n d i j x i j k \begin{array}{c} f_{1}=\sum_{k=1}^{m} \sum_{i=0}^{n} \sum_{j=0}^{n} d_{i j} x_{i j k} \\ f_{2}=\max _{1 \leq k \leq m} \sum_{i=0}^{n} \sum_{j=0}^{n} d_{i j} x_{i j k}-\min _{1 \leq k \leq m} \sum_{i=0}^{n} \sum_{j=0}^{n} d_{i j} x_{i j k} \end{array} f1=∑k=1m∑i=0n∑j=0ndijxijkf2=max1≤k≤m∑i=0n∑j=0ndijxijk−min1≤k≤m∑i=0n∑j=0ndijxijk
其中：
x i j k = { 1 推销员 k 由城市 i 到达城市 j 0 否则 s.t. ∑ k = 1 m ∑ i = 1 n x i 0 k = m ∑ k = 1 m ∑ j = 1 n x 0 j k = m ∑ k = 1 m ∑ i = 1 n x i j k = 1 ∀ j = 1 , … , n ∑ k = 1 m ∑ j = 1 n x i j k = 1 ∀ i = 1 , … , n \begin{array}{l} x_{i j k}=\left\{\begin{array}{lc} 1 & \text { 推销员 } k \text { 由城市 } i \text { 到达城市 } j \\ 0 & \text { 否则 } \end{array}\right. \\ \text { s.t. } \\ \sum_{k=1}^{m} \sum_{i=1}^{n} x_{i 0 k}=m \\ \sum_{k=1}^{m} \sum_{j=1}^{n} x_{0 j k}=m \\ \sum_{k=1}^{m} \sum_{i=1}^{n} x_{i j k}=1 \quad \forall j=1, \ldots, n \\ \sum_{k=1}^{m} \sum_{j=1}^{n} x_{i j k}=1 \quad \forall i=1, \ldots, n \\ \end{array} xijk={10 推销员 k 由城市 i 到达城市 j 否则 s.t. ∑k=1m∑i=1nxi0k=m∑k=1m∑j=1nx0jk=m∑k=1m∑i=1nxijk=1∀j=1,…,n∑k=1m∑j=1nxijk=1∀i=1,…,n
其中， f 1 f_{1} f1表示所有推销员的总路程， f 2 f_{2} f2代表推销员中最长路线与最短路线的差值（平衡度）。约束条件1和约束条件2表示编号为 0 的中心城市的入度和出度都必须为，即代表个推销员均从中心城市出发并且完成行程后都回到了中心城市。约束条件3和约束条件4表示除中心城市外的其他所有城市的出度和入度均为 1，即每个城市能且只能被推销员中的一个人访问。
参考文献：
[1]杨帅. 求解多旅行商问题的进化多目标优化和决策算法研究[D].武汉科技大学,2020.
二、多目标蜣螂优化算法
非支配排序的蜣螂优化算法（Non-Dominated Sorting Dung beetle optimizer，NSDBO）
三、NSDBO求解多旅行商问题
本文选取国际通用的TSP实例库TSPLIB中的测试集bayg29，bayg29中城市分布如下图所示：
本文采用NSDBO求解bayg29，两个目标函数分别是： f 1 f_{1} f1表示所有推销员的总路程， f 2 f_{2} f2代表推销员中最长路线与最短路线的差值（平衡度）。设置城市13作为两个旅行商的起点城市，部分代码如下：
close all;
clear ;
clc;
TestProblem=1;%1
MultiObj = GetFunInfo(TestProblem);
MultiObjFnc=MultiObj.name;%问题名
% Parameters
params.Np = 100; % Population size
params.Nr = 200; % Repository size
params.maxgen =5000; % Maximum number of generations
numOfObj=MultiObj.numOfObj;%目标函数个数
D=MultiObj.nVar;%维度
f = NSDBO(params,MultiObj);
X=f(:,1:D);%PS
Obtained_Pareto=f(:,D+1:D+numOfObj);%PF
save f f
NSDBO求解得到的Pareto前沿：
NSDBO求解得到的Pareto前沿值 f 1 f_{1} f1（总路程）与 f 2 f_{2} f2（平衡度）如下：
22102.9727798521 0.00168180065520573
22102.9727798521 0.00168180065520573
20329.0091686614 0.0896807373028423
20329.0091686614 0.0896807373028423
15456.3506535282 1.33753213901582
15456.3506535282 1.33753213901582
13950.3392331925 695.147203595617
14420.4208157020 64.1232588333942
13922.7061066601 722.780330128020
14053.9188784077 591.567558380348
13801.7974549415 843.688981846539
14400.5426626992 244.943774088882
21666.5808905080 0.0437079968960461
21666.5808905080 0.0437079968960461
13879.7557217029 765.730715085202
14219.2814238736 426.205012914498
14259.9199643935 385.566472394623
13576.1099393706 1069.37649741747
14274.5105510786 370.975885709519
14187.8876952665 457.598741521547
13592.8678000419 1052.61863674616
13405.1284596659 1240.35797712221
14057.8134204866 587.673016301442
13601.5016056637 1043.98483112440
14595.0969246490 50.3895121391015
14285.0633767127 360.423060075419
14250.8059878040 394.680448984042
13638.7940708342 1006.69236595388
13563.0504709883 1082.43596579977
14270.4999860843 374.986450703771
14034.0203420467 611.466094741369
14413.6027244188 231.883712369266
14644.0574853126 1.42895147543368
13819.0824906934 826.403946094693
13395.6431995137 1249.84323727441
14180.6060656979 464.880371090151
13870.0498645105 775.436572277533
13939.7611609484 705.725275839711
14420.0001688052 225.486267982903
14264.7872139482 380.699222839900
13405.1284596659 1240.35797712221
13962.9518331561 682.534603631974
13412.3745988936 1233.11183789447
13655.5519315055 989.934505282572
14045.2478310867 600.238605701406
13679.3964854527 966.089951335388
14026.1398964906 619.346540297437
14098.7532568577 546.733179930393
14618.5185669984 26.9678697896543
13696.8739147629 948.612522025227
13664.5719518761 980.914484912005
13936.7403600022 708.746076785834
13731.9599733822 913.526463405907
13796.4774910062 849.008945781922
14339.6906369881 305.795799799950
13755.7555255629 889.730911225152
14230.2555149456 415.230921842478
13755.7555255629 889.730911225152
14075.4804896908 570.005947097243
14288.6127867984 356.873649989633
14368.6697670641 276.816669724018
14607.6529096612 37.8335271268925
14002.5714775335 642.914959254576
14161.5141852619 483.972251526198
14382.7329591811 262.753477606963
13776.9216244865 868.564812301621
14317.4067712580 328.079665530078
14166.6544834181 478.831953370019
13738.9976648916 906.488771896463
13521.4744741303 1124.01196265782
13826.3329814449 819.153455343180
13784.7296079061 860.756828882028
13916.2244996389 729.261937149149
13976.6848214007 668.801615387379
14257.0766510210 388.409785767103
14312.3050031855 333.181433602567
14628.5102433837 16.9761934043490
14077.7837516092 567.702685178916
13887.8238203771 757.662616410972
13833.3772965324 812.109140255715
13436.7083012567 1208.77813553138
14088.0610029341 557.425433853994
13809.5751207736 835.911316014491
14113.9329148047 531.553521983379
14082.5893944871 562.897042300954
14300.6489105867 344.837526201390
13904.1650632058 741.321373582243
13853.2481981306 792.238238657443
14128.6338178830 516.852618905066
13821.0433350971 824.443101691019
13864.1807704821 781.305666305961
13896.8478118052 748.638624982852
13604.9130781688 1040.57335861927
14409.1938928293 236.292543958801
13576.1099393706 1069.37649741747
13956.5785499213 688.907886866759
13980.0588635344 665.427573253641
14363.1056599183 282.380776869733
13676.7180304290 968.768406359041
13566.4242494531 1079.06218733502
13499.2228447289 1146.26359205914
13499.2228447289 1146.26359205914
13851.4600244969 794.026412291182
13369.9522776565 1407.65153624345
13716.8037297586 928.682707029437
14375.9303533726 269.556083415446
13775.3113920826 870.175044705454
13520.3889436525 1125.09749313561
13859.2706574781 786.215779310020
13594.6303125791 1050.85612420895
14236.9511003133 408.535336474735
14198.1049883458 447.381448442302
14049.1379747565 596.348462031567
14633.5007437487 11.9856930393889
13395.6431995137 1249.84323727441
14586.7033097544 58.7831270336346
14614.5413354607 30.9451013274302
14328.8775983163 316.608838471781
14139.5171512169 505.969285571210
14137.1833694194 508.303067368681
14211.7359073611 433.750529426938
14063.4210532577 582.065383530337
14173.0376035351 472.448833253017
14020.6014731387 624.884963649392
13534.9758344685 1110.51060231963
13930.6918937885 714.794542999563
13689.5831246536 955.903312134470
13929.0756214169 716.410815371169
13670.0038946683 975.482542119748
13457.8744001802 1187.61203660785
14389.8358659876 255.650570800486
13789.7429864621 855.743450325952
14103.8422445527 541.644192235358
13844.6837666621 800.802670126003
13615.7964115027 1029.69002528542
14007.7447084211 637.741728366974
13836.2330110090 809.253425779076
13584.2165699118 1061.26986687624
13412.4010601850 1233.08537660310
14038.1089258591 607.377510928947
13971.5053321160 673.981104672085
13716.8037297586 928.682707029437
13731.9599733822 913.526463405907
14588.2825964126 57.2038403755196
13985.3186270225 660.167809765614
13994.1550223314 651.331414456641
13461.8156004230 1183.67083636509
13426.2945585894 1219.19187819868
14293.7515820089 351.734854779168
14325.7364561681 319.749980619964
13709.6557064622 935.830730325880
13805.8957068296 839.590729958496
14351.8856863840 293.600750404034
14394.4781343436 251.008302444491
13907.9095224934 737.576914294700
13738.9976648916 906.488771896463
13426.2945585894 1219.19187819868
13513.8097355449 1131.67670124316
13610.3107205101 1035.17571627802
13482.9816993465 1162.50473744156
13433.5671591085 1211.91927767957
15948.7678292123 0.108952378204776
14069.0409095266 576.445527261479
13767.6797855170 877.806651271094
13482.9816993465 1162.50473744156
14622.6276397819 22.8587970062299
14243.7207815203 401.765655267734
13643.8754436191 1001.61099316897
14307.1410311894 338.345405598670
14145.4970568141 499.989379973967
13989.5438116728 655.942625115286
14126.0632200757 519.423216712366
14216.4557735630 429.030663225126
13689.5831246536 955.903312134470
14119.8505980888 525.635838699319
14360.0618842428 285.424552545263
13883.4155821249 762.070854663193
14639.6224207000 5.86401608805682
14335.2573198184 310.229116969712
13659.9601697577 985.526267030351
13534.9758344685 1110.51060231963
14148.7146707741 496.771766013964
13369.9522776565 1407.65153624345
13513.8097355449 1131.67670124316
13584.2165699118 1061.26986687624
14155.0961810537 490.390255734353
15815.4017049468 1.13979477541398
13626.0791770923 1019.40725969574
15815.4017049468 1.13979477541398
13632.5966503624 1012.88978642570
14601.3731536162 44.1132831718569
13968.0335912706 677.452845517460
13649.5132494857 995.973187302346
14012.5704338590 632.916002929059
13765.4113625185 880.075074269558
13474.6746390399 1170.81179774814
14581.7524217132 63.7340150748669
14204.5192799572 440.967156830911
13474.6746390399 1170.81179774814
部分规划路径如下：
