蚁群算法简介(part3: 蚁群算法之更新信息素)

时间:2021-06-08 10:33:20

信息素的局部更新策略  

每只蚂蚁在构造出一条从起点到终点的路径后,蚁群算法还要求根据路径的总长度来更新这条路径所包含的每条边上信息素的浓度(在旅行商问题中每座城市是图中的一个节点,城市两两间有一条边相连)。下面给出了蚁群算法更新信息素的公式:

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" alt="" width="381" height="141" />

.

上面的第一个公式体现了信息素的更新值的计算,其中,Ck代表第k只蚂蚁所构造的路径的总长度,Q是凭经验设定的一个参数,通常置为1。component(i,j)表示从城市i到城市j的一条边,m是蚂蚁的总数。上面的第二个公式是说i和j这条边的信息素值tau等于这条边上原有的信息素加上在上一次构造路径活动中所有经过这条边的蚂蚁贡献的信息素更新值(即第一个公式)。

假设三只蚂蚁构造的路径长度如下所示:

Tour 1: 450
Tour 2: 380
Tour 1: 460

则它们的信息素的更新值相加结果为:componentNewPheromone = (Q / 450) + (Q / 380) + (Q / 460)

最后把它加到各个蚂蚁的路径中每条边原来的信息素值之上:

componentPheromone = componentPheromone + componentNewPheromone

上述信息素更新过程的伪代码如下所示:

for ant in colony do //蚁群中每条蚂蚁都逐个更新自己路径经过边上的的信息素
tour = ant.getTour(); //蚂蚁计算路径的总长度
pheromoneToAdd = getParam('Q') / tour.distance(); //蚂蚁计算信息素的更新值
for cityIndex in tour do //回溯路径中的每个城市
if lastCity(cityIndex) do // 如果当前城市是路径上的最后一个城市,取出它和第一个城市间的边
edge = getEdge(cityIndex, )
else do
edge = getEdge(cityIndex, cityIndex+) //否则,取当前城市和路径上下一个城市间的边
end if
currentPheromone = edge.getPheromone(); //获得边上之前的信息素
edge.setPheromone(currentPheromone + pheromoneToAdd)//原有信息素加上更新值后得到新信息素
end for // ant路径上所有的边的信息素更新完毕
end for //蚁群中所有蚂蚁都处理完毕

信息素的挥发

在自然界中蚂蚁遗留下来的信息素经过一段时间后会挥发,我们在算法中也模拟上述过程,具体公式如下:

aaarticlea/png;base64,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" alt="" />

rho是一个人为设定的参数,称为挥发率,一般为0-1间的数。

相应的伪代码如下所示:

for edge in edges
updatedPheromone = ( - getParam('rho')) * edge.getPheromone()
edge.setPheromone(updatedPheromone)
end for

精英蚂蚁策略

 精英蚂蚁是对蚁群算法的一种改进,所谓精英蚂蚁是指当全部蚂蚁都构造完各自的路径之后,所有路径中最短的那条路径所对应的蚂蚁。精英蚂蚁策略的公式如下所示,它和上面的信息素局部更新公式的唯一区别在于精英蚂蚁在像其它蚂蚁一样更新完自己路径上的信息素后,还要再重复一遍信息素的更新过程,只不过这次要把更新值乘上一个参数e,e通常选在0和1之间。

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6Odho2H1QtJ9vAr62c/NUTq881UVltLQn5+nMZbDrLdAUBOMclo/7y+7a+2rub2j1U+w08u2muMug/sODHb7m/SiF/tUGsVR5OgN9a+qHjJLK+w2ArJwCnuWrU1+jssiz+3PjUJsyrS/nZO7vqCk4T1uQ6K+bhCKtLOdRWdipDW0s/NCLAYwsAZOuqUV993dM0Oe0/DeqnszheMenZUL3atE1wTPocCuFbOyOtlrBfDxDVpdK5PADgZFeN+prd5nfVpjaor/j97ECy2vROB7UqNEmNSZ9g9znF6sw3HT/+WlDzTjOor0c4VlXVNM3qYe/3e9M0ehqssxNU13VmRmHbtjqO6TwYx1GjMObcPR4Ps9BAeTGVOT2Mtaoq50Tr281zop2eiUDaAGDXVaP+ltSmQ2vPwaeStNWhrUP9mAeQR1KuAzU5ddrbMznCg/p2GVZV5XQJmMWf5uP2kpDb7aZnpes4eo8Zdrnf71pFoiPoI1rBoXfqUObNSrP9ZvMeE8uV/Zi0AUCMHxL1dTPV7TipuVEvTODf6h5Q73R4Y+OfRzE1UG1Ssdi9PuFBfftQTsQ1X2c3oO3Iqpa3LjNVHfT/OlmmTW/XQpwlBkrD8nv16eJNPi3L0vxT9T9TLwmnDQBi/KioX6S3+8oLqVJgcAYvwjPafjDleqvaZPr/7WKJH9R3ylmdBKa+9Xg8VAMzNa26rud5dh79fL/fdUwlchiGsizNtziJ6fte1Q692Tm+yW/TNLfbbZ7naZp0qN20AUCMHxL102Tm3z0Vqlc3j9NNP6kViecIR311gNvFEl6pb79TzXS7EqCSN0PmbdsOw+CfO7+HwNAZN6nVV9gVCx3TOZSdBr1imEH9yLQBQBhR/4Nem8C/+pHV+X05CET91f6P8KC+fRzNy/O/azeUBqpfqoWYI6hXwJ6OV5al6SRY3ZVPWw6b2G9SmGdPD4DDEfU/6LUGuj+on+COcqdZHe8Qjd87Ad4e1J+myXxQx7F7UMqy7LpOPfZ6JaaK5myY6CjL0q5JKDGmKT9Nk/1XVRHGcTRpGMexaRrzfnt7hheqjwDgI+p/kCLNs7Pu1Ytrv6LgkdQsxdOYGW3OIvutHfjtuFvXtXmDBvXN21QhG8dR0/L14uomSM6eEGq+b03UcCp5zgGd7n2TpNvtpiqdQrs50VpbGJ82ANhF1P+g13bgdyaL+Y3U3Jg9ixT4p2lSwYb3Y9ZqeL3o96WbFXF1XZui1pR4E5jVBC/L0i58fyTeZjfuzUJBHV+h3ZlLqOmB5ivKsjR1RH3cdOnHpA0AdhH1P0g3/Rfuyxpy1rytvu/ZS05b06g8q6oahmGrTMy6x7Zt7a51p5YwTZP6VJzeAsVmM2nO/6LwkI0eA6hzZ3byKctSrXbnUPouex8etd3NfD1nFH83bQCwi6j/QS9HfSTLLKUDgCsi6n+QeoOTehQQACBnRP0PUm+zPxMNAICvIOp/lpr7uW2eDwBIE1H/s+Z51jQ0fzIXAAAnI+qfYRxHPZ712wkBAGSNqA8AQC6I+gAA5IKoDwBALoj6AADkgqgPAEAuiPoAAOSCqA8AQC6I+gAA5IKoDwBALoj6AADkgqgPAEAuiPoAAOSCqA8AQC6I+gAA5IKoDwBALoj6AADkgqgPAEAuiPoAAOSCqA8AQC6I+gAA5IKoDwBALoj6AADkgqgPAEAuiPoAAOSCqA+kaJ7nuq6Lorjf73VdN01TVVXTNMMwdF1XVVXf999OI4DrIeoDKaqqap7noiiKopjneVmWaZqKoui6blmWcRyLovh2GgFcD1EfSNE4jsuymJC//Bnp+763/wQAkYj6QKKcBn3TNOafVVXR1gfwAqI+kCg7zC/LUhRF0zTm/7uum6bpfr9/KXUALomoDyTKjOKbf6rbX30A0zTVdT1N0/cSCOB6iPpAojSBX/9/v99Nu1+z/JqmYRo/gGcR9QEAyAVRHwCAXBD1AQDIBVEfAIBcEPWBVBQf8Hg8vp0tAAkh6gOp0Mb7Utf1/Rm3263v+6ZpyrK0o/7tdvt2tgAkhKgPpOLxeNgB++UdePTAHh2kbdtjEwng0oj6QELatrWb++8cSpv5lGV5VNoidV2n9J/8vd81DIO2SWbfJCSOqA8kxDxnT7QZ38u0t8/5Q/sZRv3l9yORvp0KYAdRH0iLgoe831Jv2/apof15nvu+f/NpfsWfewmn75Bc80gkXAJRH0jLPM/2jLxhGN452jRNT3UYaELAOw/10cjCtR4L9H6ulwvWdZAnoj6QnNvtZvfzv9kGjTcMg77xnR3+zaMC9XygqqqOS+BHHJLraZp0EGXffoBCVVVN09R1TZ0AKSDqAylSd7GcEy0Ut/S95pG+LzA1FYW6xDu9j8q1xmUU7O1HJJveGtUtDkkz8A6iPvARh0zEM06YkVdV1TiOil7vzCdQEFVNpe/7d0Yo3izDGEfl2h7Ud6K+agPPDrUAH0LUB443z/P7Pdv2pj3vNENj9H2vlf0alX95WMF8/P1h8kPKMOyoXC9/DurbSxhUA1BN6JA0A28i6gPHG8fx/W75ozbt2XW/38uyVMAz49OvfZ1p5poO7ZeTfUgZBhyY62VZCmuZpamlDcOgrg4W9SEdRH08YZ5nexsZ30Hf89+/f//973/HHOusNP+h67pDtsI1O94Ub2/as0VLBuxop69zprZFFqNJp1rPy7K8nOwXyvB+v7dtq/kETdMENsw5NteLFfV1ysyLyr728HkqL8CHEPURSz2uZVnWda252bqpGW+uMfvtv3///vXrn0MOdVqaXVVVHTIS72za84lN9du2dZrUGlmwxxTii7H4s8mrpwm8lrCnynCeZ2f1ndZBbL3/2Fwvv+dhNE3TNI0ZJui6Tq/UdX3aQgwgjKiPWE3T2DfKvu8/0foc/vn169dhDf1z0uxQO/Koox27aY9jHEe/DarWqv1d5xfjs2WokGzXMExngy/ZXAMnIOojlj0DWUPOWw3lrZ5ze27zmuGfX7+Kojgs5ken+dUEb37psc+8sTfteWdNuUMdCX43uNktwLwSf+qP8lQZag6BCcnzPCsLq10jKecaOAFRH69QA3Rr3HSrX7fv+835Wf/9+/evX4EB1LCY7oFAml9J8G+RKXynmWhPjD+wk79pmtUwFp7aFj71r3mzDNXQr6pKHe/K19boQDq5Br6CqI9XqPV5+GH/+/fvoiiOGtR3fCjNvqqqDg8Phz/PbRiGQEVE8W81Op5TjE+VoVIb8/7Ecw2cgKiPp6m381MryP/79+8DB/Z/+2yaLepAPvaYhzc0H49HWZaBuXKa2ub3sYeLUVHTf/3ZgZJny3Drex2J5xo4B1EfT9Mwqt/VrHnUunXaM5bN/OroXu7hn6Pj/mqaj0vw/xvH8di6hVaOHdt5UNd1eKRAU9v8KW9bp94c9uWBEtuzZbgaX3US7dOaeK6BcxD18TTdZP02U1VV2nnUiVJVVc3zHJhTfYLVNH8iwV3XHTjn6xMh314UsMv57NapP9azZahVc+Yjms1XVZUdp9PPNXAOoj6ettUI1vTyqqrs1pLC5/LtDs/VNH8iwQcO6rdtq/rHIUcTM2ctktOK3Tr1x3qhDO/3u7OS3i63S+QaOAdRH097PB5boUi319WnjHz31rmV5sMTfNSS+k+E/PetFuPhAyWHb0vwpnNyDZyDqI8jmSeP+T20RZLLnY9NsOkneIe2hEsw5G85dqDkkDI8QcrjWUAAUR9HKoqi7/tpmpzZWLoJJhjGUkvwUSF/HMfTFpenPLLzOXnmGj8AUR9H0nbl/tTlZG+CSSVYIb9t2zdD/jRNJ3cVJDuy81F55hpXR9THGYpT1sof6PwEm5D/5nEUij7xnJ6Ay43sHCLPXOPqiPr4IDMjurjIbqbfSvA0TWVZvh/ytbL8/NlwqQ2UnCPPXOPqiPr4oK2lUMn6SoLVOn8n5GuFunlIz/mlndRAyWnyzDWujqgPfJMJ+ffnafc39TMbSQ0nX25k5xB55hpXQdQHvubZ3WNipDCScrmRnUPkmWtcDlEf+A77+blHeX9mwCFMeq4ysnOIPHONyyHqAwCQC6I+AAC5IOoDAJALoj4AALkg6gMAkAuiPpCErus0A/zN42gR/yFJAvDzEPWBVLwZ9ed5VtWBlWMAthD1gVQURfFyM12bwxzSWwDgByPqA0nQpj1vNtOJ+gDCiPpAEswjW6Zp0mNdXjjIO70FAHJA1AeSoGb6PM9N09R1bZrsdte9bxxHcwTt6s+gPoAAoj6QhKIozGNb+74fhuHZI/R9T/c+gDCiPvB99pN4Xm6s65G7xyYMwA9D1Ae+zwzqD8Og/1Hsn+c50L3vDOEXPNYdwB6iPvB9RVHUdb38bvQvy6J/xtOgvj3MDwA+oj7wfXbAVg0gvp/fme73bHUBQFaI+gAA5IKoDwBALoj6AADkgqgPAEAuiPoAAOSCqA8AQC6I+gAA5IKoDwBALoj6AADkgqgPAEAuiPoAAOSCqA8AQC6I+gAA5IKoDwBALoj6AADkgqgPAEAuiPoAAOSCqA8AQC6I+gAA5OL/AMYRwF1qFIAjAAAAAElFTkSuQmCC" 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