动态规划解TSP问题

1、薅用 动态规划 方法编程求解下面的问题羂某推销员要从城市v1出发，访问其它城市v2, v3,，v6各一次且仅一次，最后返回 v1。 D为各城市间的距离矩阵。问：该推销员应如何选择路线，才能使总的行程最短？v101020304050v212018302521v32319051015螁D =v4343240816v545271110018v656221620120腿羄1、变量设定 蚂阶段k：已遍历过k个结点，k=1,2-6,70蕿K=1表示刚从V1出发，k=7表示已回到起点V1蕿状态变量Xk=(i, Sk)：已遍历k个结点，当前位于i结点，还未遍历的结点集合为 Sh 则 X1=(1，2,3,4,5

2、,6), X6=(i,)，X7=(1，)蒄决策变量Uk=(i, j)：已遍历k个结点，当前位于i结点，下一个结点选择jo蒃状态转移方程：Xk+1 = T(XK Uk) = (j, Sk-j)蚀第k阶段的指标函数Vk = Di,jo蚈最优指标函数Fk(Xk) = FkQ Sk)：已遍历k个结点，当前从i结点出发，访问Sk 中的结点一次且仅一次，最后返回起点V1的最短距离。袃则 Fk（i, Sk） = min Di,j + F+i（j，Sk-j） 1< k< 6膃F7（X7） = F（1 ,）=0螆2、分析:薇袄(1) k=6 时，F6(i,)=minDi,1 + F7(X7) = D

3、i,1i=2,3,4,5,6葿膈X6=(i,)羆 U6=(i, j)蚄 X7=(1,)薀 V6=Di,j芇F7（1,）莅 V6 + F(X7)膀（2,）薂(2,1)蕿（1,）袅12袁0荿 12=F6(2,)螇（3,）芄(3, 1)蚁（1,）蒀23祎0蚃 23=F6(3,)莁（4,）蒂(4, 1)膈（1,）肃34肂0艿 34=F6(4,)莇（5,）螆(5, 1)袂（1,）莀45虿0芆 45=F6(5,)薃（6,）膈(6,1)螇（1,）蚅56莃0艿 56=F6(6,)羆即k=6时，对于每一种状态X6,都有唯一的决策U6肃(2) k=5 时，F5(i, S5) = minDi,j + F6(j )i

4、=2,3,4,5,6芁 X5=(i,S5)芈 U5=(i,j)薄 X6=(j,)袄 V5=Di,j肇F6(j,)莆 V5 + F6(X6)羃(2,6薄(2,6)腿（6,）蝿21蚇56肀 77=F5(2,6)賺(2,5袇(2,5)肆（5,）螁25羈45羅 70=F5(2,5)蒅(2,4薁(2,4)聿（4,）莈30羅34节 64=F5(2,4)肁(2,3(2,3)(3,)18234仁 F5(2,3)(3,6)(3,6)(6,)15567仁 F5(3,6)(3,5)(3,5)(5,)104555=F5(3,5)(3,4)(3,4)(4,)53439=F5(3,4)(3,2)(3,2)(2,)1912

5、3仁 F5(3,2)(4,6)(4,6)(6,)165672=F5(4,6)(4,5)(4,5)1 (5,)84553=F5(4,5)(4,3)(4,3)(3,)142327=F5(4,3)(4,2)(4,2)(2,)321244=F5(4,2)(5,6)(5,6)(6,)185674=F5(5,6)(5,4)(5,4)p4,)n103444=F5(5,4)(5,3)(5,3)(3,)112334=F5(5,3)(5,2)(5,2)(2,)271239=F5(5,2)(6,5)(6,5)(5,)124557=F5(6,5)(6,4)(6,4)(4,)203454=F5(6,4)(6,3)(6,

6、3)(3,)1162339=F5(6,3)(6,2)(6,2)(2,)|221234=F5(6,2)即k=时，对于每一种状态X5,都有唯一决策U5(3) k=4 时，F4(i,S4) = min(Di,j + F5(j,S5) )i=2,3,4,5,6X4=(i,S4)U4=(i,j)X5=(j,S5)V4=Di,jF5(j,S5)V4 + F5(j,S5)(2,3,4)(2,3)f(3,4)n183957=F4(2,3,4)(2,4)(4,3)302757=F4(2,3,4)(2,4,5)(2,4)(4,5)305383(2,5)f(5,4)亍254469=F4(2,4,5)(2,5,6)(

7、2,5)(5,6)257499(2,6)(6,5)215778=F4(2,5,6)(2,3,5)(2,3)(3,5)185573(2,5)(5,3)253459=F4(2,3,5)(2,3,6)(2,3)(3,6)187189(2,6)F(6,3)213960=F4(2,3,6)(2,4,6)(2,4)(4,6)3072102(2,6)(6,4)215475=F4(2,4,6)(3,2,4)(3,2)F(2,4)196483(3,4)(4,2)54449=F4(3,2,4)(3,2,5)(3,2)(2,5)197089(3,5)(5,2)103949=F4(3,2,5)(3,2,6)(3,2)

8、(2,6)197796(3,6)(6,2)153449=F4(3,2,6)(3,4,5)(3,4)(4,5)55358(3,5)(5,4)104454=F4(3,4,5)(3,4,6)(3,4)(4,6)57277(3,6)(6,4)155469=F4(3,4,6)(3,5,6)(3,5)(5,6)107484(3,6)(6,5)155772=F4(3,5,6)(4,2,3)(4,2)(2,3)324173(4,3)(3,2)43134=F4(4,2,3)(4,2,5)(4,2)(2,5)3270102(4,5)(5,2)83947=F4(4,2,5)(4,2,6)(4,2)(2,6)3277

9、109(4,6)(6,2)163450=F4(4,2,6)(4,3,5)(4,3)f(3,5)n45559(4,5)(5,3)83442=F4(4,3,5)(4,3,6)(4,3)(3,6)47175(4,6)f(6,3)1163955=F4(4,3,6)(4,5,6)(4,5)(5,6)87482(4,6)(6,5)165773=F4(4,5,6)(5,2,3)(5,2)f(2,3)1274168(5,3)(3,2)113142=F4(5,2,3)(5,2,4)(5,2)(2,4)276491(5,4)；(4,2)1104454=F4(5,2,4)(5,2,6)(5,2)(2,6)27771

10、04(5,6)(6,2)183452=F4(5,2,6)(5,3,4)(5,3)：(3,4)J113950(5,4)P(4,3)n102737=F4(5,3,4)(5,3,6)(5,3)(3,6)117182(5,6)|(6,3)183957=F4(5,3,6)(5,4,6)(5,4)f(4,6)107282(5,6)(6,4)185472=F4(5,4,6)(6,2,3)(6,2)l(2,3)1224163(6,3)：(3,2)163147=F4(6,2,3)(6,2,4)(6,2)(2,4)226486(6,4)(4,2)204464=F4(6,2,4)(6,2,5)(6,2)(2,5)2

11、27092(6,5)(5,2)12395仁 F4(6,2,5)(6,3,4)(6,3)(3,4)163955(6,4)(4,3)202747=F4(6,3,4)(6,3,5)(6,3)(3,5)165571(6,5)(5,3)123446=F4(6,3,5)(6,4,5)(6,4)(4,5)205373(6,5)(5,4)124456=F4(6,4,5)(4) k=3 时，F3(i,S3) = minDi,j + F4(j,S4)i=2,3,4,5,6X3=(i,S3)U3=(i,j)X4=(j,S4)V3=Di,jF4(j,S4)V3 + F4(j,S4)(2,3,4,5)(2,3)(3,4

12、,5)185472(2,4)(4,3,5)304272(2,5)(5,3,4)253762=F3(2,3,4,5)(2,3,4,6)(2,3)(3,4,6)186987(2,4)(4,3,6)305585(2,6)(6,3,4)214768=F3(2,3,4,6)(2,3,5,6)(2,3)(3,5,6)187290(2,5)(5,3,6)255782(2,6)(6,3,5)214667=F3(2,3,5,6)(2,4,5,6)(2,4)(4,5,6)3073:103(2,5)(5,4,6)257297(2,6)(6,4,5)215677=F3(2,4,5,6)(3,2,4,5)(3,2)(2

13、,4,5)196988(3,4)(4,2,5)54752=F3(3,2,4,5)(3,5)(5,2,4)105464(3,2,4,6)(3,2)(2,4,6)197594(3,4)(4,2,6)55055=F3(3,2,4,6)(3,6)(6,2,4)156479(3,2,5,6)(3,2)(2,5,6)197897(3,5)(5,2,6)105262=F3(3,2,5,6)(3,6)(6,2,5)155166(3,4,5,6)(3,4)(4,5,6)573:78(3,5)(5,4,6)107282(3,6)(6,4,5)15567仁 F3(3,4,5,6)(4,2,3,5)(4,2)(2,3

14、,5)325991(4,3)(3,2,5)44953(4,5)(5,2,3)84250=F3(4,2,3,5)(4,2,3,6)(4,2)(2,3,6)3260 192(4,3)(3,2,6)44953=F3(4,2,3,6)(4,6)(6,2,3)164763(4,2,5,6)(4,2)(2,5,6)3278110(4,5)(5,2,6)85260=F3(4,2,5,6)(4,6)(6,2,5)165167(4,3,5,6)(4,3)(3,5,6)47276(4,5)(5,3,6)85765(4,6)(6,3,5)164662=F3(4,3,5,6)(5,2,3,4)(5,2)(2,3,4)

15、275784(5,3)(3,2,4)114960(5,4)(4,2,3)103444=F3(5,2,3,4)(5,2,3,6)(5,2)(2,3,6)276087(5,3)(3,2,6)114960=F3(5,2,3,6)(5,6)(6,2,3)184765(5,2,4,6)(5,2)(2,4,6)2775102(5,4)(4,2,6)105060=F3(5,2,4,6)(5,6)(6,2,4)186482(5,3,4,6)(5,3)(3,4,6)116980(5,4)(4,3,6)105565=F3(5,3,4,6)(5,6)(6,3,4)184765=F3(5,3,4,6)(6,2,3,4

16、)(6,2)(2,3,4)225779(6,3)(3,2,4)164965(6,4)(4,2,3)203454=F3(6,2,3,4)(6,2,3,5)(6,2)(2,3,5)225981(6,3)(3,2,5)1649 丁65(6,5)(5,2,3)124254=F3(6,2,3,5)(6,2,4,5)(6,2)(2,4,5)226991(6,4)(4,2,5)204767(6,5)(5,2,4)125466=F3(6,2,4,5)(6,3,4,5)(6,3)(3,4,5)165470(6,4)(4,3,5)204262(6,5)(5,3,4)123749=F3(6,3,4,5)(5) k=

17、2 时，F2(i,S2) = minDi,j + F3(j,S3)i=2,3,4,5,6X2=(i,S2)U2=(i,j)X3=(j,S3)V2=Di,jF3(j,S3)V2 + F3(j,S3)(2,3,4,5,6)(2,3)(3,4,5,6)1871丁89(2,4)(4,3,5,6)306292(2,5)(5,3,4,6)256590(2,6)(6,3,4,5)214970=F2(2,3,4,5,6)(3,2,4,5,6)(3,2)(2,4,5,6)197796(3,4)(4,2,5,6)56065=F2(3,2,4,5,6)(3,5)(5,2,4,6)106070(3,6)(6,2,4,

18、5)156681(4,2,3,5,6)(4,2)(2,3,5,6)326799(4,3)(3,2,5,6)46266=F2(4,2,3,5,6)(4,5)(5,2,3,6)86068(4,6)(6,2,3,5)165470(5,2,3,4,6)(5,2)(2,3,4,6)276895(5,3)(3,2,4,6)115566(5,4)(4,2,3,6)105363=F2(5,2,3,4,6)(5,6)(6,2,3,4)185472(6,2,3,4,5)(6,2)(2,3,4,5)226284(6,3)(3,2,4,5)165268(6,4)(4,2,3,5)205070(6,5)(5,2,3,4

19、)124456=F2(6,2,3,4,5)(6) k=1 时，F1(1,S1) = minD1,j + F2(j,S2)X仁(1,S1)U1=(1,j)X2=(j,S2)V1=D1,jF2(j,S2)V1 + F2(j,S2)(1,2,3,4,5,6)(1,2)(2,3,4,5,6)107080=F1(1,2,3,4,5,6)(1,3)(3,2,4,5,6)206585(1,4)(4,2,3,5,6)306696(1,5)(5,2,3,4,6)4063103(1,6)(6,2,3,4,5)50561063、伪代码和时间复杂度为方便计算，结点编号改为 0 到 5.用一张二维表格F表示F(i,Sk)行数是n,列数是2n-1。 (2)行号表示当前所在的结点 i。列号对应的五位二进制表示表示V5,V4,V3,V2,V的一个子集，1表示在集合中，0 表示不在集合中。例如：00110表示的集合为 V3,V2， 00000表示空集再用一张n*2n-