北航计算机研究生课程算法设计与分析Assignment-1

1、欢迎下载 一、 解：设第k月的需求量为Nk(k=1,2,3,4)状态变量Xk:第k月初的库存量，X1=X5=Q 0Xkw Nk+N4决策变量 Uk：第 k 月的生产量，max0, Nk-Xk Uk min6, Nk+-+N4 - Xk 状态转移方程：Xk+i = Uk + Xk-Nk第 k 月的成本 Vk = 0.5*(Xk - Nk)Uk=03 + Uk + 0.5*(Uk + Xk - Nk)0设Fk(Xk)是由第k月初的库存量Xk开始到第4月份结束这段时间的最优成本则 Fk(Xk) = minVk + F+i(X k+i)1 k 4=min 3 + Uk + 0.5*(Uk + Xk -

2、 Nk) +kF(Uk + Xk - Nk) Uk 0min 0.5*(Xk - Nk) + Fk+i(Xk - Nk) Uk=0F5(X5)=0四个月内的最优成本为Fi(X1)=F(0)详细计算步骤如下：(i) k=4时0WX4W4, max0, 4 - X4U4Wmin6, 4-X4X4U4X5V4F5(X5)V4 + F5(X5)04077=F4(0)130606=F4(1)220505=F4 310404=F4(3)400000=F4 即对于状态X4的每个取值，都有唯一确定的决策变量U4使得F4(X4)最优(2) k=3时0X36, max0, 2 - X3戶U3min6, 6-X3X

3、3U3X4V3F4(X4)V3 + 丘(X4)0205712316.5612.5428513539.5413.56411011= F3(0)1104711215.5A11.5327512438.5412.5541010=F3(1)200077=F3(2)114.5610.522611337.5411.5449093010.5A6.5=F3（3）125510236.5410.53488402156=F3 135.549.52471 075031.545.5=F3（5）1466604302=F3 （3） k=2时0X2W9, max0, 3 - X2U2min6, 9-X2X2U2X3V2F3（X

4、3）V2 + F3（X3）03061117417.51017.5529716=F2（0）6310.56.51712051116316.51016.5428715=F2（1）539.56.516641161721041115215.51015.5327714=F2（2）438.56.51554106166511.55.51730001111= F2（3）114.51014.5226713337.56.5144496155510.55.51666122144010.51010.5=F2（4）125713236.56.513348614459.55.5155611213502178=F2 (5)13

5、5.56.512247613358.55.5144610126031.56.58=F2 (6)146612257.55.513369211704268=F2 156.55.5122682108052.55.58=F2(8)167A9906325=Fz(9)(4) k=1 时X1=0, max0, 2 U1 min6, 11X1U1X2V1F2(X2)V1 + E(X2)020521316.51521.54281422539.5M120.5=F1(0)641110.521.5由以上计算可得，4个月的总最优成本为Fi(0) = 20.5千元)从k=1回溯，可得最优结果中各阶段的状态变量 Xk和决策

6、变量Uk如下表:月份k产量Uk月初库存量Xk需求量Nk每月成本Vk15029.52033036021140440二、解：1、变量设定阶段k：已遍历过k个结点，k=1,26,7。K=1表示刚从V1出发，k=7表示已回到起点V1 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 Sk。则 X仁(1, 2,3,4,5,6), X6=(,),X7=(1,)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j。 状态转移方程：Xk+1 = T(XK Uk) = (j, Sk-j)第k阶段的指标函数Vk = Di,j。最优指标函数Fk(Xk) = F

7、k(, 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。则 Fk(, Sk) = min Di,j + F+1(j, Sk-j) 1 k 6F7(X7) = F(1,)=02、分析:(1) k=6 时，F6(,)=minDi,1 + F7(X7) = Di,1i=2,3,4,5,6X6=(,)U6=(i, j)X7=(1,)V6=Di,jF7(1,)V6 + F7(X7)(2,)(2, 1)(1,)r 120：12=F6(2,):(3,)(3, 1)(1,)23023=F6(3,)(4,)(4, 1)(1,)34034=F6(4,)(5,

8、)(5, 1)(1,)；450：45=F6(5,)(6,)(6, 1)(1,)56056=F6(6,)即k=6时，对于每一种状态X6,都有唯一的决策U6(2) k=5 时，F5(, S5) = minDi,j + F6(j )i=2,3,4,5,6X5=(i,S5)U5=(i,j)X6=(j,)V5=Di,jF6(j,)V5 + F6(X6)(2,6(2,6)(6,)215677=F5(2,6):(2,5(2,5)(5,)254570=F5(2,5)(2,4(2,4)(4,)303464=F5(2,4)(2,3(2,3)(3,)18 23仁 F5(2,3)(3,6)(3,6)(6,)15567

9、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仁 F5(3,2)(4,6)(4,6)(6,)165672=F5(4,6)(4,5)(4,5)(5,)8 4553=F5(4,5)(4,3)(4,3)(3,)42327=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)(4,)103444=F5(5,4)(5,3)(5,3)(3,)112334=F5(5,3)(5,2)(5,2)(2,)

10、271239=F5(5,2)(6,5)(6,5)(5,)124557=F5(6,5)(6,4)(6,4)(4,)203454=F5(6,4)(6,3)(6,3)(3,)162339=F5(6,3)(6,2)(6,2)(2,)2212:34=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)(3,4)183957=F4(2,3,4) 1(2,4)

11、(4,3)302757=F4(2,3,4) 1(2,4,5)(2,4)(4,5)305383(2,5)(5,4)2544:69=F4(2,4,5) 1(2,5,6)(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) 1(2,3,6)(2,3)(3,6)187189(2,6)(6,3)213960=F4(2,3,6)(2,4,6)(2,4)(4,6)3072102(2,6)(6,4)215475=F4(2,4,6) 1(3,2,4)(3,2)(2,4)196483(

12、3,4)(4,2)544:49=F4(3,2,4) 1(3,2,5)(3,2)(2,5)197089(3,5)(5,2)103949=F4(3,2,5)(3,2,6)(3,2)(2,6)19177(3,6)(6,2)1534:49=F4(3,2,6) 1(3,4,5)(3,4)(4,5)55358(3,5)(5,4)1044:54=F4(3,4,5) 1(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)r 1557r 72=F4(3,5,6) 1(4,2,3)(4,2)(2,3)32

13、4173(4,3)(3,2)43134=F4(4,2,3)(4,2,5)(4,2)(2,5)I?70102(4,5)(5,2)83947=F4(4,2,5) 1(4,2,6)(4,2)(2,6)3277109(4,6)(6,2)r 1634:50=F4(4,2,6) I(4,3,5)(4,3)(3,5)455(4,5)(5,3)83442=F4(4,3,5)(4,3,6)(4,3)(3,6)47175(4,6)(6,3)r 1639:55=F4(4,3,6) I(4,5,6)(4,5)(5,6)87482(4,6)(6,5)165773=F4(4,5,6)(5,2,3)(5,2)(2,3)27

14、4168(5,3)(3,2)113142=F4(5,2,3) 1(5,2,4)(5,2)(2,4)276491(5,4)(4,2)10 4454=F4(5,2,4) 1(5,2,6)(5,2)(2,6)2777104(5,6)(6,2)183452=F4(5,2,6)(5,3,4)(5,3)(3,4)11 39I。:(5,4)(4,3)102737=F4(5,3,4) 1(5,3,6)(5,3)(3,6)117182(5,6)(6,3)18 :3957=F4(5,3,6) 1(5,4,6)(5,4)(4,6)107282(5,6)(6,4)185472=F4(5,4,6)(6,2,3)(6,2

15、)(2,3)22 141飞3:(6,3)(3,2)1631:47=F4(6,2,3) 1(6,2,4)(6,2)(2,4)226486(6,4)(4,2)204464=F4(6,2,4) 1(6,2,5)(6,2)(2,5)227092(6,5)(5,2)12395仁 F4(6,2,5)(6,3,4)(6,3)(3,4)1639飞5(6,4)(4,3)2027:47=F4(6,3,4) 1(6,3,5)(6,3)(3,5)165571(6,5)(5,3)123446=F4(6,3,5)(6,4,5)(6,4)(4,5)2053:73n(6,5)(5,4)124456=F4(6,4,5)(4)

16、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)(3,4,5)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)

17、(6,3,5)214667=F3(2,3,5,6)(2,4,5,6)(2,4)(4,5,6)3073103(2,5)(5,4,6)257297(2,6)(6,4,5)215677=F3(2,4,5,6)(3,2,4,5)(3,2)(2,4,5)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,

18、2,5,6)(3,6)(6,2,5)155166(3,4,5,6)(3,4)(4,5,6)57378(3,5)(5,4,6)107282(3,6)(6,4,5)15567仁 F3(3,4,5,6)(4,2,3,5)(4,2)(2,3,5)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)326092(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

19、,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)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

20、,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)(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)164965(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)204

21、767(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=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)r 187189(2,4)(4,3,5,6)306292(2,5)(5,3,4,6)256590(2,6)(6,3,4,5)

22、：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)r 106070(3,6)(6,2,4,5)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) 1(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)M85472(6,2,3,4,5)(6,2)(2,3,4,5)226284(6,3)(3,2,4,5)I 165268(6,4)(4,2,3,5)：205070(6,5)(5,2,3,4)124456=F2(6,2,3,4,5)(6) k=1 时，F1