1、19 y 2002 M02 JOURNAL OF ENGINEERING MATHEMATICSVol.19 Supp.Feb.2002cI|:1005-3085(2002)05-0101-06公交车调度问题的数学模型 6;, ( bv,100084)K 1:5、y 、B 8 ,#M9 T。1oM:; ; “S?s |:AMS(2000)90C08 ms |:TB114.1 DS M :A1 问题的背景和要求5 v g 4B L= S 5。 51 pBH L, % Z 5:B, N A1 ,4 _ _ _ Zs? p;=,B EL V ; , - H$ L 4B ! () r( ) “Z,#B
2、Y f / HWV。 , 1 v /,5T ve,4 /y5。 n54 gBH1 L ,L Z_ 14_,/Z_ 13_, M L_ Z 、 _ , LBT Z_ / Z HWs。e n, HL !: L B|v Z, S Z100 ; ZL ( 20 /l H;i$#rm41 p: Z HWB1V10s,* HB1V5s, qV120%; H I n4 rm,4 qB9150%S。51 p , V ? a I nrm m “S/,L !9BL T ?(T )Z, _? H YV,i LCZ 1 ! ;Zv$ Z Z m S,V7|5B 、 ,i p ZE。2 建立模型的思路和框架5 Vs 0
3、5:1) y E H , B Z/,L V 。B1 M L=Bt ,YV E EV,%M # S |1 p “S 。 6B y ,V ATBB HWV?,B1 p/, |L Z r V。2) YV “S。5 L= B “S?, = “S1 I n:B Q Z m Z HW; 6B Q m Z q。 YV E, p“S? =5。3) , LCZ K 。 B Hs 5。3 建立确定性模型的一个例子1) # p(1) - _S:j =1,2,n; Z : H Ytrj_ Z uj(t),j =1,2,n;/ Z : H YtVj_/ Z dj(t),j =1,2,n ;_W HW:Vj -1_j_W
4、 HW( j_ HW):j ,j =2,n; Z :B; Z K B;Y H HW t;Y H HW t;(2) %M #M1M %M :? H YV,_ T =(T 0,T1, T2, , Tk,Tm),T0:B r _j =1 H Y;Tk:k R _j =1 H Y,k =1,2,mM1M :k R j_ H Y:Tk1 =Tk,Tkj =Tk1 +j-1l=1l,j =2,3,n -1;k R j_ H Z :Pk(Tkj),k =1,2,m;j =1,2,n -1;k Rj_ H,_ Zsf :Wkj(0):VTk-1jTkj H Z ;Wkj(h):k Rj_ H,_ -Vh ?
5、Z ;hkj:k Rj_ H,_ K Z ,Wkj(hkj)0,Wkj(hkj +1)=0 m(m1)n/:m:(3) M1M 9 Tk Rj_,_ Z/, =/ Z :akj =max(Pk(Tk j-1)-TkjTk-1 jdj(t)dt),0k Rj_,_ Z/,j_ V , Z :bkj = B -akj,k -1 R j_k Rj_ H=,_ Zr :102 19 m1Wkj(0)=TkjKk-1 juj(t)dt;9 k Rj_,_ L= Z Pkj:B:55 5, j_ Z(k +1) r H, 1 Kvhkj ,N /5( Am2):max0hhkjhs.t.hkjr=hWkj
6、(r)bkjm2=: hkj =0, N H_ ? ,“,103y 5 Pkj =hkjh=0Wkj(h); H 7:hk+1j =0; : hkj 0, N H_ ? ,“,Pkj =bkj; H 7:hk+1j =hkj +1,O:Wk+1 j(h +1)=Wkj(h),h =0,1,(hkj -1)Wk+1 j(hkj)=hkjh=hkjWkj(h)-bkj.“, :k R j_ H Z :Pk(Tkj)=akj +Pkj=akj +hkjh=0Wkj(h), hkj =0,B, hkj 0,k =1,2,m;j =1,2,n -1。(4) “S9 k Rj_ H,_ -h Z :Wkj
7、(h),h =1,hkj X HW :WTkj(h)=Tkj -Tk-h j,h =1,hkj. Y H : T 1, T , H T1,Tm,5Y H H q= H ,V5s9 H , 9 ,:TOver W1(T)=Tkj T1, T2Wkj(h)|WTkj(h)5,h =1,hkjTkj T 1, T2Pk(Tkj)Y H H q= H ,V10s9 H , 9 ,:TOverW2(T)=Tkj T1, T2Wkj(h)|WTkj(h)10,h =1,hkjTkj T 1, T2Pk(Tkj) q50%s1= q50%?Q (_ -1),:TCap-low(T)=kj 1 B -Pk(T
8、kj)B 0.5m (n -1)104 19 (5) pT =(T0, T1, T2, Tk, Tm), P:minT C =Over W1(T)+OverW2(T)+Cap-low(T), “ 。1 %M Ve:s Hs:5 H H , H=W?,“M 。(6) p 5:E: ZE, s/ZE。 L= p, T%M , pB“ VZE, p K,9 V |。2) # psY p 、/ Z_ HWV, V 。 VmU m3, HW, LV U ,LphLV U /; LLV U/ , LLphLV U/ 。m3 B , L= SE “。Vy A, 、/s 7,| s 7, e。V V ? A1
9、,55M,E p。3) 9 T ,v ,?W,9 “Sf ,YV1 E,4 aZ。(1) L ! H (1l H) B_ / H r_ ,YVt 9 uj(t)dj(t)。(2) _W ( ( / HW)j;B =100, B =120。(3) S?_ , *、 H :*640 940;15501850。 ?s3 H : H I1,* H I2, H I3。 H HW t =5(s); H HW t =10(s)。(4) %M 3 H ?W:J1, J2, J3,N V9 T 0,T1,Tm 。(5) /Z_ ( V9 , /T:Over W1 =* H Hq,OverW2 = H H q,C
10、ap-low = q50%s1,4 “, (1/3, “Sf C。9 V: c:Total= /Z_ ?9 ;v HW( I n Q105y 5 ) V: c: Z_ (up-bus),/Z_ (down-bus)。 S?_ ?*? - l M,?W/ :A. /Z_? HWWM:(J1, J2, J3) OverW 2 Over W1 Cap-low C Total up-bus down-bus(3,2,2) 0.0006 0.0941 0.6175 0.2374 420 22 22(4,2,3) 0.0488 0.1161 0.4140 0.1930 331 22 22(4,3,3) 0
11、.1016 0.3194 0.4031 0.2747 301 15 15(5,2,3) 0.0517 0.1336 0.3390 0.1748 295 22 22(5,3,3) 0.1204 0.3585 0.3150 0.2646 265 15 15(5,2,4) 0.1565 0.1336 0.3301 0.2067 280 22 22(6,2,2) 0.0364 0.1632 0.3414 0.1803 299 22 22(6,3,3) 0.1344 0.3704 0.2547 0.2532 240 15 15B. /Z_* HWW, (4,3,2)V U: (4,3,2),/(4,2,
12、3)。(J1, J2, J3) OverW 2 Over W1 Cap-low C Total up-bus down-bus(4,3,2) 0.0184 0.1733 0.4126 0.2014 330 45 15(4,2,3) 0.0860 0.2547 0.4507 0.2638 331 15 45(5,3,1) 0.0252 0.0842 0.5672 0.2255 385 135 15(5,3,2) 0.0250 0.1912 0.3341 0.1834 295 45 15(5,2,3) 0.1035 0.2933 0.3790 0.2586 295 15 45(5,3,3) 0.1
13、204 0.3585 0.3150 0.2646 265 15 15(5,4,2) 0.0922 0.2912 0.3260 0.2365 280 56 11(6,3,2) 0.0528 0.2203 0.2793 0.1841 269 45 15V Z “Sf C# A, A(5,2,3) (8)。K,+Y14# ,5 gM BZ 4,I#9 bv S“ 3,N V U。AMathematicalModel of BusSchedulingTANZe-guang, JIANG Qi-yuan(TsinghuaUniversity, Beijing 100084)Abstract:In this paper the background of the problem and idea of mathematical modeling are given.A specificmathemctical model and corresponding numerical results are presented.Key words:bus scheduling;mathematicalmodel106 19
