2008 10 10 :100026788 (2008) 1020076206 (, 400074) :, Inver2over,,, : ; ; ; Inver2over ; : F54015 : A Research on vehicle routing problem with stochastic demand and PSO2DP algorithm with Inver2over operator PENG Yong ( Traffic & Transportation School, Chongqing Jiaotong Univ, Chongqing 400074, China) Abstract : This paper discussed one vehicle routing problem(vrp) with stochastic demand. A PSO(Particle Swarm Optimization) 2DP(Dynamic Programming) algorithm with Inver2over operator was provided to find the priori tour with the minimal expected cost. PSO with Inver2over operator was used to optimize the priori tour. And DP was used to calculate the fitness value, that is, the expected cost of the priori. At last, a numerical example was available. Key words : VRP ; PSO ; dynamic programming ; Inver2over ; stochastic demand 1,,. (VRP). CVRP MDVRP PVRP SDVRP SVRP VRPB VRPPD VRPSF VRPTW [1 8 ].,..,,,,,,,. (PSO), Eberhart Kennedy, [9 ]. PSO,.,. 2,, [10 ]. :2007204216 : (1973 - ),,,,, :,E2mail :pyepeng @hotmail. com.
10 77 Inver2over [11 ], (DP),.. 2 G = ( V, A, D),,V = {0,1,, n} ( ) (0 ),A = { ( i, j) : i, j V, i j}, D = { d i, j : i, j V, i j}.. Q.. i ( i = 1,, n),. ( ),, 1,., ;,,,,,,,,. i Q, p i, k = Pr ob ( i = k), k = 0,1,, K Q. s = (0, s (1), s (2),, s ( n) ),. 0, s (1), s (2),, s ( n),,, (, 0, ),.,,,.. s = (0,1,, n). j, q, f j ( q) j ( ),, f 0 ( Q) ; f p j ( q), f r j ( q). j f j ( q) : f p j ( q) : f p j ( q) = (1 - p j+1,0 ) d j, j+1 + f j+1 ( q - k) p j+1, k + f j ( q) = minimum{ f p j ( q),f r j ( q) } (1) + p j+1,0 (1 - p j+2,0 ) d j, j+2 + f j+2 ( q - k) p j+2, k + (2 d j+1,0 + f j+1 ( q + Q - k) ) p j+1, k + p j+1,0 p j+2,0 (1 - p j+3,0 ) d j, j+3 + f j+3 ( q - k) p j+3, k + + + p j+1,0 p n,0 (1 - p n+1,0 ) d j, n+1 + f n+1 ( q - k) p n+1, k + (2 d n+1,0 + f n+1 ( q + Q - k) ) p n+1, k + p j+1,0 p n,0 p n+1,0 d j, n+1, (2 d j+2,0 + f j+2 ( q + Q - k) ) p j+2, k (2 d j+3,0 + f j+3 ( q + Q - k) ) p j+3, k Πj (0,1,, n) (2) f r j ( q) :
: : : f r j ( q) = d j,0 + (1 - p j+1,0 ) d 0, j+1 + FP j ( q) = : FP n + 1 ( q) = 0. : 78 2008 10 : : FR n + 1 ( q) = 0 f j+1 ( Q - + p j+1,0 (1 - p j+2,0 ) d 0, j+2 + f j+2 ( Q - + p j+1,0 p j+2,0 (1 - p j+3,0 ) d 0, j+3 + k) p j+1, k k) p j+2, k f j+3 ( Q - k) p j+3, k + + p j+1,0 p n,0 (1 - p n+1,0 ) d 0, n+1 + d i, n+1 = d i,0, Π i (0,1,2,, n) d n+1,0 = 0 p n+1, k = 1, k = 0 0, otherwise f n+1 ( q) = 0 f j+1 ( q - k) p j+1, k + f n+1 ( Q - k) p n+1, k, Πj (0,1,, n) (3) (2 d j+1,0 + f j+1 ( q + Q - k) ) p j+1, k + p j+1,0 FP j+1 ( q) (4) f p j ( q) = (1 - p j+1,0 ) d j, j+1 + p j+1,0 (1 - p j+2,0 ) d j, j+2 + p j+1,0 p j+2,0 (1 - p j+3,0 ) d j, j+3 1 f r j ( q). + + p j+1,0 p n,0 (1 - p n+1,0 ) d j, n+1 + p j+1,0 p n,0 p n+1,0 d j, n+1 + FP j ( q) (5) FR j ( q) = (1 - p j+1,0 ) d 0, j+1 + f j+1 ( Q - k) p j+1, k + p j+1,0 FR j+1 ( q) (6) f r j ( q) = d j,0 + FR j ( q) (7) f r j ( q),, d j,0 q ; FR j ( q) q.,f r j ( q) q. 2 f j ( q). f j ( q), q, j. q 1 q 2, q 2,, q = q 2 - q 1,, q 1,, q 2 j q 1 j.,f j ( q 1 ) f j ( q 2 ). 3 h j, q h j,f j ( q) = f p j ( q) ;,f j ( q) = f r j ( q). q, q,,,,,,, f p j ( q) f r j ( q) ; q = Q - q, q,,,,f p j ( q) f r j ( q)., f p j ( q) f r j ( q). h j q Q, q h j, f j ( q) = f p j ( q)., 2, q h j, f j ( q) = f r j ( q)., h j., (DP) : 1 (6) (7) f r j ( q) ;
10 79 2 (4) (5) f p j ( q) ; 3 (1) f j ( q) ; 4 1 3, j = 1 ; 5 (6) (7) f r 0 ( Q), (4) (5) f p 0 ( Q), (1) f 0 ( Q). 3 Inver2over PSO2DP PSO,,. PSO. ( ),.,.,.,.. 6, (1,2,5, 4,6,3) 0 1 2 5 4 6 3 0. (DP) : 1, DP ; 2, i ; 3 rand p, j ; p < rand p c, i j ;, i j ; 4, i j, 6 ; 5, i j Inver2 over, i = j, 3 ; 6 DP,,, ;, ; 2, ; 7 ; 8 2,. 2 Inver2over PSO2DP 4 Q = 10 12, ( 1),, 0 1 2 3 4 5 6, ( 2).,,, ( ),. 20, 200, p = 0104-0103 Π, p c = 013. p PSO,,, p,, ( p ). Inver2over PSO2DP : 0 10 12 4 11 6 9 8 1 3 5 2 7 0 40193.
80 2008 10 1 (0 ) 1 2 3 4 5 6 7 8 9 10 11 12 0 4. 12 2. 55 1. 13 5. 45 5. 26 5. 55 1. 12 1. 64 1. 31 4. 04 4. 19 3. 46 1 4. 99 1. 51 3. 81 4. 23 4. 91 5. 52 1. 34 2. 11 4. 56 5. 56 2. 06 2 5. 68 5. 72 3. 38 4. 03 1. 05 2. 72 5. 03 4. 77 3. 69 2. 77 3 4. 41 2. 31 1. 31 4. 64 2. 24 3. 94 5. 51 4. 86 2. 13 4 3. 65 2. 33 4. 59 4. 69 5. 46 2. 93 2. 40 1. 92 5 3. 81 3. 04 2. 70 3. 21 5. 44 4. 20 3. 18 6 3. 46 1. 73 1. 37 4. 39 2. 06 4. 21 7 3. 97 3. 03 2. 23 3. 03 2. 10 8 1. 65 5. 82 4. 87 4. 88 9 2. 98 4. 33 2. 82 10 2. 92 1. 09 11 2. 34 2 0 1 2 3 4 5 6 1 0. 24345 0. 054569 0. 16133 0. 12971 0. 032936 0. 097919 0. 28009 2 0. 13662 0. 027712 0. 04536 0. 12235 0. 13486 0. 22554 0. 30757 3 0. 082525 0. 090671 0. 088191 0. 25539 0. 077449 0. 2345 0. 17127 4 0. 13108 0. 20114 0. 059234 0. 16738 0. 0021825 0. 12681 0. 31217 5 0. 13954 0. 050703 0. 30254 0. 2942 0. 088107 0. 031363 0. 093553 6 0. 28669 0. 034003 0. 19568 0. 056504 0. 16103 0. 050269 0. 21583 7 0. 14629 0. 092226 0. 11449 0. 056507 0. 1848 0. 19528 0. 2104 8 0. 12249 0. 21857 0. 14159 0. 20943 0. 051093 0. 024589 0. 23223 9 0. 21392 0. 15604 0. 05198 0. 26328 0. 064535 0. 16897 0. 081277 10 0. 03009 0. 016473 0. 066693 0. 26944 0. 10769 0. 31589 0. 19373 11 0. 11571 0. 08452 0. 20838 0. 038973 0. 23886 0. 080667 0. 2329 12 0. 15502 0. 29702 0. 098522 0. 11719 0. 03415 0. 084814 0. 21329 5,.,.,,,,,,. Inver2over, (DP),.. : [ 1 ] Cordeau J 2F, Laporte G,Mercier A. A unified tabu search heuristic for vehicle routing problems with time windows[j ]. Journal of the Operational Research Society, 2001, 52 : 928-936. [ 2 ] Desaulniers G, Lavigne J, Soumis F. Multi2depot vehicle scheduling problems with time windows and waiting costs[j ]. European Journal of Operational Research, 1998, 111 : 479-494. [ 3 ] Freling R, Wagelmans A P M, Paixao J M P. Models and algorithms for single2depot vehicle scheduling [J ]. Transportation Science, 2001, 35 : 165-180. [ 4 ] Hadjar A, Marcotte O, Soumis F. A branch2and2cut algorithm for the multiple depot vehicle scheduling problem[j ]. Operations
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