Research on vehicle routing problem with stochastic demand and PSO2DP algorithm with Inver2over operator

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1 : (2008) (, ) :, 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 , 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 ]. : : ( ),,,,, :,E2mail com.

2 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) :

3 : : : f r j ( q) = d j,0 + (1 - p j+1,0 ) d 0, j+1 + FP j ( q) = : FP n + 1 ( q) = 0. : : : 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) ;

4 (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) (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),, , ( 2).,,, ( ),. 20, 200, p = Π, p c = 013. p PSO,,, p,, ( p ). Inver2over PSO2DP :

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