Traveling Salesperson Problem and Neural Networks. A Complete Algorithm in Matrix Form

Transcription

1 Procdigs of th th WSEAS Itratioal Cofrc o COMPUTERS, Agios Nikolaos, Crt Islad, Grc, July 6-8, 7 47 Travlig Salsprso Problm ad Nural Ntworks A Complt Algorithm i Matrix Form NICOLAE POPOVICIU Faculty of Mathmatics-Iformatics Dpartmt of Mathmatics Hyprio Uivrsity of Bucharst Str Călăraşilor 69, Bucharst ROMANIA MIOARA BONCUŢ Faculty of Scics Dpartmt of Mathmatics Lucia Blaga Uivrsity of Sibiu Victorii Avu, Sibiu ROMANIA Abstract Th work dscribs all th cssary stps to solv th travlig salsprso problm This optimizatio problm is vry asy to formulat -ad a lot of works do it-, but it is rathr difficult to solv it By usig [] as a mai rfrc, w formulat a algorithm i a matrix form to solv th problm Th mathmatical approach is basd o Hopfild ural tworks ad uss th rgy fuctio with th dsct gradit mthod Th algorithm i matrix form is asir to us or to writ a computatio program Th work has six sctios Th sctio 5 dscribs th algorithm to solv th travlig salsprso problm ad th sctio 6 cotai a umrical xampl Ky-Words Travlig salsprso problm, travlig salsprso algorithm, rgy fuctio, dsct gradit Itroductio Th travlig salsprso problm ( TSP ) is a optimizatio problm A salsprso must mak a closd circuit through a crtai umbr of citis, visitig ach of thm oly oc, miimizig th total distac travld ad th salsprso rturs to th startig poit at th d of th trip W dot by K = K, K = ( d XY ), d X X = th distacs matrix, whr d XY is th distac btw th citis X ad Y Rlatd with TSP problm w hav thr typs of solutios : a) th possibl solutio ( th salsprso passs may tims through crtai citis ); b) th admissibl solutio ( th salsprso passs oly oc through ach city, but th distac travld is ot miim ); c) th optimal solutio ( th solutio is admissibl ad th distac travld is miim ) W ar itrstd i fidig th optimal solutio Our task is to fid th ukow wights v, th lmts of wights matrix V V = VN N, V = ( vx ), X =, ; =, which dscribs th optimal solutio, whr th subscript X rfrs to th city ad th subscript rfrs to th positio of th city X o th tour (rout) R I ay admissibl solutio is satisfid th coditio v X J { ; }, ad th wight chags with th rout R, i V = V (R) W dot by R() all possibl tours i a -city! ( )! problm Th R( ) = = Th fuctio R() is a rapidly icrasig fuctio [] For TSP problm thr xists two cass Cas 6 Th optimal solutio ca b obtaid by a xhaustiv sarch through all admissibl routs Cas 7 I this cas th TSP problm blogs to th class kow as NPC ( o possibl complt ) problm Th solvig of TSP problm is basd o ural twork mthod, which grats a TSP algorithm I this work w dscrib th TSP algorithm i a matrix form, rathr th o compots form Th ural twork mthod has its origis i cotiuous Hopfild tworks [], pag 44 I a Hopfild twork th iput layr Sx is idtical with th output layr Sy Th ural twork for TSP has uros (procssig lmts) i layr Sx Each uro has a output fuctio of sigmoid form

2 Procdigs of th th WSEAS Itratioal Cofrc o COMPUTERS, Agios Nikolaos, Crt Islad, Grc, July 6-8, 7 48 f : R (;), f ( s) =, λ > + λs Th output fuctio is th sam for all uros Th paramtr λ is th curv slop If λ 5 th th fuctio f is almost th Havisid fuctio, with th valus ad Th wights matrix ad rgy fuctio Durig th algorithm w shall dscrib w us th colums ad th lis of wights matrix V That is why w us som spcial otatios, as follows V = ( v v v v ) = ( Vcol Vcol Vcol ) ( ) T V = Vlia Vlib Vli X Vlir, Vcol R V = ( Vli Vli Vli ) T i Vli, Vli i R = ( va vb vx vr ) T ( v v v v ) Vcol VliX = X X X Also w us th sum of lmts o li X ad o colum ad dot VSli X = = v, = X = v Usig th abov otatios w costruct th xtdd matrix Vx havig th form va va va VSlia vx vx vx VSliX Vx = vr vr vr VSli r Th mathmatical modl of TSP problm ds two supplmtary wights havig th maig [] v X ( + ) = vx, vx = vx () Ay admissibl rout R has a associatd matrix V ad a rgy fuctio dotd E = E(R) Dfiitio Th rgy fuctio is dfid by four sums, as it follows [], pag 5 ; [3] A B C D E ( R) = Σ + Σ + Σ3 + Σ4 () Σ = X = i = =, i v X ivx Σ = = X = Y =, Y X v vy 3 = Σ = = v 4 = d v v v X = Y = = XY Y, + + Y, Σ ( ) Y X A lot of paprs ad books limit th discussios at this formula ad do ot show how to us it i a solvig algorithm Propositio Th four sums of rgy fuctio ar rprstd i th followig vctor form Σ = < v k ; v > k < Σ = < Vli i ; Vli k > i< k Σ3 = = VSli X X Σ4 = dab[ va VSlib vb + va VSlib vb + + v a ( VSlib vb )] + + d ac[ va( VSlic vc ) + va ( VSlic vc ) + + v a ( VSlic vc )] +, whr th last sum is xtdd for all distacs i th uppr suprior triagular positios, i d XY = dik, i < k Th otatio < u; v > mas th scalar product < u; v >= u T v, u R, v R Proof O uss th dfiitios of sums Σ, Σ, Σ3, Σ4 with a covit associatio of th wights v (Ed) ( ) ( ) 3 Th rlatio btw cotiuous Hopfild modl ad TSP problm Th Hopfild twork with procssig lmts, attachd to TSP problm procds from th cotiuous Hopfild modl [], pag 44 Th cotiuous modl is dscribd by two diffrtial quatios ( with idpdt otatios [] )

3 Procdigs of th th WSEAS Itratioal Cofrc o COMPUTERS, Agios Nikolaos, Crt Islad, Grc, July 6-8, 7 49 dui u p = = w i i v + Ii dt (3) Ri df de ( v = i ) dvi i = p dt (4) dvi dt whr vi = f ( ui ), ui = f ( vi ), i =, Two thigs ar vry importat i th futur: th tim dlay u i / R i from quatio (3) ad E = E(v) from (4) Th variabls u i from cotiuous modl [] bcom u, X =, ; =, i TSP problm W dot U = U, U = ( u ), whr u ar iput variabls Th w comput th wights λu v = f ( u ), = + v X / V = V, V = ( vx ) (5) Accordig to th gral tchiqus of ural tworks, th variabls u ar updatd wh th algorithm passs from tim t ( th rout t ) to tim t + ( th rout t + ) Th updatig is do by a rcurrt rlatio which has two quivalt forms: a compot form or a matrix form, rspctivly u ( t + ) = u + Δu (6) U ( t + ) = U + ΔU, Δ U = ( Δu ) (7) Now th mai qustio is to fid th appropriat form of corrctios Δu ( Agai w us th gral ural tworks thory: th corrctios ar dfid by dsct gradit of rgy fuctio So w hav th followig dpdcs: E = E( R), E = E( v), v = f ( u), E = E( u) de de Th drivativ ar positiv, amly >, > dv du Du to tim dlay from (3) ad th dsct gradit, w dfi th corrctios by th rlatio de Δu = u < (8) dvx Th > is a paramtr cotrolld by th usr 4 Th xplicit corrctio form ad w matrix otatios Th formula (8) ad E = E(v) giv th followig corrctios Δu = [ u A vx k = ; k B v Y C vyk( ' Y = ; Y X Y = k = D d X Y ( vy, + + vy, ) ] Δt (9) Y = whr appar som paramtrs for usr s disposal ( ;), Δt (;), ', < ' 5 I ordr to comput th laborious formula (9) w us w otatios, as it follows VliX ; k ) = VSliX vx Vcol ; Y X ) = vx V ; ' ) = Y = k = vy k ' ( Kli ; ) d ( v v ) = S X = + + Y X Y Y, Y, () ( th maigs of th lttrs ar: S is th sum i th matrix V or K tc) Propositio Th corrctios (9) tak th form Δ u = { u AVSli [ X vx ] B[ vx ] C[ ' )] D[ KliX ; ) ]} Δt () Proof O uss th otatios () (Ed) Th formula () dtrmi us to itroduc th followig matrix V, V = ( v v ) V = Y, + + Y, () Propositio 3 All th sums from () crat a w matrix ( as a product ) KV = ( Kli X ; ) ) (3) Proof O uss () ad th dirct computatio (Ed) W ca writ th lmts Δu ( from () or quivalt th matrix ΔU ( from (7) if w itroduc th matrics ( dotd by a succssio of two or thr lttrs )

4 Procdigs of th th WSEAS Itratioal Cofrc o COMPUTERS, Agios Nikolaos, Crt Islad, Grc, July 6-8, 7 43 VS = VS, VSL = VSL, VSC = VSC Explicitly, for =4, th abov matrics hav th forms v; VS = VSlia VSlib VSL = VSli c VSlid VSC = VSlia VSlib VSlic VSlid VSlia VSlib VSlic Vslid VSlia VSlib Vsli c VSli d Propositio 4 Th corrctios () from th propositio hav th matrix form U = U, ΔU = ΔU Δ U = { U A[ VSL V ] B[ VSC V ] C( VS ) D( KV )} Δt (4) Proof W us () ad th spcial matrics VS, VSL ad VSC (Ed) Th updatig rcurrt rlatios (6) or th quivalt matrix form (7) work if w kow th iitial valus u = u () or th iitial matrix U = U () for first rout 5 Th TSP algorithm i matrix form Havig all th abov otatios, formulas ad idas w ca dscrib th TSP algorithm W choos to dscrib this algorithm i matrix form Stp W itroduc th iput data : a) - umbr of tows; K = ( d X Y ) ; N - umbr of algorithm itratios b) gral paramtrs ', λ,, Δt ; c) ihibitios paramtrs A, B, C, D d) output fuctio f ( s) = + λs ) iitial valus U = ( u ), X =, ; =, f) w dclar th dimsios for all matrics : K, U, V, V, Vx ad so o Stp W xcut th computatios i a DO loop as it follows L CONTINUE DO L3 t=, N comput th sigmoid outputs ad crat th matrix V = ( vx () DO L X=, DO L =, v = f [ ux ] L CONTINUE L CONTINUE comput th sums Σ, Σ, Σ3 from propositio Σ = < vk ; v > k < Σ = < Vlii ; Vlik > i< k 3 = Σ ( ) Y = k = vy k t comput th xtdd matrix Vx( usig K, V, Vx( w comput th sum Σ 4 from propositio comput th rgy fuctio A B C D E = Σ + Σ + Σ3 + Σ4 optioal: prit th valus t, V, E( comput th followig matrics at tim t VSL = ( VSliX, X =, VSC = ( ), =, V = ( vx, + ( + vx, ( ), for all X =, ; =, K V comput th corrctio matrix Δ U ( by usig th formula (4) from propositio 4

5 Procdigs of th th WSEAS Itratioal Cofrc o COMPUTERS, Agios Nikolaos, Crt Islad, Grc, July 6-8, 7 43 updat th iput matrix U by th rcurrt quatio U ( t + ) = U + ΔU L3 CONTINUE (th DO loop util t=n ) Stp 3 Vrify if th closd Do loop grats a admissibl TSP solutio, by th matrix V ( N) = ( vx ( N) ) Thr ar svral possibilitis ( vrsios ) Vrsio Th matrix V (N ) grats a admissibl TSP solutio Th GO TO labl L4 Vrsio Th matrix V (N ) do ot grat a admissibl solutio bcaus v ( N) { ; } i v ( ; ) Th w rplac N by N ', N ' > N ad GO TO labl L ad rsum th cycl DO loop Vrsio 3 O uss th matrix V (N ) ad comput th maxim lmt o ach li X, X =, W dot it by v ( N) = If v ( N) ( ε;), ε > 8 ( for xampl ) th w st v = ad all th othr lmts v ( N) =, o th li X (wir-tak-all) Th rsultig matrix is dotd Vl ( N ), whr l mas th work o lis Aalogous w ca comput th maxim lmt v ( N) o ach colum =, So w obtai th matrix Vc ( N ), whr th lttr c mas th work o colums Comput th routs dscribd by matrics Vl ( N ), Vc ( N ) ad tak th bst o GO TO L4 L4 CONTINUE Stp 4 Prit th fial rsults : N, Vl ( N ) or Vc ( N ), E(N) ad th rout R STOP END 6 Applicatio Lt b with th valu = 4 ad th distacs btw th tows a, b, c, d giv by th matrix K = Apply th abov algorithm to 3 5 fid th bst rout Solutio W us th paramtrs = 4, ' = 5, λ =, =, Δt, N = A =, B =, C = 4, D =, f ( s) = + s Th iitial iputs ar U = U () = For t = w obtai V () = Σ = 3595, Σ = 35 95, Σ 3 = 5 Σ 4 = ; E( ) = 7784 ad so o Rfrcs [] FREEMAN A Jams, SKAPURA M David, Nural Ntworks: Algorithms, Applicatios ad Programmig Tchiqus, Addiso-Wsly Publishig Compay, 99 [] JAIN K Ail, MAO Jiachag, MOHIUDDIN K M, Artificial Nural Ntworks: A Tutorial, IEEE, March, 996 [3] KRÖSE B, Va dr SMAGT Patrick, A Itroductio to Nural Ntworks, Chaptr 5, Uivrsity of Amstrdam, Eighth Editio, Novmbr 996 [4] POPOVICIU Nicola, BONCUT Mioara, A Complt Squtial Larig Algorithm for RBF Nural Ntworks with Applicatios, WSEAS Trasactios o Systms, Issu, Volum 6, Jauary 7, pags 4-3 [5] SYED SAAD ALZHAR A, RBF Nural Ntworks Basd Slf-Tuig Adaptiv Rgulator, WSEAS Trasactios o Systms, Issu 9, Volum 3, Nov 4