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Vehicle routing with time windows:Two optimization algorithmsMarshall L. Fisher ,Kurt O. Jornsten,Oli B. G. MadsenOperation Research,Vol.45,No3,May-June 1997,pp.488-492

(Lagrangian relaxation)K-Tree(K-tree relaxation)

VRPTW formulation(Lagrangian relaxation)K-Tree(K-tree relaxation)

VRPTWm=n=;index 0=Qk=kqi=icij=ijtij=ijsi=iei=iui=iT=N={1,.,n}N0=N{0}M={1,.,m}

VRPTWxijk= 1,ijk 0,otherwiseyik 1,ik 0,otherwiseti=it0ek=kt0uk=k

VRPTW:

cij=ijxijk= 1,ijk 0,otherwisem=,M={1,.,m}n=,N={1,.,n},N0=N{0}

(2):

xijk= 1,ijk 0,otherwisem=,M={1,.,m}n=,N={1,.,n},N0=N{0}

(3):

xijk= 1,ijk 0,otherwisem=,M={1,.,m}n=,N={1,.,n},N0=N{0}

(4):ti+si+tij-(1-xijk)Ttj

xijk= 1,ijk 0,otherwisem=,M={1,.,m}n=,N={1,.,n},N0=N{0}ti=itj=jtij=ijsi=iT=

(5):t0ek+t0j-(1-x0jk)Ttj

x0jk= 1,jk 0,otherwisem=,M={1,.,m}n=,N={1,.,n},N0=N{0}tj=jt0j=jt0ek=kT=

(6):iti+si+tt0-(1-xi0k)Tt0uk

xi0k= 1,ik 0,otherwisem=,M={1,.,m}n=,N={1,.,n},N0=N{0}ti=iti0=isi=iT=t0uk=k

(7):iieiti ui

ti=iei=iui=i

(8):e0t0ekt0uk u0

t0ek=kt0uk=ke0=u0=

(10):

xijk= 1,ijk 0,otherwisem=,M={1,.,m}n=,N={1,.,n},N0=N{0}qi=iQk=k

(11):0ti 0

(12):xijk01xijk{0,1}

(13):

(14):yik {0,1}(15):

yik 1,ik 0,otherwise

Solution Methods:(Lagrangian relaxation)K-Tree(K-tree relaxation)

1:Lagrangian relaxationLagrangian relaxationNear Optimal Solution(relax)(constrain)(primal problem)(Linear Programming Relaxation)(lower bound)(optimal)

Solution Method- Lagrangian relaxation(1/7)

(constrain)Lagrangian multiplier(subgradient)

Solution Methods- Lagrangian relaxation(2/7)

(15)ik1: min- (16) subject to (13) and (14)

Solution Methods- Lagrangian relaxation(3/7)

2: min (17) subject to (2)-(12)(shortest path problem with time windows and capacity constraints,SPTWCC)SPTWCCcyclestwo-cycle elimination(Kolen et al.1987)Solution Methods- Lagrangian relaxation(4/7)

(15): (18) Where

Solution Methods- Lagrangian relaxation(5/7)

xy(Gap=(Optimal solution-Lower bound)/Lower bound in %)case VRPTWlower bound(18)(subgradient optimization)branch-and-bound method(Variable Splitting Approach)lower boundsSolution Methods- Lagrangian relaxation(6/7)

Solution Methods- Lagrangian relaxation(7/7)branching process:(1)yik10(2)

Solution Method - K-tree Approach(1/3)Tree:Basis: one node is a treeK-tree:Basis: Kchildren is a K-tree

Solution Method - K-tree Approach(2/3)Example: 4-tree

Solution Method - K-tree Approach(3/3)2:K-treeK-tree;TSP1-treeFisher(1994)minimum K-tree method--

Conclusionpaper100(Column Generation Approach)(Desrochers et al.1992)