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

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1. (Lagrangian relaxation) 2.K-Tree ( K-tree relaxation)

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

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VRPTW m= n= ;index 0= Q k = k q i = i c ij = i j t ij = i j s i = i e i = i u i = i T= N={1,.,n} N 0 =N {0} M={1,.,m}

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VRPTW x ijk = 1, i j k 0,otherwise y ik 1, i k 0,otherwise t i = i t 0ek = k t 0uk = k

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VRPTW : c ij = i j x ijk = 1, i j k 0,otherwise m= ,M={1,.,m} n= ,N={1,.,n},N 0 =N {0}

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(2): x ijk = 1, i j k 0,otherwise m= ,M={1,.,m} n= ,N={1,.,n},N 0 =N {0}

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(3): x ijk = 1, i j k 0,otherwise m= ,M={1,.,m} n= ,N={1,.,n},N 0 =N {0}

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(4): t i +s i +t ij -(1-x ijk )Tt j x ijk = 1, i j k 0,otherwise m= ,M={1,.,m} n= ,N={1,.,n},N 0 =N {0} t i = i t j = j t ij = i j s i = i T=

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(5): t 0ek +t 0j -(1-x 0jk )Tt j x 0jk = 1, j k 0,otherwise m= ,M={1,.,m} n= ,N={1,.,n},N 0 =N {0} t j = j t 0j = j t 0ek = k T=

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(6): i t i +s i +t t0 -(1-x i0k )Tt 0uk x i0k = 1, i k 0,otherwise m= ,M={1,.,m} n= ,N={1,.,n},N 0 =N {0} t i = i t i0 = i s i = i T= t 0uk = k

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(7): i i e i t i u i t i = i e i = i u i = i

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(8): e 0 t 0ek t 0uk u 0 t 0ek = k t 0uk = k e 0 = u 0 =

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(10): x ijk = 1, i j k 0,otherwise m= ,M={1,.,m} n= ,N={1,.,n},N 0 =N {0} q i = i Q k = k

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(11): 0 t i 0 (12): x ijk 0 1 x ijk {0,1}

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(13): (14): y ik {0,1} (15): y ik 1, i k 0,otherwise

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Solution Methods : 1. (Lagrangian relaxation) 2.K-Tree ( K-tree relaxation)

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1: Lagrangian relaxation Lagrangian relaxation Near Optimal Solution (relax) (constrain) (primal problem) (Linear Programming Relaxation) (lower bound) (optimal) Solution Method- Lagrangian relaxation(1/7)

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(constrain) Lagrangian multiplier (subgradient) Solution Methods- Lagrangian relaxation(2/7)

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(15) ik 1 : min- (16) subject to (13) and (14) Solution Methods- Lagrangian relaxation(3/7)

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

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(15) : (18) Where Solution Methods- Lagrangian relaxation(5/7)

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x y (Gap=(Optimal solution-Lower bound)/Lower bound in %) case VRPTW lower bound (18) (subgradient optimization) branch-and-bound method (Variable Splitting Approach) lower bounds Solution Methods- Lagrangian relaxation(6/7)

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branching process: (1) y ik 1 0 (2) Solution Methods- Lagrangian relaxation(7/7)

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

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

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Solution Method - K-tree Approach(3/3) 2: K-tree K-tree ; TSP 1-tree Fisher(1994) minimum K-tree method - -

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Conclusion paper 100 (Column Generation Approach)(Desrochers et al.1992)