Time analysis for planning a path in a time-window network

Abstract

A systematic method is proposed to generate time information on the paths and nodes on a time-window network for planning and selecting a path under a constraint on the latest entering time at the destination node. Specifically, three algorithms are proposed to generate six basic time characteristics of the nodes, including the earliest and latest times of arriving at, entering, and departing from each node on the network. Using the basic time characteristics, we identify inaccessible nodes that cannot be included in a feasible path and evaluate the accessible nodes’ flexibilities in the waiting time and staying time. We also propose a method for measuring adverse effects of including an arc. Finally, based on the time characteristics and the proposed analyses, we develop an algorithm that can find the most flexible path in a time-window network.

Appendix

Algorithm I. Evaluation of EA _i , EE _i , and ED _i :

Step 1: Initialization

  For each node i, except the source node, set EA _i =∞, EE _i =∞, and ED _i =∞;

  EA ₁=0; EE ₁=0; ED ₁=0;

  Mark all the nodes as unexamined nodes;

Step 2: Let node i be the node with the minimum ED _i value among the unexamined nodes;

  For each unexamined successor j of node i, do

   if ED _i +dur(i, j)<EA _j , then

    EA _j=ED_i+dur(i, j);

    If EA _j > , then go to the next iteration of the for-loop;

    Find the minimal k(1≤k≤E _j ), such that EA _j ≤Eend _j ^k;

    EE _j=#128;;max{EA_j, Ebegin_j ^k};

    If EE _j > , then go to the next iteration of the for-loop;

    Find the minimal k(1≤k≤D _j ), such that EE _j ≤Dend _j ^k;

    ED_j=max{EE_j, Dbegin_j ^k};

    If ED _j has been modified, then EP _j =i;

    /* EP _j is used to store the earliest path route ^*/

   Mark node i as an examined node;

Step 3: Repeat step 2 until all the nodes have been examined;

Algorithm II. Evaluation of LE _i and LD _i :

Step 1: Initialization

  For each node i other than destination n, do LE _i =−∞ and LD _i =−∞;

  LE _n = ;

  Mark all the nodes as unexamined nodes;

Step 2: Backward computation

  Let node i be the node with the maximum LE _i among the unexamined nodes;

  For each unexamined predecessor j of i, do

   if LE _i –dur(j, i)>LD _j , then

    If LE _i –dur(j, i)<Dbegin _j ¹, then go to the next iteration of the for-loop;

    Find the maximal k(1≤k≤D _j ) such that Dbegin _j ^k≤LE _i –dur(j, i);

    LD _j =max{LD _j , min{LE _i -dur(j, i), Dend _j ^k};

    If LD _j <Dbegin _j ¹, then go to the next iteration of the for-loop;

    Find the maximal k(1≤k≤E _j ), such that Ebegin _j ^k≤LD _j ;

    LE _j =max{LE _j , min{LD _j , Eend _j ^k}};

    If LE _j is changed, then set LP _j =i;

    /* LP _j stores the latest path route */

  Mark node i as an examined node;

Step 3: Repeat step 2 until all the nodes have been examined.

Algorithm III. Evaluation of LA _i :

For each node i except the source node, do

begin

LA _i =−∞;

For each predecessor node j of i, do

 x=LE _i −dur(j, i);

if x≥LD _j then y=LD _j

else if x≥ED _j then begin

Find the maximal k such that Dbegin _j ^k≤x;

  y=min{Dend _j ^k, x}

end

 else y=−∞;

LA _i =max{LA _i , y+dur(j, i)};

if LA _i , is changed, then set LAP _i =j /^* LAP _i is used to store the node from which we

      can arrive at node i in the latest time. ^*/

end