A Branch and Bound Algorithm and Iterative Reordering Strategies for Inserting Additional Trains in Real Time: A Case Study in Germany

Procedure 1

Lower bound and critical path.

Input: existing timetable

Output: lower bound and critical path

(1)   Begin

(2)    calculate reachable matrix and for original and additional operation respectively, where 1/0 represents that it

     is (not) reachable from node to node,

(3)    while     do

(4)     ,

(5)     for   and   do

(6)     ,

(7)     ,

(8)    end for

(9)    

(10) end while

(11) for  all   and   do

(12)  if   &   then

(13)   , i.e. and are incompatible arcs

(14)  end if

(15) end for

(16) for   all     do

(17)  ,

(18)  if     then

(19)   

(20)  else  if  

(21)   

(22)  end if

(23) end for

(24) ,   , ,

(25) while     do

(26)  if     then

(27)   , that is and are incompatible arcs

(28)  end if

(29)  if   &   then

(30)   

(31)  end if

(32)  if   &   then

(33)   

(34)  end if

(35)  if     then

(36)   , ,

(37)  end if

(38)  if     then

(39)   

(40)  end if

(41) end while

(42) Return   and