Last-Train Timetabling under Transfer Demand Uncertainty: Mean-Variance Model and Heuristic Solution

Algorithm 1

TS algorithm procedure.

Input:

  The initial solution: Initial_s;

  The collected transfer demand: ;

  The risk-aversion coefficient: .

Output:

  The best solution: Best_s(Max_Iter);

  The expected value of objective function: ;

  The variance of objective function: .

(1) Present_s(0), Best_s(0) ⇐ Initial_s;

(2) Iter ⇐ 1;

(3) if Iter ≥ Max_Iter then

(4)  Calculate , ;

(5)  return Best_s(Max_Iter), , ;

(6) else

(7)   Generate the neighborhood () by adjustment on the Present_s;

(8)   Calculate ;

(9)   Choose the best neighbors according to ;

(10)   Pick the best solution which satisfies

        or among the neighbors;

(11)  Best_r ⇐ ;

(12)   if    then

(13)   Best_s(Iter) ⇐ Best_r;

(14)   Present_s(Iter) ⇐ Best_s(Iter);

(15)  end if

(16)  Add Best_r to the top of tabu list;

(17)  if Tabu list is full then

(18)   Remove the bottom of tabu list;

(19)  end if

(20)  if Best_s(Iter)=Best_s() then

(21)   Iter ⇐ ;

(22)  else

(23)   Iter ⇐ 1;

(24)  end if

(25) end if