A Multiobjective Iterated Greedy Algorithm for Truck Scheduling in Cross-Dock Problems

Algorithm 2

Modified crowding distance assignment procedure.

Procedure: Modified Crowding Distance Assignment

Working Set ≔Current set of solutions

For each  𝑚∈  objectives

    Sort Working Set using the current objective  𝑚

    Set the distance of the extreme elements on the Working Set to  ∞

    𝑓_𝑚^max≔ maximum value for objective  𝑚  in Working Set

    𝑓_𝑚^min≔ minimum value for objective  𝑚  in Working Set

    // Calculate the distance of all other elements in the Working Set

    For  𝑖∶=2,…,Working  𝑆𝑒𝑡_{𝐿𝑎𝑠𝑡𝐸𝑙𝑒𝑚𝑒𝑛𝑡−1}

         𝑊𝑜𝑟𝑘𝑖𝑛𝑔𝑆𝑒𝑡_𝑖⋅𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒∶=𝑊𝑜𝑟𝑘𝑖𝑛𝑔𝑆𝑒𝑡_𝑖+⋅𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒(𝑊𝑜𝑟𝑘𝑖𝑛𝑔𝑆𝑒𝑡_𝑖+1⋅𝑂𝑏𝑗𝑒𝑐𝑡𝑖𝑣𝑒_𝑚−𝑊𝑜𝑟𝑘𝑖𝑛𝑔𝑆𝑒𝑡_𝑖−1⋅𝑂𝑏𝑗𝑒𝑐𝑡𝑖𝑣𝑒_𝑚)(𝑓_𝑚^max−𝑓_𝑚^min)

  Endfor

Endfor

// Adjust the fitness value with the selection counter in each individual

// First obtain the extreme distance values(for normalizing)

Max Dist ≔ max(Working Set · distance)

Min Dist ≔ min(Working Set · distance)

For each  𝑖∈  Working Set

   // The infinit values of the extreme solutions should be replaced

   // by the maximum acceptable value

   If  Working  𝑆𝑒𝑡_𝑖 · distance = ∞  then

     Working  𝑆𝑒𝑡_𝑖 · fitness  1∶=(𝑊𝑜𝑟𝑘𝑖𝑛𝑔𝑆𝑒𝑡_𝑖⋅𝑆𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛𝐶𝑜𝑢𝑛𝑡𝑒𝑟+1)

   Else

     Working  𝑆𝑒𝑡_𝑖 · fitness  ∶=(𝑊𝑜𝑟𝑘𝑖𝑛𝑔𝑆𝑒𝑡_𝑖⋅𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒+Min𝐷𝑖𝑠𝑡)/(Max𝐷𝑖𝑠𝑡+Min𝐷𝑖𝑠𝑡)(𝑊𝑜𝑟𝑘𝑖𝑛𝑔𝑆𝑒𝑡_𝑖⋅𝑆𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛𝐶𝑜𝑢𝑛𝑡𝑒𝑟+1)

   Endif

Endfor