A Memetic Lagrangian Heuristic for the 0-1 Multidimensional Knapsack Problem

Algorithm 2

Local improvement heuristic.

LocalImprovementHeuristic(𝝀, 0-1MKP instance)

{

 for  𝑖←1 to  𝑁

  (𝐱^∗,𝐛^∗)←LMMKP(𝝀);

  𝐼←{𝑖∶𝑏^∗_𝑖≤𝑏_𝑖} and 𝐽←{𝑖∶𝑏^∗_𝑖>𝑏_𝑖};

  if  𝐼={1,2,…,𝑚}

   update the best solution;

   choose a random element 𝑘 among 𝐼;

   𝜆_𝑘←𝜆_𝑘−𝛿;

  else

   choose a random element 𝑘 among 𝐽;

   𝜆_𝑘←𝜆_𝑘+𝛿;

 return  𝝀 corresponding to the best solution;

}