Artificial Bee Colony Algorithm Merged with Pheromone Communication Mechanism for the 0-1 Multidimensional Knapsack Problem

Algorithm 2

ABCPUD-MKP.

1. Initialization: Initialize parameters 𝐾,𝑁,𝐿𝑖𝑚𝑖𝑡,𝛼,𝛽,𝑄,𝜏₀,𝐶_𝑖=0 (𝑖=1,…, K ) and so on;

2. Initial solutions: Randomly generated 𝐾 food sources {𝑆_𝑖(0)∣𝑖=1,…, K};

3. Loop:

 For 𝑡=1 to 𝑁 do:

    {

    (1) For 𝑖=1 to K do: (local searches performed by employed bees)

      {Associate each employed bee with a food source 𝑆_𝑖(𝑡) and compute its nectar amount;

      Find a new 𝑆^′_𝑖(𝑡) in the neighborhood of 𝑆_𝑖(𝑡) and compute its nectar amount;

      Take the better one in {𝑆^′_𝑖(𝑡), 𝑆_𝑖(𝑡)} as a new location of the employed bee;}

    (2) For𝑗=1 to K do: (further local searches performed by onlooker bees)

       {Select a food source 𝑆_𝑗(𝑡) from {𝑆_𝑖(𝑡)} for every onlooker bee;

       Find a new 𝑆^′_𝑗(𝑡) in the neighborhood of 𝑆_𝑗(𝑡) and compute its nectar amount;

       Take the better one in {𝑆^′_𝑗(𝑡), 𝑆_𝑗(𝑡)} as a new location of the corresponding bees;}

   ( 𝟑 ) Exploiting new food sources (global searches with a guidance performed by scout bees)

       For 𝑙=1 to K do: (food sources)

           {If 𝑆_𝑙(𝑡)=𝑆_𝑙(𝑡−1) then 𝐶_𝑙=𝐶_𝑙+1;

           If 𝐶_𝑙=𝐿𝑖𝑚𝑖𝑡 then

              {Abandon food source 𝑙 and the associated employed bee becomes a scout;

               𝑆_𝑙(𝑡)={}, 𝑎𝑙𝑙𝑜𝑤𝑒𝑑_𝑙=𝐽;

              Repeat (constructing a new solution)

             Select an object 𝑗 with 𝑃^𝑙_𝑗(𝑡) given by (2);

             Add the object into the current solution: 𝑆_𝑙(𝑡)=𝑆_𝑙(𝑡)+{𝑜_𝑗};

              𝑎𝑙𝑙𝑜𝑤𝑒𝑑_𝑙=𝑎𝑙𝑙𝑜𝑤𝑒𝑑_𝑙−{𝑜_𝑗};

              Until  𝑎𝑙𝑙𝑜𝑤𝑒𝑑_𝑙 is empty

             The scout bee becomes again an employed bee;

              𝐶_𝑙=0;}}

   ( 𝟒 ) Generating, diffusing and updating the pheromone

       For each object 𝑜_𝑖 in 𝐽

          Calculate the Δ𝜏_𝑖 according to (7) and (8).

       Select the Top-𝑘 solutions from this iteration, and obtain 𝐷_Top-𝑘;

       For each solution 𝑆_𝑤∈𝐷_Top-𝑘

         {For each pair of objects 𝑐_𝑖𝑗 in 𝑆_𝑤

            𝑐_𝑖𝑗.count++;

            Calculate the associated distance 𝑑_𝑟(𝑜_𝑖,𝑜_𝑗)

            Calculate Δ𝜏_𝑖𝑗 according to (9);}

       For each object 𝑜_𝑖 in 𝐽

          Update the trail level 𝜏_𝑗 on all objects according to (6) and (10);

    (5) Perform the local optimization in 1-1 or 2-1 exchange ways;

    (6) Memorize the best food source 𝑆_best found so far;

        𝑡=𝑡+1;

    }

4. Output: Return 𝑆_best while the predefined end condition is met.