Distribution Network Design for Fixed Lifetime Perishable Products: A Model and Solution Approach

Algorithm 2

Upper bound calculation.

         Generating feasible solution from the lower bound solution

        Phase 1 generating a solution that satisfy single sourcing constrain

𝑖_min=0, 𝑁 = number of DCs, 𝑀 = number of retailers

For 𝑗=1 to 𝑀

   If ∑^𝑁_𝑖=0𝑦_𝑖𝑗=0 (if a retailer exist that is assigned to no DC, allocate this retailer to all DCs)

    𝑦_𝑖𝑗=1∀𝑖∈𝐼={1,2,…,𝑁};

   End

For 𝑗=1 to 𝑀

   While there exists at least one retailer such that ∑^𝑁_𝑖=0𝑦_𝑖𝑗>1 repeat the following steps

    For 𝑖=1 to 𝑁

      Calculate 𝐶_𝑖 as follow (𝐶_𝑖 = Cost of DC_𝑖 based on the current retailers assigned to it);

      𝐶_𝑖=𝐾_𝑖∑_𝑗𝑑_𝑗𝑦_𝑖𝑗+𝐾^′_𝑖∑_𝑗𝑣_𝑗𝑦_𝑖𝑗+∑_𝑗𝑤_𝑖𝑗𝑑_𝑗𝑦_𝑖𝑗+𝐹_𝑖𝑥_𝑖

      𝐾_𝑖=2ℎ_𝑖𝑂_𝑖+𝐴_𝑖,𝐾^′_𝑖=ℎ_𝑖𝑍_𝛼lt_𝑖,𝑤_𝑖𝑗=𝑇∑_𝑗(𝑡_dc-su+dis_𝑖𝑗)

    End

    Find DC_𝑖 with minimum 𝐶_𝑖 and let 𝑖_min=𝑖

      For 𝑗=1 to 𝑀

         If 𝑦_𝑖min,𝑗=1

         For 𝑖=1 to 𝑁 and 𝑖≠𝑖_min

          If 𝑦_𝑖𝑗=1

             𝑦_𝑖𝑗=0;

          End

         End

        End

      End

   End

End

           Phase 2 generating a solution that satisfy 𝑄 constrain

While at least a DC exists with violated 𝑄 constraint (constraint (11))

   Select a DC with violated 𝑄 constraint;

   Let 𝑗_min = the last assigned retailer to DC (refer to set 𝐺_𝑖́ in lower bound calculation);

   Remove retailer_𝑗min from the set of retailers allocated to DC;

   Allocate retailer_𝑗min to a DC that leads to minimum cost and its 𝑄 constraint would not be violated;

End

Calculate current upper bound using the following formula

current upper bound =∑_𝑖ℎ_𝑖⋅𝑄_𝑖/2+𝑍_𝛼lt_𝑖∑_𝑗(𝑣_𝑗𝑦_𝑖𝑗)+∑_𝑖∑_𝑗(𝑂_𝑖+𝐴_𝑖)(𝑑_𝑗𝑦_𝑖𝑗)/𝑄_𝑖+∑_𝑖𝐹_𝑖𝑥_𝑖

+∑_𝑖∑_𝑗𝑇(dis_𝑖𝑗+𝑡_dc-su)𝑑_𝑗𝑦_𝑖𝑗;