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

Algorithm 1

Lower bound calculation.

𝑝=1, 𝜋 = Initial value for of Lagrangian multiplier, 𝑁 = number of DCs,

𝑀 = number of retailers

For 𝑖=1 to 𝑁

   For 𝑗=1 to 𝑀

      Calculate IB́_𝑖𝑗 (IB́_𝑖𝑗 = individual benefit of retailer_𝑗 if assigned to DC_𝑖)

      Make set 𝐺_𝑖, so that 𝐺_𝑖={𝑗∈𝐽s.t.IB́_𝑖𝑗<0}

   End

   Arrange members of 𝐺_𝑖 in ascending order of IB́_𝑖𝑗

   Repeat the following steps for all members of 𝐺_𝑖

      Compute 𝑆_𝑖(𝑝) as follow

      𝑆_𝑖(𝑝) = Cost of DC_𝑖 if the first 𝑝 members of set 𝐺_𝑖 are assigned to DC_𝑖

      𝑆_𝑖(𝑝)=𝐾_𝑖∑_𝑗𝑑_𝑗𝑦_𝑖𝑗+𝐾^′_𝑖∑_𝑗𝑣_𝑗𝑦_𝑖𝑗+∑_𝑗(𝑤_𝑖𝑗𝑑_𝑗−𝜋_𝑗)𝑦_𝑖𝑗+𝐹_𝑖𝑥_𝑖+∑_𝑗𝜋_𝑗,

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

      If 𝑝=1

         If 𝑆_𝑖(1)<0

           Assign the 1st member of 𝐺_𝑖 to DC_𝑖, (𝑦_𝑖𝑗=1 and 𝑥_𝑖=1)

         End

      End

      𝑝=𝑝+1;

      If 𝑆_𝑖(𝑝−1)<0 and 𝑆_𝑖(𝑝)<0;

         Assign the 𝑝th member of 𝐺_𝑖 to DC_𝑖, (𝑦_𝑖𝑗=1)

      End

   End

End

Calculate current lower bound using the following formula

current lower bound =∑_𝑖ℎ_𝑖𝑄_𝑖/2+𝑍_𝛼lt_𝑖∑_𝑗𝑣_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+∑_𝑗𝜋_𝑗1−∑_𝑖𝑦_𝑖𝑗