Modeling and Optimization of the Multiobjective Stochastic Joint Replenishment and Delivery Problem under Supply Chain Environment

Algorithm 1

Steps of calculating crowding distance.

Step 1. For each 𝑥_𝑖∈𝐼, initialize distance 𝐷_(𝑖)=0

Step 2. Calculate the crowding distance of 𝑥_𝑖

   Step 2.1. Sorting 𝐼 for each objective 𝑚,𝐼=sort(𝐼,𝑚)

       𝐷(𝐼₍₁₎)=𝐷(𝐼₍₁₎)+2(𝐼_(2)⋅𝑚−𝐼_(1)⋅𝑚)²

       𝐷(𝐼_𝐼)=𝐷(𝐼_𝐼)+2(𝐼_𝐼⋅𝑚−𝐼_{𝐼−1⋅𝑚})²

   For 𝑖=2,…,|𝐼|−1

       𝐷(𝐼_(𝑖))=𝐷(𝐼_(𝑖))+(𝐼_{(𝑖+1)⋅𝑚}−𝐼_{(𝑖−1)⋅𝑚})²

   Step 2.2. 𝐷_(𝑖)=𝑠𝑞𝑟𝑡(𝐷_(𝑖))

where |𝐼| is the number of individuals contained in 𝐼; 𝐼_{(𝑖)⋅𝑚} is the objective value of individual 𝑖.