Step 5 ��When the number of iteration reaches a predefined maximum number, output the optimal results; otherwise, repeat Steps 2�C4.4.2. Multiobjective Evolution Algorithm (MOEA) Approach for the MSJRDIn this section, a brief introduction blog of sinaling pathways of MOP is given. Then, an HDE-based procedure to handle the MSJRD using noninferior and crowding distance is designed.4.2.1. Some Definitions of MOP Definition 1 (multiobjectiveoptimizationproblems(MOP)) i=1,2,��,m.(20)A general?gi(x)��0,?��Min?F(x)=f1(x),f2(x),��,fk(x)Subject??to: MOP consists of n decision variables, k objective functions, and m constrains. In Definition 1, x refers to the decision space and gi(x) are constrains of MOP.Definition 2 (Pareto optimal solution) ��The optimal solution of MOP is often referred to as the Pareto optimal solution.

Let vector a belong to x and suppose that x* is a subset of x. If there does not exist any vector in x* that is better than a, then a is called noninferior solution (or Pareto optimal solution) of x*. Moreover, if vector a is the noninferior solution of x, then vector a is the Pareto optimal of the MOP.4.2.2. HDE-Based Procedures for MSJRD Using MOEA Approach There exist many difficulties when applying DE to solve an MOP compared with single objective problem. The main challenges for solving MOP are as follows: how to generate offspring and how to keep Pareto solutions uniformly distributed. The classical DE is not suitable for an MOP since many good solutions may be abandoned due to its one-to-one competing mechanism. This will also be confirmed by a numerical example.

Therefore, we also use an HDE which uses truncation selection to choose next generation based on front rank and crowding distance adopted by Qian and li [27]. The steps of calculating crowding distance are presented in Algorithm 1.Algorithm 1Steps of calculating crowding distance.In this algorithm, the low front rank corresponds to the high quality of a solution. As to the those individuals with the same front rank, the larger crowding distance means better distribution. Therefore, individuals with lower front rank and larger crowding distance are selected to the next generation.The first target can be divided into an inventory problem and a delivery problem. When all ki are determined, the optimal delivery cost can be calculated by solving a TSP.

In addition, for a stochastic JRP with normal distributed demand, when ki, zi, and T are known, the stochastic JRP can then be solved. With the same value of ki, zi, and T in the second target, we can obtain the corresponding value of the second target. Then change ki, zi, and T with the following steps until the termination condition is satisfied. The steps of HDE-based Brefeldin_A approach are described as follows.Step 1 ��Initialization: set related parameters (CR, F, and NP) for the HDE. Set the lower bound and the upper bound of ki, respectively; that is, kiLB = 1 and kijUB = 100.