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Recently, the first oligopolistic competition model of the closed-loop supply chain networkinvolving uncertain demand and return has been established. This model belongs to the contextof oligopolistic firms that compete noncooperatively in a Cournot-Nash framework. In this paper,we modify the above model in two different directions. (i) For each returned product fromdemand market to firm in the reverse logistics, we calculate the percentage of its optimal productflows in each individual path connecting the demand market to the firm. This modificationprovides the optimal product flow routings for each product in the supply chain and increases theoptimal profit of each firm at the Cournot-Nash equilibrium. (ii) Our model extends the methodof finding the Cournot-Nash equilibrium involving smooth objective functions to problemsinvolving nondifferentiable objective functions. This modification caters for more real-lifeapplications as a lot of supply chain problems involve nonsmooth functions. Existence of theCournot-Nash equilibrium is established without the assumption of differentiability of the givenfunctions. Intelligent algorithms, such as the particle swarm optimization algorithm and thegenetic algorithm, are applied to find the Cournot-Nash equilibrium for such nonsmoothproblems. Numerical examples are solved to illustrate the efficiency of these algorithms

mathematical problems in engineering

Intelligent Optimization Algorithms: A Stochastic Closed-Loop Supply Chain Network Problem Involving Oligopolistic Competition for Multiproducts and Their Product Flow Routings