Optimality of a Classical Two-Level Supply Chain Problem Under Interval Uncertainty via AM–GM Parametric Approach

Due to the inexactness and uncertainty, most of the parameters involved in a supply chain system are imprecise. So, there arises a question — how can one handle such imprecise parameters to formulate a supply chain model under uncertainty? To find the answer to this question, this work focuses on the optimality of a classical supply chain model under interval uncertainty using interval parametric approach. In this chain, a manufacturer and a retailer are involved. Here, inventories of both manufacturer and retailer are considered with interval valued inventory components. The average costs of all members of the chain are obtained using parametric interval mathematics. Now, there arise another question — how can interval valued average costs of the chain under interval uncertainty be minimized? To do so, an Arithmetic Mean-Geometric Mean (AM–GM) inequality-based parametric geometric optimization approach has been introduced to find the optimality conditions for minimizing the manufacturer’s, retailer’s as well as their joint interval-valued average costs in parametric form. After that, using the Stackelberg game principle, interval valued minimum average costs for both the manufacturer and retailer are obtained in parametric form. Finally, theoretical findings are supported by numerical examples demonstrating the feasibility of the model and stability of the interval parametric approach. This work combines parametric interval analysis, inequality-based parametric optimization and game theory in supply chain cost modeling and contributes novel insights into uncertain decision-making.