Optimizing joint operations decision-making involving substitute products: a Stackelberg game model and nested PSO

It is not uncommon that supply chain partners carry out cooperative advertising in green production involving green and dirty products (i.e., substitute products). Besides the advertising decisions, they need to jointly make many other decisions, such as substitute products’ production quantities, wholesale prices, and retail prices. Practice and literature have shown that manufacturers-retailers’ joint decision-making is of paramount importance yet challenging. This decision-making difficulty is compounded by governments’ carbon tax policies and financial subsidies. To facilitate firms in making decisions, this study examines the joint decision-making mechanism involving local governments’ carbon taxes and subsidies. To overcome the limitations of the relevant literature addressing one product and relatively fewer decisions, we include both dirty and green products and consider diverse decisions, including technology selection, production quantities, wholesale prices, and retail prices for both products. Additionally, we consider the retailers’ advertising investment decisions for both products and the manufacturers’ ratios of advertising investment paid to retailers. Capitalizing on decision interactions, we develop a Stackelberg game-based bilevel optimization model. Caused by the large number of decisions and their interactions, solving the game model analytically is barely possible. Consequently, we propose an algorithm of nested particle swarm optimization (NPSO). We perform numerical examples to show how the game model and the NPSO can help firms make complex joint decisions with many interactions. We also carry out sensitivity analysis based on which managerial insights are drawn.