Joint decisions of pricing and inventory of a dual-channel supply chain with demand ambiguity considering free-riding behavior under sales effort: based on a fuzzy optimization method

Market demand often exhibits ambiguity due to insufficient historical data (e.g., new product launches), complicating joint decision-making in dual-channel supply chains (DCSC), especially with free-riding on sales effort. To address this, we construct a game-theoretic model and an analytical framework, based on generalized possibility distribution theory, for joint pricing and inventory decisions under such fuzzy demand. Closed-form solutions for optimal pricing decisions, sales effort, and inventory factor are derived under both decentralization and centralization. These solutions are applicable under the decentralized mechanism with fewer or simpler constraints. When the demand distribution is unknown or not second-order differentiable, a distributionally robust fuzzy optimization (DRFO) model is developed. Furthermore, the study investigates coordination mechanisms of the wholesale-price discount and revenue-sharing contract. The research finds that: (1) Under fuzzy demand, retailers respond to risk by increasing sales effort and moderately raising prices. This finding contrasts with the conventional result in stochastic models, where uncertainty typically leads to optimal prices lower than the risk-free price. (2) To maintain competitiveness, the demand-dependent retail price and online price move in tandem. However, sales effort boosts selling prices but is moderated by free-riding and the cost of sales effort. Furthermore, ignoring demand ambiguity leads to a biased estimation of the inter-channel price gap. (3) A higher degree of free-riding harms the retail channel under decentralization, and its impact on the overall supply chain’s expected profit depends on the channel price difference. Numerical studies also reveal that the optimal price decreases as uncertainty increases; price sensitivity suppresses own-channel demand while promoting cross-channel demand; and higher free-riding does not always benefit the manufacturer and may even reduce its profits under decentralization.