Uncertain multi-item supply chain with two level trade credit under promotional cost sharing

In this paper, we have extended the Tsao's (2010) and Huang et al.'s (2012) investigations incorporating higher level of trade credit and imprecise constraints on resources after correcting the mistakes in their formulations. Here, the multi-item two level supply chain model is formulated without/with budget constraint of the retailer. Customers' credit period is introduced to boost the base demand of the items. It is established that if the supplier shares a part of the promotional cost then the channel profit as well as the individual profits increase. It is also established that the customers' credit period has sufficient significance in the supply chain. In their formulation Tsao as well as Huang et al. considered some holding costs for the supplier. But according to the model, the supplier does not hold any product, i.e., the supplier's holding cost should be zero. Also the interest earned and the interest paid by the retailer and the supplier are not properly calculated in their study. In the present study, these mistakes are corrected. In addition, when the available budget of the retailer is uncertain in the sense of fuzzy or rough, the optimum profits of the said models are evaluated and presented. Models with imprecise constraints are transferred to equivalent crisp constraints following suitable techniques. Existence of the optimal solution of the unconstrained crisp model is established analytically. These models are solved by Generalized Reduced Gradient (GRG) method using LINGO 14.0 software and/or Particle Swarm Optimization (PSO) technique. The models with imprecise inventory costs involve imprecise objectives and in those cases marketing decisions are made using PSO in two approaches - Direct approach, i.e., no crisp equivalents of the imprecise objectives are used for solving the models and another is Expected value (of the objectives) optimization approach. The models are illustrated with some hypothetical numerical examples. Some managerial implementations are also derived.