Two-layer supply chain model for Cauchy-type stochastic demand under fuzzy environment

The purpose of this paper is to deal with profit maximization problem of two-layer supply chain (SC) under fuzzy stochastic demand having finite mean and unknown variance. Buyback policy is employed from the retailer to supplier. The profit of the supplier solely depends on the order size of the retailers. However, the loss of shortage items is related to loss of profit and goodwill dependent. The authors develop the profit function separately for both the retailer and supplier, first, for a decentralized system and, second, joining them, the authors get a centralized system (CS) of decision making, in which one is giving more profit to both of them. The problem is solved analytically first, then the authors fuzzify the model and solve by fuzzy Hausdorff distance method.

The analytical models are formed for both centralized and decentralized systems under non-cooperative and cooperative environment with suitable constraints. A significant assumption on density function, namely Cauchy-type density function, is introduced for demand rate because of its wider range of the retailers’ satisfactions. Fuzzy Hausdorff metric is incorporated within the fuzzy components of the fuzzy sets itself. Using this method, the authors find out closure values of both centralized and decentralized policies, which is an essential part of any cooperative and non-cooperative two-layer SC models. Moreover, the authors take care of the profit values with corresponding ambiguities for both the systems explicitly.

It is found that the centralize policy of SC could only be able to maximize the profit of both the retailers and suppliers. All analytical results are illustrated numerically along with sensitivity analysis and side by side comparative studies between Hausdorff and Euclidean distance measure are done exclusively.

The main focus of attention of the proposed model is given to usefulness of Hausdorff distance. Unlike other distances, Hausdorff distance can take special care on the similarity measures of different fuzzy sets. Researchers have been engaged to use Hausdorff distance on the different fuzzy sets but, in this study, the authors have used it within the components of a same fuzzy set to gain more closure values than other methods.

The use of this Hausdorff distance approach is totally new as per literature survey suggested yet. However, the Cauchy-type density function has not been introduced anywhere in SC management problems by modern researchers still now. In crisp model, the sensitivity on goodwill measures really provides a special attention also.