This article develops a three-layer supply chain (SC) pollution dependent production transportation model with rework on fuzzy approximate reasoning approach. First of all, we develop a cost function of the model of reworkable items, in which pollution generates due to transportation of goods only. The aim of this study is to reduce the aggregated pollution level that simultaneously could optimize the integrated SC cost function under fuzzy approximate reasoning. Basically, the theory of two-tailed (randomized L-R fuzzy) approximate reasoning is introduced first time on demand rate in this model which is an extension of one-tailed (randomized L-fuzzy/R-fuzzy) approximate reasoning in the perspectives of the proposed model. An application of a new heart like approximate dual feasible region has also been utilized here also. However, a comparative study has been made taking numerical examples for crisp, general fuzzy, one- and two-tailed fuzzy approximate reasoning models exclusively. Finally, few numerical analyses with sensitivity and graphical discussions are done to legitimize the model.

This study explores carbon emission issues over production manufacturing system with the context of joint inventory control and sustainable trade credit financing for deteriorating items in a supplier–retailer–customer model under volumetric fuzzy system. Here, the supplier offers an upstream full trade credit to the retailer and the retailer in turn, provides a downstream partial trade credit to her/his customers. Demand rate is assumed to be functionally depends upon carbon emissions, unit selling price and the length of the credit period offered by the retailer to their customers explicitly. Taking a case study, we develop a profit maximization crisp problem first then considering the unit selling price and the carbon elasticity as fuzzy numbers we formulate an equivalent fuzzy problem. After that, the fuzzy problem has been defuzzified with the help of general fuzzy system and the new volumetric fuzzy system under two different scenarios namely with and without carbon emissions also. The aim of this study is to develop a best methodology over fuzzy system on trade credit financing problem so as to optimize the average profit function over the optimum design variables the credit period, the order quantity, the cycle time, the partial shortage time and minimum carbon emissions, respectively. Finally, numerical illustrations and its comparative study with the existing methods, sensitivity analysis and graphical illustrations are performed to justify the novelty of the proposed approach.

In this article, we develop the definition of the degree of fuzziness for a polygonal fuzzy set. First of all we take a polygon of sides 2n + 1 and then we calculate its area. Also we consider the area of an half ellipse whose length of major axis are the length of the fuzzy interval of the given fuzzy set and its semi minor axis is 1. We construct the relative error as the measure of the degree of fuzziness and present the index value of the fuzzy parameter after K years and its role in modern decision making problems. Finally real life applications are discussed to justify the novelty of the new method.