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This paper deals with an adaptation of an application of nonlinear heptagonal dense fuzzy number. The concept of linear and as well as non-linear for both symmetric and asymmetric heptagonal dense fuzzy number is introduced here. We develop a new ranking method for non-linear heptagonal dense fuzzy number also. Considering a backorder inventory model with non-linear heptagonal dense fuzzy demand rate we have utilized a modified centroid method for defuzzification. For decision maker's aspects, numerical examples, comparative study with other dense fuzzy numbers and a sensitivity analysis show the superiority of the nonlinear heptagonal dense fuzzy number. Finally, graphical illustrations are made to justify the model followed by a conclusion.Mathematics Subject Classification. 03E72, 90B50.

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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.carbon emission, default risk, new defuzzification method, two-level trade credit, volumetric fuzzy number | INTRODUCTIONHere we shall discuss the existing related literatures into two parts, the general overview of the basic model and motivation and specific study, respectively. | General overview of the basic modelIn the classical economic order quantity (EOQ) model, it was assumed that the produced items are nonperishable and have an indefinite useful life. But in the real world, most of the items are subject to some risks such as decay, dryness, damage, evaporation, pilferage, and obsolescence and so on. Products like blood bags, chemicals, pharmaceuticals, food products and volatile liquids, and so on deteriorate quickly over time. So, these products cannot serve its original purpose after a period of time and they are disposed of. Therefore, the impact of loss from deterioration cannot be ignored. Ghare and Schrader 1 first established a revised form of the EOQ model for deteriorating items. They believed that products begin to deteriorate as soon as they arrive at the system. But in practice, many products (i.e., fruits, vegetables, medicines, and electronic products) preserve their freshness for a specific period of time. This event is known as noninstantaneous deterioration. This concept was first investigated by Wu et al. 2 for determining the optimal ordering policy for noninstantaneous deteriorating items (NIDIs) with stock-dependent demand and partial backordering.Moreover, trade credit is a widespread common practice...

The present article was developed for the economic order quantity (EOQ) inventory model under daytime, non-random, uncertain demand. In any inventory management problem, several parameters are involved that are basically flexible in nature with the progress of time. This model can be split into three different sub-models, assuming the demand rate and the cost vector associated with the model are non-randomly uncertain (i.e., fuzzy), and these may include some of the retained learning experiences of the decision-maker (DM). However, the DM has the option of revising his/her decision through the application of the appropriate key vector of the fuzzy locks in their final state. The basic novelty of the present model is that it includes a computer-based decision‐making process involving flowchart algorithms that are able to identify and update the key vectors automatically. The numerical study indicates that when all parameters are assumed to be fuzzy, the double keys of the fuzzy lock provide a more accurate optimum than other methods. Sensitivity analysis and graphical illustrations are made for better justification of the model.

In this article we develop an economic order quantity (EOQ) model with backlogging where the decision is made jointly from two decision maker supposed to view one of them as the industrialist (developer) and the other one as the responsible manager. The problem is handled under dense fuzzy environment. In fuzzy set theory the concept of dense fuzzy set is quite new which is depending upon the number of negotiations/ turnover made by industrial developers with the supplier of raw materials and/or the customers. Moreover, we have discussed the preliminary concept on dense fuzzy sets with their corresponding membership functions and appropriate defuzzification method. The numerical study explores that the solution under joint decision maker giving the finer optimum of the objective function. A sensitive analysis, graphical illustration and conclusion are made for justification the new approach.

In this article, the parabolic dense fuzzy set is defined, and its basic arithmetic operations are studied with graphical illustration. The lock set concept is incorporated in a parabolic dense fuzzy set. Then, it is applied to the problems of fishery culture via the modeling of an economic order quantity model. Here, the fingerlings are fed to reach the ideal size to fulfill the customer’s demand. The growth rate of the fingerlings is assumed as a linear function. After the sales of all fish, the pond is cleaned properly for a new cycle. Here, the model is solved in a crisp sense first. Then, we fuzzify the model considering the demand rate as a parabolic dense lock fuzzy number and obtain the result in a fuzzy environment. The main aim of our study was to find the quantity of the ordering items such that the total inventory cost gets a minimum value. Lastly, sensitivity analysis and graphical illustrations were added for better justification of our model.