The Delphi method is a process where subjective data are transformed into quasi-objective data using statistical analysis and are converged to stable points. The Delphi method was developed by the RAND Corporation at Santa Monica, California, and is widely used for long-range forecasting in management science. It is a method by which the subjective data of experts are made to converge using some statistical analyses. This article proposes a variation of the Delphi method using triangular fuzzy numbers, where the communication method with the experts is the same, but the estimation procedure is different. The utility of the method is illustrated by a numerical example.

In the current situation, environmental pollution is one of the vital issues affecting every country. In this research paper, we have developed a production inventory model based on carbon emissions, level of greenness, and the warranty of a product. However, very little research has reported on the topics mentioned above. To set up a sustainable imperfect production inventory model, the following suppositions are made: (a) that carbon is released during the production process and that it can be mitigated by imposing technology preventing carbon release directly into the environment; (b) that manufacturers provide a price discount on the selling price of the product to attract customers; and (c) that manufacturers also give attention to the warranty on the goods. This paper assesses the effects of the greenness, warranty of an item, and technology preventing carbon release into the environment on overall profit to help decision-makers make more effective decisions about pricing and replenishment. Three decision variables will need to have their optimal values determined using an algorithm. To justify the proposed model, one numerical example is solved. Finally, a sensitivity analysis is performed to determine how various factors affect total profit.

Multigranulation rough set (MGRS) based on soft relations is a very useful technique to describe the objectives of problem solving. This MGRS over two universes provides the combination of multiple granulation knowledge in a multigranulation space. This paper extends the concept of fuzzy set Shabir and Jamal in terms of an intuitionistic fuzzy set (IFS) based on multi-soft binary relations. This paper presents the multigranulation roughness of an IFS based on two soft relations over two universes with respect to the aftersets and foresets. As a result, two sets of IF soft sets with respect to the aftersets and foresets are obtained. These resulting sets are called lower approximations and upper approximations with respect to the aftersets and with respect to the foresets. Some properties of this model are studied. In a similar way, we approximate an IFS based on multi-soft relations and discuss their some algebraic properties. Finally, a decision-making algorithm has been presented with a suitable example.

In the current study, a neutrosophic quadratic fractional programming (NQFP) problem is investigated using a new method. The NQFP problem is converted into the corresponding quadratic fractional programming (QFP) problem. The QFP is formulated by using the score function and hence it is converted to the linear programming problem (LPP) using the Taylor series, which can be solved by LPP techniques or software (e.g., Lingo). Finally, an example is given for illustration.

This paper introduces a time efficient deep learning-based solution to the traveling salesman problem with time window (TSPTW). Our goal is to reduce the total tour length traveled by -*the agent without violating any time limitations. This will aid in decreasing the time required to supply any type of service, as well as lowering the emissions produced by automobiles, allowing our planet to recover from air pollution emissions. The proposed model is a variation of the pointer networks that has a better ability to encode the TSPTW problems. The model proposed in this paper is inspired from our previous work that introduces a hybrid context encoder and a multi attention decoder. The hybrid encoder primarily comprises the transformer encoder and the graph encoder; these encoders encode the feature vector before passing it to the attention decoder layer. The decoder consists of transformer context and graph context as well. The output attentions from the two decoders are aggregated and used to select the following step in the trip. To the best of our knowledge, our network is the first neural model that will be able to solve medium-size TSPTW problems. Moreover, we conducted sensitivity analysis to explore how the model performance changes as the time window width in the training and testing data changes. The experimental work shows that our proposed model outperforms the state-of-the-art model for TSPTW of sizes 20, 50 and 100 nodes/cities. We expect that our model will become state-of-the-art methodology for solving the TSPTW problems.

Scheduling involves planning and arranging jobs across a coordinated set of events to satisfy the customer’s demands. In this article, we present a simple approach for the flow-shop (FS) scheduling problem under fuzzy environment in which processing time of jobs are represented by pentagonal fuzzy numbers. This study is intended to reduce the rental cost of the machine in compliance with the rental policy. The fuzzy FS scheduling problem is solved without converting the processing time into its equivalent crisp numbers using a robust ranking technique and a fuzzy arithmetic pentagonal fuzzy numbers. A numerical illustration indicates that the approach is workable, accurate, and relevant.