Sustainable food supplier selection (SFSS) can be handled as an uncertain decision-making issue. The Pythagorean fuzzy set (PFS), a type of non-standard fuzzy set, offers an expanded description space for articulating fuzzy and uncertain data. Accordingly, this paper proposes a Pythagorean fuzzy synthetic decision method-based selection framework for solving the SFSS problem within a subjective context. Then, the weighted distance measures for the PFS are introduced to derive the importance degrees of the experts, which can provide a more objective decision result. Then, an information fusion method with a PFS-weighted power average (WPA) operator is introduced to form a group decision matrix competent to accommodate the deviation effect. Next, an extended PF-measurement of alternatives and ranking according to compromise solution (MARCOS) method integrating PF-criteria importance through inter-criteria correlation (CRITIC) is presented to calculate the priority of each supplier, which can capture the inter-correlations between criteria. Finally, a numerical example of SFSS is implemented to show the application of the proposed synthetic decision approach. Subsequently, the sensitivity analysis of distance parameters and comparison analysis among different SFSS approaches were conducted to test the rationality and advantages of the proposed framework for resolving the SFSS problem. The results show that the reported method can provide a practical way to resolve the SFSS problems with uncertain data.

The purpose of this study is to offer requisite models for unravelling the complex issues that arise in a supplier selection-order allocation problem by soothing the risk and disturbances. To do this, a new uncertain environment, interval type-2 Pythagorean fuzzy set (IT2PFS) is introduced to assist the experts for assuring secure and reliable outcomes in hesitant situations. The operational laws on IT2PFS under Dombi t-norm and t-conorm are defined, which are further utilized to develop geometric Bonferroni mean and Bonferroni mean operators based on Choquet integral (CI) under IT2PFS. The aggregation operators can ennoble the pliability of the information blending process through the adjustment of several parameters and can make good interactions among the criteria with their weights. Thereafter, to determine the weights of the criteria, an empirical method, namely the decision making trial and evaluation laboratory (DEMATEL) is amplified by conjoining CI and IT2PFS into it. A multicriteria decision making method, namely, the multiattribute border approximation area comparison (MABAC) and subsequently the CI based grey relational analysis method are then employed to classify the suppliers and derive their corresponding weights with reference to the sustainable criteria. Over and above, a new multiobjective optimization model is exhibited to uphold the purchasing managers for assembling suppliers keeping in mind the sustainability aspects and overall risk of suppliers by following Markowitz portfolio theory. A real-life supply chain management problem is demonstrated to elucidate the aptness and efficiency of the proposed work. Furthermore, the effectiveness and importance of the study are validated via the comparative analysis with existing approaches. The main contribution of the study is in handling the difficulty and confusion that arise during information gathering, information aggregation, suppliers evaluation and order allocation phases.

This paper describes a study on bi-level multi-objective transportation problem in fuzzy environment. To alleviate a difficulty of transporting goods within nodes for long distances, we venture to formulate a transportation problem using bi-level criteria. Again, in this paper, we incorporate the real-life uncertain situation in bi-level transportation system through fuzziness. It produces a new class of transportation problem, namely fuzzy bi-level multi-objective transportation problem (FBMOTP). We solve FBMOTP with and without priority of the 1st level of transportation problem (TP) using the Vogal Approximation Method (VAM) which are called respectively Approach 1 and Approach 2. A special emphasis of this paper is to introduce a new approach namely Approach 3 for solving the FBMOTP in addition to the considered two approaches. In this regard, we incorporate an algorithm for Approach 3 to find compromise solution of the FBMOTP using VAM. Thereafter, the proposed algorithm is illustrated to show the usefulness by taking a numerical example. A comparative analysis with the existing studies is provided to understand the effectiveness of the proposed methodology. At last, conclusions are described with the lines of future study of the paper.

We are observing more often extreme climate incidents because of global warming. There is a dire requirement for governments, enterprises, the overall population, and academics to take facilitated activities so as to handle the difficulties forced by environmental change. The most important strategic issue is to design an effective and environmentally concerned logistics system as transportation is one of the fundamental reasons for carbon emanations. Having this goal and mentioned highly important contributions in the field, this article introduces an unprecedented integrated mathematical model for a green solid transportation system with dwell time to execute the carbon tax, cap, and offset regulation. Due to market fluctuations, the supply and demand parameters are not always of crisp nature. Hence, a twofold (type-2 intuitionistic) uncertainty is incorporated in this article to provide a realistic transportation system. A new ranking defuzzification technique is presented for conversion into a deterministic form. After that, a fuzzy technique and a nonfuzzy technique are used to get the Pareto-optimal solution of the proposed problem. The performances of our findings are discussed with industrial-based application examples. Moreover, a comparative study is explored among the other relevant existing techniques. Managerial insights, conclusions, and avenues of future scopes are offered at the end of this article.

This paper considers a multi-objective fixed-charge transportation problem (MOFCTP) in which the parameters of the objective functions are random rough variables, while the supply and the demand parameters are rough variables. In real-life situations, the parameters of a multi-objective fixed-charge transportation problem may not be defined precisely, because of globalization of the market, uncontrollable factors, etc. As such, the multi-objective fixed-charge transportation problem is proposed under rough and random rough environments. To tackle uncertain (rough and random rough) parameters, the proposed model employs an expected value operator. Furthermore, a procedure is developed for converting the uncertain multi-objective fixed-charge transportation problem into a deterministic form and then solving the deterministic model. Three different methods, namely, the fuzzy programming, global criterion, and ϵ-constrained methods, are used to derive the optimal compromise solutions of the suggested model. To provide the preferable optimal solution of the formulated problem, a comparison is drawn among the optimal solutions that are extracted from different methods. Herein, the ϵ-constrained method derives a set of optimal solutions and generates an exact Paretofront. Finally, in order to show the applicability and feasibility of the proposed model, the paper includes a real-life example of a multi-objective fixed-charge transportation problem. The main contribution of the paper is that it deals with MOFCTP using two types of uncertainties, thus making the decision making process more flexible.

In this paper, we propose a new mathematical model to analyze bimatrix goal game under entropy environment. This new approach incorporates the entropy function of the players as an objective to the bimatrix game. The solution concepts behind this game are based on getting the probability to achieve some specified goals. We consider the solution, not only by the payoff values, but also by the probability to get them. Numerical examples are presented to illustrate the methodology in the paper.