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The new definition of a fuzzy vertex cover of a fuzzy graph depending on the fuzzy covering radius is introduced in this paper. A fuzzy covering problem for a fuzzy network in modification of the fuzzy covering radius as per requirement is considered and solved in an optimization sense. The covering problem is customized with the help of a set of programming problems, and they are solved by using mathematical software “LINGO”. Two relevant algorithms are designed in this paper. Also, one of the burning issues of today’s world, “Climate Action” (one of the most important sustainable developmental goals), together with various weather parameters, is taken in the application part. The fuzzy environment for this issue of India to save our nature is modeled for making some decisions to improve the situations for this purpose, and the whole methodology reflects the importance of the applicability of our proposed model.

This paper presents a general software reliability growth model (SRGM) based on non-homogeneous Poisson process (NHPP) and optimal software release policy with cost and reliability criteria. The main motive of this study is to develop a software release time decision model considering maintenance cost and warranty cost under fuzzy environment. In previous studies, maintenance cost has been defined either in terms of warranty cost or fault debugging cost. In reality, maintenance cost includes the cost of free patches, updates, technical support and future enhancement. Also, it is possible that maintenance process causes the removal of software faults in the operational phase including the faults which occur outside the warranty period or warranty definition. In other words, warranty action may be included the maintenance action, but not the converse. Considering this fact, maintenance cost and warranty cost are defined separately in the proposed study. Initially, an SRGM has been proposed with the revised concept of imperfect debugging phenomenon considering fault removal efficiency (FRE). Furthermore, the effect of changes in various environmental factors on models parameters has been taken into account. Numerical examples based on real software failure data sets have been given to analyze the performance of the proposed models.

This article proposes a novel algorithm for optimizing a nonlinear dynamic system subjected to flexible path constraints. Each flexible constraint is fuzzified by using the concept of degree-of-acceptability, and the fuzzy degree-of-satisfaction for the objective is then derived. The numerical method of Integrated Controlled Random Search for Dynamic System (ICRS/DS) is applied here for solving the resulting fuzzy decision problem. One numerical example is supplied, demonstrating the applicability of the proposed algorithm.

In both practical applications and theoretical analysis, there are many fuzzy chance-constrained optimization problems. Currently, there is short of real-time algorithms for solving such problems. Therefore, in this paper, a continuous-time neurodynamic approach is proposed for solving a class of fuzzy chance-constrained optimization problems. Firstly, an equivalent deterministic problem with inequality constraint is discussed, and then a continuous-time neurodynamic approach is proposed. Secondly, a sufficient and necessary optimality condition of the considered optimization problem is obtained. Thirdly, the boundedness, global existence and Lyapunov stability of the state solution to the proposed approach are proved. Moreover, the convergence to the optimal solution of considered problem is studied. Finally, several experiments are provided to show the performance of proposed approach.

This paper presents a method for ranking a set of decision making units according to their level of efficiency and which takes into account uncertainty in the data. Efficiency is analysed using fuzzy DEA techniques and the ranking is based on the statistical analysis of cases that include representative situations. The method enables the removal of the sometimes unrealistic hypothesis of a perfect trade-off between increased inputs and outputs. This model is compared with other DEA models that work with imprecise or fuzzy data. As an illustration, we apply our ranking method to the evaluation of a group of Spanish seaports, as well as teams playing in the Spanish football league. We compare the results with other methods and we show that our method enables a total ranking of the seaports, and that the ranking of football teams is found to be more consistent with final league positions.

This paper considers the identification of problems which generate anomalies at firms through the observed symptoms on the basis of fuzzy relations and Zadeh's compositional rule of inference. A procedure for determining the fuzzy cause vector of an economic and financial diagnosis problem is proposed, which consists of the design of fuzzy relational matrix and the resolution of a system of fuzzy relational equations. An efficient algorithm for solving fuzzy relational equations in terms of the associated set covering problem is introduced. It utilizes a back-tracking method to generate each minimal covering, where no duplicate or non-minimal coverings exist. A numerical example of firms' insolvency causes diagnosis is also included.

This work considers providing a common base for measuring the relative efficiency of a group of homogeneous decision making units in a fuzzy environment. The principle of compromise of the technique for order preference by similarity ideal solution is employed for solving the data envelopment analysis model with fuzzy objectives and constraints. An algorithm with the entropic regularization implementation for finding the compromise solution of the fuzzy data envelopment analysis model is developed. An illustrative example verifying the idea of this paper is provided. The contribution of this work is represented by the improvement of the discriminatory power of the fuzzy DEA, gained through the common weight evaluation.

A crucial challenge for global query optimization in a multidatabase system (MDBS) is that some local optimization information, such as local cost parameters, may not be accurately known at the global level because of local autonomy. Traditional query optimization techniques using a crisp cost model may not be suitable for an MDBS because precise information is required. In this paper we present a new approach that performs global query optimization using a fuzzy cost model that allows fuzzy information. We suggest methods for establishing a fuzzy cost model and introduce a fuzzy optimization criterion that can be used with a fuzzy cost model. We discuss the relationship between the fuzzy optimization approach and the traditional (crisp) optimization approach and show that the former has a better chance to find a good execution strategy for a query in an MDBS environment, but its complexity may grow exponentially compared with the complexity of the later. To reduce the complexity, we suggest to use so-called k-approximate fuzzy values to approximate all fuzzy values during fuzzy query optimization. It is proven that the improved fuzzy approach has the same order of complexity as the crisp approach.

The optimization techniques used in commercial-off-the-shelf (COTS) selection process faces challenges to deal with uncertainty in many important selection parameters, for example, cost, reliability and delivery time. In this paper, we propose a fuzzy optimization model for selecting the best COTS product among the available alternatives for each module in the development of modular software systems. The proposed model minimizes the total cost of the software system satisfying the constraints of minimum threshold on system reliability, maximum threshold on the delivery time of the software, and incompatibility among COTS products. In order to deal with uncertainty in real-world applications of COTS selection, the coefficients of the cost objective function, delivery time constraints and minimum threshold on reliability are considered fuzzy numbers. The fuzzy optimization model is converted into a pair of mathematical programming problems parameterized by possibility (feasibility) level α using Zadeh's extension principle. The solutions of the resultant problems at different α-cuts provide lower and upper bounds of the fuzzy minimum total cost which helps in constructing the membership function of the cost objective function. The solution approach provide fuzzy solutions instead of a single crisp solution thereby giving decision maker enough flexibility in maintaining cost-reliability trade-off of COTS selection besides meeting other important system requirements. A real-world case study is discussed to demonstrate the effectiveness of the proposed model in fuzzy environment.

Application Package Software (APS) has emerged as a ready-to-use solution for the software industry. The software system comprises of a number of components which can be either purchased from the vendor in the form of COTS (Commercial Off-the-Shelf) or can be built in-house. Such a decision is known as Build-or-Buy decision. Under the situations wherein the software has the responsibility of supervising life-critical systems, the inception of errors in software due to inadequate or incomplete testing, is not acceptable. Such life-critical systems enforces upon meeting the quality standards of the software as unforbiddenable. This can be achieved by incorporating a fault-tolerant design that enables a system to continue its intended operation rather than failing completely when some part of the system fails. Moreover, while designing a fault-tolerant system, it must be apprehended that 100% fault tolerance can never be achieved and the closer we try to get to 100%, the more costly the system will be. The proposed model shall incorporate consensus recovery block scheme of fault tolerant techniques. Through this paper, we shall focus on build-or-buy decision for an APS in order to facilitate optimal component selection thereby, maximizing the reliability and minimizing the overall cost and source lines of code of the entire system. Further, since the proposed problem has incompleteness and unreliability of input information such as execution time and cost, hence, the environment in the proposed model is taken as fuzzy.

In this paper, we study a decision-making problem related to software creation using commercial-off-the-shelf (COTS) products in a modular software system. The optimal selection of COTS products is difficult due to the variations in various critical parameters such as cost, reliability, execution time, and delivery time. Further, it is difficult to estimate precisely the values of these parameters since sufficient data may not be available and also there could be measurement errors. We present a fuzzy 0–1 optimization model of the multiobjective COTS products selection problem using exponential membership functions that simultaneously minimize the total cost, size, execution time and delivery time and maximize the reliability of a modular software system subject to many realistic constraints. The fuzzy goals are defined for each selection criterion as per the preferences of the decision maker and are aggregated using product operator to obtain an equivalent optimization model for optimal COTS selection. A real-world case study is discussed to demonstrate the effectiveness of the proposed model and the solution methodology.

Performance of a software is an important feature to determine the quality of the software developed. Performance testing of modular software is a time consuming and costly task. Several performance testing tools (PTTs) are available in the market which help software developers to test their software performance. In this paper, we propose an integrated multiobjective optimization model for evaluation and selection of best-fit PTT for modular software system. The total performance tool cost is minimized and the fitness evaluation score of the PTTs is maximized. The fitness evaluation of PTT is done based on various attributes by making use of the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). The model allows the software developers to select the number of PTTs as per their requirement. The individual performance of the modules is considered based on some performance properties. The reusability constraints are considered, as a PTT can be used in the same module to test different properties and/or it can be used in different modules to test same or different performance properties. A real-world case study from the domain of enterprise resource planning (ERP) is used to show the working of the suggested optimization model.

This study surveys the use of fuzzy numbers in classic optimization models, and its effects on making decisions. In a wide sense, mathematical programming is a collection of tools used in mathematical optimization to make good decisions. There are many sectors of economy that employ it. Finance and government, logistics and manufacturing, the distribution of the electrical power are worth to be first mentioned.

When real life problems are modeled mathematically, there is always a trade-off between model’s accuracy and complexity. By this survey, we aim to present in a concise form some mathematical models from the literature together with the methods to solve them. We will focus mainly on fuzzy fractional programming problems. We will also refer to but not describe in detail the multi-criteria decision-making problems involving fuzzy numbers and linear fractional programming models.

This paper presents an extended Branch-and-Bound algorithm for solving fuzzy linear bilevel programming problems. In a fuzzy bilevel programming model, the leader attempts to optimize his/her fuzzy objective with a consideration of overall satisfaction, and the follower tries to find an optimized strategy, under himself fuzzy objective, according to each of possible decisions made by the leader. This paper first proposes a new solution concept for fuzzy linear bilevel programming. It then presents a fuzzy number based extended Branch-and-bound algorithm for solving fuzzy linear bilevel programming problems.