Crisp Case Research Articles

Estimation of Reservoir Storage Capacity Using the Gould-Dincer Formula with the Aid of Possibility Theory

This paper presents a method for estimating reservoir storage capacity using the Gould–Dincer normal formula (G-DN), enhanced by the possibility theory. The G-DN equation is valuable for regional studies of reservoir reliability, particularly under climate change scenarios, using regional statistics. However, because the G-DN formula deals with measured data, it introduces a degree of uncertainty and fuzziness that traditional probability theory struggles to address. Possibility theory, an extension of fuzzy set theory, offers a suitable framework for managing this uncertainty and fuzziness. In this study, the G-DN formula is adapted to incorporate fuzzy logic, and the possibilistic nature of reservoir capacity is translated into a probabilistic framework using α-cuts from the possibility theory. These α-cuts approximate probability confidence intervals with high confidence. Applying the proposed methodology, in the present crisp case with the storage capacity D = 0.75, the value of the capacity C was found to be 1271×106 m3, and that for D = 0.5 was 634.5×106 m3. On the other hand, in the fuzzy case using the possibility theory, the value of the capacity for D = 0.75 is the internal [315,5679]×106 m3 and for D = 0.5 the value is interval [158,2839]×106 m3, with a probability of ≥95% and a risk level of α = 5% for both cases. The proposed approach could be used as a robust tool in the toolkit of engineers working on irrigation, drainage, and water resource projects, supporting informed and effective engineering decisions.

Study on uncertain impact of thermal radiation on micropolar hybrid nanofluid flow between porous channel walls with distinct nanoparticles

ABSTRACT This work considers the micropolar hybrid nanofluid flows between porous channel walls with distinct nanoparticles such as Copper (Cu) and Alumina ( A l 2 O 3 ) with fuzzy extension. The volume fraction of hybrid nanoparticles is regarded as an uncertain parameter and represented using a triangular fuzzy number varying between 0 % and 2 % to enhance the uncertainty in the current study. Thermal radiation is a kind of heat transmission and one of several strategies to improve thermal efficiency in various systems. The proposed micropolar hybrid nanofluid flow examines the thermal radiation effect that enhances the thermal efficiency of the flow between the porous channels. A novel generalized double parameter-based homotopy approach is used to study and analyse the uncertain impact of different parameters on the thermal and velocity profiles of the micropolar hybrid nanofluid flow. Finally, the results obtained are validated in a crisp case to test the accuracy of the proposed technique.

Study on MHD hybrid nanofluid flow with heat transfer in a stretchable converging/diverging channels embedded in a porous medium with uncertain volume fraction

This study considers a hybrid nanofluid flow in a non-parallel plate embedded in permeable stretchable convergent and divergent channels. It investigates the impact of the magnetic field and heat transmission on the velocity and temperature profiles of the hybrid nanofluid flow. Here, the walls of the stationary channels are considered to be allowed to stretch or shrink, and the hybrid nanofluids are Cu-Ag, with water as the base fluid. The nanoparticle volume fraction of hybrid nanofluid is considered an uncertain parameter, represented with a triangular fuzzy number ranging from 0.0 − 0.04 . A novel double parametric homotopy approach is used to study the impacts of the relevant parameters on the velocity and temperature profiles and validate the obtained result by comparing it with previously published literature in the crisp case without the porous medium. Finally, the fuzzy velocity and temperature profiles at distinct channel locations are graphically illustrated, accompanied by numerical representations of velocity and temperature bounds at various positions along the stretchable convergent/divergent channels.

SOLUTION OF FUZZY VOLTERRA INTEGRAL EQUATIONS USING COMPOSITE METHOD

With the rapid development of of fuzzy Integral equations, the problem of finding its solution becomes increasingly important for knowledge propagation and putting researchers wisdom to work. A recent development trend of fuzzy integral equations is modeling many problems in the field of science and technology. If the unknown function in the considered equation has a solution in terms of infinite series expansion, this proposed method becomes more accurate to find the exact solution. According on the parametric form of a fuzzy number, a fuzzy integral equation converts to two systems of integral equations in the crisp case, and then proposed method is achieved to obtain the exact fuzzy solutions of fuzzy Volterra integral equations. To validate the effectiveness of proposed method, some examples of considered equations are solved and the results show that it achieved the best performance.

Numerical approach on time-fractional Sawada-Kotera equation based on fuzzy extension of generalized dual parametric homotopy algorithm

This paper develops a fuzzy non-linear fifth-order time-fractional Sawada-Kotera model with a singular kernel and a non-singular Mittag-Leffler kernel. The proposed fractional differential equation is discussed with the Caputo and ABC fractional derivative under strongly generalized results and with fuzzy modelling. A novel double parametric scheme, i.e., q-homotopy analysis generalized transform approach (q-HAGTM), is considered to find the solution of the proposed model with Caputo and ABC fractional derivatives. The problem’s uniqueness and convergence analysis are investigated using Banach’s fixed point theorem. Finally, the numerical results are validated by comparing them with the available results in Caputo and ABC sense under strongly generalized derivatives in the crisp case.

Fuzzy Finite Elements Solution Describing Recession Flow in Unconfined Aquifers

In this work, a novel fuzzy FEM (Finite Elements Method) numerical solution describing the recession flow in unconfined aquifers is proposed. In general, recession flow and drainage problems can be described by the nonlinear Boussinesq equation, while the introduced hydraulic parameters (Conductivity K and Porosity S) present significant uncertainties for various reasons (e.g., spatial distribution, human errors, etc.). Considering the general lack of in situ measurements for these parameters as well as the certain spatial variability that they present in field scales, a fuzzy approach was adopted to include the problem uncertainties and cover the disadvantage of ground truth missing data. The overall problem is encountered with a new approximate fuzzy FEM numerical solution, leading to a system of crisp boundary value problems. To prove the validity and efficiency of the new fuzzy FEM, a comparative analysis between the proposed approach and other well-known and tested approximations was carried out. According to the results, the proposed FEM numerical solution agrees with Karadinumerical method for the crisp case and is in close agreement with the original analytical solution proposed by Boussinesq in 1904 with the absolute reduced error to be 4.6‰. Additionally, the possibility theory is applied, enabling the engineers and designers of irrigation, drainage, and water resources projects to gain knowledge of hydraulic properties (e.g., water level, outflow volume) and make the right decisions for rational and productive engineering studies.

STUDY ON TEMPORAL-FUZZY FRACTIONAL P-KDV EQUATION WITH NON-SINGULAR MITTAG LEFFLER KERNEL

This work discusses the solution of temporal-fuzzy fractional non-linear p-KdV equations employing a singular kernel and a non-singular Mittag Leffler kernel. A novel q-homotopy analysis approach with a generalised transform is proposed to study the fuzzy time-fractional model with two distinct fractional operators, and the behaviour of the solution is studied in both crisp and uncertain cases. Consequently, the efficiency and accuracy of the proposed method have been obtained by comparing the obtained numerical results with the available results under the assumption of crisp case for α = 1 that validate the obtained results. Finally, the efficiency of the proposed fractional orders is checked with distinct fractional operators.

Legendre wavelet method based solution of fractional order prey–predator model in type-2 fuzzy environment

Researchers have recently examined several fractional-order prey–predator population models in a crisp or type-1 fuzzy environment. But incompleteness in the membership may also be present. In this regard, type-2 fuzzy numbers are employed in this inquiry to address the system’s imprecision since they are more realistic than type-1 fuzzy numbers in light of the uncertain membership grades. To numerically estimate the solution, the Legendre wavelet method (LWM) is used in conjunction with the r2-cut of the r1-plane form of the type-2 fuzzy parameters. To test the applicability of the present method, three cases are examined. The present solutions are quantitatively compared with the RK4 solutions in the crisp case. Approximations of the type-2 fuzzy prey and predator populations are also included. In addition, two- and three-dimensional plots are depicted to examine the behaviour of prey and predator populations in crisp and type-2 fuzzy cases. To assess the LWM’s efficacy, comparison graphs of the acquired solutions and the results of the RK4 approach are also provided. Numerical validation against the RK4 method in crisp cases demonstrates strong agreement, while revealing contrasting population behaviours in first two scenarios, whereas in the third case, both populations reduce with time. Further, the behaviour of the fuzzy solution has also been shown graphically using the footprint of uncertainty (FOU).

Approaching the square of opposition in terms of the f-indexes of inclusion and contradiction

We continue our research line on the analysis of the properties of the f-indexes of inclusion and contradiction; in this paper, specifically, we show that both notions can be related by means of the, conveniently reformulated, Aristotelian square of opposition. We firstly show that the extreme cases of the f-indexes of inclusion and contradiction coincide with the vertexes of the Aristotelian square of opposition in the crisp case; then, we allocate the rest of f-indexes in the diagonals of the extreme cases and we prove that the Contradiction, Contrariety, Subcontrariety, Subalternation and Superalternation relations also hold between the f-indexes of inclusion and contradiction.

On fuzzification and optimization problems of clustering indices

Results of clustering are qualitatively evaluated by quantities called clustering indices. While many clustering indices are proposed, in B. Desgraupes [Clustering Indices (2016)], Desgraupes reviewed 27 indices, most of which are applicable only to crisp clustering and only one of which is applicable to fuzzy clustering. In the previous article [D. R. Sope and M. Fujio, On a fuzzification of clustering indices, in Proc. Fuzzy System Symposium of the Japan Intelligent Information Fuzzy Society, Vol. 35 (2002), pp. 443–448], the authors gazed on analytic 3 indices among them and modified them to fit fuzzy clustering by regarding the membership degrees as distribution functions of objects over clusters. In this paper, this method called fuzzification, is applied to all the indices of Desgraupes. This investigation also includes optimization of indices. A significant benefit of fuzzy clustering is that the membership degrees allow the optimization problems to be treated as continuous, whereas the ones in the crisp case are discrete, making it generally easier to solve. After giving a precise description of fuzzification which is briefly given in D. R. Sope and M. Fujio [On a fuzzification of clustering indices, in Proc. Fuzzy System Symposium of the Japan Intelligent Information Fuzzy Society, Vol. 35 (2002), pp. 443–448], the authors fuzzify all of 27 indices of Desgraupes and then solve the fuzzy clustering problems having 13 analytic indices among them as the objective functions by the gradient method.

Fuzzy computational study on the generalized fractional smoking model with caputo gH-type derivatives

This work considers a generalized fuzzy fractional smoking model with Caputo [Formula: see text]-types fractional derivatives upon considering the case of uncertainty quantification. The disease-free equilibrium point and stability of the equilibrium point have been discussed for the fuzzy nonlinear fractional smoking model. The analytical proofs for the existence and uniqueness of the proposed model are concerned with the help of the fixed-point theorem, Banach contraction, and Schauder theorem. A robust double parametric approach with a generalized transform is used to study the behavior of the fuzzy fractional model in an uncertain context and obtain the convergence analysis of the study in a crisp context. Finally, the obtained results of the proposed model have been validated with the Runge–Kutta method of fourth order in crisp case [Formula: see text].

Application of the Esscher Transform to Pricing Forward Contracts on Energy Markets in a Fuzzy Environment

The paper is dedicated to modeling electricity spot prices and pricing forward contracts on energy markets. The underlying dynamics of electricity spot prices is governed by a stochastic mean reverting diffusion with jumps having mixed-exponential distribution. Application of financial mathematics and stochastic methods enabled the derivation of the analytical formula for the forward contract's price in a crisp case. Since the model parameters' incertitude is considered, their fuzzy counterparts are introduced. Utilization of fuzzy arithmetic enabled deriving an analytical expression for the futures price and proposing a modified method for decision-making under uncertainty. Finally, numerical examples are analyzed to illustrate our pricing approach and the proposed financial decision-making method.

Fuzzy Inventory Model for Imperfect Items with Price Discount and Penalty Maintenance Cost

The present paper considers the fuzzy economic manufacturing model (FEMM) for an inventory model with an imperfect production process that has been studied along with rework. During the pandemic, it is evident that the products accumulated without a sale, which has increased the maintenance cost of the products. This research paper compares a special sale of products with discount and without discount prices both in the fuzzy environment and in the crisp case. New computing methods based on fuzzy logic are being utilized to enhance identification, decision making, and optimization. A triangular fuzzy number is applied in the economic production quantity to emphasize the importance of optimal manufacturing. The EPQ model’s optimal total cost is obtained in the crisp version. It is to be noted that this model is developed in the fuzzy sense by using the deterioration as a triangular fuzzy number. The applications of this model in the fields are constructing customized industrial machinery or heavy‐duty construction equipment, specific chemicals, and processed food. By using MATLAB R2021, a numerical example of the optimal solution is provided. Finally, the present research discusses how changing several parameters affects the optimum total cost.

Fuzzy topological analysis of pizza graph

<abstract><p>Fuzzy topological indices are getting attention these days due to their vast applications in daily life. In crisp case, topological indices are beneficial in chemical graph theory but as far as fuzzy graph theory is concerned, fuzzy topological indices are useful in identifying human trafficking, and multi-criteria decision-making environments. In this paper, we have computed the fuzzy topological indices such as the first and second fuzzy Zagreb indices, Randic index, and Harmonic index for the $ Pz_{n} $ pizza graph. We have found generalized results for the above-mentioned structure.</p></abstract>

Solution of generalised fuzzy fractional Kaup–Kupershmidt equation using a robust multi parametric approach and a novel transform

This work considers a fuzzy fractional model of the Kaup–Kupershmidt equation with uncertain initial conditions. The uncertain behaviour of capillary gravity and dispersive waves is studied using a robust double parametric fuzzy approach termed the q-Homotopy analysis method with a generalised novel transform. The proposed method’s accuracy and efficiency have been explained by comparing the observation of fuzziness in the solution with several prior accessible results for the crisp case using the double parametric approach. This methodology provides a viable alternative to existing methods for various fuzzy fractional models.

An inventory model with uncertain demand under preservation strategy for deteriorating items

The capacity of a firm to accomplish its goals is financially compromised by degeneration of goods. A suitable preservation strategy to reduce degradation is a vital part of the managerial decisions. This study employs preservation technologies under uncertain demand to frame a continuous review inventory model with full back-ordering and the influence of promotional efforts. Survey of existing research finds few models with synchronised optimization over this entire scenario with all factors.The best values of the preservation cost and the two fractions of the cycle period when inventory is kept against the backorder part are determined to lower the total average cost. A mathematical model is built to incorporate these elements and numerical scenarios are presented to compare three possible approaches. In both crisp and fuzzy contexts, the sensitivity of the solution and decision variables concerning various inventory characteristics is investigated. Backorder duration is inversely proportional to the presence of preservation. The coefficient of preservation has a tipping point below which accepting the impact of undamped deterioration becomes more cost-effective. The total cost at the optimal point is more elastic to a reduction in base deterioration rate and relatively inelastic to its increase. Finally, this study proves that the preservation strategy converges over deterioration for the crisp case rather than the fuzzy case. It is expected the fuzzy case can provide better results, however, the crisp case provides lower total cost than the fuzzy case though it is slightly less efficient in per unit cost.

On chromatic number and perfectness of fuzzy graph

Perfect graphs have a significant role in graph theory in terms of structure as well as application. This article embarked a comprehensive study on perfect fuzzy graphs attempted to obtain the fuzzy analog of the famous weak and strong perfect graph theorems following due classification of fuzzy graphs in terms of perfectness. The inter-relationship between chromatic numbers of a fuzzy graph and its complement graph has been studied in detail deriving important theories and results. It has been shown that unlike crisp case, for fuzzy graphs there exists one and only one perfect fuzzy graph theorem satisfying both the conditions of strong and weak perfect graph theorems. A detailed study of the perfectness of different types of fuzzy graphs has been carried out comparing the results with those obtained on crisp graphs. It has been shown through illustrative examples how the perfectness of a fuzzy graph and that of its underlying crisp graph is independent of each other. Finally, the use of perfect fuzzy graph on application grounds has been evaluated by using it in analyzing how a huge percentage of population in a city is unknowingly living under a big threat due exposure to electromagnetic radiation emitted from cell-phone towers.

Aggregation of Weak Fuzzy Norms

Aggregation is a mathematical process consisting in the fusion of a set of values into a unique one and representing them in some sense. Aggregation functions have demonstrated to be very important in many problems related to the fusion of information. This has resulted in the extended use of these functions not only to combine a family of numbers but also a family of certain mathematical structures such as metrics or norms, in the classical context, or indistinguishability operators or fuzzy metrics in the fuzzy context. In this paper, we study and characterize the functions through which we can obtain a single weak fuzzy (quasi-)norm from an arbitrary family of weak fuzzy (quasi-)norms in two different senses: when each weak fuzzy (quasi-)norm is defined on a possibly different vector space or when all of them are defined on the same vector space. We will show that, contrary to the crisp case, weak fuzzy (quasi-)norm aggregation functions are equivalent to fuzzy (quasi-)metric aggregation functions.

Numerical Study of MHD Third-Grade Fluid Flow through an Inclined Channel with Ohmic Heating under Fuzzy Environment

The uncertainties or fuzziness occurs due to insufficient knowledge, experimental error, operating conditions, and parameters that give the imprecise information. In this article, we discuss the combined effects of the gravitational and magnetic parameters for both crisp and fuzzy cases in the three basic flow problems (namely, Couette flow, Poiseuille flow, and Couette–Poiseuille flow) of a third-grade fluid over an inclined channel with heat transfer. The dimensionless governing equations with the boundary conditions are converted into coupled fuzzy differential equations (FDEs). The fuzzified forms of the governing equations along with the boundary conditions are solved by employing the numerical technique bvp4c built in MATLAB for both cases, which is very efficient and has a less computational cost. In the first case, proposed problems are analyzed in a crisp environment, while in the second case, they are discussed in a fuzzy environment with the help of α -cut approach, which controls the fuzzy uncertainty. It is observed that the fuzzy gravitational and magnetic parameters are less sensitive for a better flow and heat situation. Using triangular fuzzy numbers (TFNs), the left, right, and mid values of the velocity and temperature profile are presented due to various values of the involved parameters in tabular form. For validation, the present results are compared with existing results for some special cases, viz., crisp case, and they are found to be in good agreement.

Stochastic approach to model spot price and value forward contracts on energy markets under uncertainty

The paper deals with a model of electricity spot prices. The proposed dynamics of electricity spot prices is driven by a mean reverting diffusion with jumps having hyperexponential distribution. The analytical formula for the forward contract’s price is derived in a crisp case. Inasmuch as the model parameters are considered to be evaluated imprecisely, their fuzzy counterparts are introduced. With usage of the fuzzy arithmetic, the analytical expression for the forward contract’s price is derived. Several numerical examples highlighting attributes of the fuzzy forward electricity prices are brought out.