Triangular Fuzzy Parameters Research Articles

A scenario-based mathematical approach to a robust project portfolio selection problem under fuzzy uncertainty

As a major concern of chief managers in each organization, project portfolio selection has a special place in their responsibilities. To assist managers in making decisions, applicable optimization models play an essential role in such processes. In this regard, this paper provides a stochastic optimization model for a project portfolio selection problem under different scenarios. Providing the novelty in the model along with making it closer to reality, the interdependency between revenue and cost of projects is considered. Due to the inherent uncertainty of parameters, the revenue and cost of each project, as well as contributed capital, follow triangular fuzzy parameters. Contrary to the previous model, the appreciation of assets is considered in the proposed model as the other novelty of the proposed model. To tackle the uncertainty of parameters, a robust possibilistic approach is used, which has been first-ever devised in such problems. Being both optimistic and pessimistic approaches available for decision-makers, a new measure is introduced to make the model inclusive. Moreover, by considering the confidence level as both parameter and decision variables, the robust possibilistic programming approach is adopted to solve the proposed model. Using the new proposed measure, the optimal average value of robust model are obtained under different confidence level. Finally, solving the optimization model, the results are provided by implementing the realization for uncertain parameters, and regarding the obtained results, discussions are made to provide some insights to the managers.

Sustainable Integrated Fuzzy Optimization for Multimodal Petroleum Supply Chain Design with Pipeline System: The Case Study of Vietnam

Over the years, oil-related energy sources have played an irreplaceable role in both developed and developing countries. Therefore, the efficiency of petroleum supply chains is a key factor that significantly affects the economy. This research aimed to optimize the configuration of the uncertainty multimodal petroleum supply chain in terms of economy, energy and environment (3E assessment). This study proposes a novel integration methodology between a heuristic algorithm and exact solution optimization. In the first stage, this study determines the facilities’ potential geographical coordinates using heuristic algorithm. Then, the fuzzy min-max goal programming model (FMMGPM) was developed to find the multi-objective solutions. In particular, this model allows analysis of supply chain uncertainty through simultaneous factors such as demand, resource, cost and price. These uncertainty factors are expressed as triangular fuzzy parameters that can be analyzed in terms of both probability and magnitude. Moreover, the model is applied to the entire petroleum supply chain in Vietnam, including downstream and upstream activities. In addition, another novelty is that for the first time, pipeline systems in logistics activities are considered in Vietnam’s petroleum supply chain optimization study. The results also show the short-term and long-term benefits of developing a pipeline system for oil transportation in Vietnam’s petroleum supply chain. To evaluate the effects of uncertainty on design decisions, this study also performed a sensitivity analysis with scenarios constructed based on different magnitudes and probabilities of uncertainty.

Fuzzy linear programming problem with fuzzy decision variables: A geometrical approach

Decision making problems force the decision maker to consider fuzzy decision variables in a fuzzy linear programming problem. Therefore, the proposed fuzzy linear programming problem considers a triangular fuzzy decision variable with triangular fuzzy parameters. A defuzzyfication method based on incenter of a triangle, which is formed by joining the vertices of the triangular fuzzy parameters and fuzzy decision variables. The coordinate of the incenter is obtained by using the concept of geometry. In next step, the incenter is considered as a vector quantity. Using the concept of obtaining the magnitude of a vector, the final mathematical model is formulated. The final mathematical programming problem is a crisp non-linear programming problem. The resultant crisp mathematical programming problem is solved by appropriate mathematical software. A numerical example is presented to illustrate the methodology.

Three stages algorithm for finding optimal solution of balanced triangular fuzzy transportation problems

In the literature, the fuzzy optimal solution of balanced triangular fuzzy transportation problem is negative fuzzy number. This is contrary to the constraints that must be non-negative. Therefore, the three stages algorithm is proposed to overcome this problem. The proposed algorithm consist of segregated method with segregating triangular fuzzy parameters into three crisp parameters. This method avoids the ranking technique. Next, total difference method is used to get initial basic feasible solution (IBFS) value based on segregating triangular fuzzy parameters. While, modified distribution algorithm is used to determine optimal solution based on IBFS velue. In order to illustrate the proposed algorithm is given the numerical example and based on the result comparison, the proposed algorithm equality to the two existing algorithms and better then the one existing algorithm. The proposed algorithm can solve in the fuzzy decision-making problems and can also be extended to an unbalanced fuzzy transportation problem.

Fixed charge 4D-TP for a breakable item under hybrid random type-2 uncertain environments

The concept of hybrid random type-2 uncertainty is yet to be used in decision-making problems. Here, under this category, random type-2 triangular fuzzy, random type-2 Gaussian fuzzy and random type-2 zigzag uncertain variables are presented. For illustration, a fixed charge four dimension transportation problem (FC4D-TP) for a breakable item is formulated with its selling prices, purchasing costs, transportation costs, fixed charges, availability, demands and conveyances’ capacities as hybrid random type-2 uncertain parameters. The model is de-randomized using probability chance constraint. The reduced type-2 triangular and Gaussian fuzzy models are de-fuzzified by CV based reduction technique and the type-2 zigzag uncertain model is transformed to deterministic one by expected value-based reduction method. In addition to these, the model with type-2 triangular fuzzy parameters is also de-fuzzified by two other methods such as CV-centroid based defuzzification and nearest interval approximation methods. All these reduced deterministic problems along with some particular models are solved by Generalized Reduced Gradient method using LINGO 18.0 and numerically illustrated. Some sensitivity analyses are presented. Expressions of a previous model are deduced. Importance of FC4D-TP formulation is pointed out. Effects of hybrid uncertainties are also discussed. The proposed nature of parameters’ uncertainty is illustrated.

An effective solution approach for multi-objective fractional fixed charge problem with fuzzy parameters

A multi-objective fixed charge problem in existence of several fractional objective functions with triangular fuzzy parameters is considered in this study. The problem previously has been tackled only by Upmanyu and Saxena (2016) with a method containing wrong mathematical concepts (see the commentary of Kaur and Kumar (2017)). To overcome the shortcomings of the literature, an effective solution approach based on a typical goal programming approach is proposed to solve the problem for obtaining a Pareto-optimal solution. The proposed approach considers the shortcomings of the method of Upmanyu and Saxena (2016) and applies no ranking function of fuzzy numbers. In addition, the goal programming stage considers no preference from decision maker. The computational experiments provided by an example of the literature, prove the effectiveness of the proposed approach.

A possibilistic closed-loop supply chain: pricing, advertising and remanufacturing optimization

This paper addresses pricing and reverse channel selection decisions in a closed-loop supply chain (CLSC) under fuzziness of demand function’s parameters. Despite numerous studies in which the demand is sensitive to selling price, in this paper demand function is considered as a function of both selling price and advertising level. Decisions are made in a CLSC consisting of a manufacturer, a retailer and a third party under centralized and decentralized decision making structures. In the decentralized structure, three different models are examined which differ on the member who collects used products from customers. Collection process is conducted by the manufacturer, the retailer or the third party. The problem is formulated as a Stackelberg game model in which the manufacturer acts as a leader. Moreover, various collection structures are deeply studied through a numerical analysis in which a real case study is provided and useful managerial insights are presented based on numerical results. The results show that the centralized structure outperforms the decentralized one in achieving the highest total expected profit, attaining highest demand by setting lowest selling price, and also by considering the environmental viewpoint and resource usage (achieving highest collection rate). Finally, sensitivity analysis on triangular fuzzy parameters is conducted to examine impact of triangular fuzzy parameters on model’s outputs.

An interval-valued triangular fuzzy modified multi-attribute preference model for prioritization of groundwater resources management

This study aims at developing an inexact interval-valued triangular fuzzy based multi-attribute preference model (IVTF-MAPM) method for supporting the selection of remediation strategies of groundwater remediation. The introduction of interval-valued triangular fuzzy parameters into the attributes makes it possible for dealing with multiple uncertainties existing in many real-world cases, considering more possible values and expressing the decision making information more precisely. A attribute system which consists of fifteen remediation alternatives with respect to ten attributes in each period were taken into consideration assessment of the technologies for groundwater remediation. The pairwise comparisons between the alternatives were expressed by the valued preference model and the weights of the attributes were scored by interval Analytic Hierarchy Process. After the modeling formulation, a chlorhydrocarbons-contaminated groundwater management system was provided to demonstrate the applicability of the developed method. Final results showed that A14, A3, A9 and A1 were placed at the top ranking levels for 5-, 10-, 15- and 20-years of remediation, respectively. Compared to the results conducted form FMAPM and MAPM, it is indicated that the significant impact of multiple uncertainties existing in the system is specified.

Pricing European Options with Triangular Fuzzy Parameters: Assessing Alternative Triangular Approximations in the Spanish Stock Option Market

The aim of this paper is contributing from a practical and empirical perspective to option pricing under fuzziness. When we evaluate Black–Scholes option pricing formula with triangular fuzzy numbers, we obtain a non-triangular price that may be slightly difficult to use in practical applications. We improve the applicability of the fuzzy version of that formula by proposing and testing three triangular approximations when the subjacent asset price, its volatility and free interest rate are triangular fuzzy numbers. To check the goodness of these approximations, we firstly evaluate their closeness to the actual values of fuzzy Black and Scholes model. We find that the quality of those approximations depends on options maturity and moneyness grade and if we are pricing call or put options. We also assess the capability of those approximating methods to reflect satisfactorily real market prices and obtain good results. To develop all empirical applications, we use a sample of options on IBEX35 traded in the Spanish derivatives market on 3/1/2017.

Optimizing a Just-In-Time logistics network problem under fuzzy supply and demand: two parameter-tuned metaheuristics algorithms

Just-In-Time (JIT) is a popular philosophy in many industrial practices. The concept of JIT in early studies concerned with improving operational efficiency and waste minimization. In recent decades, however, JIT principles have also connected to logistics efficiency particularly for distribution of raw materials and finished goods. In the literature, several attempts have been made to optimize JIT logistics networks. On the one hand, most studies have typically focused on deterministic and small-scale problems which have been solved by exact algorithms. On the other hand, when large-scale problems were considered and usually were solved by metaheuristics algorithms, uncertainty sources and fine-tuning of the metaheuristics parameters were generally ignored. In this paper, we develop a mixed-integer linear optimization model to investigate a large-scale JIT logistics problem with 15 different sizes. To deal with different uncertainty sources, the customers demand and suppliers’ capacity as the two main sources of uncertainty in practice are considered as triangular fuzzy parameters. The proposed model aims to minimize total logistics cost including costs of transportation, inventory holding and backorders. A particle swarm optimization algorithm is applied to solve the problem, and its results are then validated by a harmony search algorithm. Both algorithms parameters are tuned using response surface methodology and Taguchi method. Finally, the conclusion and some directions for future research are proposed.

Potential Health Risk of Arsenic in Groundwater near Tongyu County, Western of Jilin Province: A Case Study for Health Risk Assessment Based on Triangular Fuzzy Number

Based on the fuzzy uncertainty of assessment system, the fuzzy mathematics theory was applied to denote the groundwater health risk. On the basis of defining the parameters as triangular fuzzy numbers, uncertain models with triangular fuzzy parameters were established. As a case study, the fuzzy model was applied to evaluate the health risk of arsenic in groundwater in Tongyu County, Jilin Province. According to these calculations, interval values of health risk of arsenic under different confidence levels were obtained. The calculated results indicated that interval values of health risk of arsenic in phreatic water, confined water in Quaternary and Tertiary were equal to 2.43×10-5～105.74×10-5a-1, 3.45×10-5～98.444×10-5a-1 and 2.01×10-5～117.41×10-5a-1, and the most probable values were 23.72×10-5a-1, 63.53×10-5a-1 and 34.39×10-5a-1respectively. The assessment results showed groundwater in study area had great health risks and the fuzzy models more scientifically reveal the health risks state of hazardous substances in groundwater than the traditional ones.

Simulating continuous fuzzy systems for fuzzy solution surfaces

Our approach to simulation of a fuzzy ODE system is to evaluate the system with triangular fuzzy parameters, evaluating at the left/vertex/right supports and values in between. Solutions are presented as a graph(s) of the variable(s) of interest, with respect to time. With multiple fuzzy parameters, a solution graph is likely to become overloaded with superfluous information. Because of the cost of processing and plotting the superfluous information, critical support values may be skipped. We implement a reduction algorithm for determining solution boundaries, as trajectories are computed. Keeping aware of membership values, quantization considerations, and solution neighborhoods, we are able to collect a fraction of the data collected by brute force methods, and yet provide better coverage of fuzzy parameters. Three dimensional fuzzy solution trajectories are the second improvement we investigate. By including membership computation in our simulations, we produce fuzzy solution surface boundaries. These surfaces enclose the possible solution space. To make this process manageable, we combine this with boundary determination. In this paper we describe our method and present 3D solutions of two classical systems of ODE.