Equivalent Crisp Model Research Articles

Stability analysis of fuzzy differential equation for prey–predator system underZ-number fuzzy and granular differentiability concept

The stability analysis is one of the basic problems in the field of system, prey–predator, production inventory control, signal processing etc. The goal of stability analysis for prey–predator and production inventory control model is to determine the region of the different variables and parameters at which the system is stable or unstable. This paper explores an uncertain prey–predator model where both prey and predator populations are fuzzy variables. Also we have considered all the prey–predator parameters of the system are imprecise. This fuzzy prey–predator model is converted to an equivalent crisp model using granular differentiability approach. The condition of local stability is obtained mathematically by analysis the eigenvalues of characteristic equation. And also we introduced a production inventory model as a case study of fuzzy dynamical system to illustrate the efficiency of the result and practical aspects.

An uncertain support vector machine based on soft margin method.

Traditional support vector machines (SVMs) play an important role in the classification of precise data. However, due to various reasons, available data are sometimes imprecise. In this paper, uncertain variables are adopted to describe the imprecise data, and an uncertain support vector machine (USVM) is built for linearly alpha-nonseparable sets based on soft margin method, where a penalty coefficient is utilized as the trade-off between the maximum margin and the sum of slack variables. Then the equivalent crisp model is derived based on the inverse uncertainty distributions. Numerical experiments are designed to illustrate the application of the soft margin USVM. Finally, metrics, such as accuracy, precision, and recall are used to evaluate the robustness of the proposed model.

A Bi-Objective Model for Scheduling Construction Projects Using Critical Chain Method and Interval-Valued Fuzzy Sets

Numerous constraints affect construction projects, and lack of management may lead to schedule deviation. In the execution phase of the project, due to the lack of timely access to the required resources and the existence of uncertainty, the project activities do not progress following the schedule, and as a result, schedule deviation occurs. The scheduling addresses resource constraints by the critical chain method and deals with delays in activities by placing buffers that have emerged as a method for scheduling construction projects. This paper presents a new bi-objective mathematical model which aims to reduce delay and increase quality, based on the critical chain method and resource constraint for scheduling construction projects. In the proposed model, the activities have been considered multi-mode ones. Moreover, this paper has assumed each activity to be executed in a normal or crashing way. Due to the uncertainty in real-world problems, the duration of the activity is expressed using triangular interval-valued fuzzy numbers. A new interval-valued fuzzy solution process is presented in this paper using a two-step approach. First, the equivalent crisp model is given; then in the second step, a goal programming approach is applied for transforming the bi-objective model into the single-objective one. Finally, the mathematical model is implemented on a case study adapted from the literature, and sensitivity analysis of the results is conducted.

A multi-objective integrated procurement, production, and distribution problem of supply chain network under fuzziness uncertainties

In this paper, we devoted a design under uncertainty of a four-echelon supply chain network including multiple suppliers, multiple plants, multiple distributors and multiple customers. The proposed model is a bi-objective mixed integer linear programming which considers several constraints and aims to minimize the total costs including the procurement, production, storage and distribution costs as well as to maximize on-time deliveries (OTD). To bring the model closer to real-world planning problems, the objective function coefficients (e.g. procurement cost, production cost, inventory holding and transport costs) and other parameters (e.g., demand, production capacity and safety stock level), are all considered triangular fuzzy numbers. Besides, a hybrid mathematical model-based on credibility approach is constructed for the problem, i.e., expected value and chance constrained models. Moreover, to build the crisp equivalent model, we use different property of the credibility measure. The resulted crisp equivalent model is a bi-objective mixed integer linear programs (BOMILP). To transform this crisp BOMILP into a single objective mixed integer linear programs (MILP) model, we apply three different aggregation functions. Finally, numerical results are reported for a real case study to demonstrate the efficiency and applicability of the proposed model.

The Production Routing Problem Under Uncertain Environment

To reduce waste and cost as well as to promote the efficient and sustainable use of resources in supply chains, a common focus of practice and research is to integrate decisions of various operational processes such as production, inventory, distribution, and routing. The question of which is more representative of the production routing problem (PRP) then arises. In this paper, the focus is on uncertainty in the PRP, and demand and cost uncertainty are considered simultaneously. According to different decision criteria, three uncertain programming models are correspondingly formulated. Then, through researching the crisp equivalence and conversion of the proposed uncertain models, a series of crisp equivalent models are proposed under the assumptions of particular uncertainty distributions, such as linear, zigzag, and normal distributions. To verify the accuracy and usefulness of the proposed models put forward in this paper, a series of experiments are conducted. Finally, several interesting managerial aspects with respect to the relationship between the confidence level and variance of uncertain variables that are gained from the numerical experiments are highlighted. First, the overall cost of PRP grows as the confidence level increases under uncertain environment. Second, the probability that the optimal total cost of the PRP is less than or equal to a given threshold strictly increases as the threshold increases under uncertain environment. Third, in circumstances such as higher confidence level that decision makers generally pay more attention to, the growth of the variance of uncertain variables may lead to the increase of the total cost.

A novel Approach for Fully Intuitionistic Fuzzy Multi-Objective Fractional Transportation Problem

This paper exhibits a novel approach for solving fully intuitionistic fuzzy multi-objective fractional transportation problem (FIF-MOFTP). Because of the change of market policies, we expect that the transportation cost, the shipped quantity, the source, and the destination parameters are not generally exact. In this paper, a theorem which demonstrate that FIF-MOFTP is ever solvable was presented. In our approach, the problem is converted into a linear one based on some transformations. Then the linearized model is reduced to a crisp multi-objective transportation problem utilizing the accuracy function for each objective. Moreover, various theorems that set up the relation among the FIF-MOFTP and its equivalent crisp model using linear, hyperbolic, and parabolic membership functions are also proofed. To approve the proposed approach, a numerical example is incorporated. Deductions and future exploration of this paper are depicted at end.

The stochastic multi-modal hub location problem with direct link strategy and multiple capacity levels for cargo delivery systems

This paper introduces the stochastic multi-modal hub location problem with direct link strategy and multiple capacity levels for cargo delivery systems under demand uncertainty. For capturing the uncertain nature of demand, we present a stochastic optimization model to formulate this problem formally via the expected value and the chance-constrained programming techniques. Under some mild assumptions, we propose a computationally tractable approach to turn the original model into a crisp equivalent integer second-order cone programming model, which can be efficiently solved by CPLEX for only small instances. Hence, we design a memetic algorithm incorporating genetic search and local intensification for realistic size instances. Furthermore, we provide extended analyses of the original model by considering the mode-specific hub and incorporating the fixed transportation cost. To demonstrate the superiority of the proposed model and the effectiveness of the solution approach, we conduct a series of computational experiments based on Turkish network data set.

Optimal Pricing in Recycling and Remanufacturing in Uncertain Environments

With the increasing awareness of environmental protection, firms pay much more attention to the recycling and remanufacturing of used products. This paper addresses the problem of the optimal pricing in recycling and remanufacturing in uncertain environments. We consider two strategies of remanufacturing products, by which a recycled product can be repaired and sold as a second-hand product or dissembled into materials for production of new products according to its quality. As the market demand for products and the quantities of recycled products, such as fashion products and mobile phones, usually lack historical data, this paper adopts uncertainty theory to depict uncertainty in establishing the pricing model. An uncertain programming model and a series of crisp equivalent models are proposed under the assumptions of particular uncertainty distribution. Finally, numerical experiments are performed to show how various parameters influence the results of the proposed model.

A new uncertain DEA model and application to scientific research personnel

Data envelopment analysis (DEA), which has been widely used since it was introduced by Charnes et al. (J Econom 30:91–107, 1985), is an effective method for evaluating the relative efficiency of decision-making units. DEA models require accurate input data and output data; however, if no sample is available to estimate accurate data, then uncertain DEA is introduced. This paper reports on several new studies on uncertain DEA using the Hurwicz criterion, which attempts to find the intermediate area between extremes. Some uncertain DEA models, as well as their crisp equivalent models, are presented. Then, the Hurwicz ranking method is proposed based on these models, which can give an evaluation to all the decision-making units. By varying the parameter $$\beta $$ in the Hurwicz criterion, which reflects the optimism of the decision maker, the new ranking method can exhibit various forms. Finally, an application to scientific personnel is provided to prove the advantage of the proposed method.

A transportation planning problem with transfer costs in uncertain environment

As a generalization of existing uncertain transportation models, this paper proposes a new uncertain transportation model with transfer costs, of which the demands and the transportation costs as well as the transfer costs are uncertain variables. The model is presented in a form with expected-value objective and chance constraints. Based on the operational laws of uncertain variables, the presented model is transformed into an equivalent crisp model. After that, a numerical experiment is performed to illustrate the application of the model.

Production planning in industrial townships modeled as hub location–allocation problems considering congestion in manufacturing plants

In this paper, we aim to develop optimal production plans in industrial townships modeled as hub location–allocation problems (HLAP) taking congestion into account. In the proposed model, hub nodes are considered as industrial townships where manufacturing plants and a central distribution warehouse are located, and two objectives are targeted. The first is to minimize the total costs, which includes the cost of hub deployment, factories and warehouses, transportation, and so forth. The second is to minimize the total elapsed time of products in manufacturing plants and warehouses modeled as queues. Due to the ambiguity in estimating the model’s parameters, they are considered as fuzzy parameters to make model closer to reality. The fuzzy model is then converted into an equivalent crisp model by combining the expected value (EV) and the fuzzy chance constrained programming (FCCP) approaches. Subsequently, the bi-objective crisp model is converted into a single aggregated objective model. In order to validate the proposed model, six numerical examples are solved, and the sensitivity of the proposed model with regard to changes in model’s parameters is investigated.

An uncertain parallel machine problem with deterioration and learning effect

An uncertain uniform parallel machine scheduling problem with job deterioration and a learning effect is considered. Job processing times, due dates, deterioration rates and learning rates are assumed to be uncertain variables. The objective functions are the total weight earliness, tardiness and makespan. Three mathematical programming models are presented, i.e., expected value model, pessimistic value model and measure chance model. These models can be converted into equivalent crisp models by the inverse uncertainty distribution method. A hybrid algorithm mixed with dispatching rules based on structural features is employed to solve the problem. Finally, computational experiments are presented to illustrate the effectiveness of proposed algorithms.

Rough‐fuzzy quadratic minimum spanning tree problem

AbstractA quadratic minimum spanning tree problem determines a minimum spanning tree of a network whose edges are associated with linear and quadratic weights. Linear weights represent the edge costs whereas the quadratic weights are the interaction costs between a pair of edges of the graph. In this study, a bi‐objective rough‐fuzzy quadratic minimum spanning tree problem has been proposed for a connected graph, where the linear and the quadratic weights are represented as rough‐fuzzy variables. The proposed model is formulated by using rough‐fuzzy chance‐constrained programming technique. Subsequently, three related theorems are also proposed for the crisp transformation of the proposed model. The crisp equivalent models are solved with a classical multi‐objective solution technique, the epsilon‐constraint method and two multi‐objective evolutionary algorithms: (a) nondominated sorting genetic algorithm II (NSGA‐II) and (b) multi‐objective cross‐generational elitist selection, heterogeneous recombination, and cataclysmic mutation (MOCHC) algorithm. A numerical example is provided to illustrate the proposed model when solved with different methodologies. A sensitivity analysis of the example is also performed at different confidence levels. The performance of NSGA‐II and MOCHC are analysed on five randomly generated instances of the proposed model. Finally, a numerical illustration of an application of the proposed model is also presented in this study.

Uncertainty based genetic algorithm with varying population for random fuzzy maximum flow problem

AbstractThis paper investigates the uncertain maximum flow of a network whose capacities are random fuzzy variables. We have developed the expected value model (EVM) and the chance‐constrained model (CCM) for maximum flow problem (MFP) under random fuzzy environment and formulated their crisp equivalent models. To solve these models, we have proposed a varying population genetic algorithm with indeterminate crossover (VPGAwIC). In VPGAwIC, selection of a chromosome depends on its lifetime. An improved lifetime allocation strategy (iLAS) has also been proposed to determine the lifetime of the chromosome. The ages of the chromosomes are defined linguistically as Young, Middle, and Old, which follow some uncertainty distributions. The crossover probability is indeterminate, and it depends on the ages of the parents, which is defined by an uncertain rule base. The number of offspring, generated from a population of parents, is determined by the reproduction ratio. The population is updated in 2 ways: (i) All the chromosomes with ages greater than their lifetimes are discarded from the population, and (ii) the offspring are combined with their parents for the next generation. The proposed VPGAwIC is compared with the genetic algorithm developed by Gen, Cheng, and Lin (2008) for maximum flow problem. Wilcoxon signed‐rank test has been performed to show the superiority of the proposed VPGAwIC.

Tripartite equilibrium strategy for a carbon tax setting problem in air passenger transport.

Carbon emissions in air passenger transport have become increasing serious with the rapidly development of aviation industry. Combined with a tripartite equilibrium strategy, this paper proposes a multi-level multi-objective model for an air passenger transport carbon tax setting problem (CTSP) among an international organization, an airline and passengers with the fuzzy uncertainty. The proposed model is simplified to an equivalent crisp model by a weighted sum procedure and a Karush-Kuhn-Tucker (KKT) transformation method. To solve the equivalent crisp model, a fuzzy logic controlled genetic algorithm with entropy-Bolitzmann selection (FLC-GA with EBS) is designed as an integrated solution method. Then, a numerical example is provided to demonstrate the practicality and efficiency of the optimization method. Results show that the cap tax mechanism is an important part of air passenger trans'port carbon emission mitigation and thus, it should be effectively applied to air passenger transport. These results also indicate that the proposed method can provide efficient ways of mitigating carbon emissions for air passenger transport, and therefore assist decision makers in formulating relevant strategies under multiple scenarios.

Linear programming problems with some multi-choice fuzzy parameters

In this paper, we consider some Multi-choice linear programming (MCLP) problems where the alternative values of the multi-choice parameters are fuzzy numbers. There are some real-life situations where we need to choose a value for a parameter from a set of different choices to optimize our objective, and those values of the parameters can be imprecise or fuzzy. We formulate these situations as a mathematical model by using some fuzzy numbers for the alternatives. A defuzzification method based on incentre point of a triangle has been used to find the defuzzified values of the fuzzy numbers. We determine an equivalent crisp multi-choice linear programming model. To tackle the multi-choice parameters, we use Lagranges interpolating polynomials. Then, we establish a transformed mixed integer nonlinear programming problem. By solving the transformed non-linear programming model, we obtain the optimal solution for the original problem. Finally, two numerical examples are presented to demonstrate the proposed model and methodology.

Fish and broiler optimal harvesting models in imprecise environment

In this paper, a two-species harvesting model has been considered and developed a solution procedure which is able to calculate the equilibrium points of the model where some biological parameters of the model are interval numbers. A parametric mathematical program is formulated to find the biological equilibrium of the model for different values of parameters. This interval-valued problem is converted into an equivalent crisp model using interval mathematics. The main advantage of the proposed procedure is that different characteristics of the model can be presented in a single framework. Analytically, the existence of steady state and stabilities are looked into. Using mathematical software, the model is illustrated and the results are obtained and presented in tabular and graphical forms.

Some new ranking criteria in data envelopment analysis under uncertain environment

Data Envelopment Analysis (DEA) is a very effective method to evaluate the relative efficiency of decision-making units (DMUs), which has been applied extensively to education, hospital, finance, etc. However, in real-world situations, the data of production processes cannot be precisely measured in some cases, which leads to the research of DEA in uncertain environments. This paper will give some researches to uncertain DEA based on uncertainty theory. Due to the uncertain inputs and outputs, we will give three uncertain DEA models, as well as three types of fully ranking criteria. For each uncertain DEA model, its crisp equivalent model is presented to simplify the computation of uncertain models. Finally, a numerical example is presented to illustrate the three ranking criteria.

A Fuzzy Order Promising Model With Non-Uniform Finished Goods

Traditionally, the homogeneity of available units of the same finished good (FG $$_i$$ ) to be promised to customers has been assumed. However, contexts with lack of homogeneity in the product (LHP) are characterised by units of the same FG $$_i$$ , which differ in some characteristics that are relevant for customers and give rise to different subtypes. For instance, in the ceramic industry, tiles are classified based on quality, tone and gage, because of functional and aesthetical reasons related to their joint installation. LHP imposes new constraints in the order promising process because customers need homogeneous units. However, the final amount of the homogeneous units in planned lots is uncertain when promising orders, because they will only be known once produced and classified. In this sense, we introduce homogeneity constraints including fuzzy sets; specifically, we address the interaction among fuzzy homogeneity coefficients that represent the fraction of each homogeneous sublot. Therefore, modelling uncertainty in interdependent technological coefficients in a dynamic context is one of the main novelties of our proposal. Thus, in this paper, in order to reliably meet the homogeneity required by customers, a fuzzy model is proposed to support the promising process in LHP contexts after taking into account uncertainty in planned homogeneous sublots. The fuzzy model is translated into an alpha-parametric equivalent crisp model. Here, it is important to highlight another important novelty of the paper related to the proposed methodology to analyse the suitability of the minimum degree of possibility (the $$\alpha$$ -cut), by an adapted TOPSIS-based fuzzy procedure. Finally, the experimental design, which is inspired in the ceramic sector, proves both the validity of the model and a better performance of the fuzzy model compared to the deterministic one, in several executions with forecasts of the real size of homogeneous sublots.

Pricing and Design of After-Sales Service Contract: The Value of Mining Asymmetric Sales Cost Information

In this paper, we study a pricing and after-sales service contract design problem, where a retailer purchases products from a manufacturer and then sells to the consumers. The sales cost is the retailer’s private information and might be mined by the manufacturer via advanced learning algorithms and related big data techniques. We first develop a crisp equivalent model, based on which the optimal contracts and the supply chain parties’ profits under asymmetric information are derived. We show that, compared with the optimal wholesale price and after-sales service level with symmetric information, asymmetric cost information makes the wholesale price distorted upward when the retailer’s sales cost is low. When the retailer’s cost is high, the after-sales service level is distorted downward. We characterize the manufacturer’s loss and the retailer’s gains due to asymmetric sales cost information. This helps the manufacturer make the investment decision of big data techniques. Interestingly, we find that the retailer might voluntary disclose the sales cost information, which results in a win-win situation for the manufacturer and the retailer. This makes the manufacturer less favor big data techniques. Finally, we conduct extensive sensitivity analysis with respect to the retailer’s sales cost, the consumer’s sensitivity to the retailer’s after-sale service level, and the fraction of high-type retailers in the market.