Crisp Model Research Articles

A study of an EOQ model with public-screened discounted items under cloudy fuzzy demand rate

This article deals with an economic order quantity inventory model of imperfect items under non-random uncertain demand. Here we consider the customers screen the imperfect items during the selling period. After a certain period of time, the imperfect items are sold at a discounted price. We split the model into three cases, assuming that the demand rate increases, decreases, and is constant in the discount period. Firstly, we solve the crisp model, and then the model is converted into a fuzzy environment. Here we consider the dense fuzzy, parabolic fuzzy, degree of fuzziness and cloudy fuzzy for a comparative study. The basic novelty of this paper is that a computer-based algorithm and flow chart have been given for the solution of the proposed model. Finally, sensitivity analysis and graphical illustration have been given to check the validity of the model.

On Solving Bi-level Multi-Objective Fully Quadratic Fractional Optimization Model with Fuzzy Demands

This paper has been designed to introduce the method for solving the Bi-level Multi-objective (BL-MO) Fully Quadratic Fractional Optimization Model through Fuzzy Goal Programming (FGP) approach by utilising non-linear programming. In Fully Quadratic Fractional Optimization Model, the objective functions are in fractional form, having quadratic functions in both numerator and denominator subject to quadratic constraints set. The motive behind this paper is to provide a solution to solve the BL-MO optimization model in which number of decision-makers (DM) exists at two levels in the hierarchy. First, the fractional functions with fuzzy demand, which are in the form of fuzzy numbers, are converted into crisp models by applying the concept of α-cuts. After that, membership functions are developed which are corresponding to each decision-maker’s objective and converted into simpler form to avoid complications due to calculations. Finally, the model is simplified by applying FGP approach, and a compromised solution to the initial model is obtained. An algorithm, flowchart and example are also given at the end to explain the study of the proposed model.

Solution of an Economic Production Quantity model using the generalized Hukuhara derivative approach

In this study, an economic production quantity (EPQ) model with deterioration is developed where the production rate is stock dependent and the demand rate is unit selling price and stock dependent. The low unit selling price and more stocks correspond high demand but more stock corresponds to slow production because of the avoidance of unnecessary stocks. First of all, we develop the production model by solving some ordinary differential equations having deterministic profit function under some specific assumptions. Later, we develop the fuzzy model by solving the fuzzy differential equations using Generalized Hukuhara derivative. In fact, the differential equation of the model has been split into two parts namely gH(L-R) and gH(R-L) on the basis of left(L) and right(R) α- cuts of fuzzy numbers for which the problem itself is transformed into multi-objective EPQ problem. A new formula of aggregation of several objective values obtained at different aspiration levels has been discussed to defuzzify the fuzzy multi-objective problems. We solve the crisp and fuzzy models using LINGO software. Numerical and graphical illustrations confirm that the model under Generalized Hukuhara derivative of (R-L) type contributes more profit which is one of the basic novelties of the proposed approach.

Biological control of infection pervasive via pest: a study of prey–predator model incorporating prey refuge under fuzzy impreciseness

ABSTRACT The prey–predator systems exist in almost all ecological systems and the refuge stage is highly concerned with such ecological systems in which the prey goes out of reach of the attacking region of the predator. In this paper, dynamical behaviour of predator and prey species with Holling type III response function is investigated. We add refuge in prey species and also additional predators are introduced into the system. Firstly, the crisp model is formulated with these effects and the boundedness, existence, conditions of local and global stability, local bifurcation and Hopf bifurcation are discussed. The crisp model is transformed into the fuzzy model to incorporate the unpredictable behaviour and using the Root Mean Square (RMS) defuzzification method, the fuzzy model is reduced to the defuzzified model in order to avoid the complexity of the analysis of the solution. Further, numerical simulations are performed with the help of MATLAB, MATCONT package and we examine solution curves and phase portraits in two different cases, in the presence and in the absence of additional predators along with the Hopf bifurcation scenarios. Finally, the sensitivities of the controlling parameters are plotted graphically and presented their significances in controlling the infection of the prey.

Rough approximation‐based inventory optimization with random prices for B2C companies during the promotion period

Most B2C e-commerce giants such as Amazon, Bestbuy, Alibaba, periodically or occasionally conduct promotional activities that put significant stress on their inventory systems. The sharp demand and price fluctuations caused by these promotional activities distress stocks over an inventory period if customer satisfaction is maintained. This article develops a chance-constrained multiobjective programming model to obtain optimal order quantities in an e-commerce inventory system with stochastic demand and prices. Traditional techniques for coping with the stochastic coefficients using an expected value operator or probability measures are usually inefficient in estimating quantities. With rational assumptions regarding market behavior, stochastic demand and price are characterized with a normal distribution based on historical data. A novel method called rough approximation is designed to flexibly formulate the feasible region, from which a crisp model is obtained that has no uncertain information. A proxy ideal point-based genetic algorithm (PIP-GA) is then developed to solve the model, allowing decision makers to fix the optimal order quantities for different promotional activity scales. Finally, the results are illustrated using a practical example from the Chinese B2C e-commerce giant “JD.COM.” Some special features are discussed to show the availability of the proposed model and the solution methodology.

A fuzzy leakage inventory model with shortage using triangular and trapezoidal fuzzy numbers

Inventory refers to any kind of economically valued resources that are in various stages of being made ready for sale. With the development of the Economic Order Quantity (EOQ) model by Ford Harris, many models under different conditions and assumptions are proposed. Leakage is a common phenomenon whose occurrence will reduce the profitability of the firm by increasing the minimum operational cost. In majority of real-life inventory problems we face many uncertainties in the key parameters of the corresponding model. These impreciseness and uncertainties in crisp model are improved by using fuzzy set theory. So, in this context, a fuzzy leakage inventory order level is developed by taking holding cost and shortage cost as Triangular as well as Trapezoidal Fuzzy Numbers. Defuzzification is done using Signed Distance Method. A relevant numerical example is illustrated along with sensitivity analysis to justify the proposed notion that the optimal values are improved in fuzzy environment as compared to that of in crisp environment.

An epidemic model through information-induced vaccination and treatment under fuzzy impreciseness.

In this work, we propose a nonlinear susceptible (S), vaccinated (V), infective (I), recovered (R), information level (U) (SVIRUS) model for the dynamical behavior of the contagious disease in human beings. We mainly consider the spread of information during the course of epidemic in the population. Different rate equations describe the dynamics of the information. We have developed the proposed model in crisp and fuzzy environments. In the fuzzy model, to describe the uncertainty prevailed in the dynamics, all the parameters are taken as triangular fuzzy numbers. Using graded mean integration value (GMIV) method, the fuzzy model is transformed into defuzzified model to represent the solutions avoiding the difficulties. The positivity and the boundedness of the crisp model are discussed elaborately and also the equilibrium analysis is accomplished. The stability analysis for both the infection free and the infected equilibrium are established for the crisp model. Application of optimal control of the crisp system is explored. Using Pontryagin’s Maximum Principle, the optimal control is explained. The effect of vaccination is analyzed which leads the model to be complex in nature. The effect of saturation constant for information is described for the crisp model and also the effects of weight constants on control policy are discussed. Finally, it is concluded that the treatment is more fruitful and information related vaccination is more effective during the course of epidemic.

Multi‐objective optimization for sustainable production planning

AbstractSustainable production planning can be achieved by including several key factors along with production which affect the environment and economy. Carbon emission and energy consumption are essential factors that need to be reduced for achieving sustainable development. In this article, a methodology is proposed for production planning via several process options using available flexibility in the crisp model formulation. A fuzzy multi‐objective model is formulated for satisfying imprecise goals with a view of minimizing carbon emission and energy consumption in industries. This article addresses the decision making in order to develop a better compromise solution for production planning. For practical planning problems, linear and non‐linear membership functions are used to characterize fuzziness. For nonlinear case, two approaches are proposed with multi‐criteria programming models to deal with empirical data points that is, quadratic approximation and piecewise linear approximation. Three examples are illustrated to show the applicability and purpose of the presented methodologies.

Fuzzy group identification problems

The Group Identification Problem (“Who is a J?”) introduced by Kasher and Rubinstein [14] assumes a finite class of agents, each one with an opinion about the membership to a group J of the members of the society, consisting in a function that indicates for each agent, including herself, the degree of membership to J. The problem is that of aggregating those functions, satisfying different sets of axioms and characterizing different aggregators. The literature has already considered fuzzy versions of this problem. In this paper we consider alternative fuzzy presentations of the axiomatic of the original problem. While some results are analogous to those of the original crisp model, we show that our fuzzy version is able to overcome some of the main impossibility results of Kasher and Rubinstein.

Fuzzy reverse logistics inventory model of smart items with two warehouses of a retailer considering carbon emissions

Running the business smoothly for protecting the environment is a significant challenge, on which industries are trying something to do at their level best. Reverse logistics play an important role in system design by reducing environmental consequences and increasing economic and social impacts. Given the recent fluctuations of the market, the production cost and ordering cost are considered triangular fuzzy numbers in this study. Customers’ demand is met at the right time, and there is no shortage of items; thus, attention can be paid to two warehouses of a retailer. The setup costs Purchasing costs and deterioration costs of this system are affected by the learning effects, which lead to a decrease in the total cost. Inflation is a significant problem in the market because manufacturing, remanufacturing, and retailers are all affected. This study proposes a reverse logistics system model so that customers can resolve their complaints about defective items and carbon emissions under two warehouses. Numerical results show that the fuzzy model is more economically beneficial than the crisp model, finds that the crisp and fuzzy model saw a difference of 0.34% in total cost. Two numerical examples illustrate this study, and a sensitivity analysis is performed using tables and graph.

Solution of a pollution sensitive supply chain model under fuzzy approximate reasoning

This article develops a three-layer supply chain (SC) pollution dependent production transportation model with rework on fuzzy approximate reasoning approach. First of all, we develop a cost function of the model of reworkable items, in which pollution generates due to transportation of goods only. The aim of this study is to reduce the aggregated pollution level that simultaneously could optimize the integrated SC cost function under fuzzy approximate reasoning. Basically, the theory of two-tailed (randomized L-R fuzzy) approximate reasoning is introduced first time on demand rate in this model which is an extension of one-tailed (randomized L-fuzzy/R-fuzzy) approximate reasoning in the perspectives of the proposed model. An application of a new heart like approximate dual feasible region has also been utilized here also. However, a comparative study has been made taking numerical examples for crisp, general fuzzy, one- and two-tailed fuzzy approximate reasoning models exclusively. Finally, few numerical analyses with sensitivity and graphical discussions are done to legitimize the model.

A credibility approach on fuzzy slacks based measure (SBM) DEA model

Data Envelopment Analysis (DEA) is a multi-criteria technique based on linear programming to deal with many real-life problems, mostly in nonprofit organizations. The slacks-based measure (SBM) model is one of the DEA model used to assess the relative efficiencies of decision-making units (DMUs). The SBM DEA model directly used input slacks and output slacks to determine the relative efficiency of DMUs. In order to deal with qualitative or uncertain data, a fuzzy SBM DEA model is used to assess the performance of DMUs in this study. The credibility measure approach, transform the fuzzy SBM DEA model into a crisp linear programming model at different credibility levels is used. The results came from the fuzzy DEA model are more rational to the real-world situation than the conventional DEA model. In the end, the data of Indian oil refineries is collected, and the efficiency behavior of the companies obtained by applying the proposed model for its numerical illustration.

Neutrosophic Theory and Its Application in Various Queueing Models: Case Studies

Queueing theory is an important technique to study and evaluate the performance of system. Queueing theory is applied in many applications such as logistics, finance, emergency services and project management, etc. In this research we apply neutrosophic philosophy in queueing theory. We deal withseveral queue models such as M/M/1 queue, M/M/s queue and M/M/1/b queue. We illustrate, solve, and find the performance measures of M/M/1, M/M/s, and M/M/1/b crisp queue models via examples with exact arrival rate and service rate. Queueing models affect by many factors such as arrival rate, service rate, number of servers, etc. These factors are not constantly expressed by accurate times; hence we express the parameters of queueing system by the neutrosophic. We express arriving rates and serving rates by neutrosophic values. We also illustrate, solve, and find the performance measures of NM/NM/1, NM/NM/s, and NM/NM/1/b neutrosophic queue models via examples. We concluded that the performance measures of neutrosophic queue models is more accurate than crisp queue models.

(2005-5916) A novel defuzzification approach of Type-2 fuzzy variable to solving matrix games: An application to plastic ban problem

The novelty of the type-2 fuzzy variable (T2FV) is its fuzzy secondary possibility distribution. Here, we aim to formulate a new defuzzification model of triangular type-2 fuzzy variables (TT2FVs) based on type reduction. We also show its superiority compared to other existing models by providing some numerical examples. Later, we develop a matrix game problem in the type-2 fuzzy environment to show the applicability of the proposed defuzzification method in a real-world situation. Based on this defuzzification method, two crisp linear programming models are derived, which are subsequently solved by the simplex method using LINGO 17.0 software. Also, the expected pay-off for the maximizing player is calculated as TT2FV, which is desirable. Finally, to show the validity and applicability, the proposed methodology is illustrated with the plastic ban problem.

An axiomatic design-based mathematical programming method for heterogeneous multi-criteria group decision making with linguistic fuzzy truth degrees

This paper aims to develop a new axiomatic design-based mathematical programming method for heterogeneous multi-criteria group decision making (HMCGDM) problems with linguistic fuzzy truth degrees (LFTDs). The main contributions of this paper are summarized in five aspects: (1) The information content definitions for six types of fuzzy numbers are initially provided according to axiomatic design . (2) Considering the authority of experts on different criteria and group consensus, a bi-objective programming model is constructed to derive experts’ weights by maximizing individual deviation and minimizing group discordance. (3) Each alternative is assessed on the basis of its information content to a fuzzy positive ideal solution. Information content is firstly used to define the linguistic fuzzy consistency and inconsistency indices. (4) A bi-objective linguistic fuzzy mathematic programming model is built to determine the criteria weights, which considers consistency and inconsistency indices simultaneously. This model can be dexterously transformed into a crisp linear programming model for resolution by the linguistic scale function. (5) The group information content of each alternative to fuzzy positive ideal solution is calculated to determine the ranking order of alternatives. Finally, an example of blockchain service provider selection is given to validate the proposed method.

Adjustment of credit period and stock-dependent demands in a supply chain model with variable imperfectness

In this model, a supply chain model has been developed where a manufacturer continuously transports perfect quality items at no cost to a distribution center to maintain the market demand. At the beginning of the process, the production process is in an “in-control” state, but after a random time, it goes to an “out-of-control” state. Thus the production process produces perfect as well as imperfect items in this state. The rate of imperfect production depends on production rate and production-run length in the “out-of-control” state. The unit production cost of the manufacturer depends on the production rate which indicates that the higher production rate is the cause of higher production cost if it exceeds the normal production rate set by the manufacturer. The manufacturer provides a credit period to the distributor to increase his sales growth. In the competitive business world, credit period affects the demand rate of a retailer/buyer due to its late payment facility. So, here, credit period dependent demand function is introduced for the distributor. During the festive season, demand factor of products is usually affected by stock level. Also, for some types of products (such as clothing), the demand factor depends on its stock level. Thus we have considered stock-dependent demand function for the customers. Now, in an infinite time horizon, the problem is how they adjust their demands to sustain the whole system’s financial flow. The novelty of this model is to analyze the compatibility of this two-type of demand in their businesses so that both can maintain their profits. Also, there is another credit length which is offered by the distributor. The proposed model has been discussed in type-2 fuzzy environment due to the uncertainty of the credit period. The purpose of this model is to optimize the integrated profit of the system by optimizing production rate and production-run time. Finally, numerical examples have been provided to illustrate the feasibility of both crisp and fuzzy model and some conclusions are derived conducting a sensitivity analysis of different parameters.

Forward and inverse techniques for fuzzy fractional systems applied to radon transport in soil chambers

This targeted work deals with the forward and inverse approaches to handle fractional radon diffusion problems in an uncertain environment. In general, fractional differential equations may model certain problems more efficiently than differential equations of integer order. As such, this work proposes Mittage-Leffler function method to handle fuzzy fractional radon transport mechanism. Further, inverse methods are used to estimate the unknown parameters that characterize the physical systems when data from experimental measurements are already known. As such, a fractional Euler type inverse approach is also proposed in this work to estimate the involved fuzzy parameters. Finally, for the validation of the results, we have used the experimental data and crisp models.

Multi-objective robust optimisation model for MDVRPLS in refined oil distribution

ABSTRACT At depots with refined oil shortage, arranging a reasonable distribution scheme with limited supply affects operation costs, demand satisfaction rate of gasoline stations (hereafter, ‘station satisfaction’), and overtime penalty. This study considers the refined oil distribution problem with shortages using a multi-objective optimisation approach from the perspective of decision makers of oil marketing companies. The modelling and solving process involves (i) formulation of a crisp multi-depot vehicle routing model with limited supply (MDVRPLS) which considers station priority and soft time windows, (ii) development of a robust optimisation model (ROM) to manage uncertainty in demand, and (iii) the proposal of a multi-objective particle swarm optimisation (MOPSO)algorithm. Results of numerical experiments show that (i) the crisp model can better balance operation costs, station satisfaction, and overtime penalty, which produces 3.33% and 4.60% increase in station satisfaction at an increased unit cost and overtime penalty respectively; (ii) ROM successfully addresses uncertainty in demand compared to the crisp model, which requires an additional 8.81% in cost and 12.85% in penalty; and (iii) the MOPSO manages these MDVRPLS models more effectively than other heuristic algorithms. Therefore, applying ROM of refined oil supply shortage to the management significantly improves the efficiency and resists the disturbance caused by external uncertainties, providing scope for efficient distribution of scarce resources.

Portfolio Selection Using Data Envelopment Analysis Cross-Efficiency Evaluation with Undesirable Fuzzy Inputs and Outputs

In this paper, we discuss a portfolio selection problem based on fuzzy data envelopment analysis cross-efficiency evaluation wherein both undesirable fuzzy inputs and outputs have been considered. We first propose a data envelopment analysis cross-efficiency model, in which both undesirable inputs and outputs are considered. Furthermore, considering the imprecision of the data, we extend the crisp model to the fuzzy environment and propose a fuzzy data envelopment analysis cross-efficiency model with coexisting undesirable input and output data. We then apply the proposed model to the portfolio selection problem and present a novel mean-semivariance portfolio selection model based on fuzzy data envelopment analysis cross-efficiency scores, in which several realistic constraints are considered, including a budget, cardinality, buy-in thresholds, and no short selling constraints. After that, we employ the genetic algorithm (GA) to solve the proposed model. Finally, a real-life case study is presented to demonstrate the effectiveness of the proposed approaches.

A Credibility-Based MPS/MRP Integrated Programming Model Under Complex Uncertainty

Master production schedules (MPS) and Material requirements planning (MRP) are two important planning modules in the production planning system. In this paper, a new fuzzy credibility-based double-sided chance-constrained programming (FCDCP) model is proposed for the capacitated MPS/MRP integrated programming problem under complex uncertainty. The proposed model supports simultaneous lot sizing and scheduling decisions and can deal with uncertainty presented as fuzziness in both right-hand and left-hand sides of constraints. To tackle the FCDCP model, the corresponding defuzzification methods based on credibility measure are proposed for transforming the fuzzy MPS/MRP integrated programming model into a crisp equivalent mixed-integer programming model. When decision-makers decide on MPS/MRP integrated programming problem, the optimal lot sizing and scheduling solutions from the FCDCP model can not only provide an effective method for production planners to deal with the uncertainty in the production system, making the production plan more reasonable and accurate, but it can also help decision-makers obtain the optimal production plans and determine the best allocation of production resources, effectively reducing the total cost of the production system. Finally, an industrial example from a transmission manufacturer is provided to show the applicability and flexibility of the model.