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A Study of a Back Order EOQ Model under cloud type Intuitionistic dense fuzzy demand rate
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The present article deals with a backorder Economic Order Quantity (EOQ) model for natural leisure/closing time system where the demand rate depends upon the total shortage period and the seasonal effect. A cost minimization problem is developed by trading off setup cost, inventory cost, cost for idle time and shortage cost. The intuitionistic step order fuzzy number for optimization has been developed, assuming all parameters as fuzzy numbers. Ranking is done by employing score function, accuracy value on the centre of gravity and Euclidean distance function over the objective function. Finally numerical examples are considered to justify the model.
In the traditional economic order quantity/economic production quantity model, most of the items considered are of perfect type. But this situation rarely takes place in practice. Thus, in this paper, an economic order quantity model with imperfect-quality items is developed. screening process is performed, and the items of imperfect quality are sold as a single batch. A proportionate rate of discount for the items of imperfect quality has also been studied. Moreover, a case study has been incorporated to comprehend the model. To nullify the issues of non-random uncertainties of demand rate in business scenario, cloudy fuzzy method has been utilized here. Numerical study reveals that cloud model along with its new defuzzification methods can give maximum profit of the model all the time instead of deterministic ones. Finally, sensitivity analysis and graphical illustrations are made to justify the novelty of the model.
This article deals with a classical economic order quantity (EOQ) model under monsoon type fuzzy demand rate. It is nothing but the generalization of cloudy fuzzy model. We split the EOQ model into three parts according to the real-time fuzzy components of the demand rate. To defuzzify the model we develop an algorithm and the solution is obtained with the help of a nonlinear optimization technique that requires maximum aspiration level of the fuzzy membership of the objective function. Moreover, for comparative study we take numerical results of the crisp, general fuzzy and cloudy fuzzy model also. By this study we have shown that the decision maker might have to choose the monsoon type fuzzy environment all the time to control the proposed inventory. Moreover, sensitivity analysis and graphical illustrations are made to justify the new fuzzy approach.
In this article we develop an economic order quantity (EOQ) model with backlogging where the
decision is made jointly from two decision maker supposed to view one of them as the industrialist (developer)
and the other one as the responsible manager. The problem is handled under dense fuzzy environment. In fuzzy set
theory the concept of dense fuzzy set is quite new which is depending upon the number of negotiations/ turnover
made by industrial developers with the supplier of raw materials and/or the customers. Moreover, we have discussed the preliminary concept on dense fuzzy sets with their corresponding membership functions and appropriate defuzzification method. The numerical study explores that the solution under joint decision maker giving the finer optimum of the objective function. A sensitive analysis, graphical illustration and conclusion are made for justification the new approach.
This article deals with a triangular dense fuzzy set having special property on Cauchy sequence. In this set,
the normality will never be attained unless we unlock by a special key on triangular dense fuzzy set at its
final defuzzified state. We give several definitions on triangular dense fuzzy lock sets first and then discuss
its locking unlocking property for single-key, double-key, and multiple keys environments with special
reference to the convergence of Cauchy sequence. The non-membership function of the proposed lock set
has also been studied. The graphical representations of the (non-)membership functions are developed, and
the defuzzifications are done by existing methods of dense fuzzy sets as well as cloudy fuzzy sets
implicitly. However, we have extended this fuzzy lock set into fuzzy lock matrix to generalize the concept.
Finally, we discuss the fields of its practical application and draw a conclusion for better motivation.
This paper deals with the classical backorder economic order quantity (EOQ) inventory model under cloudy fuzzy environment. The traditional concepts on fuzziness of the fuzzy parameters remains the same in all the time, but in practice, due to the human experience and precision the fuzziness began to remove from the system over time. However, we take the crisp model first, then fuzzifying the model to get a decision under the cloudy fuzzy (extension of dense fuzzy) demand rate followed by its practical application. A new defuzzification method has been utilized for ranking the fuzzy numbers. Finally, comparative analysis between the crisp, general fuzzy as well as cloudy fuzzy solutions are done extensively. The graphical illustrations and numerical examples are studied to justify the usefulness of the new approach in the model itself.
In this study we introduce the concept of denser property in fuzzy membership function used in neutrosophic sets. We present several new definitions and studied their properties. Defuzzification methods over neutrosophic triangular dense fuzzy sets and neutrsophic triangular intuitionistic dense fuzzy sets are then given. Finally practical applicability of the methods have been discussed with graphical implications in
recent times.
This paper investigates a hill type economic production-inventory quantity (EPIQ) model with variable lead-time, order size and reorder point for uncertain demand. The average expected cost function is formulated by trading off costs of lead-time, inventory, lost sale and partial backordering. Due to the nature of the demand function, the frequent peak (maximum) and valley (minimum) of the expected cost function occur within a specific range of lead time. The aim of this paper is to search the lowest valley of all the valley points (minimum objective values) under fuzzy stochastic demand rate. We consider Intuitionistic fuzzy sets for the parameters and used Intuitionistic Fuzzy Aggregation Bonferroni mean for the defuzzification of the hill type EPIQ model. Finally, numerical examples and graphical illustrations are made to justify the model.
This article deals with a new novel defuzzification method for the dense fuzzy sets. In our study, we first define the dense fuzzy set for triangular fuzzy numbers. Then new defuzzification methods have been formulated with crisp convergence tests. Cauchy sequence has been utilised for better illustrations. We show the usefulness and the global justification of newly introduced methods. Applications of new methods in decision making, psychological test, military selection, cryptography, photography, crime research, filtering of noisy environment and risk analysis have been given as its key indexes.
This article deals with the bi-objective manufacturing inventory problems. We know that the present world economy is very much competitive and nobody leaves a pace to the other company for their profit-seeking enterprises. However, every company has a specific demand curve for selling their commodities or products which may intersect each other or not. Under these circumstances it is troublesome to get an optimum solution for bi-objective problems in the business arena
that maximises their individual average total profit. To overcome these situations, we have used seven methods, namely
goal programming, goal attainment, utility function, individual optimisation, geometric mean, global criteria, and mini–max
method extensively. Moreover, we have studied the cases of two book producer’s problems along with their parabolic and linear several demand positions, respectively. Numerical examples and graphical illustrations are made to justify the model.
This paper deals with a new approach of linguistic dichotomous fuzzy variables for a classical backordered EOQ (Economic Order Quantity) model with PE (Promotional Effort) and selling price dependent demand rate. In practice, we have observed that the demand rate during a shortage period decreases with time. Based on these assumptions, we have developed a cost minimization problem (a crisp model) by trading off setup cost, inventory cost, backordering cost and cost for promotional effort. Then, we have studied a fuzzy model by considering the coefficient vectors as pentagon fuzzy numbers associated with some co-ordinates. Defuzzification is made with the help of the center-of-gravity method followed by a ranking index and the Euclidean distance of the objective function. Considering a numerical example, phi- (ϕ-)coefficients have been computed for each method and a decision is made according to the natural characteristics of the decision variables. Finally, conclusions are drawn, explaining the justification of the model.
This paper considers an inventory model for items with imperfect quality in fuzzy environment by assuming that learning occurs in setting the fuzzy parameters. This implies that inventory planners collect the information and build up knowledge from the previous shipments, and thus learning process occurs in estimating fuzzy parameters. So, it is hypothesized that the fuzziness of each fuzzy parameter reduces by learning of inventory planner. In doing so, the study developed total profit function with fuzzy parameter, where triangular fuzzy number is used to quantify the fuzziness of the data. Next, the learning curve is incorporated into the fuzzy model to account for learning in fuzziness. Subsequently, the optimal policy, including the batch size and total profit are derived using the classical approach. Finally, numerical examples and a comparison among fuzzy learning, fuzzy and crisp cases are provided to highlight the importance of using learning in fuzzy model.
The traditional concept of economic order quantity (EOQ) inventory model is that, the demand rate is continuous throughout the cycle time. But through clinical inquiry anybody can notice something different from that assumption. In reality, for any inventory we usually observe that each day has an unavoidable natural leisure/closing time and at that time no demand rate is viewed. Considering this phenomenon in an EOQ model we may split a day into two parts, one is the opening time period and the other one is closing time period. However, the demand rate which is out-flowed from the system has a measuring error. In the basis of low, zero and excessive measuring error, the model itself is divided into three parts, for a finite cycle time (counted in days only). Due to the flexible nature of the opening and closing time period of the model, a general Fuzzy set and Intuitionistic Fuzzy Optimization (IFO) technique have been analyzed to illustrate the model. Numerical examples and sensitivity analysis have been made with the help of LINGO software and got a wonderful optimum solution near 12-hour opening time duration in a day. Finally a conclusion is made keeping some scope of future work.
Keywords: Inventory, Fuzzy sets, Natural idle time, Intuitionistic Fuzzy Optimization.
Maji\cite{maj-13}, firstly proposed neutrosophic soft sets can handle the
indeterminate information and inconsistent information which exists commonly in
belief systems. In this paper, we have firstly redefined complement, union and
compared our definitions of neutrosophic soft with the definitions given by
Maji. Then, we have introduced the concept of neutrosophic soft matrix and
their operators which are more functional to make theoretical studies in the
neutrosophic soft set theory. The matrix is useful for storing an neutrosophic
soft set in computer memory which are very useful and applicable. Finally,
based on some of these matrix operations a efficient methodology named as
NSM-decision making has been developed to solve neutrosophic soft set based
group decision making problems.
An intuitionistic fuzzy economic order quantity (EOQ) inventory model with backlogging is investigated using the score functions for the member and non-membership functions. The demand rate is varying with selling price and promotional effort (PE). A crisp model is formulated first. Then, intuitionistic fuzzy set and score function (or net membership function) are applied in the proposed model, considering selling price and PE as fuzzy numbers. To obtain the best inventory policy, ranking index method has been adopted, showing that the score function can maintain the ranking rule also. Moreover, optimization is made under the general fuzzy optimal (GFO) and intuitionistic fuzzy optimal (IFO) policy. Finally, a graphical illustration, numerical examples with sensitivity analysis and conclusion is made to justify the model.
In this article, an alternative method to evaluate the arithmetic operations on fuzzy number has been developed, on the assumption that the Dubois-Prade left and right reference functions of a fuzzy number are distribution function and complementary distribution function respectively. Using the method, the arithmetic operations of fuzzy numbers can be done in a very simple way. This alternative method has been demonstrated with the help of numerical examples.
We provide a self contained survey of the state of art of the fuzzy binary relations and some of their applications.
The present fuzzy arithmetic based on Zadeh's possibilistic extension principle and on the classic definition of a fuzzy set has many essential drawbacks. Therefore its application to the solution of practical tasks is limited. In the paper a new definition of the fuzzy set is presented. The definition allows for a considerable fuzziness decrease in the number of arithmetic operations in comparison with the results produced by the present fuzzy arithmetic.
In this article, it has been shown that the Dubois-Prade left and right reference functions of a fuzzy number viewing as a distribution function and a complementary distribution function respectively, leads to a very simple alternative method of finding the membership of any function of a fuzzy number. This alternative method has been demonstrated with the help of different examples.
The paper considers an inventory model with backorders in a fuzzy situation by employing two types of fuzzy numbers, which are trapezoidal and triangular. A full-fuzzy model is developed where the input parameters and the decision variables are fuzzified. The optimal policy for the developed model is determined using the Kuhn–Tucker conditions after the defuzzification of the cost function with the graded mean integration (GMI) method. Numerical examples and a sensitivity analysis study are provided to highlight the differences between crisp and the fuzzy cases.
Online product reviews significantly impact the online purchase decisions of consumers. However, extant decision support models have neglected the randomness and fuzziness of online reviews and the interrelationships among product features. This study presents an integrated decision support model that can help customers discover desirable products online. This proposed model encompasses three modules: information acquisition, information transformation, and integration model. We use the information acquisition module to gather linguistic intuitionistic fuzzy information in each review through sentiment analysis. We also apply the information transformation module to convert the linguistic intuitionistic fuzzy information into linguistic intuitionistic normal clouds (LINCs). The integration module is employed to obtain the overall LINCs for each product. A ranked list of alternative products is determined. A case study on Taobao.com is then provided to illustrate the effectiveness and feasibility of the proposal, along with sensitivity and comparison analyses, to verify its stability and superiority. Finally, conclusions and future research directions are suggested.
Nowadays, databases expand rapidly due to electronically generated information from different fields like bioinformatics, census data, social media, business transactions, etc. Hence, feature selection/attribute reduction in databases is necessary in order to reduce time, cost, storage and noise for better accuracy. For this purpose, the rough set theory has been played a very significant role, but this theory is inefficient in case of real-valued data set due to information loss through discretization process. Hybridization of rough set with intuitionistic fuzzy set successfully dealt with this issue, but it may radically change the outcome of the approximations by adding or ignoring a single element. To handle this situation, we reconsider the hybridization process by introducing intuitionistic fuzzy quantifiers into the idea of upper and lower approximations. Supremacy of intuitionistic fuzzy quantifier over VPRS and VQRS is presented with the help of some examples. A novel process for feature selection is given by using the degree of dependency approach with intuitionistic fuzzy quantifier-based lower approximation. A greedy algorithm along with two supportive examples is presented in order to demonstrate the proposed approach. Finally, proposed algorithm is implemented on some benchmark datasets and classification accuracies for different classifiers are compared.
In this note, to overcome the limitation of Wei’s method (Int J Fuzzy Syst 17(3):484–489, 2015), a modified linear programming model is proposed for finding the optimal weight vector of the attributes.
Hesitant fuzzy linguistic preference relation (HFLPR) is a new preference structure that the decision makers (DMs) are hesitant about several possible linguistic terms of preference information for pairwise comparison between alternatives each other. The purpose of this paper is to investigate the additive consistency of hesitant fuzzy linguistic preference relation (HFLPR) with a new expansion principle for hesitant fuzzy linguistic term sets (HFLTSs). In order to normalize HFLTSs with different numbers of linguistic terms, a least common multiple extended (LCME) principle is proposed. According to the LCME principle, the additive consistent index of a HFLPR is defined to measure the consistency level of the HFLPR. For improving the unacceptable additive consistency of a HFLPR, a pure integer programming model is constructed to derive an acceptable additive consistent HFLPR. The similarities between DMs are calculated based on their individual HFLPR with acceptable additive consistency. Subsequently, the confidence degrees of DMs are defined to derive DMs' weights, and some examples are analyzed to verify the effectiveness of the proposed method.
Interval-valued hesitant fuzzy linguistic set, as an extension of linguistic term set and interval-valued hesitant fuzzy set, can effectively describe uncertainty, hesitancy, and inconsistency inherent in decision-making process. In this paper, a method based on comprehensive cloud aggregation operators is developed for solving multi-attribute group decision-making (MAGDM) problems with interval-valued hesitant fuzzy linguistic numbers (IVHFLNs). First, a comprehensive cloud of an IVHFLN is defined according to the cloud model. Using the comprehensive cloud, a distance measure between two comprehensive clouds is proposed and a distance measure for comprehensive cloud matrices is developed. Then, some generalized geometric aggregation operators of comprehensive clouds for IVHFLNs are introduced, including a comprehensive cloud generalized weighted geometric operator, a comprehensive cloud generalized ordered weighted geometric operator, and a comprehensive cloud generalized hybrid geometric operator. Subsequently, two approaches are presented to compute the deviations among attributes. Then, a programming model of maximizing deviation is established to derive the attribute weights objectively. To determine DMs’ weights, a technique for order preference by similarity to ideal solution-based approach is put forward in interval-valued hesitant fuzzy linguistic environment. The individual order of alternatives for each decision maker is obtained by the comprehensive cloud aggregation operator. To generate the collective order of alternatives, a multi-objective assignment model is constructed and transformed into a single-objective program to resolve. Thus, a method based on comprehensive cloud aggregation operators is proposed for MAGDM with IVHFLNs. Finally, two examples of application are given to demonstrate the applicability and effectiveness of the proposed method.
In recent times, environmental pollution is one of the most critical issues of any industrial set up throughout the globe. The excess of marginal pollution and risk for unsold goods (like chemical and plastic items etc.) and their decomposition play a vital role in any decision-making process. Also the uses of automation/technology in pollution control measures contributes to another kind of pollution drastically. Moreover, the waste products as well as factory outlet sewage water itself have some negative externalities on the environment. Keeping this fact in mind, we first define a new functional relation between environmental pollution rate and production rate. Then we describe an economic production quantity model with deterioration for cycle time dependent production rate. We include preservation technology cost to restrict the normal deterioration. Considering a real data set obtained from the case study of a sponge iron industry, we analyze the crisp model first, then we go for its fuzzification. Inspiring from the learning effects we have utilized a new fuzzy inference scheme based on the dense fuzzy rule. However, our findings suggest that to restrict pollution and financial loss, the decision-maker might have to control the gross production of the industry. Finally, sensitivity analysis and graphical illustrations are also provided to justify the model.
The study of learning effect on inventory models with imprecise parameters is a research topic that has recently emerged. The research papers have published so far studied this aspect from a theoretical point of view and thus the literature lacks the investigation of this topic from a practical standpoint. To close this research gap, we conducted a semi-structured interview with a number of industry experts to gain insights into the prevalence of learning and forgetting in real applications. Based on the insights gained from the interviews, we have developed a recently published model by countering the assumption of full transfer of learning. The model developed herein proposes a situation where the knowledge gained by the operator in setting imprecise parameters deteriorates over the planning cycles due to intermittent planning process. A numerical study suggests that accounting for the effect of knowledge depreciation/forgetting on imprecise parameters leads to reduction in maximum inventory, which consequently reduces the total cost of the system.
The main aim of this paper is to characterize the set of real parameters associated to a fuzzy number, represented in a general form which include the most important characteristics, with the following property: for any given fuzzy number there exists at least a trapezoidal fuzzy number which preserves a fixed parameter. The uniqueness of the nearest trapezoidal fuzzy number with this property is proved, the average Euclidean distance being considered. As an important property, each resulting trapezoidal approximation operator is continuous. The main results are illustrated by examples.
This article fuzzyfies EOQ crisp model with backlogged while demand of the customers varies with selling price and promotional effort. The demand rate in stock out situation decreases with duration of the shortage period and it comes back to its initial rate since promotional effort continues. The cost function consists of set up cost, inventory cost, shortage cost and cost for promotional effort. The coefficient vectors of the concerned minimization cost function are considered as fuzzy numbers which are transformed into interval numbers. Then, the interval objective function has been transformed into a classical multi-objective EOQ model using intuitionistic fuzzy technique. A comparative study on Pareto optimality and optimality under Lagrange's interpolating polynomial function has been made illustrating numerical example.
In this paper, we propose a new mathematical formalization of the concept of polygonal fuzzy numbers and an extension of this notion to fuzzy sets on RnRn. We study the mathematical structure of these families of fuzzy sets and show that each family is a complete and separable metric space when endowed with the generalized Hausdorff metric. Moreover, we show that for n=1, the families of polygonal fuzzy numbers are isomorphic to some convex and closed convex cone of a finite dimensional space. We obtain generalizations and extensions of some previous results on polygonal fuzzy numbers and simplified proofs of some well-known results about approximation of fuzzy n-dimensional quantities. Finally, some developments about the approximation of families of fuzzy sets are introduced.
The article deals with a backorder EOQ (Economic Order Quantity) model with promotional index for fuzzy decision variables. Here, a profit function is developed where the function itself is the function of m-th power of promotional index (PI) and the order quantity, shortage quantity and the PI are the decision variables. The demand rate is operationally related to PI variables and the model has been split into two types for the multiplication and addition operation. First the crisp profit function is optimized, letting it free from fuzzy decision variable. Yager (1981) ranking index method is utilized here to have a best inventory policy for the fuzzy model. Finally, a graphical presentation of numerical illustrations and sensitivity analysis are done to justify the general model.
Este libro es una introducción a la aritmética difusa.
A fuzzy number is a fuzzy subset of the real line whose highest membership values are clustered around a given real number called the mean value ; the membership function is monotonia on both sides of this mean value. In this paper, the usual algebraic operations on real numbers are extended to fuzzy numbers by the use of a fuzzification principle. The practical use of fuzzified operations is shown to be easy, requiring no more computation than when dealing with error intervals in classic tolerance analysis. The field of applications of this approach seems to be large, since it allows many known algorithms to be fitted to fuzzy data.
In this short note we propose a novel fuzzy complement functional. This functional is different from other functionals known in the literature. However, it turns out to be an alternative characterization of the well-known negation function.
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