mrp with flexible constraints a fuzzy mathematical programming approach in manufacturing environments with complex product structures and multiple production stages material requirements planning mrp systems are the most commonly used for production planning and material supply decision making however in practice diverse difficulties arise such as uncertainty in market demand resources with limited available capacities uncertainty in capacity data or uncertain costs likewise classical procedures of resolution applied in mrp environments do not optimize production decisions this paper provides a new linear programming model for medium term production planning in a capacity constrained mrp multi product multi level and multi period manufacturing environment then this model is transformed into three fuzzy models with flexibility in the objective function in the market demand and in the available capacity of resources the main goal is to determine the master production schedule the mrp for each raw component in each period stock levels delayed demand and capacity usage levels over a given planning horizon in such a way as to hedge against uncertainty finally the model is tested using real data from an automobile seat manufacturer

a new possibilistic programming approach for solving fuzzy multiobjective assignment problem in this paper we propose a new possibilistic programming approach to solve a fuzzy multiobjective assignment problem in which the objective function coefficients are characterized by triangular possibility distributions the proposed solution approach simultaneously minimizes the best scenario the likeliest scenario and the worst scenario for the imprecise objective functions using level sets the level sets are used to define the confidence level of the fuzzy judgments of the decision maker additionally we provide a systematic framework in which the decision maker controls the search direction by updating both the membership values and aspiration levels until a set of satisfactory solutions is obtained numerical examples with dataset from realistic situations are provided to demonstrate the effectiveness of the proposed approach

fuzzy optimization for supply chain planning under supply demand and process uncertainties in today s global marketplace individual firms do not compete as independent entities rather as an integral part of a supply chain this paper proposes a fuzzy mathematical programming model for supply chain planning which considers supply demand and process uncertainties the model has been formulated as a fuzzy mixed integer linear programming model where data are ill known and modelled by triangular fuzzy numbers the fuzzy model provides the decision maker with alternative decision plans for different degrees of satisfaction this proposal is tested by using data from a real automobile supply chain

an interactive possibilistic programming approach for multiple objective supply chain master planning providing an efficient production plan that integrates the procurement and distribution plans into a unified framework is critical to achieving competitive advantage in this paper we consider a supply chain master planning model consisting of multiple suppliers one manufacturer and multiple distribution centers we first propose a new multi objective possibilistic mixed integer linear programming model mopmilp for integrating procurement production and distribution planning considering various conflicting objectives simultaneously as well as the imprecise nature of some critical parameters such as market demands cost time coefficients and capacity levels then after applying appropriate strategies for converting this possibilistic model into an auxiliary crisp multi objective linear model molp we propose a novel interactive fuzzy approach to solve this molp and finding a preferred compromise solution the proposed model and solution method are validated through numerical tests computational results indicate that the proposed fuzzy method outperforms other fuzzy approaches and is very promising not only for solving our problem but also for any molp model especially multi objective mixed integer models

the design of reverse distribution networks models and solution procedures abstract reverse distribution or the management of product return flows induced by various forms of reuse of products and materials has received growing attention throughout this decade in this paper we discuss reverse distribution and propose a mathematical programming model for a version of this problem due to the complexity of the proposed model we introduce a heuristic solution methodology for this problem the solution methodology complements a heuristic concentration procedure where sub problems with reduced sets of decision variables are iteratively solved to optimality based on the solutions from the sub problems a final concentration set of potential facility sites is constructed and this problem is solved to optimality the potential facility sites are then expanded in a greedy fashion to obtain the final solution this heuristic expansion was also performed using the solution found with a greedy heuristic to provide a short list of potential facility sites computational tests demonstrate a great deal of promise for this solution method as high quality solutions are obtained while expending modest computational effort

possibilistic linear programming for managing interest rate risk abstract in this study we consider the linear programming problem with the objective of the investment discounted value where the interest rate is imprecise and has a triangular possibilistic distribution an crisp auxiliary bi objective linear programming model is proposed to resolve this possibilistic nature furthermore we develop an extended zimmermann approach called augmented max min approach for solving this auxiliary bi objective linear programming problem and other multiple objective linear programming problems finally a numerical bank balance sheet problem where interest rates price of futures contract loan demand deposit supply and ratio of desired loan to deposit are assumed to be fuzzy is solved for illustrating the new approach

some new properties of the structured singular value j c doyle et al 1982 have shown that a necessary and sufficient condition for robust stability or robust performance in the h sup infinity frame work may be formulated as a bound on the structured singular value mu of a specific matrix m which includes information on the system model the controller the model uncertainty and the performance specifications often it is desirable to express the robust stability and performance conditions as norm bounds on transfer matrices t which are of direct interest to the engineer e g sensitivity or complementary sensitivity the present paper shows how to derive bounds on sigma t from bounds on mu m

fuzzy scheduling modelling flexible constraints vs coping with incomplete knowledge abstract an overview of some fuzzy set based approaches to scheduling is proposed emphasizing two distinct uses of fuzzy sets representing preference profiles and modelling uncertainty distributions the first setting leads to a valued non compensatory generalization of constraint directed scheduling the other setting yields a possibility theoretic counterpart of pert where probability distributions of activity durations are changed into possibility distributions for the purpose of modelling incomplete information it is pointed out that a special case of the latter interval valued pert is a difficult ill known problem regarding the determination of critical activities latest starting times and floats lastly when flexible constraints and uncertain processing times are to be jointly considered the use of possibilistic decision theory leads to the computation of robust schedules

solving a multiobjective possibilistic problem through compromise programming abstract real decision problems usually consider several objectives that have parameters which are often given by the decision maker in an imprecise way it is possible to handle these kinds of problems through multiple criteria models in terms of possibility theory here we propose a method for solving these kinds of models through a fuzzy compromise programming approach to formulate a fuzzy compromise programming problem from a possibilistic multiobjective linear programming problem the fuzzy ideal solution concept is introduced this concept is based on soft preference and indifference relationships and on canonical representation of fuzzy numbers by means of their cuts the accuracy between the ideal solution and the objective values is evaluated handling the fuzzy parameters through their expected intervals and a definition of discrepancy between intervals is introduced in our analysis

a new possibilistic optimization model for multiple criteria assignment problem this paper presents a new multiple criteria optimization model of an assignment problem with imprecise coefficients besides minimizing the total cost total time of finishing jobs and maximization of the overall achieved quality we introduce a new criterion that minimizes the number of workers employed to finish all jobs it contributes significantly in multi job assignment to adjust the number of workers assigned to at least one job for balancing work allocation among the workers furthermore we employ new diversification constraints to obtain a reasonable tradeoff between the number of workers employed and number of jobs assigned a new interactive possibilistic programming approach is developed for trapezoidal possibility distributions which uses alpha level sets to incorporate confidence levels of the decision maker in his fuzzy judgments leading to alpha efficient solutions numerical experiments are conducted using data coming from a manpower planning problem to demonstrate working of the proposed multiple criteria assignment model and effectiveness of the fuzzy interactive approach

ranking and defuzzification methods based on area compensation abstract we present some interesting properties related to the area compensation procedure to compare fuzzy numbers it has been proved that this method produces more than a fuzzy interval order it induces a ranking of fuzzy numbers some further results are given about the transitivity property and about computational aspects extensions to non normal fuzzy numbers and fuzzy quantities are also proposed

a procedure for ordering fuzzy subsets of the unit interval abstract we introduce a function to help in the ordering of fuzzy subsets of the unit interval this function is the integral of the mean of the level sets associated with the fuzzy subsets various properties of this function are studied

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

possibilistic linear programming a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem in this paper we review some fuzzy linear programming methods and techniques from a practical point of view in the rst part the general history and the approach of fuzzy mathematical programming are introduced using a numerical example some models of fuzzy linear programming are described in the second part of the paper fuzzy mathematical programming approaches are compared to stochastic programming ones the advantages and disadvantages of fuzzy mathematical programming approaches are exemplied in the setting of an optimal portfolio selection problem finally some newly developed ideas and techniques in fuzzy mathematical programming are briey reviewed c 2000 elsevier science b v all rights reserved

a novel robust fuzzy stochastic programming for closed loop supply chain network design under hybrid uncertainty abstract in today s business environments the high importance of economic benefits and environmental impacts of using scrapped products has caused most companies to move to designing the closed loop supply chain network this paper considers the closed loop supply chain network design problem under hybrid uncertainty while there are two sources of uncertainty for most parameters thus require fortifying of the robustness of the decision the first source is that some uncertain parameters may be based on the future scenarios which are considered according to the probability of their occurrence the second source is that the values of these parameters in each scenario are usually imprecise and can be specified by possibilistic distributions in this case the best robust decision has some additional properties in terms of mean value and variability of the objective function we introduced two types of the variability named scenario variability and possibilistic variability possibility theory is used to choose a solution in such a problem and a novel robust fuzzy stochastic programming approach is proposed that has significant advantages the performance of the proposed model is also compared with that of other models in term of the mean cost and variability by simulation

a new approach to some possibilistic linear programming problems abstract in practice the unit costs profits of new products or new projects the lending and borrowing interest rates and cash flows are always imprecise we provide an auxiliary multiple objective linear programming model to solve a linear programming problem with imprecise objective and or constraint coefficients our strategy is to maximize the most possible value of the imprecise profit at the same time we would like to minimize the risk of obtaining lower profit and maximize the possibility of obtaining higher profit this strategy is equivalent to the practical considerations of financial problems in this paper a numeric investment problem is solved for illustrating the new approach

a possibilistic programming approach for closed loop supply chain network design under uncertainty the design of closed loop supply chain networks has attracted more attention in recent years according to business and environmental factors the significance of accounting for uncertainty and risk in such networks spurs an interest to develop appropriate decision making tools to cope with uncertain and imprecise parameters in closed loop supply chain network design problems this paper proposes a bi objective possibilistic mixed integer programming model to deal with such issues the proposed model integrates the network design decisions in both forward and reverse supply chain networks and also incorporates the strategic network design decisions along with tactical material flow ones to avoid the sub optimalities led from separated design in both parts to solve the proposed possibilistic optimization model an interactive fuzzy solution approach is developed by combining a number of efficient solution approaches from the recent literature numerical experiments are conducted to demonstrate the significance and applicability of the developed possibilistic model as well as the usefulness of the proposed solution approach

a study of the ranking function approach through mean values abstract the average value was introduced to help in the ordering of fuzzy numbers and it was defined by means of an integrating process of a parametric function representing the position of every cut in the real line by considering the relation between a particular case of the average value and the notion of expectation for fuzzy numbers we interpret the general ranking function approach as a ranking process using a mean value and an interval relation some well known indices are included in this schema using the general average value a more general mean value for a fuzzy number is defined finally some properties of the general mean value are studied as well as its behaviour on several particular cases

numerical methods for differential equations in random domains physical phenomena in domains with rough boundaries play an important role in a variety of applications often the topology of such boundaries cannot be accurately described in all of its relevant detail due to either insufficient data or measurement errors or both this topological uncertainty can be efficiently handled by treating rough boundaries as random fields so that an underlying physical phenomenon is described by deterministic or stochastic differential equations in random domains to deal with this class of problems we propose a novel computational framework which is based on using stochastic mappings to transform the original deterministic stochastic problem in a random domain into a stochastic problem in a deterministic domain the latter problem has been studied more extensively and existing analytical numerical techniques can be readily applied in this paper we employ both a stochastic galerkin method and monte carlo simulations to solve the transformed stochastic problem we demonstrate our approach by applying it to an elliptic problem in single and double connected random domains and comment on the accuracy and convergence of the numerical methods

the mean value of a fuzzy number abstract the notion of expectation is defined for fuzzy numbers of intervals viewing them as consonant random sets this expectation is interval valued and remains additive in the sense of fuzzy interval addition as a by product some light is shed on the debate between probability and possibility theory and the problem of how to carry measure theoretic notions into the field of possibility theory

strategic robust supply chain design based on the pareto optimal tradeoff between efficiency and risk abstract the strategic design of a robust supply chain has to determine the configuration of the supply chain so that its performance remains of a consistently high quality for all possible future conditions the current modeling techniques often only consider either the efficiency or the risk of the supply chain instead we define the strategic robust supply chain design as the set of all pareto optimal configurations considering simultaneously the efficiency and the risk where the risk is measured by the standard deviation of the efficiency we model the problem as the mean standard deviation robust design problem msd rdp since the standard deviation has a square root expression which makes standard maximization algorithms based on mixed integer linear programming non applicable we show the equivalency to the mean variance robust design problem mv rdp the mv rdp yields an infinite number of mixed integer programming problems with quadratic objective miqo when considering all possible tradeoff weights in order to identify all pareto optimal configurations efficiently we extend the branch and reduce algorithm by applying optimality cuts and upper bounds to eliminate parts of the infeasible region and the non pareto optimal region we show that all pareto optimal configurations can be found within a prescribed optimality tolerance with a finite number of iterations of solving the miqo numerical experience for a metallurgical case is reported

linear programming with fuzzy parameters an interactive method resolution this paper proposes a method for solving linear programming problems where all the coefficients are in general fuzzy numbers we use a fuzzy ranking method to rank the fuzzy objective values and to deal with the inequality relation on constraints it allows us to work with the concept of feasibility degree the bigger the feasibility degree is the worst the objective value will be we offer the decision maker dm the optimal solution for several different degrees of feasibility with this information the dm is able to establish a fuzzy goal we build a fuzzy subset in the decision space whose membership function represents the balance between feasibility degree of constraints and satisfaction degree of the goal a reasonable solution is the one that has the biggest membership degree to this fuzzy subset finally to illustrate our method we solve a numerical example

the expected value of a fuzzy number abstract the notions of expected interval and expected value of fuzzy number are introduced the expected interval is defined as the expected value of an interval random set generated by the fuzzy number and the expected value of this number is defined as the centre of the expected interval the linearity of the expected interval is studied