calculus for interval valued functions using generalized hukuhara derivative and applications this paper is devoted to studying differential calculus for interval valued functions by using the generalized hukuhara differentiability which is the most general concept of differentiability for interval valued functions conditions examples and counterexamples for limit continuity integrability and differentiability are given special emphasis is set to the class f t c g t where c is an interval and g is a real function of a real variable here the emphasis is placed on the fact that f and g do not necessarily share their properties underlying the extra care that must be taken into account when dealing with interval valued functions two applications of the obtained results are presented the first one determines a delta method for interval valued random elements in the second application a new procedure to obtain solutions to an interval differential equation is introduced our results are relevant to fuzzy set theory because the usual fuzzy arithmetic extension functions and mathematical analysis are done on a cuts which are intervals

min max min robust combinatorial optimization the idea of k adaptability in two stage robust optimization is to calculate a fixed number k of second stage policies here and now after the actual scenario is revealed the best of these policies is selected this idea leads to a min max min problem in this paper we consider the case where no first stage variables exist and propose to use this approach to solve combinatorial optimization problems with uncertainty in the objective function we investigate the complexity of this special case for convex uncertainty sets we first show that the min max min problem is as easy as the underlying certain problem if k is greater than the number of variables and if we can optimize a linear function over the uncertainty set in polynomial time we also provide an exact and practical oracle based algorithm to solve the latter problem for any underlying combinatorial problem on the other hand we prove that the min max min problem is np hard for every fixed number k even when the uncertainty set is a polyhedron given by an inner description for the case that k is smaller or equal to the number of variables we finally propose a fast heuristic algorithm and evaluate its performance

min max and min max regret versions of combinatorial optimization problems a survey abstract min max and min max regret criteria are commonly used to define robust solutions after motivating the use of these criteria we present general results then we survey complexity results for the min max and min max regret versions of some combinatorial optimization problems shortest path spanning tree assignment min cut min s t cut knapsack since most of these problems are np hard we also investigate the approximability of these problems furthermore we present algorithms to solve these problems to optimality

minimax regret solution to linear programming problems with an interval objective function in this paper a linear programming problem with an interval objective function is treated first the previous approaches to this problem are reviewed and the drawbacks are pointed out to improve the drawbacks a new approach to this problem is proposed by introducing the minimax regret criterion as used in decision theory the properties of minimax regret solution are investigated in order to obtain the minimax regret solution a method of solution by a relaxation procedure is proposed it is shown that the solution is obtained by repetitional use of the simplex method a numerical example is given to illustrate the proposed solution method

violation analysis on two step method for interval linear programming in many real world problems system parameters or model coefficients may be bounded between lower and upper bounds due to a variety of uncertainties over the past decades intensive research efforts have focused on interval linear programming ilp to tackle such uncertainties as one of the most popular methods for solving ilp problems two step method tsm proposed by huang et al in 1995 allows uncertain information to be directly communicated into the optimization process and resulting solutions such that decision alternatives could be generated through the interpretation of interval solutions however part of optimum solution points obtained through tsm may go beyond the decision space in some cases this phenomenon referred to as solution violation may mislead decision makers to unreasonable policies plans or strategies which play important roles in the social and economic development therefore this study first investigates and identifies the essential cause for the solution violation of tsm following that an improved solution method namely itsm is proposed to avoid resulting violation by introducing extra constraints in the solving process a numeric example is then presented to demonstrate the effectiveness of itsm in handling solution violation

binomial confidence intervals abstract for x with binomial n p distribution section 1 gives a one page table of 95 and 99 confidence intervals for p for n 1 2 30 this interval is equivariant under x n x and p 1 p has approximately equal probability tails is approximately unbiased has crow s property of minimizing the sum of the n 1 possible lengths and each of its ends is increasing in x and decreasing in n with about as regular steps as possible sections 2 and 3 consider the usual approximate confidence intervals calculations and asymptotic results show the need for the continuity correction in these even when n is large because of the nonuniformity of the binomial normal convergence as p 0 these intervals fail to have their stated asymptotic confidence coefficients simple corrections are given for this in the approximate interval x n c n x n 1 x n 1 2n it is shown that the factor c n should be replaced by c n c 2 2c n 1 n

linear programming with fuzzy objectives abstract this paper presents a new method for solving linear programming problems with fuzzy parameters in the objective function to determine a compromise solution the authors do not reduce the set of infinitely many objective functions to a single compromise objective function as done in classical deterministic or stochastic procedures here the problem is only reduced to a few extreme objective functions the information contained in the membership functions can be used to any extent by a method called level related pair formation moreover if different level sets are considered the stability of the optimal solution can be tested

minmax regret combinatorial optimization problems with ellipsoidal uncertainty sets we consider robust counterparts of uncertain combinatorial optimization problems where the difference to the best possible solution over all scenarios is to be minimized such minmax regret problems are typically harder to solve than their nominal non robust counterparts while current literature almost exclusively focuses on simple uncertainty sets that are either finite or hyperboxes we consider problems with more flexible and realistic ellipsoidal uncertainty sets we present complexity results for the unconstrained combinatorial optimization problem the shortest path problem and the minimum spanning tree problem to solve such problems two types of cuts are introduced and compared in a computational experiment

algorithms for linear programming problems with interval objective function coefficients this paper presents three algorithms for solving linear programming problems in which some or all of the objective function coefficients are specified in terms of intervals which algorithm is applicable depends upon a the number of interval objective function coefficients b the number of nonzero objective function coefficients and c whether or not the feasible region is bounded the algorithms output all extreme points and unbounded edge directions that are multiparametrically optimal with respect to the ranges placed on the objective function coefficients the algorithms are most suitable to linear programs in which the objective function coefficients are deterministic but are likely to vary from time period to time period as for example in blending problems

on fuzzy strong semicontinuity and fuzzy precontinuity abstract using the notion of fuzzy sets azad in 1981 introduced and studied the concept of semiopen semiclosed sets semi continuous mappings almost continuous mappings and weakly continuous mappings in the fuzzy setting in the same spirit defining fuzzy open closed sets and fuzzy preopen preclosed sets we introduce and make a preliminary study of fuzzy strongly semicontinuous and fuzzy precontinuous mappings in this paper

fuzzy weak totally semicontinuous and fuzzy weak totally precontinuous mappings abstract the aim of this paper is to introduce some new classes of mappings called fuzzy weak totally continuous fuzzy totally precontinuous fuzzy weak totally semicontinuous fuzzy totally m precontinuous fuzzy totally m precontinuous and fuzzy totally irresolute mappings their compositions their relationships with other mappings preservations of some fuzzy spaces under these mappings and some examples are studied we also prove that the property that the mapping is fuzzy totally semicontinuous and fuzzy totally precontinuous is equivalent to the property that the mapping is fuzzy totally continuous

on fuzzy semi irresolute and strongly irresolute functions abstract in this paper two classes of functions between fuzzy topological spaces are introduced under the terminologies fuzzy semi irresolute and strongly irresolute functions each of which is independent of the fuzzy continuity of chang 1968 the class of fuzzy irresolute functions as recently introduced by mukherjee and sinha 1989 is seen to lie strictly between these two classes several characterizations of these functions along with their relationships with certain other types of functions are investigated

mappings in fuzzy topological spaces abstract in this paper some characterizations of fuzzy strongly semicontinuous and fuzzy precontinuous mappings are given by using the notions of quasi coincident and quasi neighbourhood due to pu and liu j math anal appl 76 1980 571 599 the definitions of open and preopen mappings which are defined by mashhour et al acta math hung 41 1983 213 218 are extended to fuzzy sets

confidence intervals for a binomial proportion thirteen methods for computing binomial confidence intervals are compared based on their coverage properties widths and errors relative to exact limits the use of the standard textbook method x n 1 96 square root of x n 1 x n n or its continuity corrected version is strongly discouraged a commonly cited rule of thumb stating that alternatives to exact methods may be used when the estimated proportion p is such that np and n 1 p both exceed 5 does not ensure adequate accuracy score limits are easily calculated from closed form solutions to quadratic equations and can be used at all times based on coverage functions the continuity corrected score method is recommended over exact methods its conservative nature should be kept in mind as should the wider fluctuation of actual coverage that accompanies omission of the continuity correction

on some near fuzzy continuous functions between fuzzy topological spaces abstract the concepts of fuzzy almost strongly continuous fuzzy continuous fuzzy weekly continuous and fuzzy weakly continuous are introduced and studied in the light of the concept of q coincidence in a fuzzy setting together with the concept of q neighbourhoods the notions of fuzzy neighbourhoods and fuzzy neighbourhoods are defined to investigate the characterizations and properties of these functions finally a comparative study regarding the mutual interrelations among these maps along with fuzzy continuous maps is made

fuzzy totally continuous and totally semi continuous functions abstract the aim of this paper is to introduce two new classes of functions called fuzzy totally continuous and fuzzy totally semicontinuous functions their characterizations examples composition of these functions their relationships with other fuzzy functions and preservation of some fuzzy spaces under these functions are studied

irresolute and lmost open functions between fuzzy topological spaces abstract the aim of this paper is to introduce and characterize the concept of fuzzy irresolute functions which is strictly stronger than the concept of fuzzy semi continuous functions and independent of the concept of fuzzy continuous functions the independence of the two definitions of fuzzy almost open functions as introduced in 5 and 3 and some of their characterizations are also established

semi continous and semi closed mappings and semi connectedness in fuzzy setting abstract the aim of the paper is mainly to study and to find different characterizations of semi continuous and semi closed mappings between fuzzy topological spaces the notion of fuzzy semi open sets is used to define fuzzy ti spaces i 0 1 2 and these spaces are investigated under fuzzy semi continuity finally fuzzy semi connectedness is introduced and studied to some extent

approximate binomial confidence limits abstract this article examines the accuracy of normal approximations to confidence limits for the binomial n p parameter p the method of getting exact values from an f table is too slow for practical use and useless for large n the two usual approximations have unsatisfactory accuracy both for small n and asymptotically and the simpler one lacks an invariance property of the confidence limit hall s 1982 approximation also lacks invariance and even when symmetrized has unsatisfactory accuracy pratt s 1968 approximation has excellent accuracy for all n inversion of a molenaar 1973 normal approximation to the binomial gives better asymptotic accuracy than pratt s approximation but poor values for observations close to n

goal programming problems with interval coefficients and target intervals abstract in conventional goal programming the coefficients of objective functions and constraints and target values are determined as crisp values however it is not frequent that the coefficients and the target values are known precisely in such cases the coefficients and target values should be represented by intervals reflecting the imprecision this paper treats goal programming problems in which coefficients and target values are given the intervals it is shown that four formulations of the problems can be considered the properties of the four formulated problems are investigated an example is given to demonstrate the differences between the four formulations

linear bilevel programming with interval coefficients in this paper we address linear bilevel programs when the coefficients of both objective functions are interval numbers the focus is on the optimal value range problem which consists of computing the best and worst optimal objective function values and determining the settings of the interval coefficients which provide these values we prove by examples that in general there is no precise way of systematizing the specific values of the interval coefficients that can be used to compute the best and worst possible optimal solutions taking into account the properties of linear bilevel problems we prove that these two optimal solutions occur at extreme points of the polyhedron defined by the common constraints moreover we develop two algorithms based on ranking extreme points that allow us to compute them as well as determining settings of the interval coefficients which provide the optimal value range

fuzzy preopen sets and fuzzy preseparation axioms abstract in this paper the concept of preopen sets due to a s mashhour m e abd el monsef and s n el debb 7 has been generalised to the fuzzy setting further fuzzy separation axioms have been introduced and investigated with the help of fuzzy preopen sets

minmax regret approach and optimality evaluation in combinatorial optimization problems with interval and fuzzy weights this paper deals with a general combinatorial optimization problem in which closed intervals and fuzzy intervals model uncertain element weights the notion of a deviation interval is introduced which allows us to characterize the optimality and the robustness of solutions and elements the problem of computing deviation intervals is addressed and some new complexity results in this field are provided possibility theory is then applied to generalize a deviation interval and a solution concept to fuzzy ones

robust regret for uncertain linear programs with application to co production models abstract this paper considers the regret optimization criterion for linear programming problems with uncertainty in the data inputs the problems of study are more challenging than those considered in previous works that address only interval objective coefficients and furthermore the uncertainties are allowed to arise from arbitrarily specified polyhedral sets to this end a safe approximation of the regret function is developed so that the maximum regret can be evaluated reasonably efficiently by leveraging on previous established results and solution algorithms the proposed approach is then applied to a two stage co production newsvendor problem that contains uncertainties in both supplies and demands computational experiments demonstrate that the proposed regret approximation is reasonably accurate and the corresponding regret optimization model performs competitively well against other optimization approaches such as worst case and sample average optimization across different performance measures

enhanced interval linear programming abstract an enhanced interval linear programming eilp model and its solution algorithm have been developed that incorporate enhanced interval uncertainty e g a b and c in a linear optimization framework as a new extension of linear programming the eilp model has the following advantages its solution space is absolutely feasible compared to that of interval linear programming ilp which helps to achieve insight into the expected value oriented trade off between system benefits and risks of constraint violations the degree of uncertainty of its enhanced interval objective function eiof would be lower than that of ilp model when the solution space is absolutely feasible and the eiof s expected value could be used as a criterion for generating the appropriate alternatives which help decision makers obtain non extreme decisions moreover because it can be decomposed into two submodels eilp s computational requirement is lower than that of stochastic and fuzzy lp models the results of a numeric example further indicated the feasibility and effectiveness of eilp model in addition ei nonlinear programming models hybrid stochastic or fuzzy eilp models as well as risk based trade off analysis for ei uncertainty within decision process can be further developed to improve its applicability

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

on small sample confidence intervals for parameters in discrete distributions summary the traditional definition of a confidence interval requires the coverage probability at any value of the parameter to be at least the nominal confidence level in constructing such intervals for parameters in discrete distributions less conservative behavior results from inverting a single two sided test than inverting two separate one sided tests of half the nominal level each we illustrate for a variety of discrete problems including interval estimation of a binomial parameter the difference and the ratio of two binomial parameters for independent samples and the odds ratio

maximum excess dominance identifying impractical solutions in linear problems with interval coefficients abstract in this paper we propose the concept of maximum excess dominance med and illustrate how it can be used to compare solutions in linear optimization problems with interval objective coefficients when a solution dominates another solution with med the expected outcome of the former is guaranteed to be better than that of the latter across a wide range of probability distributions hence med can be used to eliminate dominated solutions from consideration furthermore we provide an efficient way to check if a solution is dominated by another feasible solution in binary optimization problems and illustrate how dominated solutions can be pruned away by introducing med constraints to the original binary optimization formulation we also propose an algorithm to find the best non dominated solution and conduct computational experiments to evaluate its performance

almost compactness in fuzzy topological spaces abstract we introduce and study almost compactness for fuzzy topological spaces we show that the almost continuous image of an almost compact fuzzy topological space is almost compact moreover we show that generally almost compactness for fuzzy topological spaces is not product invariant but if x and y are almost fuzzy topological spaces and x is product related to y then their fuzzy topological product is almost compact

a relationship between probability interval and random sets and its application to linear optimization with uncertainties abstract random sets and probability intervals could be represented as uncertain parameters in a linear program however they are not the same this paper presents a method by which an intersection of random sets provides an approximation of a given probability interval two specific random sets generating this approximation are used to construct probability density mass functions that provide the smallest and the largest expected values with respect to the probability interval uncertainty we show that our construction is computationally more efficient than computing directly from probability intervals these results are applied to linear programs with random set and probability interval uncertainties to obtain optimal solutions with respect to pessimistic optimistic and minimax regret approaches

a robust lot sizing problem with ill known demands the paper deals with a lot sizing problem with ill known demands modeled by fuzzy intervals whose membership functions are possibility distributions for the values of the uncertain demands optimization criteria in the setting of possibility theory that lead to choose robust production plans under fuzzy demands are given some algorithms for determining optimal robust production plans with respect to the proposed criteria and for evaluating production plans are provided some computational experiments are presented

optimality conditions for generalized differentiable interval valued functions in this paper we study the optimal solutions set for a generalized differentiable interval valued function necessary and sufficient optimality conditions are established for gh differentiable functions convexity assumptions that are necessary or required to ensure the characterization of the optimal solutions are weaker or less strict than those presented in previous works these convexity assumptions are the weakest to characterize the optimal solutions set known results for classical non interval valued optimization are particular cases of the ones proved here

ensemble confidence intervals for binomial proportions we propose two measures of performance for a confidence interval for a binomial proportion p the root mean squared error and the mean absolute deviation we also devise a confidence interval for p based on the actual coverage function that combines several existing approximate confidence intervals this ensemble confidence interval has improved statistical properties over the constituent confidence intervals software in an r package which can be used in devising and assessing these confidence intervals is available on cran

possible and necessary optimality tests in possibilistic linear programming problems in this paper two kinds of optimality concepts are introduced into fuzzy linear programming problems in order to discuss the rationality of an obtained solution one is possible optimality evaluating the possibility degree to which the solution is optimal the other the dual concept is necessary optimality evaluating the necessity degree to which the solution is optimal when possibility distributions are defined by closed intervals degrees of possible and necessary optimalities become binary necessary and sufficient conditions for possible and necessary optimalities are induced in this case extending the results methods of calculating the degrees of possible and necessary optimalities are proposed some numerical examples are given to demonstrate the proposed methods

fuzzy strongly semiopen sets and fuzzy strong semicontinuity abstract in this paper we first introduce and study the fuzzy strongly semiopen and fuzzy strongly semiclosed sets in fuzzy space then we introduce the strong semi interior and strong semiclosure of a fuzzy set and kuratowski s 14 sets theorem of general topology is generalized finally we introduce the fuzzy strongly semicontinuous fuzzy strongly semiopen and fuzzy strongly semiclosed mappings on fuzzy topological spaces and also establish some of their characteristic properties and discuss relations between fuzzy continuous fuzzy semicontinuous and fuzzy strongly semicontinuous mappings

the accuracy of approximate intervals for a binomial parameter abstract this article considers the uniform convergence of the coverage probabilities of some approximate confidence intervals for the binomial parameter p induced by central limit arguments a uniform upper bound on the coverage probabilities of any such interval obtained by transformation of the sample proportion pcirc n is derived the coverage probability of the interval induced by the arcsine transformation turns out to be very close to the upper bound replacing pcirc n by the bayes estimate x n 2 for some however gives even better uniform asymptotic properties the choice z 2 2 is recommended where z is the 1 th quantile of the standard normal distribution with such numerical results show that this interval is satisfactory even in the uniform sense since it possesses uniform asymptotic coefficients that are very close to the nominal ones furthermore the best k for of the form kz 2 such that the uniform asymptotic coefficient is as close to the nomin

choosing robust solutions in discrete optimization problems with fuzzy costs in this paper a wide class of discrete optimization problems which can be formulated as a 0 1 linear programming problem is discussed it is assumed that the objective function costs are not precisely known this uncertainty is modeled by specifying a finite set of fuzzy scenarios under every fuzzy scenario the costs are given as fuzzy intervals possibility theory is then applied to chose a solution in such a problem and mixed integer linear programming models are proposed to compute this solution

portfolio selection under independent possibilistic information this paper deals with a portfolio selection problem with independently estimated possibilistic return rates under such a circumstance a distributive investment has been regarded as a good solution in the traditional portfolio theory however the conventional possibilistic approach yields a concentrated investment solution considering the reason why a distributive investment is advocated a new approach to the possibilistic portfolio selection is proposed

multiobjective programming in optimization of the interval objective function abstract in this paper the mathematical programming problem whose objective function has interval coefficients is investigated to maximize the interval objective function the order relations which represent the decision maker s preference between interval profits are defined by the right limit the left limit the center and the width of an interval the maximization problem with the interval objective function is converted into a multiobjective problem using the order relations the minimization problem is also converted into a multiobjective problem using the order relations which represent the decision maker s preference between interval costs