optimal ordering policy for inventory systems with quantity dependent setup costs we consider a continuous review inventory system in which the setup cost of each order is a general function of the order quantity and the demand process is modeled as a brownian motion with a positive drift assuming the holding and shortage cost to be a convex function of the inventory level we obtain the optimal ordering policy that minimizes the long run average cost by a lower bound approach to tackle some technical issues in the lower bound approach under the quantity dependent setup cost assumption we establish a comparison theorem that enables one to prove the global optimality of a policy by examining a tractable subset of admissible policies since the smooth pasting technique does not apply to our brownian inventory model we also propose a selection procedure for computing optimal policy parameters when the setup cost is a step function

amenable groups and stabilizers of measures on the boundary of a hadamard manifold let x be a symmetric space of noncompact type or more generally a hadamard manifold i e a complete simply connected riemannian manifold of nonpositive sectional curvature we consider a finite positive borel measure on the ideal boundary x see sect 2 for the definitions let g be the isometry group of x and g the subgroup which stabilizes the measure in the case of a symmetric space we obtain the following result

a solvable one dimensional model of a diffusion inventory system we analyse the optimal ordering policy for impulse control of a one product inventory system subject to a demand modelised by a diffusion process the purpose is to minimize the expected discounted cost that includes a fixed set up cost and linear costs of purchase storage and shortage the optimal cost is explicitly obtained as the smoothest solution of a quasi variational inequality derived from the optimal principle of dynamic programming the optimal s s policy is determined as the unique solution of a system of algebraic equations

pseudo integral and generalized choquet integral abstract due to many applications the choquet integral as a powerful tool for modeling non deterministic problems needs to be further extended therefore the paper is devoted to a generalization of the choquet integral as a basis the pseudo integral for bounded integrand is extended to the case for nonnegative integrands at first and then the generalized choquet integral is defined as special cases pseudo choquet stieltjes integrals pseudo fuzzy stieltjes integrals g choquet integrals pseudo n fuzzy integrals and pseudo s fuzzy integrals are obtained and various kinds of properties and convergence theorems are shown meanwhile markov jensen minkowski and holder inequalities are proved in the end the generalized discrete choquet integral is discussed the obtained results for the generalized choquet integral cover some previous results on different types of nonadditive integrals

a possibility measure is not a fuzzy measure abstract in this note we show that a possibility measure is not a particular type of fuzzy measure except in trivial cases

lebesgue theorems in non additive measure theory in this paper the sufficient and necessary conditions for different kinds of lebesgue theorem in non additive measure theory are presented respectively the equivalence among the lebesgue theorem the monotone convergence theorems of fuzzy and of choquet integral are shown as direct results of the lebesgue theorems the necessary conditions for egoroff s theorem are given

autocontinuity and convergence theorems for the choquet integral we provide convergence theorems for the choquet integral for various notions of convergence monotone almost uniform convergence from above below is associated to monotone autocontinuity from above below the dominated almost uniform convergence is associated to monotone autocontinuity autocontinuity can be associated to the dominated convergence in measure in mean or in strict measure

a unified approach to the monotone convergence theorem for nonlinear integrals abstract we present a unified approach to the monotone convergence theorem for nonlinear integrals such as the choquet the sipos the sugeno and the shilkret integral a nonlinear integral may be viewed as a nonlinear functional defined on a set of pairs of a nonadditive measure and a measurable function we thus formulate our general type of monotone convergence theorem for such a functional the key tool is a perturbation of functional that manages not only the monotonicity of the functional but also the small change of the functional value arising as a result of adding small amounts to a measure and a function in the domain of the functional our approach is also applicable to the lebesgue integral when a nonadditive measure is additive

amenable isometry groups of hadamard spaces one of the ever recurring themes in the theory of nonpositively curved spaces is the relation between geometry and topology since the universal covering space of a complete space of nonpositive curvature is contractible this amounts to relations between the geometry of the space or its universal covering space and the algebraic structure of its fundamental group a paradigm is the well known result of avez av that the fundamental group of a compact riemannian manifold m of nonpositive sectional curvature has exponential growth if and only if m is not flat gromov observed that the proof of avez yields a somewhat stronger statement the fundamental group of m is amenable if and only if m is flat see g1 p 93 zimmer generalized this to the case where m is complete and of finite volume zi further generalizations were obtained by anderson an and burger and schroeder bs the objective of this paper is the discussion of these results under the weaker assumption that the underlying spaces are alexandrov spaces of nonpositive curvature for the investigation of relations between the fundamental group and the geometry of a space of nonpositive curvature it is convenient to study the isometric action of the fundamental group on the universal covering space in many situations it is not necessary to assume that a group action arises in this particular way and one studies isometric actions on complete simply connected spaces of nonpositive curvature this is the case in most of our arguments below let x be a hadamard space that is a complete simply connected geodesic space of nonpositive curvature in the sense of alexandrov a flat in x is a

optimal impulse control when control actions have random consequences we consider a generalised impulse control model for controlling a process governed by a stochastic differential equation the controller can only choose a parameter of the probability distribution of the consequence of his control action which is therefore random we state optimality results relating the value function to quasi variational inequalities and a formal optimal stopping problem we also remark that the value function is a viscosity solution of the quasi variational inequalities which could lead to developments and convergence proofs of numerical schemes further we give some explicit examples and an application in financial mathematics the optimal control of the exchange rate

some quantities represented by the choquet integral this paper shows that the choquet integral can represent some useful quantities such as supremum infimum essential supremum essential infimum mean median quantile 100 percentile and l estimators including trimmed mean

the autocontinuity of set function and the fuzzy integral abstract the autocontinuity and some other concepts of a set function are introduced and sugeno s fuzzy measure with some annexed conditions is studied on a fuzzy measure space egoroffs theorem is proved it is also proved that f mean convergence is equivalent to convergence in measure for a sequence of measurable functions finally some convergence theorems of sequence of fuzzy integrals are proved especially using the concept of autocontinuity a necessary and sufficient condition for the convergence in measure theorem is given

relationship among continuity conditions and null additivity conditions in non additive measure theory the paper shows all the implication relationship among six continuity conditions and two null additivity conditions with respect to non additive measures the six continuity conditions are continuity from above order continuity strong order continuity exhaustivity continuity from below and null continuity and the two null additivity conditions are null additivity and weak null additivity the implication relationship is summarized in a diagram

on egoroff s theorems on fuzzy measure spaces in this paper we show that egoroff s theorem which is well known in classical measure theory remains valid for a fuzzy measure without any additional condition previous results obtained by wang zhenyuan and others are improved egoroff s theorem on fuzzy measure spaces is formulated in full generality taylor s theorem which concerns convergence almost everywhere in classical measure theory is also extended to fuzzy measure spaces

convergence theorems for monotone measures in classical measure theory there are a number of convergence theorems such as the egorov the riesz and the lusin theorem among others we consider monotone measures i e monotone set functions vanishing in the empty set and defined on a measurable space and discuss how and to which extent classical convergence theorems can be carried over to this more general case

impulse control of brownian motion consider a storage system such as an inventory or cash fund whose content fluctuates as a 2 brownian motion in the absence of control holding costs are continuously incurred at a rate proportional to the storage level and we may cause the storage level to jump by any desired amount at any time except that the content must be kept nonnegative both positive and negative jumps entail fixed plus proportional costs and our objective is to minimize expected discounted costs over an infinite planning horizon a control band policy is one that enforces an upward jump to q whenever level zero is hit and enforces a downward jump to q whenever level s is hit 0 q q s we prove the existence of an optimal control band policy and calculate explicitly the optimal values of the critical numbers q q s

on sufficient conditions for the egoroff theorem of an ordered topological vector space valued non additive measure in this paper we consider an ordered vector space endowed with a locally full topology which is called an ordered topological vector space we show that the egoroff theorem remains valid for the ordered topological vector space valued non additive measure in the following four cases the first case is that the measure is strongly order totally continuous the second case is that the measure is strongly order continuous and possesses an additional continuity property suggested by sun in 1994 when the ordered topological vector space has a certain property the third case is that the measure is continuous from above and below when the topology is locally convex the fourth case is that the measure is uniformly autocontinuous from above continuous from below and strongly order continuous when the topology is locally convex our results are applicable to several ordered topological vector spaces

order continuous of monotone set function and convergence of measurable functions sequence in this paper we introduce the concept of strongly order continuity of a set function and use it to investigate the convergence of measurable functions sequence and of integral sequence on fuzzy measure space several equivalent conditions of strongly order continuity are presented it is shown that a necessary and sufficient condition that lebesgue s theorem remains valid for a monotone set function is that the set function possesses strongly order continuity

autocontinuity convergence in measure and convergence in distribution abstract it is shown that the autocontinuity is equivalent to the property that the convergence in measure implies the convergence in distribution this result is applied to denneberg s dominated convergence theorem for the choquet integral

property s of fuzzy measure and riesz s theorem abstract in this note a sufficient and necessary condition for riesz s theorem to hold in the fuzzy measure space is given

explicit solution of a two dimensional deterministic inventory problem we determine the optimal ordering policy for impulse control of a deterministic two product inventory system subject to constant demand rates linear storage and shortage costs and economies of joint ordering the optimal cost is explicitly obtained as the smoothest solution of a two dimensional quasi variational inequality exact construction of the boundary of the continuation set leads to the optimal ordering policy extensions to independent products average cost and n product case n 2 are also briefly considered

the vitali convergence in measure theorem of nonlinear integrals abstract the vitali convergence in measure theorem for the abstract lebesgue integral is fundamental in lebesgue integration theory and yields the bounded convergence theorem and the dominated convergence theorem as its applications in this paper in a unified way using the perturbation method the vitali convergence in measure theorem is established for nonlinear integrals such as the choquet sipos sugeno and shilkret integrals and their symmetric and asymmetric extensions it is derived from the fatou and the reverse fatou type lemmas for perturbative nonlinear integral functionals

on egoroff s theorems on finite monotone non additive measure space in this note we give four versions of egoroff s theorem in non additive measure theory by using condition e the pseudo condition e of set function and the duality relations between the conditions these conditions offered are not only sufficient but also necessary for the four kinds of egoroff s theorems

conditions for egoroff s theorem in non additive measure theory abstract this paper gives necessary and or sufficient conditions for egoroff s theorem in non additive measure theory a necessary and sufficient condition described without measurable functions two sufficient conditions and a necessary condition one of the two sufficient conditions is strong order total continuity continuity at measurable sets of measure zero with respect to net convergence and the other is strong order continuity sequential continuity at measurable sets of measure zero together with property s the necessary condition is strong order continuity in addition the paper shows the following continuity from above and below which is a known sufficient condition and the above mentioned two sufficient conditions are independent of each other the disjunction of these three sufficient conditions is not a necessary condition if the underlying set is at most countable then strong order continuity is necessary and sufficient and generally strong order continuity is not sufficient

on the null additivity and the autocontinuity of a fuzzy measure abstract in this paper further research on the null additivity and autocontinuity of fuzzy measures is made we prove that autocontinuity from below and autocontinuity from above of a fuzzy measure are equivalent when this fuzzy measure is finite furthermore if the space x is countable then the null additivity is equivalent to the autocontinuity

optimal policies for brownian inventory systems with a piecewise linear ordering cost we consider a stochastic inventory system with general piecewise linear ordering cost the cumulative demand is modeled as a brownian motion process the ordering cost function is neither convex nor concave it may not be monotone and it is not even necessarily continuous and it includes most ordering cost functions studied in the literature e g economies of scale or dis economies of scale all unit discount or incremental discount and multiple setup costs as special cases in addition to ordering cost the system incurs the usual holding shortage cost and the objective is to minimize the average system cost per unit of time despite the complexity in the ordering cost function we show that an optimal control policy is very simple it is either an s s policy or a one sided singular control policy

generalized convergence theorems for monotone measures abstract in this paper we propose three types of absolute continuity for monotone measures and present some of their basic properties by means of these three types of absolute continuity we establish generalized egoroff s theorem generalized riesz s theorem and generalized lebesgue s theorem in the framework involving the ordered pair of monotone measures the egoroff theorem the riesz theorem and the lebesgue theorem in the traditional sense concerning a unique monotone measure are extended to the general case these three generalized convergence theorems include as special cases several previous versions of egoroff like theorem riesz like theorem and lebesgue like theorem for monotone measures respectively

asymptotic structural characteristics of fuzzy measure and their applications abstract in fuzzy measure theory as sugeno s fuzzy measures lose additivity in general the concept almost which is well known in classical measure theory splits into two different concepts almost and pseudo almost in order to replace the additivity it is quite necessary to investigate some asymptotic behaviors of a fuzzy measure at sequences of sets which are called waxing and waning and to introduce some new concepts such as autocontinuity converse autocontinuity and pseudo autocontinuity these concepts describe some asymptotic structural characteristics of a fuzzy measure in this paper by means of the asymptotic structural characteristics of fuzzy measure we also give four forms of generalization for both egoroff s theorem riesz s theorem and lebesgue s theorem respectively and prove the almost everywhere pseudo almost everywhere convergence theorem the convergence in measure pseudo in measure theorem of the sequence of fuzzy integrals in the last two theorems the employed conditions are not only sufficient but also necessary

impulse control of brownian motion the constrained average cost case when a manufacturer places repeated orders with a supplier to meet changing production requirements he faces the challenge of finding the right balance between holding costs and the operational costs involved in adjusting the shipment sizes we consider an inventory whose content fluctuates as a brownian motion in the absence of control at any moment a controller can adjust the inventory level by any positive or negative quantity but incurs both a fixed cost and a cost proportional to the magnitude of the adjustment the inventory level must be nonnegative at all times and continuously incurs a linear holding cost the objective is to minimize long run average cost we show that control band policies are optimal for the average cost brownian control problem and explicitly calculate the parameters of the optimal control band policy this form of policy is described by three parameters q q s 0 q q s when the inventory falls to zero rises to s the controller expedites curtails shipments to return it to q q employing apparently new techniques based on methods of lagrangian relaxation we show that this type of policy is optimal even with constraints on the size of adjustments and on the maximum inventory level we also extend these results to the discounted cost problem

sequentially continuous non monotonic choquet integrals schmeidler proc amer math soc 97 2 1986 255 261 established an integral representation theorem through the choquet integral for functionals satisfying monotonicity and a weaker condition than additivity namely comonotonic additivity murofushi sugeno machida fuzzy sets and systems 64 1994 73 86 generalize this representation to the case of bounded variation functionals omitting the monotonicity condition we give an alternative approach which is based on sequential continuity and tolerates non monotonicity this later condition is equivalent to s additivity in measure theory

optimal inventory control with jump diffusion and nonlinear dynamics in the demand in this paper we consider an inventory control problem with a nonlinear evolution and a jump diffusion demand model this work extends the earlier inventory model proposed by benkherouf and johnson math methods oper res 76 2012 pp 377 393 by including a general jump process however as those authors note their techniques are not applicable to models with demand driven by jump diffusion processes with drift therefore the combination of diffusion and general compound poisson demands is not completely solved from the dynamic programming principle we transform the problem into a set of quasi variational inequalities q v i the difficulty arises when solving the q v i because the second derivative and integration term appear in the same inequality our technique is to construct a set of coupled auxiliary functions then an analytical study of the q v i implies the existence and uniqueness of an s s policy

a set valued egoroff type theorem in this paper an egoroff type theorem for a fuzzy multimeasure defined on a s algebra of subsets of an abstract space and taking values in the family of all nonvoid closed subsets of a real normed space is established some necessary and or sufficient conditions under which egoroff s theorem is valid are also presented

optimal control of a brownian production inventory system with average cost criterion in this paper we study a stochastic production inventory system with finite production capacity and random demand the cumulative production and demand are modeled by a two dimensional brownian motion process there is a setup cost for switching on the production and a convex holding and shortage cost and our objective is to find the optimal production inventory control that minimizes the average cost both lost sales and backlog cases are studied for the lost sales model we show that within a large class of policies the optimal production strategy is either to produce according to an s s policy or never turn on the machine at all thus it is optimal for the firm to not enter the business whereas for the backlog model we prove that the optimal production policy is always of the s s type our approach first develops a lower bound for the average cost among a large class of nonanticipating policies and then shows that the value function of the desired policy reaches the lower bound the results offer insights on the structure of the optimal control policies as well as the interplays between system parameters

on a stochastic inventory model with a generalized holding costs this paper is concerned with finding the optimal replenishment policy for an inventory model that minimizes the total expected discounted costs over an infinite planning horizon the demand is assumed to be driven by a brownian motion with drift and the holding costs inventory and shortages are assumed to take some general form this generalizes the earlier work where holding costs were assumed linear it turns out that problem of finding the optimal replenishment schedule reduces to the problem of solving a quasi variational inequality problem qvi this qvi is then shown to lead to an s s policy where s and s are determined uniquely as a solution of some algebraic equations

smooth fit principle for impulse control of multidimensional diffusion processes value functions of impulse control problems are known to satisfy quasi variational inequalities qvis a bensoussan and j l lions impulse control and quasivariational inequalities heyden son philadelphia 1984 translation of controle impulsionnel et inequations quasi variationnelles gauthier villars paris 1982 this paper proves the smooth fit c 1 property of the value function for multidimensional controlled diffusions using a viscosity solution approach we show by examples how to exploit this regularity property to derive explicitly optimal policy and value functions

inventory management with stochastic lead times this article analyzes a continuous time back ordered inventory system with stochastic demand and stochastic delivery lags for placed orders this problem in general has an infinite dimensional state space and is hence intractable we first obtain the set of minimal conditions for reducing such a system s state space to one dimension and show how this reduction is done next by modeling demand as a diffusion process we reformulate the inventory control problem as an impulse control problem we simplify the impulse control problem to a quasi variation inequality qvi based on the qvi formulation we obtain the optimality of the s s policy and the limiting distribution of the inventory level we also obtain the long run average cost of such an inventory system finally we provide a method to solve the qvi formulation using a set of computational experiments we show that significant losses are incurred in approximating a stochastic lead time system with a fixed lead time system thereby highlighting

explicit solution of inventory problems with delivery lags we consider a continuous time inventory system with fixed delivery lag subject to a demand modelled as a diffusion process with drift excess demand is backlogged we prove that the optimal ordering policy is a function of the sum of the stock on hand and the stock ordered but not yet delivered moreover we state a relation linking the value function when orders are pending with the value function when no order is pending as a consequence the a priori infinite dimensional quasi variational inequality qvi satisfied by the value function reduces to a finite dimensional one the one product inventory problem is then solved explicitly in the case of linear holding and shortage costs with fixed and proportional ordering cost the optimal policy is determined it is an s s policy applied to the sum of stock on hand and orders pending which means that when this sum decays below a critical level s an order to level s is placed

a diffusion inventory model for deteriorating items this paper is concerned with finding the optimal impulse control schedule for a stochastic inventory model for deteriorating items that minimizes the total expected discounted cost over an infinite horizon it turns out that the optimal impulse control policy can be found as a solution of a quasi variational inequality qvi problem of which details are provided further the problem of finding the optimal impulse control policy that minimizes the total expected costs per unit time is addressed

structure of manifolds of nonpositive curvature clearly 1 rank m dim m we may regard rank m as a measurement of the flatness of m since a parallel field x along yv and perpendicular to yv is a jacobi field if and only if the sectional curvatures of the planes i t a x t are zero for all t e r in particular m is flat if rank m dim m hence a compact surface of nonpositive curvature has rank one unless it is a torus or a klein bottle it also follows that rank m1 x m2 rank ml rank m2 finally it is important that the definition of rank m above coincides with the usual one if m is locally symmetric

an interpretation of fuzzy measures and the choquet integral as an integral with respect to a fuzzy measure abstract in this paper the non additivity of sugeno s fuzzy measures is interpreted in terms of addition and the rationality of the choquet integral is discussed it is pointed out that a fuzzy measure on a set x expresses the interaction between the subsets of x and can be represented by an additive measure it is shown through concrete examples that the choquet integral is reasonable as an integral with respect to a fuzzy measure it is also found that the choquet integral is closely related with the representation of a fuzzy measure

autocontinuity of set valued fuzzy measures and applications the main object of this paper is to further the study of set valued fuzzy measures introduced by gavrilut in 2009 some asymptotic structural characteristics of set valued fuzzy measures such as null additivity and autocontinuity are investigated in terms of applications convergence theorems for sequences of measurable functions are established

a theory of fuzzy measures representations the choquet integral and null sets abstract some theorems of t murofushi and m sugeno fuzzy sets and systems 29 1989 201 227 concerning representation of fuzzy measures and the choquet integral are generalized it is shown that if there is a certain relation between two measurable functions then the choquet integral is additive for these two functions in addition this article discusses null sets with respect to fuzzy measures and also fuzzy measures defined on a class which is not a algebra