Increase In Difference Research Articles

Approximate solution of the p-median minimization problem

A version of the facility location problem (the well-known p-median minimization problem) and its generalization—the problem of minimizing a supermodular set function—is studied. These problems are NP-hard, and they are approximately solved by a gradient algorithm that is a discrete analog of the steepest descent algorithm. A priori bounds on the worst-case behavior of the gradient algorithm for the problems under consideration are obtained. As a consequence, a bound on the performance guarantee of the gradient algorithm for the p-median minimization problem in terms of the production and transportation cost matrix is obtained.

Core-based criterion for extreme supermodular functions

We give a necessary and sufficient condition for extremality of a supermodular function based on its min-representation by means of (vertices of) the corresponding core polytope. The condition leads to solving a certain simple linear equation system determined by the combinatorial core structure. This result allows us to characterize indecomposability in the class of generalized permutohedra. We provide an in-depth comparison between our result and the description of extremality in the supermodular/submodular cone achieved by other researchers.

MINIMIZATION OF SUPERMODULAR FUNCTIONS

In this paper we consider three principles of discard to nd the minimum of a supermodular function which is dened on the family of subsets of a given nite set.

Supermodular covering knapsack polytope

The supermodular covering knapsack set is the discrete upper level set of a non-decreasing supermodular function. Submodular and supermodular knapsack sets arise naturally when modeling utilities, risk and probabilistic constraints on discrete variables. In a recent paper Atamtürk and Narayanan (2009) study the lower level set of a non-decreasing submodular function.In this complementary paper we describe pack inequalities for the supermodular covering knapsack set and investigate their separation, extensions and lifting. We give sequence-independent upper bounds and lower bounds on the lifting coefficients. Furthermore, we present a computational study on using the polyhedral results derived for solving 0–1 optimization problems over conic quadratic constraints with a branch-and-cut algorithm.

Optimal dynamic trade credit and preservation technology allocation for a deteriorating inventory model

In this paper, we examine an optimal dynamic decision-making problem for a retailer selling a single deteriorating product, the demand rate of which varies simultaneously with time and the length of credit period that is offered to the customers. The deterioration rate is time dependent and can be reduced by an investment in preservation technology. In addition, the risk of default increases with the credit period length. A generalized model is presented to determine the optimal trade credit, preservation technology investment and replenishment strategies that maximize the retailer’s total profit after the default risk occurs over a finite planning horizon. Under certain conditions, we first establish a single period deteriorating inventory model, and then provide a comparative statics analysis that characterizes the impacts of key parameters on the retailer’s optimal trade credit and preservation technology investment decisions by using the properties of the supermodular function. Dynamic programming is then used to solve the proposed model by employing the obtained theoretical results. At the end of this paper, some numerical examples and the results of a sensitivity analysis are used to illustrate the features of the proposed model; we then offer our concluding remarks.

Bequest games with unbounded utility functions

The existence of Markov perfect equilibria for a deterministic bequest game with bounded from below utility function was established independently by Bernheim and Ray [2] and Leininger [8]. The aim of this paper is to provide a simpler proof that, in contrast to [2,8], also works for models with unbounded, e.g., isoelastic utility functions. Our approach employs the concept of strict single crossing property, introduced by Milgrom and Shannon in [10], and includes supermodular functions as special cases.

Studying mixability with supermodular aggregating functions

We introduce the concepts of ϕ-complete mixability and ϕ-joint mixability and we investigate some necessary and sufficient conditions to the ϕ-mixability of a set of distribution functions for some supermodular functions ϕ. We give examples and numerical verifications which confirm our findings.

Basic properties of lattices of cubes, algorithms for their construction, and application capabilities in discrete optimization

The basic properties of a new type of lattices—a lattice of cubes—are described. It is shown that, with a suitable choice of union and intersection operations, the set of all subcubes of an N-cube forms a lattice, which is called a lattice of cubes. Algorithms for constructing such lattices are described, and the results produced by these algorithms in the case of lattices of various dimensions are illustrated. It is proved that a lattice of cubes is a lattice with supplements, which makes it possible to minimize and maximize supermodular functions on it. Examples of such functions are given. The possibility of applying previously developed efficient optimization algorithms to the formulation and solution of new classes of problems on lattices of cubes.

Characterizing Mixability of Supermodular Functions

We introduce the concepts of φ-complete mixability and φ-joint mixability and we investigate some necessary and sufficient conditions to the φ-mixability of a set of distribution functions for some supermodular functions φ. We give examples and numerical verifications which confirm our findings.

Computation of Sharp Bounds on the Expected Value of a Supermodular Function of Risks with Given Marginals

We show that the rearrangement algorithm (RA) introduced in Puccetti and Rüschendorf (2012) to compute distributional bounds can be used also to compute sharp lower and upper bounds on the expected value of a supermodular function of d random variables having fixed marginal distributions. Compared to the analytical methods existing in the literature the algorithm is widely applicable, more easily obtained and gives insight into the dependence structures attaining the bounds.

On the relationship between modular functions and copulas

Copulas are nothing else but supermodular functions with absorbing element 0 and neutral element 1. Although a copula C cannot be modular on the unit square itself, it is effectively so on every rectangle with zero C-volume. In this paper, we show that an appropriate modular function on the unit square can be transformed into a copula by performing a simple truncation operation (either from below or above to ensure the compatibility with the Fréchet–Hoeffding bounds). Modular functions on the unit square admit an additive representation in terms of two univariate functions. We specifically focus on cases where these univariate functions are sections (horizontal, vertical, diagonal) of a given copula. Ample illustrations are provided.

Hub Location as the Minimization of a Supermodular Set Function

This paper highlights how a general class of hub location problems can be modeled as the minimization of a real-valued supermodular set function. Well-known problems such as uncapacitated hub location, p-hub median, and hub arc location, among others, are shown to be particular cases of this class. Two integer programming formulations are introduced and compared. One uses path-based variables, frequently employed in hub location, whereas the other exploits properties of supermodular functions. In addition, several worst case bounds for a greedy and a local improvement heuristic are obtained for the general class and for some particular cases in which sharper bounds can be devised. Computational experiments are performed to compare both formulations when used with a general purpose solver. Computational results obtained on benchmark instances confirm the superiority of the supermodular formulation.

Testing the Supermodular-Cut Condition

For a function $f:2^{V} \to \mathbb {Z}_{+}$ on a finite set V with f(?)=f(V)=0, a digraph D=(V,A) is called f-connected if it satisfies the f-cut condition, that is, ? D (X)?f(X) for any X⊆V, where ? D (X) is the number of arcs from X to V?X. We show that, for any crossing supermodular function f, the f-connectivity can be tested with a constant number of queries in the general digraph model with average degree bound. As immediate corollaries, we obtain constant-time testers for k-edge-connectivity, rooted-(k,l)-edge-connectivity, and the property of having k arc-disjoint arborescences. We also give a corresponding result for the undirected case.

Stochastic Comparisons of Symmetric Supermodular Functions of Heterogeneous Random Vectors

Consider random vectors formed by a finite number of independent groups of independent and identically distributed random variables, where those of the last group are stochastically smaller than those of the other groups. Conditions are given such that certain functions, defined as suitable means of supermodular functions of the random variables of the vectors, are supermodular or increasing directionally convex. Comparisons based on the increasing convex order of supermodular functions of such random vectors are also investigated. Applications of the above results are then provided in risk theory, queueing theory, and reliability theory, with reference to (i) net stop-loss reinsurance premiums of portfolios from different groups of insureds, (ii) closed cyclic multiclass Gordon-Newell queueing networks, and (iii) reliability of series systems formed by units selected from different batches.

Stochastic Comparisons of Symmetric Supermodular Functions of Heterogeneous Random Vectors

Consider random vectors formed by a finite number of independent groups of independent and identically distributed random variables, where those of the last group are stochastically smaller than those of the other groups. Conditions are given such that certain functions, defined as suitable means of supermodular functions of the random variables of the vectors, are supermodular or increasing directionally convex. Comparisons based on the increasing convex order of supermodular functions of such random vectors are also investigated. Applications of the above results are then provided in risk theory, queueing theory, and reliability theory, with reference to (i) net stop-loss reinsurance premiums of portfolios from different groups of insureds, (ii) closed cyclic multiclass Gordon-Newell queueing networks, and (iii) reliability of series systems formed by units selected from different batches.

Enactment of Stochastic Model Stream Mechanism on Multi class Queueing Network

Over the last few decades stream control mechanism has more attention in the early stages of research in communication network. The motivation of this work is deep understanding on the stream control mechanism problem in multi-class networks. In this paper, game theoretic perspectives are presented and the appropriate frame work for the study of stream control mechanism problem is analysed. Consider a min-max routing problem, where the control mechanism has to decide to which of N queues that the arriving customer should be sent. The service rate in each queue is dependent on the state of the system, may change in time and is unknown to the control mechanism. The goal of the control mechanism is to design an efficient policy which guarantees the best performance under the worst case service conditions. ie., Arriving customers are routed by a control mechanism, with the purpose of minimizing the total discounted holding cost under worst-case service conditions. Then the problem is viewed as a zero-sum (Stochastic)Markov game between the routing control mechanism and a service control mechanism. In zero-sum (Stochastic)Markov game, where the server acts as player 1 and the stream control mechanism acts as a player 2. Each player is assumed to have the information of all the previous actions of both players as well as the current and the past states of the system. The main results obtained are to identify the optimal strategy for both players. A value iteration technique is used to establish properties of the value of the game, which are related to super modularity and convexity.

Complementarities between innovation policies in emerging economies. The case of Argentina's software sector

Fil: Motta, Jorge Jose. Universidad Nacional de Cordoba. Facultad de Ciencias Economicas; Argentina

Pricing and Lead Time Competition in the Presence of Process Variability

Managing variability in both arrival and production processes has been a central challenge for both manufacturing and service industries. Models have been proposed to study the negative impact of higher process variability on operational performance. In the make to order context, the variability can influence the lead time decision and hence both the tardiness costs and the chances of winning customer orders. In this paper, we investigate the strategic role of process variability. Specifically, we consider two make to order firms that have different variability characteristics and study how they affect the firm’s strategic decisions namely lead time and price and subsequently profits. We first show that the game has a unique Nash equilibrium by using super modularity and contraction-mapping methods. We then propose an algorithm that efficiently computes the Nash equilibrium. Numerical experiments demonstrate how the strategic influence of variability depends upon business and operational characteristics.

Explicit convex and concave envelopes through polyhedral subdivisions

In this paper, we derive explicit characterizations of convex and concave envelopes of several nonlinear functions over various subsets of a hyper-rectangle. These envelopes are obtained by identifying polyhedral subdivisions of the hyper-rectangle over which the envelopes can be constructed easily. In particular, we use these techniques to derive, in closed-form, the concave envelopes of concave-extendable supermodular functions and the convex envelopes of disjunctive convex functions.

Supermodular programming on finite lattices

There is a discription of the problems of minimization of supermodular functions on the different types of lattices: Boolean lattices, lattices with relative supplements (division lattices, lattices of vector subspaces of finite-dimensional vector space, geometrical lattices), lattices equal to Cartesian product of chains. The previously obtained theoretical results, on the basis of which the problems of minimization of supermodular functions on these lattices have been solved, are shown. A new type of lattices, lattice of Cubes, is defined and described. The problems of minimization and maximization of supermodular functions are considered on it. Particular examples of such functions are given. Optimization algorithms and the possibilities of setting and solving a new class of problems on the lattices of Cubes are discussed.