Competence set expansion using an efficient 0-1 programming model

Competence set expansion is an application of habitual domain, through which a decision-maker acquires the skills required to make satisfactory decisions. The methods of Li (Operations Research 47, 1999, 209–220) and Li et al. (European Journal of Operational Research 120, 2000, 511–524) use numerous 0-1 and integer variables. This study presents a more compact model for the competence set expansion problem, which is as effective as the two methods yet more efficient.     

Optimal expansion and design of competence sets with asymmetric acquiring costs

This paper provides a method of finding the optimal expansion process and discusses the marginal analysis for expansion of the competence set when the cost functions are asymmetric. The concept of tree expansion process is introduced, and a method of finding the optimal tree expansion process is given. The paper also shows a way to identify the optimal competence set when both the expected return and cost are considered.     

Marginal analysis for competence set expansion

Given a set of decision environments, we explore the selection of the optimal competence set so as to maximize the expected net return. Marginal analysis is used to formulate the concepts. An elegant theory and subsequent algorithms are derived for special classes of problems.This research was supported in part by a Summer Research Grant from Hagan School of Business, Iona College, New Rochelle, New York.     

Optimal competence set expansion using deduction graphs

A competence set is a collection of skills used to solve a problem. Based on deduction graph concepts, this paper proposes a method of finding an optimal process so as to expand a decision maker's competence set to enable him to solve his problem confidently. Using the concept of minimum spanning tree, Yu and Zhang addressed the problem of the optimal expansion of competence sets. In contrast, the method proposed here enjoys the following advantages: it can deal with more general problems involving intermediate skills and compound skills; it can find the optimal solution by utilizing a 0–1 integer program; and it can be directly extended to treat multilevel competence set problems, and thus is more practically useful.This work was supported by the National Science Council, Taipei, Taiwan, Republic of China, Grant No. NSC-81-0301-H-009-501.     

A foundation for competence set analysis

For each decision problem, there is a competence set consisting of knowledge, information and skills for its effective solution. How the decision-maker acquires and expands his/her competence set plays a key role in the process and quality of decision-making. This paper provides a mathematical foundation for studying competence sets, their expansion processes and stability. Time and cost functions for expansion, reachable domains, effective expansion using minimal spanning trees, random set decomposition of competence sets, marginal analysis, connectivity, metrization and stability of competence sets are some key concepts introduced.     

Minimum Spanning Table and Optimal Expansion of Competence Set

Each directed graph with asymmetric costs defined over its arcs can be represented by a matrix or table, called an expansion table. We explore first the basic properties of cycles and spanning tables of expansion tables, which correspond to the cycles and spanning trees of the directed graph. Then, we derive an algorithm to find a minimum spanning table which corresponds to a minimum spanning tree in the directed graph. Finally, we discuss how to use the algorithm to find the optimal competence set expansion and also discuss related problems.     

Optimal Expansion of Competence Sets with Intermediate Skills and Compound Nodes

This paper provides a method for finding the optimal expansion process of the competence set in generalized situations in which the cost functions are asymmetric and the directed graphs, which depict all possible expansion processes, have cycles and contain intermediate skills and compound nodes.     

A vague set based decision support approach for evaluating research funding programs

Scientific Research Assessment (SRA) is receiving increasing attention in both academic and industry. More and more organizations are recognizing the importance of SRA for the optimal use of scarce resources. In this paper, a vague set theory based decision support approach is proposed for SRA. Specifically, a family of parameterized S-OWA operator is developed for the aggregation of vague assessments. The proposed approach is introduced to evaluate the research funding programs of the National Natural Science Foundation of China (NSFC). It provides a soft and expansive way to help the decision maker in NSFC to make his decisions. The proposed approach can also be used for some other agencies to make similar assessment.     

A Maclaurin-series expansion approach to multiple paired queues

Motivated by kitting processes in assembly systems, we consider a Markovian queueing system with K$K$ paired finite-capacity buffers. Pairing means that departures from the buffers are synchronised and that service is interrupted if any of the buffers is empty. To cope with the inherent state-space explosion problem, we propose an approximate numerical algorithm which calculates the first L$L$ coefficients of the Maclaurin series expansion of the steady-state probability vector in O(KLM)$O\left(KLM\right)$ operations, M$M$ being the size of the state space.     

Rough set approach to multiple criteria classification with imprecise evaluations and assignments

Dominance-based Rough Set Approach (DRSA) has been introduced to deal with multiple criteria classification (also called multiple criteria sorting, or ordinal classification with monotonicity constraints), where assignments of objects may be inconsistent with respect to dominance principle. In this paper, we consider an extension of DRSA to the context of imprecise evaluations of objects on condition criteria and imprecise assignments of objects to decision classes. The imprecisions are given in the form of intervals of possible values. In order to solve the problem, we reformulate the dominance principle and introduce second-order rough approximations. The presented methodology preserves well-known properties of rough approximations, such as rough inclusion, complementarity, identity of boundaries and precisiation. Moreover, the meaning of the precisiation property is extended to the considered case. The paper presents also a way to reduce decision tables and to induce decision rules from rough approximations.     

Optimal expansion of subspaces for eigenvector approximations

We consider the problem of how to expand a given subspace for approximating an eigenvalue and eigenvector of a matrix A. Specifically, we consider which vector in the subspace, after multiplied by A, provides optimal expansion of the existing subspace for the eigenvalue problem. We determine the optimal vector, when the quality of subspace for approximation is measured by the angle between the subspace and the eigenvector. We have also derived some characterization of the angle that might lead to more practically useful choice of the expansion vector.     

Definable predicates of standardness in internal set theory

It is shown that Nelson’s internal set theory IST has no definable predicate that is a proper extension of the standardness predicate and satisfies the carry-over, idealization, and standardization principles. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 803–809, December, 1999.     

Capacity switching options under rivalry and uncertainty

Deregulated infrastructure industries exhibit stiff competition for market share. Firms may be able to limit the effects of competition by launching new projects in stages. Using a two-stage real options model, we explore the value of such flexibility. We first demonstrate that the value of investing in a sequential manner for a monopolist is positive but decreases with uncertainty. Next, we find that a typical duopoly firm’s value relative to a monopolist’s decreases with uncertainty as long as the loss in market share is high. Intriguingly, this result is reversed for a low loss in market share. We finally show that this loss in value is reduced if a firm invests in a sequential manner and specify the conditions under which sequential capacity expansion is more valuable for a duopolist firm than for a monopolist.     

Stochastic capacity expansion with multiple sources of capacity

In this paper, we consider the multi-period single resource stochastic capacity expansion problem with three sources of capacity: permanent, contract, and spot market. The problem is modeled as a multi-stage stochastic integer program. We show that the problem has the totally unimodular property and develop polynomial-time primal and dual algorithms to solve the problem.     

On the chromatic number of set systems

An (r, l)‐system is an r‐uniform hypergraph in which every set of l vertices lies in at most one edge. Let m_k(r, l) be the minimum number of edges in an (r, l)‐system that is not k‐colorable. Using probabilistic techniques, we prove that where b_{r, l} is explicitly defined and a_{r, l} is sufficiently small. We also give a different argument proving (for even k) where a′_{r, l}=(r?l+1)/r(2^r?1re)^?l/(l?1). Our results complement earlier results of Erd?s and Lovász [10] who mainly focused on the case l=2, k fixed, and r large. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 87–98, 2001     

Structure and stability of triangle-free set systems

We define the notion of stability for a monotone property of set systems. This phenomenon encompasses some classical results in combinatorics, foremost among them the Erdos-Simonovits stability theorem. A triangle is a family of three sets such that , , are each nonempty, and . We prove the following new theorem about the stability of triangle-free set systems.

Fix . For every , there exist and such that the following holds for all : if and is a triangle-free family of -sets of containing at least members, then there exists an -set which contains fewer than members of .

This is one of the first stability theorems for a nontrivial problem in extremal set theory. Indeed, the corresponding extremal result, that for every triangle-free family of -sets of has size at most , was a longstanding conjecture of Erdos (open since 1971) that was only recently settled by Mubayi and Verstraëte (2005) for all .

     

Optimal control of harvest and bifurcation of a prey-predator model with stage structure

This paper describes a prey-predator model with stage structure for prey. The adult prey and predator populations are harvested in the proposed system. The dynamic behavior of the model system is discussed. It is observed that singularity induced bifurcation phenomenon is appeared when variation of the economic interest of harvesting is taken into account. State feedback controller is incorporated to stabilize the model system in case of positive economic interest. Harvesting of prey and predator population are used as controls to develop a dynamic framework to investigate the optimal utilization of the resource, sustainability properties of the stock and the resource rent earned from the resource. The Pontryagin’s maximum principle is used to characterize the optimal controls. The optimality system is derived and then solved numerically using an iterative method with Runge-Kutta fourth order scheme. Simulation results show that the optimal control scheme can achieve sustainable ecosystem.     

The strength of Mac Lane set theory

Saunders Mac Lane has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, , of Transitive Containment, we shall refer as . His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasises, one that is adequate for much of mathematics. In this paper we show that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke–Platek set theory and even the Axiom of Constructibility to Mac Lane's axioms; our method requires a close study of Axiom H, which was proposed by Mitchell; we digress to apply these methods to subsystems of Zermelo set theory , and obtain an apparently new proof that is not finitely axiomatisable; we study Friedman's strengthening of , and the Forster–Kaye subsystem of , and use forcing over ill-founded models and forcing to establish independence results concerning and ; we show, again using ill-founded models, that proves the consistency of ; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret and Boffa that proves a weak form of Stratified Collection, and that is a conservative extension of for stratified sentences, from which we deduce that proves a strong stratified version of ; we analyse the known equiconsistency of with the simple theory of types and give Lake's proof that an instance of Mathematical Induction is unprovable in Mac Lane's system; we study a simple set theoretic assertion—namely that there exists an infinite set of infinite sets, no two of which have the same cardinal—and use it to establish the failure of the full schema of Stratified Collection in ; and we determine the point of failure of various other schemata in . The paper closes with some philosophical remarks.