Capturing decision maker preference: Experimental comparison of decision analysis and MCDM techniques

This paper describes the results of a laboratory study which investigates preference in decision making under certainty with multiple, conflicting objectives and continuous decision variables. Techniques for solving such problems are taken from the fields of decision analysis and optimization: the SMART technique for the former and both the NAIVE and Zionts-Wallenius techniques for the latter. The purpose of the experiment is to determine the ability of each technique to correctly capture decision maker preference. In addition, the relative preference of the decision maker for each technique was obtained. The experiment was conducted on a random sample of business school undergraduates and involved a decision with three criteria. The results give insight into the use of several techniques when confronted with decisions with multiple criteria.     

Attitudes and preferences: Approaches to representing decision maker desires

Informative interrelationships between attitudes and preferences when formally studying a conflict within the framework of the Graph Model for Conflict Resolution are specified through the development of formal theorems. Specifically, the parallel nature of attitudes and preferences is illuminated to show how attitudes can provide a simple way to more effectively represent complex preference structures when decision makers hold concern for other decision makers’ interests. To illustrate this important property of attitudes, the Graph Model for Conflict Resolution is applied to the negotiation over the sale of a brownfield property both with and without attitudes.     

Set choice problems with incomplete information about the preferences of the decision maker

We consider the problem of selecting a predetermined number of objects from a given finite set. It is assumed that the preferences of the decision maker on this set are only partially known. Our solution approach is based on the notions of optimal and non-dominated subsets. The properties of such subsets and the objects they contain are investigated. The implementation of the developed approach is discussed and illustrated by various examples.     

On the finiteness property for rational matrices

We analyze the periodicity of optimal long products of matrices. A set of matrices is said to have the finiteness property if the maximal rate of growth of long products of matrices taken from the set can be obtained by a periodic product. It was conjectured a decade ago that all finite sets of real matrices have the finiteness property. This “finiteness conjecture” is now known to be false but no explicit counterexample is available and in particular it is unclear if a counterexample is possible whose matrices have rational or binary entries. In this paper, we prove that all finite sets of nonnegative rational matrices have the finiteness property if and only if pairs of binary matrices do and we state a similar result when negative entries are allowed. We also show that all pairs of 2×2 binary matrices have the finiteness property. These results have direct implications for the stability problem for sets of matrices. Stability is algorithmically decidable for sets of matrices that have the finiteness property and so it follows from our results that if all pairs of binary matrices have the finiteness property then stability is decidable for nonnegative rational matrices. This would be in sharp contrast with the fact that the related problem of boundedness is known to be undecidable for sets of nonnegative rational matrices.     

A degree property of rational matrices

In this paper, the existence of a relationship between the degrees of polynomials in any given rational matrix and the degrees of the polynomials in the inverse of that matrix is explained and proved.     

On a property of Cauchy-like matrices

We introduce in this Note a property connecting the generators of a Cauchy-like matrix to the generators of its Schur complements. This property (formulas (6)) yields to a recursive process that can be used to devise a fast and direct inversion algorithm for Cauchy-like matrices (Section 4). The approach considered here is substantially simpler than those developed in some recent works on the subject ([2], [4], [6], [7], [8]).     

On a spectral property of Jacobi matrices

Let be a Jacobi matrix with elements on the main diagonal and elements on the auxiliary ones. We suppose that is a compact perturbation of the free Jacobi matrix. In this case the essential spectrum of coincides with , and its discrete spectrum is a union of two sequences 2, x^-_j<-2$">, tending to . We denote sequences and by and , respectively.

The main result of the note is the following theorem.

Theorem.     Let be a Jacobi matrix described above and be its spectral measure. Then if and only if

-\infty,\qquad {ii)} \sum_j(x^\pm_j\mp2)^{7/2}<\infty. \end{displaymath}">

     

A combinatorial property for semigroups of matrices

In this paper we study the semigroups of matrices over a commutative semiring. We prove that a semigroup of matrices over a tropical semiring satisfies a combinatorial property called weak permutation property. We consider an application of this result to the Burnside problem for groups.     

A property of matrices with positive determinants

Let $\text{P}$ be the set of all n × n real matrices which have a positive determinant. We show here that at least 2^{n ? 1} matrices are needed to “see” each matrix in $\text{P}$. Also, any finite subset of $\text{P}$ can be “seen” from a class of at most 2^{n ? 1} matrices in $\text{P}$.     

Upper triangular operator matrices with the single-valued extension property

If A=(Aij)_1?i,j?n∈B(X) is an upper triangular Banach space operator such that AiiAij=AijAjj for all 1?i?j?n, then A has SVEP or satisfies (Dunford's) condition (C) or (Bishop's) property (β) or (the decomposition) property (δ) if and only if Aii, 1?i?n, has the corresponding property.     

λ型传递矩阵与多因素模糊决策

基于已知常用的模糊矩阵传递性概念定义了λ型传递。给出了它的几个等价条件。研究它的图论特征，指出λ型传递矩阵的圈都过强对边二元圈。随后证明了与全传递模糊矩阵的等价性。进一步研究与截矩阵的性质一致问题，证明了λ型传递满足一致性。最后给出λ型传递在模糊排充中的应用，表明它是一种新的实用的多因素模糊决策的数学模型。     

A new characterization of matrices with the consecutive ones property

We consider the following constraint satisfaction problem: Given a set F of subsets of a finite set S of cardinality n, and an assignment of intervals of the discrete set {1,…,n} to each of the subsets, does there exist a bijection f:S→{1,…,n} such that for each element of F, its image under f is same as the interval assigned to it. An interval assignment to a given set of subsets is called feasible if there exists such a bijection. In this paper, we characterize feasible interval assignments to a given set of subsets. We then use this result to characterize matrices with the Consecutive Ones Property (COP), and to characterize matrices for which there is a permutation of the rows such that the columns are all sorted in ascending order. We also present a characterization of set systems which have a feasible interval assignment.     

Stochastic matrices and a property of the infinite sequences of linear functionals

Our starting point is the proof of the following property of a particular class of matrices. Let T={T_i,j} be a n×m non-negative matrix such that ∑_jT_i,j=1 for each i. Suppose that for every pair of indices (i,j), there exists an index l such that T_i,l≠T_j,l. Then, there exists a real vector k=(k₁,k₂,…,k_m)^T,k_i≠k_j,i≠j;0<k_i?1, such that, if i≠j.Then, we apply that property of matrices to probability theory. Let us consider an infinite sequence of linear functionals , corresponding to an infinite sequence of probability measures {μ_(·)(i)}_i∈N, on the Borel σ-algebra such that, . The property of matrices described above allows us to construct a real bounded one-to-one piecewise continuous and continuous from the left function f such that