On some applications of Wald's identity to dams

It is known that the main difficulty in applying the Markovian analogue of Wald's Identity is the presence, in the Identity, of the last state variable before the random walk is terminated. In this paper we show that this difficulty can be overcome if the underlying Markov chain has a finite state space. The absorption probabilities thus obtained are used, by employing a duality argument, to derive time-dependent and limiting probabilities for the depletion process of a dam with Markovian inputs.The second problem that is considered here is that of a non-homogeneous but cyclic Markov chain. An analogue of Wald's Identity is obtained for this case, and is used to derive time- dependent and limiting probabilities for the depletion process with inputs forming a non- homogeneous (cyclic) Markov chain.     

A model of the executive's decision processes in new product development

This paper introduces the concept of fuzziness to deal quantitatively with the imprecision of the meaning of the executive's judgment stated in a natural language and presents a model of the executive's decision processes for the new product introduction which contain fuzzy-2 states, fuzzy-2 information systems, fuzzy-2 information signals, fuzzy-2 strategy are presented. The committee decision problem under fuzzy-2 constraints is dealt with.     

Markowitz’s mean-variance asset-liability management with regime switching: A continuous-time model

This paper considers an asset-liability management (ALM) problem under a continuous-time Markov regime-switching model. By adopting the techniques of [Zhou, X.Y., Yin, G., 2003. Markowitz’s mean-variance portfolio selection with regime switching: A continuous-time model. SIAM J. Control Optim. 42, 1466-1482], we investigate the feasibility, obtain the optimal strategy, delineate the efficient frontier, and establish the associated mutual fund theorem.     

A model of dynamics of group structures of human institutions

Given is a review of the substantial theories of cognitivist's trend and also of static and dynamic models of group structures that are based on them. A model of human institutiosn is constructed that explains the origin of balanced structures. This model is realized by a system of locally interacting automatons which are connected by a system of relations and endowed with the ability to react.     

A note on Fung-Fu's theorem

The axiomatic approach to rational decision malting in a fuzzy environment proposed by Fung and Fu (1975) lays down a too restrictive family of aggregation rules. Because of such axioms, the independence of the size of the groups under aggregation is assumed; moveover, in a general problem, these groups are fuzzy sets.     

The rate of convergence in Orey's theorem for Harris recurrent Markov chains with applications to renewal theory

We derive sufficient conditions for ∝ λ (dx)6Pⁿ(x, ·) - π6 to be of order o(ψ(n)^-1), where Pⁿ (x, A) are the transition probabilities of an aperiodic Harris recurrent Markov chain, π is the invariant probability measure, λ an initial distribution and ψ belongs to a suitable class of non-decreasing sequences. The basic condition involved is the ergodicity of order ψ, which in a countable state space is equivalent to $\text{Σ ψ(n)}\text{P}{}_{\mathrm{i}}\text{{τ}{}_{\mathrm{i}}\text{?n} <∞}$ for some i, where τ_i is the hitting time of the tate i. We also show that for a general Markov chain to be ergodic of order ψ it suffices that a corresponding condition is satisfied by a small set.We apply these results to non-singular renewal measures on $\text{R}$ providing a probabilisite method to estimate the right tail of the renewal measure when the increment distribution F satisfies ∝ tF(dt) 0; > 0 and ∝ ψ(t)(1- F(t))dt< ∞.     

Offshore outsourcing decision making: A policy-maker’s perspective

This research aims at finding the best governing policy for offshore outsourcing of business activities. We use Analytical Network Process, a multicriteria decision making methodology, to create the evaluation framework. From the perspective of decision makers, stakeholders, and influence groups, four policy options are evaluated with respect to approximately 50 economic, political, technological and other factors. The model provides both long-term and short-term views of the outsourcing issue concerned to all parties. The all-inclusive approach helps policy makers to decide on the best policy and has the potential to ease tension between proponents and opponents of offshore outsourcing.     

A proof of Wright's conjecture

Wright's conjecture states that the origin is the global attractor for the delay differential equation $y^{\prime }(t)=?\alpha y(t?1)[1+y(t)]$ for all $\alpha \in (0,\frac{\pi }{2}]$ when $y(t)>?1$. This has been proven to be true for a subset of parameter values α. We extend the result to the full parameter range $\alpha \in (0,\frac{\pi }{2}]$, and thus prove Wright's conjecture to be true. Our approach relies on a careful investigation of the neighborhood of the Hopf bifurcation occurring at $\alpha =\frac{\pi }{2}$. This analysis fills the gap left by complementary work on Wright's conjecture, which covers parameter values further away from the bifurcation point. Furthermore, we show that the branch of (slowly oscillating) periodic orbits originating from this Hopf bifurcation does not have any subsequent bifurcations (and in particular no folds) for $\alpha \in (\frac{\pi }{2},\frac{\pi }{2}+6.830\times {10}^{?3}]$. When combined with other results, this proves that the branch of slowly oscillating solutions that originates from the Hopf bifurcation at $\alpha =\frac{\pi }{2}$ is globally parametrized by $\alpha >\frac{\pi }{2}$.     

A case of hadwiger's conjecture

Sufficient conditions are established for Hadwiger's Conjecture to be valid in thecase r = 5.     

A generalization of Hiraguchi's: Inequality for posets

For a poset X, Dim(X) is the smallest positive integer t for which X is isomorphic to a subposet of the cartesian product of t chains. Hiraguchi proved that if | X | ? 4, then Dim(X) ? [| X |/2]. For each k ? 2, we define Dim_k(X) as the smallest positive integer t for which X is isomorphic to a subposet of the cartesian product of t chains, each of length k. We then prove that if | X | ? 5, Dim₃(X) ? {| X |/2} and if | X | ? 6, then Dim₄(X) ? [| X |/2].     

A Markov decision process model for equitable distribution of supplies under uncertainty

Many individuals suffering from food insecurity obtain assistance from governmental programs and nonprofit agencies such as food banks. Much of the food distributed by food banks come from donations which are received from various sources in uncertain quantities at random points in time. This paper presents a model that can assist food banks in distributing these uncertain supplies equitably and measure the performance of their distribution efforts. We formulate this decision problem as a discrete-time, discrete state Markov decision process that considers stochastic supply, deterministic demand and an equity-based objective. We investigate three different allocation rules and describe the optimal policy as a function of available inventory. We also provide county level estimates of unmet need and determine the probability distribution associated with the number of underserved counties. A numerical study is performed to show how the allocation policy and unmet need are impacted by uncertain supply and deterministic, time-varying demand. We also compare different allocation rules in terms of equity and effectiveness.     

Note on “A note on single-machine group scheduling problems with position-based learning effect”

In a recent paper, Lee and Wu [W.-C. Lee, C.-C. Wu, A note on single-machine group scheduling problems with position-based learning effect, Appl. Math. Model. 33 (2009) 2159–2163] proposed a new group scheduling learning model where the learning effect not only depends on the job position, but also depends on the group position. They investigate the makespan and the total completion time minimization problems on a single-machine. As for the total completion time minimization problem, they assumed that the numbers of jobs in each group are the same and the group normal setup and the job normal processing times are agreeable. Under the assumption conditions, they showed that the total completion time minimization problem can be optimally solved in polynomial time solution. However, the assumption conditions for the total completion time minimization problem do not reflect actual practice in many manufacturing processes. Hence, in this note, we propose other agreeable conditions and show that the total completion time minimization problem remains polynomially solvable under the agreeable conditions.     

A fuzzy-based decision support model for monitoring on-time delivery performance: A textile industry case study

This paper investigates uncertainties in complex supply chain situations and proposes a fuzzy-based decision support model for determining the chance of meeting on-time delivery in a complex supply chain environment. It integrates fuzzy logic principles and unitary structure-based supply chain model and enables addressing uncertainties associated with key inputs of on-time delivery performance for effective decision making process. The proposed pragmatic model deals with the fuzziness of the key inputs including, variations in demand forecasting, materials shortages and distribution lead time, and combines a fuzzy reasoning approach for monitoring on-time delivery of finished products. In systematically dealing with the uncertainties of complex supply chains, this model supports the minimizing of business losses that result from penalties and customer dissatisfaction, and the consequent reduced market share. Application of the proposed model is illustrated using a textile industry case study.     

A note on levene's tests for equality of variances

Consider testing for equality variances in a one-way analysis of variance. Levene's test is the usual F-test for equality of means computed on psuedo-observations, which one defines as the absolute deviations of the data points from an estimate of the group ‘center’. We show that, asymptotically, Levene's test has the correct level whenever the estimate of group ‘center’ is an estimate of group median. This explains why published Monte-Carlo studies have found that Levene's original proposal of centering at the sample mean has the correct level only for symmetric distributions, while centering at the sample median has correct level even for asymmetric distributions. Generalizations are discussed.     

A new proof of Korzhik's result on burst distributions

We obtain a simple derivation of Korzhik's result on the burst distribution of a linear code.     

Phase-type distribution of customer relationship with Markovian response and marketing expenditure decision on the customer lifetime value

In recent years, more and more companies have adopted relationship marketing (RM). At the core of RM is the development and maintenance of long-term relationships with valuable customers. In RM, the customer lifetime value (CLV) is the discounted profit streams of a customer across the entire customer life cycle. The CLV plays a key role in customer acquisition and retention decisions. In this paper, we present a general mathematical framework for RM, and introduce a Markov chain model which is appropriate in modeling RM because of its flexibility and probabilistic nature. We also develop the life distribution of the customer relationship, which is a phase-type distribution since it is the distribution of the first arrival stopping state. And we obtain the expectation of the CLV, which is an important statistic for good decision-making. Finally, we illustrate how to find the optimal remarketing policy numerically. The framework developed for RM systems in this paper should be seen as a practical approach to RM where one can directly apply the results of phase-type distribution and expectation of CLV to marketing decisions.     

A method based on PSO and granular computing of linguistic information to solve group decision making problems defined in heterogeneous contexts

Group decision making is a type of decision problem in which multiple experts acting collectively, analyze problems, evaluate alternatives, and select a solution from a collection of alternatives. As the natural language is the standard representation of those concepts that humans use for communication, it seems natural that they use words (linguistic terms) instead of numerical values to provide their opinions. However, while linguistic information is readily available, it is not operational and thus it has to be made usable though expressing it in terms of information granules. To do so, Granular Computing, which has emerged as a unified and coherent framework of designing, processing, and interpretation of information granules, can be used. The aim of this paper is to present an information granulation of the linguistic information used in group decision making problems defined in heterogeneous contexts, i.e., where the experts have associated importance degrees reflecting their ability to handle the problem. The granulation of the linguistic terms is formulated as an optimization problem, solved by using the particle swarm optimization, in which a performance index is maximized by a suitable mapping of the linguistic terms on information granules formalized as sets. This performance index is expressed as a weighted aggregation of the individual consistency achieved by each expert.     

Quadratic forms and Klauder's phenomenon: A remark on very singular perturbations

We give a general analysis of a class of pairs of positive self-adjoint operators A and B for which A + λB has a limit (in strong resolvent sense) as λ ↓ 0 which is an operator A_p ≠ A!