On differential equations with interval coefficients

In this study, a new approach is developed to solve the initial value problem for interval linear differential equations. In the considered problem, the coefficients and the initial values are constant intervals. In the developed approach, there is no need to define a derivative for interval-valued functions. All derivatives used in the approach are classical derivatives of real functions. The reason for this is that the solution of the problem is defined as a bunch of real functions. Such a solution concept is compatible also with the robust stability concept. Sufficient conditions are provided for the solution to be expressed analytically. In addition, on a numerical example, the solution obtained by the proposed approach is compared with the solution obtained by the generalized Hukuhara differentiability. It is shown that the proposed approach gives a new type of solution. The main advantage of the proposed approach is that the solution to the considered interval initial value problem exists and is unique, as in the real case.     

On the Variational Approach to Systems of Quasilinear Conservation Laws

The paper contains results concerning the development of a new approach to the proof of existence theorems for generalized solutions to systems of quasilinear conservation laws. This approach is based on reducing the search for a generalized solution to analyzing extremal properties of a certain set of functionals and is referred to as a variational approach. The definition of a generalized solution can be naturally reformulated in terms of the existence of critical points for a set of functionals, which is convenient within the approach proposed. The variational representation of generalized solutions, which was earlier known for Hopf-type equations, is generalized to systems of quasilinear conservation laws. The extremal properties of the functionals corresponding to systems of conservation laws are described within the variational approach, and a strategy for proving the existence theorem is outlined. In conclusion, it is shown that the variational approach can be generalized to the two-dimensional case.     

Adaptive Rejection Metropolis Simulated Annealing for Detecting Global Maximum Regions

A finite mixture model has been used to fit the data from heterogeneous populations to many applications. An Expectation Maximization (EM) algorithm is the most popular method to estimate parameters in a finite mixture model. A Bayesian approach is another method for fitting a mixture model. However, the EM algorithm often converges to the local maximum regions, and it is sensitive to the choice of starting points. In the Bayesian approach, the Markov Chain Monte Carlo (MCMC) sometimes converges to the local mode and is difficult to move to another mode. Hence, in this paper we propose a new method to improve the limitation of EM algorithm so that the EM can estimate the parameters at the global maximum region and to develop a more effective Bayesian approach so that the MCMC chain moves from one mode to another more easily in the mixture model. Our approach is developed by using both simulated annealing (SA) and adaptive rejection metropolis sampling (ARMS). Although SA is a well-known approach for detecting distinct modes, the limitation of SA is the difficulty in choosing sequences of proper proposal distributions for a target distribution. Since ARMS uses a piecewise linear envelope function for a proposal distribution, we incorporate ARMS into an SA approach so that we can start a more proper proposal distribution and detect separate modes. As a result, we can detect the maximum region and estimate parameters for this global region. We refer to this approach as ARMS annealing. By putting together ARMS annealing with the EM algorithm and with the Bayesian approach, respectively, we have proposed two approaches: an EM-ARMS annealing algorithm and a Bayesian-ARMS annealing approach. We compare our two approaches with traditional EM algorithm alone and Bayesian approach alone using simulation, showing that our two approaches are comparable to each other but perform better than EM algorithm alone and Bayesian approach alone. Our two approaches detect the global maximum region well and estimate the parameters in this region. We demonstrate the advantage of our approaches using an example of the mixture of two Poisson regression models. This mixture model is used to analyze a survey data on the number of charitable donations.     

Copula方法与相依违约研究

目前信用风险研究的重点已经从单笔债务的违约概率研究转移到多笔债务的相依违约(Dependent Defaults)研究。Copula方法是研究相依违约的重要方法。这种方法是最近几年才被应用到信用领域研究中的一种新方法。本结合代表性献对Copula方法在相依违约研究中的应用进行了探讨。探讨的内容包括Copula方法被应用于相依违约研究的原因、该方法对于相依违约建模理论的改进以及在实证应用中使用Copula方法应该注意的问题。     

A finite element recovery approach to Green’s function approximations with applications to electrostatic potential computation

In this paper, a finite element recovery approach is proposed to improve the accuracy of finite element approximations for Green’s functions in three dimensions. This recovery approach is based on some simple postprocessing. It is proved by both theory and numerics that the recovery approach is very efficient. In particular, the approach is successfully applied to some electrostatic potential computations.     

An efficient decision making approach in incomplete soft set

Zou et al. (2008) [21] presented weighted-average of all possible choice values approach of soft sets under incomplete information system in decision making. However, the approach is hard to understand and involves a great amount of computation. In order to simplify the approach, we present the simplified probability to directly instead of the incomplete information, and demonstrate the equivalence between the weighted-average of all possible choice values approach and the simplified probability approach. Finally, comparison results show that the proposed approach involves relatively less computation and is easier to implement and understand as compared with the weighted-average of all possible choice values approach.     

Using bounds on the data in linear programming: The tolerance approach to sensitivity analysis

In contrast to traditional sensitivity analysis in linear programming, the tolerance approach considers simultaneous and independent variations in a number of parameters. A primary focus of this approach is to determine a maximum tolerance percentage for selected right-hand-side terms in which the same basis is optimal as long as each term is accurate to within that percentage of its estimated value. Similarly, the approach yields a maximum tolerance percentage for selected objective function coefficients. This paper shows how the tolerance approach can exploit information on the range of possible values over which terms and coefficients can vary to yield larger maximum tolerance percentages.     

Optimal aggregation of fuzzy preference relations with an application to broadband internet service selection

This paper investigates the aggregation of multiple fuzzy preference relations into a collective fuzzy preference relation in fuzzy group decision analysis and proposes an optimization based aggregation approach to assess the relative importance weights of the multiple fuzzy preference relations. The proposed approach that is analytical in nature assesses the weights by minimizing the sum of squared distances between any two weighted fuzzy preference relations. Relevant theorems are offered in support of the proposed approach. Multiplicative preference relations are also incorporated into the approach using an appropriate transformation technique. An eigenvector method is introduced to derive the priorities from the collective fuzzy preference relation. The proposed aggregation approach is tested using two numerical examples. A third example involving broadband internet service selection is offered to illustrate that the proposed aggregation approach provides a simple, effective and practical way of aggregating multiple fuzzy preference relations in real-life situations.     

Determination of Stackelberg–Nash equilibria using a sensitivity based approach

A sensitivity based approach is presented to determine Nash solution(s) in multiobjective problems modeled as a non-cooperative game. The proposed approach provides an approximation to the rational reaction set (RRS) for each player. An intersection of these sets yields the Nash solution for the game. An alternate approach for generating the RRS based on design of experiments (DOE) combined with response surface methodology (RSM) is also explored. The two approaches for generating the RRS are compared on three example problems to find Nash and Stackelberg solutions. For the examples presented, it is seen that the proposed sensitivity based approach (i) requires less computational effort than a RSM-DOE approach, (ii) is less prone to numerical errors than the RSM-DOE approach, (iii) has the ability to find multiple Nash solutions when the Nash solution is not a singleton, (iv) is able to approximate nonlinear RRS, and (v) on one example problem, found a Nash solution better than the one reported in the literature.     

Portfolio value-at-risk optimization for asymmetrically distributed asset returns

We propose a new approach to portfolio optimization by separating asset return distributions into positive and negative half-spaces. The approach minimizes a newly-defined Partitioned Value-at-Risk (PVaR) risk measure by using half-space statistical information. Using simulated data, the PVaR approach always generates better risk-return tradeoffs in the optimal portfolios when compared to traditional Markowitz mean–variance approach. When using real financial data, our approach also outperforms the Markowitz approach in the risk-return tradeoff. Given that the PVaR measure is also a robust risk measure, our new approach can be very useful for optimal portfolio allocations when asset return distributions are asymmetrical.     

Greedy Gauss-Newton algorithms for finding sparse solutions to nonlinear underdetermined systems of equations

We consider the problem of finding sparse solutions to a system of underdetermined non-linear system of equations. The methods are based on a Gauss–Newton approach with line search where the search direction is found by solving a linearized problem using only a subset of the columns in the Jacobian. The choice of columns in the Jacobian is made through a greedy approach looking at either maximum descent or an approach corresponding to orthogonal matching for linear problems. The methods are shown to be convergent and efficient and outperform the l1 approach on the test problems presented.     

Supply chain network design: partner selection and production/distribution planning using a systematic model

In this paper, a novel multi-phase mathematical approach is presented for the design of a complex supply chain network. From the point of network design, customer demands, and for maximum overall utility, the important issues are to find suitable and quality companies, and to decide upon an appropriate production/distribution strategy. The proposed approach is based on the genetic algorithm (GA), the analytical hierarchy process (AHP), and the multi-attribute utility theory (MAUT) to satisfy simultaneously the preferences of the suppliers and the customers at each level in the network. A case study with a good quality solution is provided to confirm the efficiency and effectiveness of the proposed approach. Finally, to demonstrate the performance of the proposed approach, a comparative numerical experiment is performed by using the proposed approach and the common single-phase genetic algorithm (SGA). Empirical analysis results demonstrate that the proposed approach can outperform the SGA in partner selection and production/distribution planning for network design.     

Heuristic Methods for Solving Large Scale Network Routing Problems: The Telpaking Problem

An approach to a large scale network routing problem with nonlinear cost function is described, along with an example of its application. The approach to the problem involves a multistage construction process. This approach is applied to the telpaking problem. Results are obtained in applying this method to a 53 node sample problem.     

Theoretical insights into the augmented-neural-network approach for combinatorial optimization

The augmented-neural-network (AugNN) approach has been applied lately to some NP-Hard combinatorial problems, such as task scheduling, open-shop scheduling and resource-constraint project scheduling. In this approach the problem of search in the solution-space is transformed to a search in a weight-matrix space, much like in a neural-network approach. Some weight adjustment strategies are then used to converge to a good set of weights for a locally optimal solution. While empirical results have demonstrated the effectiveness of the AugNN approach vis-à-vis a few other metaheuristics, little theoretical insights exist which justify this approach and explain the effectiveness thereof. This paper provides some theoretical insights and justification for the AugNN approach through some basic theorems and also describes the algorithm and the formulation with the help of examples.     

Possibilistic signal processing: How to handle noise?

We propose a novel approach for noise quantifier at each location of a signal. This method is based on replacing the conventional kernel-based approach extensively used in signal processing by an approach involving another kind of kernel: a possibility distribution. Such an approach leads to interval-valued resulting methods instead of point-valued ones. We propose a theoretical justification to this approach and we show, on real and artificial data sets, that the length of the obtained interval and the local noise level are highly correlated. This method is non-parametric and has an advantage over other methods since no assumption about the nature of the noise has to be made, except its local ergodicity. Besides, the propagation of the noise in the involved signal processing method is direct and does not require any additional computation.     

Generating unconstrained two-dimensional non-guillotine cutting patterns by a Recursive Partitioning Algorithm

In this study, a dynamic programming approach to deal with the unconstrained two-dimensional non-guillotine cutting problem is presented. The method extends the recently introduced recursive partitioning approach for the manufacturer's pallet loading problem. The approach involves two phases and uses bounds based on unconstrained two-staged and non-staged guillotine cutting. The method is able to find the optimal cutting pattern of a large number of pro blem instances of moderate sizes known in the literature and a counterexample for which the approach fails to find known optimal solutions was not found. For the instances that the required computer runtime is excessive, the approach is combined with simple heuristics to reduce its running time. Detailed numerical experiments show the reliability of the method.     

A Dynamic Programming Approach for Product Selection and Supermarket Shelf-Space Allocation

A dynamic programming approach is proposed to select optimally among a given set of products and allocate integer shelf-space units to the selected products in supermarkets. The approach is designed to consider general objective-function specifications that account for space elasticity, costs of sales, and potential demand-related marketing variables. The optimization is subject to constraints due to product supply availability, 'block' product allocation and operational requirements. A primary focus is on the development of a tractable model approach that can effectively be implemented on a microcomputer. A discussion of applications and computational experience on a microcomputer is provided to support the practical applicability of the optimization approach.     

Closed-loop supply chain network design: A financial approach

The more common approaches used in the SCM consider only the physical logistic operations and ignore the financial aspects of the chain. This paper presents a financial approach to model a closed-loop supply chain design in which financial aspects are explicitly considered as exogenous variables. The model decides to determine the strategic decisions as well as the tactical decisions. The main contribution of this paper is to incorporate the financial aspects (i.e. current and fixed assets and liabilities) and a set of budgetary constraints representing balances of cash, debt, securities, payment delays, and discounts in the supply chain planning. Moreover, the financial approach applies the change in equity (instead of the measure of profit/cost in traditional approaches) as the objective function to be optimized in the presented model.To show the advantages of the presented approach, the results attributed to the financial approach and the traditional approach are compared, where the latter firstly decides on operations and fits finances afterwards. The results indicate that the traditional approach leads to lower change in equity compared to the financial approach. This fact illustrates the inadequacy of treating process operations and finances in isolated environments and pursuing as objective myopic performance indicators such as profit or cost. Moreover, a sensitivity analysis of the parameters using ANOVA for different levels of the parameters under different customer order patterns is performed to enhance the managerial insights of the study. The results clearly reveal the better improvement of using the financial approach over the traditional approach, and convince the decision makers to take advantage of the proposed approach.     

A novel two-phase group decision making approach for construction project selection in a fuzzy environment

This paper considers a construction project problem under multiple criteria in a fuzzy environment and proposes a new two-phase group decision making (GDM) approach. This approach integrates a modified analytic network process (ANP) and an improved compromise ranking method, known as VIKOR. To take uncertainty and risk into account, a new decision making approach is presented with multiple fuzzy information by a group of experts, and a risk attitude for each expert is incorporated that can be expressed linguistically. First, a modified fuzzy ANP method is introduced to address the problem of dependence as well as feedback among conflicting criteria and to determine their relative importance. Then, a fuzzy VIKOR method is extended to rank potential projects on the basis of their overall performance. An illustrative example from the literature is provided for the construction project problem to demonstrate the effectiveness and feasibility of the proposed approach. The computational results show that the proposed two-phase GDM approach is suitable to cope with imprecision and subjectivity for the complicated decision making problem. Finally, the associated results of the proposed approach with risk attitudes and without risk attitudes are compared with the results reported by Cheng and Li [1], and the merits are highlighted.