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Numerical Algebra, Control & Optimization

June 2018 , Volume 8 , Issue 2

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Fuzzy target-environment networks and fuzzy-regression approaches
Erik Kropat and Gerhard Wilhelm Weber
2018, 8(2): 135-155 doi: 10.3934/naco.2018008 +[Abstract](160) +[HTML](124) +[PDF](310.66KB)
Abstract:

In systems sciences, the role of the environment is considered as a key factor for a deeper understanding of interconnected complex systems. The framework of target-environment networks allows for an investigation of regulatory systems under various kinds of uncertainty. Parameter-dependent models are applied to predict the future states of the system with respect to uncertain observations. In particular, fuzzy possibilistic regression models have been introduced that are based on crisp measurements. In this study, the concept of fuzzy target-environment networks is further extended towards fuzzy-regression models with fuzzy data sets. Regression models for various shapes of fuzzy coefficients and fuzzy model outputs are presented.

A three echelon revenue oriented green supply chain network design
Ashkan Mohsenzadeh Ledari, Alireza Arshadi Khamseh and Mohammad Mohammadi
2018, 8(2): 157-168 doi: 10.3934/naco.2018009 +[Abstract](172) +[HTML](153) +[PDF](400.52KB)
Abstract:

Green supply chain network designing has been studied during last decades. As carbon emissions considered as a major index in today's activities around the world, here a three echelon-multi product network including manufacturer, distributor, retailer have been provided and tried to minimize the pollution gathered from manufacturing and distribution of products all over the chains which causes extra costs as penalty to the system.

As we faced with these penalties, the model determines selling prices of products for manufacturer and distribution center simultaneously by locating these centers in order to maximize the profits all around the network. Finally, the proposed model is solved through the numerical examples and the sensitivity analysis and important parameters are reported to find some management insights.

An integrated inventory model with variable holding cost under two levels of trade-credit policy
Magfura Pervin, Sankar Kumar Roy and Gerhard Wilhelm Weber
2018, 8(2): 169-191 doi: 10.3934/naco.2018010 +[Abstract](180) +[HTML](201) +[PDF](900.43KB)
Abstract:

This paper presents an integrated vendor-buyer model for deteriorating items. We assume that the deterioration follows a constant rate with respect to time. The vendor allows a certain credit period to buyer in order to promote the market competition. Keeping in mind the competition of modern age, the stock-dependent demand rate is included in the formulated model which is a new policy to attract more customers. Shortages are allowed for the model to give the model more realistic sense. Partial backordering is offered for the interested customers, and there is a lost-sale cost during the shortage interval. The traditional parameter of holding cost is considered here as time-dependent. Henceforth, an easy solution procedure to find the optimal order quantity is presented so that the total relevant cost per unit time will be minimized. The mathematical formation is explored by numerical examples to validate the proposed model. A sensitivity analysis of the optimal solution for important parameters is also carried out to modify the result of the model.

An optimal control problem by parabolic equation with boundary smooth control and an integral constraint
Alexander Arguchintsev and Vasilisa Poplevko
2018, 8(2): 193-202 doi: 10.3934/naco.2018011 +[Abstract](142) +[HTML](79) +[PDF](267.51KB)
Abstract:

In the paper, we consider an optimal control problem by differential boundary condition of parabolic equation. We study this problem in the class of smooth controls satisfying certain integral constraints. For the problem under consideration we obtain a necessary optimality condition and propose a method for improving admissible controls. For illustration, we solve one numerical example to show the effectiveness of the proposed method.

New bounds for eigenvalues of strictly diagonally dominant tensors
Yining Gu and Wei Wu
2018, 8(2): 203-210 doi: 10.3934/naco.2018012 +[Abstract](137) +[HTML](108) +[PDF](280.07KB)
Abstract:

In this paper, we prove that the minimum eigenvalue of a strictly diagonally dominant Z-tensor with positive diagonal entries lies between the smallest and the largest row sums. The novelty comes from the upper bound. Moreover, we show that a similar upper bound does not hold for the minimum eigenvalue of a strictly diagonally dominant tensor with positive diagonal entries but with arbitrary off-diagonal entries. Furthermore, other new bounds for the minimum eigenvalue of nonsingular M-tensors are obtained.

Asymptotic properties of an infinite horizon partial cheap control problem for linear systems with known disturbances
Valery Y. Glizer and Oleg Kelis
2018, 8(2): 211-235 doi: 10.3934/naco.2018013 +[Abstract](131) +[HTML](93) +[PDF](373.13KB)
Abstract:

An infinite horizon quadratic control of a linear system with known disturbance is considered. The feature of the problem is that the cost of some (but in general not all) control coordinates in the cost functional is much smaller than the costs of the other control coordinates and the state cost. Using the control optimality conditions, the solution of this problem is reduced to solution of a hybrid set of three equations, perturbed by a small parameter. One of these equations is a matrix algebraic Riccati equation, while two others are vector and scalar differential equations subject to terminal conditions at infinity. For this set of the equations, a zero-order asymptotic solution is constructed and justified. Using this asymptotic solution, a relation between solutions of the original problem and the problem, obtained from the original one by replacing the small control cost with zero, is established. Based on this relation, the best achievable performance in the original problem is derived. Illustrative examples are presented.

Experiments with sparse Cholesky using a sequential task-flow implementation
Iain Duff, Jonathan Hogg and Florent Lopez
2018, 8(2): 237-260 doi: 10.3934/naco.2018014 +[Abstract](135) +[HTML](99) +[PDF](454.37KB)
Abstract:

We describe the development of a prototype code for the solution of large sparse symmetric positive definite systems that is efficient on parallel architectures. We implement a DAG-based Cholesky factorization that offers good performance and scalability on multicore architectures. Our approach uses a runtime system to execute the DAG. The runtime system plays the role of a software layer between the application and the architecture and handles the management of task dependencies as well as the task scheduling. In this model, the application is expressed using a high-level API, independent of the hardware details, thus enabling portability across different architectures. Although widely used in dense linear algebra, this approach is nevertheless challenging for sparse algorithms because of the irregularity and variable granularity of the DAGs arising in these systems. We assess the ability of two different Sequential Task Flow implementations to address this challenge: one implemented with the OpenMP standard, and the other with the modern runtime system StarPU. We compare these implementations to the state-of-the-art solver HSL_MA87 and demonstrate comparable performance on a multicore architecture.

On the extension of an arc-search interior-point algorithm for semidefinite optimization
Behrouz Kheirfam and Morteza Moslemi
2018, 8(2): 261-275 doi: 10.3934/naco.2018015 +[Abstract](161) +[HTML](112) +[PDF](348.15KB)
Abstract:

This paper concerns an extension of the arc-search strategy that was proposed by Yang [26] for linear optimization to semidefinite optimization case. Based on the Nesterov-Todd direction as Newton search direction it is shown that the complexity bound of the proposed algorithm is of the same order as that of the corresponding algorithm for linear optimization. Some preliminary numerical results indicate that our primal-dual arc-search path-following method is promising for solving the semidefinite optimization problems.

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