scholarly journals Algebraic Solution to Constrained Bi-Criteria Decision Problem of Rating Alternatives through Pairwise Comparisons

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 303
Author(s):  
Nikolai Krivulin
Keyword(s):  
Decision Problem ◽  
Optimization Problem ◽  
Pareto Frontier ◽  
Sufficient Conditions ◽  
Approximation Error ◽  
Optimal Solution ◽  
Algebraic Solution ◽  
Pairwise Comparisons ◽  
Logarithmic Scale ◽  
Idempotent Algebra

We consider a decision-making problem to evaluate absolute ratings of alternatives from the results of their pairwise comparisons according to two criteria, subject to constraints on the ratings. We formulate the problem as a bi-objective optimization problem of constrained matrix approximation in the Chebyshev sense in logarithmic scale. The problem is to approximate the pairwise comparison matrices for each criterion simultaneously by a common consistent matrix of unit rank, which determines the vector of ratings. We represent and solve the optimization problem in the framework of tropical (idempotent) algebra, which deals with the theory and applications of idempotent semirings and semifields. The solution involves the introduction of two parameters that represent the minimum values of approximation error for each matrix and thereby describe the Pareto frontier for the bi-objective problem. The optimization problem then reduces to a parametrized vector inequality. The necessary and sufficient conditions for solutions of the inequality serve to derive the Pareto frontier for the problem. All solutions of the inequality, which correspond to the Pareto frontier, are taken as a complete Pareto-optimal solution to the problem. We apply these results to the decision problem of interest and present illustrative examples.

scholarly journals Solution of a bi-criteria problem of rating alternatives using tropical optimization

2019 ◽  
pp. 15-32
Author(s):  
Nikolai K. Krivulin ◽  
◽  
Margarita A. Tsobenko ◽  
Keyword(s):  
Pareto Front ◽  
Sufficient Conditions ◽  
Optimal Solution ◽  
Pairwise Comparisons ◽  
Logarithmic Scale ◽  
Third Order ◽  
Tropical Mathematics ◽  
All Solutions ◽  
Computational Procedures ◽  
Necessary And Sufficient

A problem is considered to evaluate scores (priorities, weights) of alternatives through the results of pairwise comparisons according to two criteria. A formal derivation and computational procedures of the solution to the problem are described, using methods of tropical mathematics, which studies algebraic systems with specially defined operations of addition and multiplication. The problem is reduced to simultaneous approximation of two matrices of pairwise comparisons by a common consistent matrix, in the Chebyshev metric in logarithmic scale. First, auxiliary variables are introduced to represent the minima of the objective functions, and a parameterized inequality is derived, which determines the set of solutions to the original optimization problem. The necessary and sufficient conditions for the existence of solutions of the inequality are used to evaluate the values of parameters, which correspond to the Pareto front of the problem. All solutions of the inequality under the obtained values are taken as a Pareto-optimal solution for the problem. To illustrate the computational procedures used, numerical examples of evaluating scores of alternatives are given for problems with matrices of the third order.

A new algorithm for finding CRM-model coefficients

2019 ◽  
Vol 5 (3) ◽  
pp. 164-185 ◽  
Author(s):  
Alexander D. Bekman ◽  
Sergey V. Stepanov ◽  
Alexander A. Ruchkin ◽  
Dmitry V. Zelenin
Keyword(s):  
Approximate Solution ◽  
Optimization Problem ◽  
Optimal Solution ◽  
Complex Form ◽  
Approximate Solutions ◽  
The Other ◽  
Programming Algorithm ◽  
Stochastic Algorithms ◽  
Large Set ◽  
Flow Rates

The quantitative evaluation of producer and injector well interference based on well operation data (profiles of flow rates/injectivities and bottomhole/reservoir pressures) with the help of CRM (Capacitance-Resistive Models) is an optimization problem with large set of variables and constraints. The analytical solution cannot be found because of the complex form of the objective function for this problem. Attempts to find the solution with stochastic algorithms take unacceptable time and the result may be far from the optimal solution. Besides, the use of universal (commercial) optimizers hides the details of step by step solution from the user, for example&nbsp;— the ambiguity of the solution as the result of data inaccuracy.<br> The present article concerns two variants of CRM problem. The authors present a new algorithm of solving the problems with the help of “General Quadratic Programming Algorithm”. The main advantage of the new algorithm is the greater performance in comparison with the other known algorithms. Its other advantage is the possibility of an ambiguity analysis. This article studies the conditions which guarantee that the first variant of problem has a unique solution, which can be found with the presented algorithm. Another algorithm for finding the approximate solution for the second variant of the problem is also considered. The method of visualization of approximate solutions set is presented. The results of experiments comparing the new algorithm with some previously known are given.

scholarly journals Quit When You Can: Efficient Evaluation of Ensembles by Optimized Ordering

2021 ◽  
Vol 17 (4) ◽  
pp. 1-20
Author(s):  
Serena Wang ◽  
Maya Gupta ◽  
Seungil You
Keyword(s):  
Optimization Problem ◽  
Optimal Solution ◽  
Combinatorial Optimization Problem ◽  
Joint Optimization ◽  
Early Stopping ◽  
Time Approximation ◽  
Approximation Bound ◽  
Evaluation Time ◽  
Speed Up ◽  
Real World Problems

Given a classifier ensemble and a dataset, many examples may be confidently and accurately classified after only a subset of the base models in the ensemble is evaluated. Dynamically deciding to classify early can reduce both mean latency and CPU without harming the accuracy of the original ensemble. To achieve such gains, we propose jointly optimizing the evaluation order of the base models and early-stopping thresholds. Our proposed objective is a combinatorial optimization problem, but we provide a greedy algorithm that achieves a 4-approximation of the optimal solution under certain assumptions, which is also the best achievable polynomial-time approximation bound. Experiments on benchmark and real-world problems show that the proposed Quit When You Can (QWYC) algorithm can speed up average evaluation time by 1.8–2.7 times on even jointly trained ensembles, which are more difficult to speed up than independently or sequentially trained ensembles. QWYC’s joint optimization of ordering and thresholds also performed better in experiments than previous fixed orderings, including gradient boosted trees’ ordering.

scholarly journals Robust Beamforming Design for Secure V2X Downlink System with Wireless Information and Power Transfer under a Nonlinear Energy Harvesting Model

Sensors ◽  
2018 ◽  
Vol 18 (10) ◽  
pp. 3294 ◽  
Author(s):  
Shidang Li ◽  
Chunguo Li ◽  
Weiqiang Tan ◽  
Baofeng Ji ◽  
Luxi Yang
Keyword(s):  
Energy Harvesting ◽  
Traffic Safety ◽  
Optimization Problem ◽  
Optimal Solution ◽  
Base Station ◽  
Power Splitting ◽  
Power Transfer ◽  
Vehicular Communication ◽  
Downlink System ◽  
Nonlinear Energy

Vehicle to everything (V2X) has been deemed a promising technology due to its potential to achieve traffic safety and efficiency. This paper considers a V2X downlink system with a simultaneous wireless information and power transfer (SWIPT) system where the base station not only conveys data and energy to two types of wireless vehicular receivers, such as one hybrid power-splitting vehicular receiver, and multiple energy vehicular receivers, but also prevents information from being intercepted by the potential eavesdroppers (idle energy vehicular receivers). Both the base station and the energy vehicular receivers are equipped with multiple antennas, whereas the information vehicular receiver is equipped with a single antenna. In particular, the imperfect channel state information (CSI) and the practical nonlinear energy harvesting (EH) model are taken into account. The non-convex optimization problem is formulated to maximize the minimum harvested energy power among the energy vehicular receivers satisfying the lowest harvested energy power threshold at the information vehicular receiver and secure vehicular communication requirements. In light of the intractability of the optimization problem, the semidefinite relaxation (SDR) technique and variable substitutions are applied, and the optimal solution is proven to be tight. A number of results demonstrate that the proposed robust secure beamforming scheme has better performance than other schemes.

scholarly journals Topology Optimization of Automotive sheet metal part using Altair Inspire

2020 ◽  
Vol 5 (3) ◽  
pp. 143-150
Author(s):  
Netsanet Ferede
Keyword(s):  
Topology Optimization ◽  
Sheet Metal ◽  
Optimization Problem ◽  
Optimal Solution ◽  
Sheet Metal Part ◽  
Metal Part ◽  
Solution Quality ◽  
Design Engineers ◽  
Automotive Sheet ◽  
Reduced Costs

In an optimization problem, different candidate solutions are compared with each other, and then the best or optimal solution is obtained which means that solution quality is fundamental. Topology optimization is used at the concept stage of design. It deals with the optimal distribution of material within the structure. Altair Inspire software is the industry's most powerful and easy-to-use Generative Design/Topology Optimization and rapid simulation solution for design engineers. In this paper Topology optimization is applied using Altair inspire to optimize the Sheet metal Angle bracket. Different results are conducted the better and final results are fulfilling the goal of the paper which is minimizing the mass of the sheet metal part by 65.9%  part and Maximizing the stiffness with Better Results of Von- Miss Stress Analysis,  Displacement, and comparison with different load cases.  This can lead to reduced costs, development time, material consumption, and product less weight.

Fast Filtering of Search Results Sorted by Attribute

2022 ◽  
Vol 40 (2) ◽  
pp. 1-24
Author(s):  
Franco Maria Nardini ◽  
Roberto Trani ◽  
Rossano Venturini
Keyword(s):  
Heuristic Algorithms ◽  
State Of The Art ◽  
Optimal Algorithm ◽  
Computational Cost ◽  
Approximation Error ◽  
Optimal Solution ◽  
Exact Algorithm ◽  
Performance Bounds ◽  
Search Results ◽  
Filtering Problem

Modern search services often provide multiple options to rank the search results, e.g., sort “by relevance”, “by price” or “by discount” in e-commerce. While the traditional rank by relevance effectively places the relevant results in the top positions of the results list, the rank by attribute could place many marginally relevant results in the head of the results list leading to poor user experience. In the past, this issue has been addressed by investigating the relevance-aware filtering problem, which asks to select the subset of results maximizing the relevance of the attribute-sorted list. Recently, an exact algorithm has been proposed to solve this problem optimally. However, the high computational cost of the algorithm makes it impractical for the Web search scenario, which is characterized by huge lists of results and strict time constraints. For this reason, the problem is often solved using efficient yet inaccurate heuristic algorithms. In this article, we first prove the performance bounds of the existing heuristics. We then propose two efficient and effective algorithms to solve the relevance-aware filtering problem. First, we propose OPT-Filtering, a novel exact algorithm that is faster than the existing state-of-the-art optimal algorithm. Second, we propose an approximate and even more efficient algorithm, ϵ-Filtering, which, given an allowed approximation error ϵ, finds a (1-ϵ)–optimal filtering, i.e., the relevance of its solution is at least (1-ϵ) times the optimum. We conduct a comprehensive evaluation of the two proposed algorithms against state-of-the-art competitors on two real-world public datasets. Experimental results show that OPT-Filtering achieves a significant speedup of up to two orders of magnitude with respect to the existing optimal solution, while ϵ-Filtering further improves this result by trading effectiveness for efficiency. In particular, experiments show that ϵ-Filtering can achieve quasi-optimal solutions while being faster than all state-of-the-art competitors in most of the tested configurations.

Design of Low Order Controllers for Decoupled MIMO Systems With Time Response Specifications

2018 ◽  
pp. 90-128
Author(s):  
Maher Ben Hariz ◽  
Wassila Chagra ◽  
Faouzi Bouani
Keyword(s):  
Convex Optimization ◽  
Optimization Problem ◽  
Closed Loop ◽  
Mimo Systems ◽  
Design Method ◽  
Optimal Solution ◽  
Time Response ◽  
Convex Optimization Problem ◽  
Tito Systems ◽  
Low Order

The design of a low order controller for decoupled MIMO systems is proposed. The main objective of this controller is to guarantee some closed loop time response performances such as the settling time and the overshoot. The controller parameters are obtained by resolving a non-convex optimization problem. In order to obtain an optimal solution, the use of a global optimization method is suggested. In this chapter, the proposed solution is the GGP method. The principle of this method consists of transforming a non-convex optimization problem to a convex one by some mathematical transformations. So as to accomplish the fixed goal, it is imperative to decouple the coupled MIMO systems. To approve the controllers' design method, the synthesis of fixed low order controller for decoupled TITO systems is presented firstly. Then, this design method is generalized in the case of MIMO systems. Simulation results and a comparison study between the presented approach and a PI controller are given in order to show the efficiency of the proposed controller. It is remarkable that the obtained solution meets the desired closed loop time specifications for each system output. It is also noted that by considering the proposed approach the user can fix the desired closed loop performances for each output independently.

An Approximate Algorithm for Triangle TSP with a Four-Vertex-Three-Line Inequality

2015 ◽  
Vol 6 (1) ◽  
pp. 35-46 ◽  
Author(s):  
Yong Wang
Keyword(s):  
Time Complexity ◽  
Optimization Problem ◽  
Nearest Neighbor ◽  
Optimal Solution ◽  
Simple Algorithm ◽  
Exact Algorithms ◽  
Combinatorial Optimization Problem ◽  
Traveling Salesman ◽  
Approximate Algorithm ◽  
Nearest Neighbor Algorithm

Traveling salesman problem (TSP) is a classic combinatorial optimization problem. The time complexity of the exact algorithms is generally an exponential function of the scale of TSP. This work gives an approximate algorithm with a four-vertex-three-line inequality for the triangle TSP. The time complexity is O(n2) and it can generate an approximation less than 2 times of the optimal solution. The paper designs a simple algorithm with the inequality. The algorithm is compared with the double-nearest neighbor algorithm. The experimental results illustrate the algorithm find the better approximations than the double-nearest neighbor algorithm for most TSP instances.

scholarly journals Caching and Placement for In-Network Caching in Device-to-Device Communications

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Somayeh Soleimani ◽  
Xiaofeng Tao
Keyword(s):  
Optimization Problem ◽  
Optimal Solution ◽  
Content Delivery ◽  
Base Stations ◽  
D2d Communications ◽  
Dual Decomposition ◽  
Placement Problem ◽  
Content Discovery ◽  
Device To Device ◽  
Promising Solution

Caching content by users constitutes a promising solution to decrease the costly transmissions with going through the base stations (BSs). To improve the performance of in-network caching in device-to-device (D2D) communications, caching placement and content delivery should be jointly optimized. To this end, we jointly optimize caching decision and content discovery strategies by considering the successful content delivery in D2D links for maximizing the in-network caching gain through D2D communications. Moreover, an in-network caching placement problem is formulated as an integer nonlinear optimization problem. To obtain the optimal solution for the proposed problem, Lagrange dual decomposition is applied in order to reduce the complexity. Simulation results show that the proposed algorithm has a near-optimal performance, approaching that of the exhaustive search method. Furthermore, the proposed scheme has a notable in-network caching gain and an improvement in traffic offloading compared to that of other caching placement schemes.

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