A Novel Method for Solving Multi-objective Shortest Path Problem in Respect of Probability Theory

Transportation process or activity can be considered as a multi-objective problem reasonably. However, it is difficult to obtain an absolute shortest path with optimizing the multiple objectives at the same time by means of Pareto approach. In this paper, a novel method for solving multi-objective shortest path problem in respect of probability theory is developed, which aims to get the rational solution of multi-objective shortest path problem. Analogically, each objective of the shortest path problem is taken as an individual event, thus the concurrent optimization of many objectives equals to the joint event of simultaneous occurrence of the multiple events, and therefore the simultaneous optimization of multiple objectives can be solved on basis of probability theory rationally. The partial favorable probability of each objective of every scheme (routine) is evaluated according to the actual preference degree of the utility indicator of the objective. Moreover, the product of all partial favorable probabilities of the utility of objective of each scheme (routine) casts the total favorable probability of the corresponding scheme (routine), which results in the decisively unique indicator of the scheme (routine) in the multi-objective shortest path problem in the point of view of system theory. Thus, the optimum solution of the multi-objective shortest path problem is the scheme (routine) with highest total favorable probability. Finally, an application example is given to illuminate the approach

Hybrid of "Intersection" Algorithm for Multi-Objective Optimization with Response Surface Methodology and its Application

Recently, a new "intersection" method for multi-objective optimization was developed in the points of view set theory and probability theory, which introduces a new idea of favorable probability to reflect the favorable degree of the utility of performance indicator in multi-objective optimization, and the product of all partial favorable probabilities of entire utilities of performance indicators makes the overall / total favorable probability of the candidate. Here, in this paper, the new "intersection" algorithm for multi-objective optimization is combined effectively with response surface methodology (RSM) by taking each response as one objective, which transfers the multi-response optimization problem into a single response one with the help of the overall / total favorable probability of each scheme. The overall / total favorable probability is the uniquely decisive index of the scheme in the optimization. Applications of the hybrid approach with two examples in material technology are given, proper predictions are obtained

Hybrids of Uniform Test and Sequential Uniform Designs with "Intersection" Method for Multi- objective Optimization

For multi-objective optimization under condition of complicated objective function, the data processing in the evaluation is sometime tediously long, special algorithm is needed to be adopted. Since the remarkable features of uniform distribution of test points within the test domain and the small number of tests, fully representative of each point, and easy to perform regression analysis, the uniform test design method is hybrid with the “intersection” method for multi-objective optimization to simplify the complicated data process in evaluation first. Furthermore, the "intersection" multi-objective optimization methodology is combined with sequential uniform design so as to get a more precise approximation for solving multi-objective optimization problem, the procedure for searching optimum of the “intersection“ multi-objective optimization methodology with sequential uniform design algorithm is put forward. A multi-objective optimization of linear programming problem with three variables is taken as our example, which involves a maximum for one objective and a minimum for another objective. The result for applying the novel approach to the example indicates the effectiveness of current hybrids

Extension of Intersection Method for Multi-Objective Optimization in Case of Interval Number and its Application

This paper aims to develop the extension of intersection method for multi-objective optimization under condition of interval number. Based on the linear correlation of partial favourable probability and the corresponding performance indicator, and the assumption of uniform distribution of the actual value of performance indicator within the range of its lower and upper limits in case of interval number, it derives that the actual partial favourable probability of a performance indicator is the arithmetic mean value of the partial favourable probabilities of the arithmetic mean value and the variation value of the interval index of the corresponding performance indicator for each candidate, or their desired sum. Furthermore, according to the rule of algorithm for the total favourable probability quantitatively, all candidates are ranked according to their total favourable probabilities to complete the multi- objective optimization in case of interval number. As applications, the quantitative assessments of multi-criteria selections for effective dwelling house walls, project managers and contractor for construction works are given in detail, satisfied results are obtained

A New "Intersection" Method for Multi-Objective Optimization in Material Selection

Till now the previous methods for multi-objective optimization adopt the "additive" algorithm for the normalized evaluation indexes, which has the inherent shortcoming of taking the form of "union" in the viewpoint of set theory. In fact, "simultaneous optimization of multiple indexes" should be more appropriate to take the form of "intersection" for the normalized evaluation indexes in the respects of set theory and "joint probability" in probability theory. In this paper, a new concept of favorable probability is proposed to reflect the favorable degree of the candidate material in the selection; All material property indicators are divided into beneficial or unbeneficial types to the material selection; Each material property indicator correlates to a partial favorable probability quantitatively, and the total favorable probability of a candidate material is the product of all partial favorable probabilities in the viewpoints of "intersection" of set theory and "joint probability" in probability theory, which is the sole decisive index in the competitive selection process. Results of the application examples indicate the validity of the new method

Approach of Solving Multi-objective Programming Problem by Means of Probability Theory and Uniform Experimental Design

In this paper, an approach to deal with the multi-objective programming problem is regulated by means of probability-based multi-objective optimization, discrete uniform experimental design, and sequential algorithm for optimization. The probability-based method for multi-objective optimization is used to conduct conversion of the multi-objective optimization problem into a single-objective optimization one in the viewpoint of probability theory. The discrete uniform experimental design is used to supply an efficient sampling to simplify the conversion. The sequential algorithm for optimization is employed to carry out further optimization. The corresponding treatments reveal the essence of the multi-objective programming, and consideration of the simultaneous optimization of each objective of multi-objective programming problem rationally. Two examples are conducted to illuminate the rationality of the approach

Error Analysis for Designed Test and Numerical Integral by Using UED in Material Research

In this article, the error analysis for designed test and definite integral in employing uniform experimental design is analogically developed on basis of midpoint rule in rectangle method for assessing definite integral through conducting discretized sampling approximately. It concluded that the discrete sampling-point by means of good lattice point in the evaluations of definite integral and maximum value of a function is promised with higher accuracy, and the predicted entire error of this uniform sampling point method for above problems decreases with the number of sampling points significantly

A new solution for solving a multi-objective integer programming problem with probabilistic multi-objective optimization

Introduction/purpose: In this paper, a new solution for solving a multiobjective integer programming problem with probabilistic multi – objective optimization is formulated. Furthermore, discretization by means of the good lattice point and sequential optimization are employed for a successive simplifying treatment and deep optimization. Methods: In probabilistic multi – objective optimization, a new concept of preferable probability has been introduced to describe the preference degree of each performance utility of a candidate; each performance utility of a candidate contributes a partial preferable probability and the product of all partial preferable probabilities deduces the total preferable probability of a candidate; the total preferable probability thus transfers a multi-objective problem into a single-objective one. Discretization by means of the good lattice point is employed to conduct discrete sampling for a continuous objective function and sequential optimization is used to perform deep optimization. At first, the requirements of integers in the treatment could be given up so as to simply conduct above procedures. Finally, the optimal solutions of the input variables must be rounded to the nearest integers. Results: This new scheme is used to deal with two production problems, i.e., maximizing profit while minimizing pollution and determining a purchasing plan for spending as little money as possible while getting as large amount of raw materials as possible. Promising results are obtained for the above two problems from the viewpoint of the probability theory for simultaneous optimization of multiple objectives. Conclusion: This method properly considers simultaneous optimization of multiple objectives in multi-objective integer programming, which naturally reflects the essence of multi-objective programming, and opens a new way of solving multi-objective problems

Portfolio investment based on probabilistic multi-objective optimization and uniform design for experiments with mixtures

Introduction/purpose: In this paper, a new approach to solving the portfolio investment problem is formulated to handle simultaneous optimization of both maximizing the rate of return and minimizing the variance of the rate of return. Probability - based multi – objective optimization is combined with uniform design for experiments with mixtures to conduct processing. Methods: Preliminarily, probability - based multi – objective optimization is employed to synthesize the bi-objective problem of simultaneous optimization of both maximizing the rate of return and minimizing the variance of the rate of return into a single objective one of the total preferable probability of each alternative scenario. The total preferable probability is the product of all partial preferable probabilities of each performance utility; subsequently, the method of uniform design for experiments with mixtures is used to create a set of effective sampling points for the portfolio investment problem to provide discretization in data processing and simplifying treatment, of which the proportion xi follows the constraint condition of xl + x2 + x3．．．+ xs = 1 with the total number of variables s for xi. Results: The new approach is used to deal with the portfolio Investment problem that is, in essence, simultaneous optimization of both maximizing the rate of return and minimizing the variance of the rate of return, which leads to reasonable consequences. The results are with the quality of rationality from the respect of the probability theory for simultaneous optimization of multiple objectives. Conclusion: This method naturally reflects the essence of the portfolio investment problem and opens a new way of solving the relevant problem

Stochastic evolutionary-based optimization for rapid diagnosis and energy-saving in pilot- and full-scale Carrousel oxidation ditches

Energy consumption is a primary issue needed to be considered for wastewater treatment targeting qualified effluent. In this paper, a hybrid model is proposed for rapid diagnosis of operational conditions meeting requirements of discharge standards and energy saving in the pilot-and full-scale Carrousel Oxidation Ditches (ODs). Based on a three-dimensional (3D) three-phase computational fluid dynamics (CFD) model, we developed an artificial neural network (ANN) model with back propagation algorithm and an accelerating genetic algorithm (AGA) model to achieve real-time simulation and system optimization in the Carrousel ODs. By incorporating the 3D-CFD and multi-site ANN models, the hybrid model provided reasonable predictions of liquid flow, sludge sedimentation and water quality in the Carrousel ODs. With help of the AGA model based on evolution theory, system optimization could be reached to meet multiple purposes such as energy saving, water-quality improving and normal sludge distribution, which was demonstrated that a 31% saving in total energy could possibly be made under an optimum operating condition compared to the existing operating condition in a full-scale OD