Solving Multi-Objective Linear Programming Problem by Statistical Averaging Method with the Help of Fuzzy Programming Method

A multi-objective linear programming problem is made from fuzzy linear programming problem. It is due the fact that it is used fuzzy programming method during the solution. The Multi objective linear programming problem can be converted into the single objective function by various methods as Chandra Sen’s method, weighted sum method, ranking function method, statistical averaging method. In this paper, Chandra Sen’s method and statistical averaging method both are used here for making single objective function from multi-objective function. Two multi-objective programming problems are solved to verify the result. One is numerical example and the other is real life example. Then the problems are solved by ordinary simplex method and fuzzy programming method. It can be seen that fuzzy programming method gives better optimal values than the ordinary simplex method.

Principal-subordinate hierarchical multi-objective programming model of initial water rights allocation

The principal-subordinate hierarchical multi-objective programming model of initial water rights allocation was developed based on the principle of coordinated and sustainable development of different regions and water sectors within a basin. With the precondition of strictly controlling maximum emissions rights, initial water rights were allocated between the first and the second levels of the hierarchy in order to promote fair and coordinated development across different regions of the basin and coordinated and efficient water use across different water sectors, realize the maximum comprehensive benefits to the basin, promote the unity of quantity and quality of initial water rights allocation, and eliminate water conflict across different regions and water sectors. According to interactive decision-making theory, a principal-subordinate hierarchical interactive iterative algorithm based on the satisfaction degree was developed and used to solve the initial water rights allocation model. A case study verified the validity of the model.

Approach for uncertain multi-objective programming problems with correlated objective functions under C_(EV) criterion

An uncertain multi-objective programming problem is a special type of mathematical multi-objective programming involving uncertain variables. This type of problem is important because there are several uncertain variables in real-world problems.Therefore, research on the uncertain multi-objective programming problem is highly relevant, particularly those problems whose objective functions are correlated. In this paper, an approach that solves an uncertain multi-objective programming problem under the expected-variance value criterion is proposed. First, we define the basic framework of the approach and review concepts such as a Pareto efficient solution and expected-variance value criterion using an order relation between various uncertain variables.Second, the uncertain multi-objective problem is converted into an uncertain single-objective programming problem via a linear weighted method or ideal point method. Then the problem is transformed into a deterministic single objective programming problem under the expected-variance value criterion. Third, four lemmas and two theorems are proved to illustrate that the optimal solution of the deterministic single-objective programming problem is an efficient solution to the original uncertainty problem. Finally, two numerical examples are presented to validate the effectiveness of the proposed approach.

New approach for uncertain random multi-objective programming problems based on C_(ESD) criterion

To overcome the defects that the traditional ap-proach for multi-objective programming under uncertain ran-dom environment(URMOP)neglects the randomness and uncer-tainty of the problem and the volatility of the results,a new ap-proach is proposed based on expected value-standard devi-ation value criterion(C_(ESD) criterion).Firstly,the effective solution to the URMOP problem is defined;then,by applying sequence relationship between the uncertain random variables,the UR-MOP problem is transformed into a single-objective program-ming(SOP)under uncertain random environment(URSOP),which are transformed into a deterministic counterpart based on the C_(ESD) criterion.Then the validity of the new approach is proved that the optimal solution to the SOP problem is also effi-cient for the URMOP problem;finally,a numerical example and a case application are presented to show the effectiveness of the new approach.

MULTI-OBJECTIVE PROGRAMMING MODEL OF TROPICAL CROPS IN HAINAN ISLAND

According to Hainan Island's biological characteristics, and existing structure of productivity of tropical crops and local climatic conditions, this paper carries on regional division of tropical crops by fuzzy mathematics. Based on calculation of basic parameters for tl1e formation of production, near-tem optimum models of tropical crops structure of each region was established by means of multi-objective programming, and a far-term grey programming model was set up through the above-mentioned near-term model and prediction of future parameters. Conclusion shows that the near-term programming may raise the profit by 5. 1-55.7 percent and far-tem programming by 54-90 percent, both gainingobvious economic benefits.

MULTI-OBJECTIVE PROGRAMMING FOR AIRPORT GATE REASSIGNMENT

To improve the efficiency of gate reassignment and optimize the plan of gate reassignment,the concept of disruption management is introduced,and a multi-objective programming model for airport gate reassignment is proposed.Considering the interests of passengers and the airport,the model minimizes the total flight delay,the total passengers′walking distance and the number of flights reassigned to other gates different from the planned ones.According to the characteristics of the gate reassignment,the model is simplified.As the multi-objective programming model is hard to reach the optimal solutions simultaneously,a threshold of satisfactory solutions of the model is set.Then a simulated annealing algorithm is designed for the model.Case studies show that the model decreases the total flight delay to the satisfactory solutions,and minimizes the total passengers′walking distance.The least change of planned assignment is also reached.The results achieve the goals of disruption management.Therefore,the model is verified to be effective.

Compactness, Contractibility and Fixed Point Properties of the Pareto Sets in Multi-Objective Programming

This paper presents the Pareto solutions in continuous multi-objective mathematical programming. We discuss the role of some assumptions on the objective functions and feasible domain, the relationship between them, and compactness, contractibility and fixed point properties of the Pareto sets. The authors have tried to remove the concavity assumptions on the objective functions which are usually used in multi-objective maximization problems. The results are based on constructing a retraction from the feasible domain onto the Pareto-optimal set.

Estimating the Macroeconomic Costs of CO_2 Emission Reduction in China Based on Multi-objective Programming

This paper estimates the macroeconornic costs of CO2 emission reduction in China employing the input-output analysis with the multi-objective programming approach. The results show that the effect of reducing CO2 emissions on China＇s economy is significant. Under the present conditions, the estimated macroeconomic costs of CO2 emission reduction in 2010 for China are approximately 3,100-4,024 RMB t-1. The stronger the abatement actions, the higher the macroeconomic costs of per unit emission reduction would be. Excavation industry, oil industry, chemical industry, and metal smelting industry have high potential to abate their CO2 emissions.

Path Selection of Multimodal Transport Based on Multi-Objective Mixed Integer Programming

Based on “One Belt and One Road”, this paper studies the path selection of multimodal transport by using the method of multi-objective mixed integer programming. Therefore, this paper studies the factors of transportation time, transportation cost and transportation safety performance, and establishes a mathematical model. In addition, the method of multi-objective mixed integer programming is used to comprehensively consider the different emphasis and differences of customers on cargo transportation. Then we use planning tools of Microsoft Excel to solve path selection and to determine whether the chosen path is economical and reliable. Finally, a relatively complex road network is built as an example to verify the accuracy of this planning method.

Determining Efficient Solutions of Multi-Objective Linear Fractional Programming Problems and Application

In this paper, a modified method to find the efficient solutions of multi-objective linear fractional programming (MOLFP) problems is presented. While some of the previously proposed methods provide only one efficient solution to the MOLFP problem, this modified method provides multiple efficient solutions to the problem. As a result, it provides the decision makers flexibility to choose a better option from alternatives according to their financial position and their level of satisfaction of objectives. A numerical example is provided to illustrate the modified method and also a real life oriented production problem is modeled and solved.

Roughly <i>B</i>-invex Multi-Objective Programming Problems

In this paper, we shall be interested in characterization of efficient solutions for special classes of problems. These classes consider roughly B-invexity of involved functions. Sufficient and necessary conditions for a feasible solution to be an efficient or properly efficient solution are obtained.

Generating Efficient Solutions in Bilevel Multi-Objective Programming Problems

In this paper, we address bilevel multi-objective programming problems (BMPP) in which the decision maker at each level has multiple objective functions conflicting with each other. Given a BMPP, we show how to construct two artificial multiobjective programming problems such that any point that is efficient for both the two problems is an efficient solution of the BMPP. Some necessary and sufficient conditions for which the obtained result is applicable are provided. A complete procedure of the implementation of an algorithm for generating efficient solutions for the linear case of BMPP is presented. A numerical example is provided to illustrate how the algorithm operates.

A Penalty Function Algorithm with Objective Parameters and Constraint Penalty Parameter for Multi-Objective Programming

In this paper, we present an algorithm to solve the inequality constrained multi-objective programming (MP) by using a penalty function with objective parameters and constraint penalty parameter. First, the penalty function with objective parameters and constraint penalty parameter for MP and the corresponding unconstraint penalty optimization problem (UPOP) is defined. Under some conditions, a Pareto efficient solution (or a weakly-efficient solution) to UPOP is proved to be a Pareto efficient solution (or a weakly-efficient solution) to MP. The penalty function is proved to be exact under a stable condition. Then, we design an algorithm to solve MP and prove its convergence. Finally, numerical examples show that the algorithm may help decision makers to find a satisfactory solution to MP.

Optimality for Multi-Objective Programming Involving Arcwise Connected d-Type-I Functions

This paper deals with the optimality conditions and dual theory of multi-objective programming problems involving generalized convexity. New classes of generalized type-I functions are introduced for arcwise connected functions, and examples are given to show the existence of these functions. By utilizing the new concepts, several sufficient optimality conditions and Mond-Weir type duality results are proposed for non-differentiable multi-objective programming problem.

Solving Fuzzy Multi-Objective Linear Programming Problem by Applying Statistical Method

In this paper, the statistical averaging method and the new statistical averaging methods have been used to solve the fuzzy multi-objective linear programming problems. These methods have been applied to form a single objective function from the fuzzy multi-objective linear programming problems. At first, a numerical example of solving fuzzy multi-objective linear programming problem has been provided to validate the maximum risk reduction by the proposed method. The proposed method has been applied to assess the risk of damage due to natural calamities like flood, cyclone, sidor, and storms at the coastal areas in Bangladesh. The proposed method of solving the fuzzy multi-objective linear programming problems by the statistical method has been compared with the Chandra Sen’s method. The numerical results show that the proposed method maximizes the risk reduction capacity better than Chandra Sen’s method.

Nonconvex Quadratic Programming Method for k－Coloring Problem：Algorithm and Computation

In this paper, we consider the socalled k-coloring problem in general case.Firstly, a special quadratic 0-1 programming is constructed to formulate k-coloring problem. Secondly, by use of the equivalence between above quadratic0-1 programming and its relaxed problem, k-coloring problem is converted intoa class of (continuous) nonconvex quadratic programs, and several theoreticresults are also introduced. Thirdly, linear programming approximate algorithmis quoted and verified for this class of nonconvex quadratic programs. Finally,examining problems which are used to test the algorithm are constructed andsufficient computation experiments are reported.

Modeling and Multi-objective Optimization of Refinery Hydrogen Network

The demand of hydrogen in oil refinery is increasing as market forces and environmental legislation, so hydrogen network management is becoming increasingly important in refineries. Most studies focused on single-objective optimization problem for the hydrogen network, but few account for the multi-objective optimization problem. This paper presents a novel approach for modeling and multi-objective optimization for hydrogen network in refineries. An improved multi-objective optimization model is proposed based on the concept of superstructure. The optimization includes minimization of operating cost and minimization of investment cost of equipment. The proposed methodology for the multi-objective optimization of hydrogen network takes into account flow rate constraints, pressure constraints, purity constraints, impurity constraints, payback period, etc. The method considers all the feasible connections and subjects this to mixed-integer nonlinear programming （MINLP）. A deterministic optimization method is applied to solve this multi-objective optimization problem. Finally, a real case study is intro-duced to illustrate the applicability of the approach.

Multi-objective Evolutionary Algorithms for MILP and MINLP in Process Synthesis

Steady-state non-dominated sorting genetic algorithm (SNSGA), a new form of multi-objective genetic algorithm, is implemented by combining the steady-state idea in steady-state genetic algorithms (SSGA) and the fitness assignment strategy of non-dominated sorting genetic algorithm (NSGA). The fitness assignment strategy is improved and a new self-adjustment scheme of is proposed. This algorithm is proved to be very efficient both computationally and in terms of the quality of the Pareto fronts produced with five test problems including GA difficult problem and GA deceptive one. Finally, SNSGA is introduced to solve multi-objective mixed integer linear programming (MILP) and mixed integer non-linear programming (MINLP) problems in process synthesis.

Integrating Conjugate Gradients Into Evolutionary Algorithms for Large-Scale Continuous Multi-Objective Optimization

Large-scale multi-objective optimization problems(LSMOPs)pose challenges to existing optimizers since a set of well-converged and diverse solutions should be found in huge search spaces.While evolutionary algorithms are good at solving small-scale multi-objective optimization problems,they are criticized for low efficiency in converging to the optimums of LSMOPs.By contrast,mathematical programming methods offer fast convergence speed on large-scale single-objective optimization problems,but they have difficulties in finding diverse solutions for LSMOPs.Currently,how to integrate evolutionary algorithms with mathematical programming methods to solve LSMOPs remains unexplored.In this paper,a hybrid algorithm is tailored for LSMOPs by coupling differential evolution and a conjugate gradient method.On the one hand,conjugate gradients and differential evolution are used to update different decision variables of a set of solutions,where the former drives the solutions to quickly converge towards the Pareto front and the latter promotes the diversity of the solutions to cover the whole Pareto front.On the other hand,objective decomposition strategy of evolutionary multi-objective optimization is used to differentiate the conjugate gradients of solutions,and the line search strategy of mathematical programming is used to ensure the higher quality of each offspring than its parent.In comparison with state-of-the-art evolutionary algorithms,mathematical programming methods,and hybrid algorithms,the proposed algorithm exhibits better convergence and diversity performance on a variety of benchmark and real-world LSMOPs.