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.

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.

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.

Multi-objective linear fractional inventory model with possibility and necessity constraints under generalised intuitionistic fuzzy set environment

This study presented a multi-objective linear fractional inventory (LFI) problem with generalised intuitionistic fuzzy numbers. In modelling, the authors have assumed the ambiances where generalised trapezoidal intuitionistic fuzzy numbers (GTIFNs) used to handle the uncertain information in the data. Then, the given multi-objective generalised intuitionistic fuzzy LFI model was transformed into its equivalent deterministic linear fractional programming problem by employing the possibility and necessity measures. Finally, the applicability of the model is demonstrated with a numerical example and the sensitivity analysis under several parameters is investigated to explore the study.

Linear-in-Parameter Models Based on Parsimonious Genetic Programming Algorithm and Its Application to Aero-Engine Start Modeling

A novel Parsimonious Genetic Programming （PGP） algorithm together with a novel aero-engine optimum data-driven dynamic start process model based on PGP is proposed. In application of this method, first, the traditional Genetic Programming（GP） is used to generate the nonlinear input-output models that are represented in a binary tree structure; then, the Orthogonal Least Squares algorithm （OLS） is used to estimate the contribution of the branches of the tree （refer to basic function term that cannot be decomposed anymore according to special rule） to the accuracy of the model, which contributes to eliminate complex redundant subtrees and enhance GP＇s convergence speed; and finally, a simple, reliable and exact linear-in-parameter nonlinear model via GP evolution is obtained. The real aero-engine start process test data simulation and the comparisons with Support Vector Machines （SVM） validate that the proposed method can generate more applicable, interpretable models and achieve comparable, even superior results to SVM.

Max-plus-linear model-based predictive control for constrained hybrid systems: linear programming solution

In this paper, a linear programming method is proposed to solve model predictive control for a class of hybrid systems. Firstly, using the （max, ＋） algebra, a typical subclass of hybrid systems called max-plus-linear （MPL） systems is obtained. And then, model predictive control （MPC） framework is extended to MPL systems. In general, the nonlinear optimization approach or extended linear complementarity problem （ELCP） were applied to solve the MPL-MPC optimization problem. A new optimization method based on canonical forms for max-min-plus-scaling （MMPS） functions （using the operations maximization, minimization, addition and scalar multiplication） with linear constraints on the inputs is presented. The proposed approach consists in solving several linear programming problems and is more efficient than nonlinear optimization. The validity of the algorithm is illustrated by an example.

A Weighted Goal Programming Model for the DASH Diet Problem: Comparison with the Linear Programming DASH Diet Model

A Linear Programming DASH diet model for persons with hypertension has previously been formulated and daily minimum cost diet plans that satisfy the DASH diets’ tolerable intake level of the nutrients for 1500 mg a day Sodium level and different daily calorie levels were obtained using sample foods from the DASH diet eating plan chart. But the limitation in the use of linear programming model in selecting diet plans to meet specific nutritional requirements which normally results in the oversupply of certain nutrients was evident in the linear programming DASH diet plan obtained as the nutrient level of the diet plans obtained had wide deviations of from the DASH diets’ tolerable upper and lower intake level for the given calorie and sodium levels. Hence the need for a model that gives diet plans with minimized nutrients’ level deviations from the DASH diets’ tolerable intake level for different daily calorie and sodium level at desired cost. A weighted Goal Programming DASH diet model that minimizes the daily cost of the DASH eating plan as well as deviations of the diets’ nutrients content from the DASH diet’s tolerable intake levels is hereby presented in this work. The formulated weighted goal programming DASH diet model is further illustrated using chosen sample foods from the DASH food chart as used in the work on the linear programming DASH diet model for a 1500 mg sodium level and 2000 calories a day diet plan as well as for 1800, 2200, 2400, 2600, 2800 and 3000 daily calorie levels. A comparison of the DASH nutrients’ composition of the weighted Goal Programming DASH diet plans and those of the linear programming DASH diet plans were carried out at this sodium level and the different daily calorie levels. It was evident from the results of the comparison that the weighted goal programming DASH diet plans has minimized deviations from the DASH diet’s tolerable intake levels than those of the linear programming DASH diet plans.

Application of Linear Programming Model to Refugee Migrating Problem

The refugee immigration problem can be considered as a special “transportation problem”. Linear Programming Model is built, where two objectives with weight in the objective function, for the shortest routes that the refugees go along and the minimum number of refugees stayed in each country. An example of EU is introduced and calculated on Lingo software. The results show that the model is available to solve the refugee immigration problem in different scale.

Approximate Solutions for a Class of Fractional-Order Model of HIV Infection via Linear Programming Problem

In this paper, we provide a new approach to solve approximately a system of fractional differential equations (FDEs). We extend this approach for approximately solving a fractional-order differential equation model of HIV infection of CD4+T cells with therapy effect. The fractional derivative in our approach is in the sense of Riemann-Liouville. To solve the problem, we reduce the system of FDE to a discrete optimization problem. By obtaining the optimal solutions of new problem by minimization the total errors, we obtain the approximate solution of the original problem. The numerical solutions obtained from the proposed approach indicate that our approximation is easy to implement and accurate when it is applied to a systems of FDEs.

Managing Tourism Development in the Dutch City of Utrecht using a Linear Programming Model of the Tourist Carrying Capacity

As one part of the Landstad International Metropolitan Area,Utrecht has become known as one of the best tourist destinations in recent years,with the rapid growth of worldwide travel trends.In this paper,through the adaption of the linear programming model,the paper intends to not only quantify the optimum number of visitors to Utrecht but also formulate a number of policy recommendations based on the reconstruction of this optimum.The paper draws the following conclusions:(1) tourist carrying capacity of Utrecht is not yet exceeded;(2) restrictive accommodation policy does not currently seem necessary;(3) the cultural-historical attractions are not yet optimally used;(4) investing in strategic provisions is currently not necessary.And from the conclusion,the paper further puts forward the following suggestions on the city’s tourism development strategy:(1) to identify "tourist flood plains";(2) to encourage the tourist disclosure of these alternatives;(3) to invest even more explicitly in residential tourism and,where possible,curb day tourism;(4) to introduce a new business model.

Multi-objective planning model for simultaneous reconfiguration of power distribution network and allocation of renewable energy resources and capacitors with considering uncertainties

This research develops a comprehensive method to solve a combinatorial problem consisting of distribution system reconfiguration, capacitor allocation, and renewable energy resources sizing and siting simultaneously and to improve power system's accountability and system performance parameters. Due to finding solution which is closer to realistic characteristics, load forecasting, market price errors and the uncertainties related to the variable output power of wind based DG units are put in consideration. This work employs NSGA-II accompanied by the fuzzy set theory to solve the aforementioned multi-objective problem. The proposed scheme finally leads to a solution with a minimum voltage deviation, a maximum voltage stability, lower amount of pollutant and lower cost. The cost includes the installation costs of new equipment, reconfiguration costs, power loss cost, reliability cost, cost of energy purchased from power market, upgrade costs of lines and operation and maintenance costs of DGs. Therefore, the proposed methodology improves power quality, reliability and security in lower costs besides its preserve, with the operational indices of power distribution networks in acceptable level. To validate the proposed methodology's usefulness, it was applied on the IEEE 33-bus distribution system then the outcomes were compared with initial configuration.

Linear Programs in Cost Accounting： A Linear Cost Model

This paper aims to find unit cost of a product for firms. It establishes a linear cost model to find unit cost. Linear goal programs assume a direct relationship between independent variable and dependent variable. Dependent variable of linear model is unit cost. Independent variables are cost accounting variables. They are supply cost, labor cost, and administration cost. This study assumes a direct relationship between supply-labor-administration costs and unit cost. Therefore, it establishes a linear cost model. The major research question of this study is to apply linear goal programming to cost accounting. The goal of this linear program is to find unit cost of product. This study uses quantitative method and human capital method. The main research result is linear costing model itself.

Application of Linear Programming Algorithm in the Optimization of Financial Portfolio of Golden Guinea Breweries Plc, Nigeria

In this study, Simplex Method, a Linear Programming technique was used to create a mathematical model that optimized the financial portfolio of Golden Guinea Breweries Plc, Nigeria. This work was motivated by the observed and anticipated miscalculations which Golden Guinea Breweries was bound to face if appropriate linear programming techniques were not applied in determining the profit level. This study therefore aims at using Simplex Method to create a Mathematical Model that will optimize the production of brewed drinks for Golden Guinea Breweries Plc. The first methodology involved the collection of sample data from the company, analyzed and the relevant coefficients were deployed for the coding of the model. Secondly, the indices collected from the first method were deployed in the software model called PHP simplex, an online software for solving Linear Programming Problem to access the profitability of the organization. The study showed that Linear Programming Model would give a high profit coefficient of N9,190,862,833 when compared with the result obtained from the manual computation which gave a profit coefficient of N7,172,093,375. Also, Bergedoff Lager, Eagle Stout and Bergedoff Malta were found not to contribute to overall profitability of the company and it was therefore recommended that their productions should be discontinued. It also recommends that various quantities of Golden Guinea Lager (1 × 12) and Golden Guinea Lager (1 × 24) should be produced.

Goal Programming for Solving Fractional Programming Problem in Fuzzy Environment

This paper is comprised of the modeling and optimization of a multi objective linear programming problem in fuzzy environment in which some goals are fractional and some are linear. Here, we present a new approach for its solution by using α-cut of fuzzy numbers. In this proposed method, we first define membership function for goals by introducing non-deviational variables for each of objective functions with effective use of α-cut intervals to deal with uncertain parameters being represented by fuzzy numbers. In the optimization process the under deviational variables are minimized for finding a most satisfactory solution. The developed method has also been implemented on a problem for illustration and comparison.

Sustainable Multi-Objective Multi-Reservoir Optimization Considering Environmental Flow

Increasing demand for water from all sectors presents a challenge for policy makers to improve water allocation policies for storage reservoirs. In addition, there are many other organisms and species present in river waters that also require water for their survival. Due to the lack of awareness many times the minimum required quantity and quality of water for river ecosystem is not made available at downstream of storage reservoirs. So, a sustainable approach is required in reservoir operations to maintain the river ecosystem with environmental flow while meeting the other demands. Multi-objective, multi-reservoir operation model developed with Python programming using Fuzzy Linear Programing method incorporating environmental flow requirement of river is presented in this paper. Objective of maximization of irrigation release is considered for first run. In second run maximization of releases for hydropower generation is considered as objective. Further both objectives are fuzzified by incorporating linear membership function and solved to maximize fuzzified objective function simultaneously by maximizing satisfaction level indicator (λ). The optimal reservoir operation policy is presented considering constraints including Irrigation release, Turbine release, Reservoir storage, Environmental flow release and hydrologic continuity. Model applied for multi-reservoir system consists of four reservoirs, i.e., Jayakwadi Stage-I Reservoir (R1), Jayakwadi Stage-II Reservoir (R2), Yeldari Reservoir (R3), Siddheshwar Reservoir (R4) in Godavari River sub-basin from Marathwada region of Maharashtra State, India.

Integer Programming Formulations for Maximum Lifetime Broadcasting Problems in Wireless Sensor Networks

Approaches based on integer linear programming have been recently proposed for topology optimization in wireless sensor networks. They are, however, based on over-theoretical, unrealistic models. Our aim is to show that it is possible to accommodate realistic models for energy consumption and communication protocols into integer linear programming. We analyze the maximum lifetime broadcasting topology problem and we present realistic models that are also shown to provide efficient and practical solving tools. We present a strategy to substantially speed up the convergence of the solving process of our algorithm. This strategy introduces a practical drawback, however, in the characteristics of the optimal solutions retrieved. A method to overcome this drawback is discussed. Computational experiments are reported.

Analysis of Energy Storage Operation Configuration of Power System Based on Multi-Objective Optimization

Driven by the goal of“carbon neutrality,”the increase in use of renewable energy power systems will be inevitable in the future.Uncontrolled output power and random volatility make it difficult to balance power in real time during system operation.Therefore,energy storage is considered to be an effective way to ensure the real-time balance of system power.However,cost of energy storage is relatively expensive.As a solution,energy storage can be used to balance the system power in order to reduce system operating costs.Taking the high proportion of wind power systems as an example,the impact of the“supply side”low-carbon transformation on the economics and reliability of power system operation is explored.In order to solve the problem of power system operation configuration optimization under the background of“carbon neutrality,”this paper establishes a multi-objective programming model.

基于改进云模型和线性规划优化算法的山区泥石流危险性评价

泥石流是一种由暴雨或地震诱发常发生在山区的破坏力强大的地质灾害,为了提前预防和减少其破坏能力,使用特定算法评价其危险性十分必要。采用云模型雾化特性对各评价指标权重进行一致性检验后耦合,得到合理的权重阈值区间。以线性规划优化算法通过设定目标函数后对各评价指标在权重区间内动态取值,从而得到各个泥石流沟渠最危险评分。改进了单指标的云数字特征公式,构建覆盖更全面的综合云模型,输入最大危险评分对研究区泥石流沟渠进行危险性分级。将该研究应用在汶川地震后北川县产生的72条泥石流沟渠实例中,其评价结果显示与研究区实际地形环境情况相吻合,为相似类型泥石流灾害的预防提供了新思路。

梅钢转炉合金模型开发

为降低转炉合金成本,采用线性规划为核心算法,开发了转炉合金模型。模型通过设计碳冗余质量分数和磷冗余质量分数、确定合金收得率和LF增碳质量分数等主要参数、基于优先加足低价合金的原则分配转炉合金用量,在满足钢水成分的前提下降低了合金成本,可产生经济效益3.27元/t。