Interval Objective Function Research Articles

A recurrent neural network approach for nonconvex interval quadratic programming

In the paper, a new approach is presented to tackle an interval optimization problem with nonconvex interval objective function and affine interval equality and inequality constraints utilizing a recurrent neural network. The approach involves converting the nonconvex interval quadratic programming into a cubic optimization via convex combinations and considering its weight as a decision variable. The recurrent neural network is then established to solve the cubic optimization through subgradient techniques and an exterior penalty function method. Moreover, it is proven that there is a unique global solution to the recurrent neural network, and its trajectory can reach the feasible region in a finite time and remain there. Additionally, it is demonstrated that the trajectory of recurrent neural network converges exponentially or in finite time towards a singleton belonging to the set of critical points of the cubic optimization. This approach is beneficial for interval quadratic programming problems where the weight is unknown or challenging to determine in practical applications. Finally, the effectiveness of this approach is verified through two numerical examples.

The impact of R&D expenditures and screening in an economic production rate (EPR) inventory model for a flawed production system with imperfect screening under an interval-valued environment

In this paper, a single product of a single manufacturing unit for an imperfect production inventory system is modelled to minimize the total cost of a production cycle. A large fraction of defective items is passed to customers by manufacturers due to imperfect screening method so manufacturer replace them with new ones while incurring penalties. This fraction depends on the investment of research and development (R&D). This paper optimizes the R&D cost to minimize the defective items produced in a production line. The demand function depends on the selling price, advertisement cost, and product quality. The quality of the produce items is measured by screening time effort and investment in R&D. Further, we considered all the costs of the production cycle in interval-valued form and elaborately discuss the application of interval metric in the total cost interval objective function. A necessary and sufficient conditions are derived for the proposed model with the help of modified KKT conditions. Results show that applying screening and R&D costs significantly reduces the number of defective items and simultaneously reduces the total cost. A numerical example is presented along with sensitivity analyses on different parameters with the help of MATLAB software.

Optimality conditions for interval valued optimization problems

This work addresses constrained optimization problems in which the objective function is interval-valued while the inequality constraints functions are real-valued. Both necessary and sufficient optimality conditions are derived. They are given through the gH-gradient and the gH-directional derivative of the interval objective function. The necessary ones are of KKT-type. The sufficient conditions are of generalized convexity type. The developed theory is illustrated by means of some numerical examples.

A credibility-based interval multi-objective crop area planning model for agricultural and ecological management

This study presents a credibility-based interval multi-objective crop area planning model for agricultural and ecological management under uncertainty. The model is developed through the integrations of interval multi-objective programming and fuzzy credibility-constrained programming by considering economic and ecological benefits of the study system, and dealing with interval and fuzzy uncertainties. It focuses on crop area optimization with the interval objective function of maximizing the system benefits, maximizing the watershed area, and fractional interval objective function of maximizing the land productivity (i.e., system benefits per unit area). Additionally, the relationship between watershed area and ecological discharge is expressed as interval linear function, which is estimated by using interval regression analysis method. Then, the model is applied to a case study in Minqin County in the lower reaches of Shiyang River Basin for crop area optimization in view of the increasing shortage of water resources and the competition between agricultural and ecological water use. Optimal solutions for different scenarios can be generated based on fuzzy credibility constraints when the predefined credibility levels are set as 0.6, 0.7 and 0.8, 0.9, respectively. The results show that the crop area optimization model can better balance the regional agricultural and ecological benefits while ensuring grain production and promoting water saving. Moreover, a higher level of credibility corresponds to a lower system benefit but a larger watershed area. When credibility level is raised from 0.6 to 0.9, the upper bounds of system benefits are decreased from 13.87×108 Yuan to 13.81×108 Yuan whereas the watershed area is increased from 41.01 km2 to 42.12 km2. These various optimal results can help decision makers weigh system benefits and constraint-violation risks. Therefore, the results of the model can provide decision-making basis for the crop area optimization and provide valuable basis for the sustainable development in Minqin County and similar study areas.

Optimal Planning of Intelligent Parking IoT Using Interval Particle Swarm Optimization Algorithm in the Presence of Fuel cell and Electrolyzer as Hydrogen Storage System

Recently, the application of the intelligent parking lot (IPL) in the power market has been exponentially increasing to decrease the greenhouse gasses, the pollution, and to decrease the deviation cost of the energy production based on electric vehicles (EV). IPLs uses charge and discharge features of EVs to exchange the energy in the upstream grid. This paper study on a new interval-analysis based optimal solution of an IPL by considering the interval uncertainties for the price of upstream gird value. The method based on using an interval-based particle swarm optimization algorithm to optimize an interval objective function with lower and upper limitations with a single-valued output. Simulation results of the presented procedure are compared with a deterministic mixed-integer linear programming to show its superiority. The results show that deviation cost has been decreased up to 10.74% while average cost has been raised into 5.17% which demonstrates the methods high performance in decreasing the average cost of IPL and the reliability of the intelligent parking lot in the presence of uncertainties derived from the upstream grid.

On the representativeness of approximate solutions of discrete optimization problems with interval objective function

On the representativeness of approximate solutions of discrete optimization problems with interval objective function

Подходы к решению задач дискретной оптимизации с интервальной целевой функцией

В работе дается обзор подходов к решению задач дискретной оптимизации с интервальной целевой функцией. Эти подходы рассматриваются в общем контексте исследований оптимизационных задач с неопределенностями в постановках. Приводятся варианты концепций оптимальности решений для задач дискретной оптимизации с интервальной целевой функцией - робастные решения, множества решений, оптимальных по Парето, слабые и сильные оптимальные решения, объединенные множества решений и др. Оценивается предпочтительность выбора той или иной концепции оптимальности при решении задач и отмечаются ограничения для применения использующих их подходов Optimization problems with uncertainties in their input data have been investigated by many researchers in different directions. There are a lot of sources of the uncertainties in the input data for applied problems. Inaccurate measurements and variability of the parameters with time are some of such sources. The interval of possible values of uncertain parameter is the natural and the only possible way to represent the uncertainty for a wide share of applied problems. We consider discrete optimization problems with interval uncertainties in their objective functions. The purpose of the paper is to provide an overview of the investigations in this field. The overview is given in the overall context of the researches of optimization problems with uncertainties. We review the interval approaches for the discrete optimization problem with interval objective function. The approaches we consider operate with the interval values and are focused on obtaining possible solutions or certain sets of the solutions that are optimal according to some concepts of optimality that are used by the approaches. We consider the different concepts of optimality: robust solutions, the Pareto sets, weak and strong solutions, the united solution sets, the sets of possible approximate solutions that correspond to possible values of uncertain parameters. All the approaches we consider allow absence of information on probabilistic distribution on intervals of possible values of parameters, though some of them may use the information to evaluate the probabilities of possible solutions, the distribution on the interval of possible objective function values for the solutions, etc. We assess the possibilities and limitations of the considered approaches

A New Method to Solve Fuzzy Interval Flexible Linear Programming Using a Multi-Objective Approach

Recently fuzzy interval flexible linear programs have attracted many interests. These models are an extension of the classical linear programming which deal with crisp parameters. However, in most of the real-world applications, the nature of the parameters of the decision-making problems is generally imprecise. Such uncertainties can lead to increased complexities in the related optimisation efforts. Simply ignoring these uncertainties is considered undesired as it may result in inferior or wrong decisions. Therefore, inexact linear programming methods are desired under uncertainty. In this paper, we concentrate a fuzzy flexible linear programming model with flexible constraints and the interval objective function and then propose a new solving approach based on solving an associated multi-objective model. Finally, numerical example is included to illustrate the mentioned solving process.

Enhancement of Capacitated Transportation Problem in Fuzzy Environment

This research work aims to study a capacitated transportation problem (CTP) with penalty cost, supplies, and demands represented by hexagonal fuzzy numbers. Based on ranking function, the supplies and demands are converted to the crisp form. Through the use of the α‐level, the problem is converted into interval linear programming. To optimize the interval objective function, we define the order relations represented by policy maker’s choice between intervals. The maximization (minimization) problem considering the interval objective function is transformed to multiobjective optimization problem based on order relations introduced by the preference of policy makers between interval profits (costs). A numerical example is given for illustration and to check the validity of the suggested approach.

New conditions for testing necessarily/possibly efficiency of non-degenerate basic solutions based on the tolerance approach

In this paper, a specific type of multiobjective linear programming problem with interval objective function coefficients is studied. Usually, in such problems, it is not possible to obtain an optimal solution which optimizes simultaneously all objective functions in the interval multiobjective linear programming (IMOLP) problem, requiring the selection of a compromise solution. In conventional multiobjective programming problems these compromise solutions are called efficient solutions. However, the efficiency cannot be defined in a unique way in IMOLP problems. Necessary efficiency and possible efficiency have been considered as two natural extensions of efficiency to IMOLP problems. In this case, necessarily efficient solutions may not exist and the set of possibly efficient solutions usually has an infinite number of elements. Furthermore, it has been concluded that the problem of checking necessary efficiency is co-NP-complete even for the case of only one objective function. In this paper, we explore new conditions for testing necessarily/possibly efficiency of basic non-degenerate solutions in IMOLP problems. We show properties of the necessarily efficient solutions in connection with possibly and necessarily optimal solutions to the related single objective problems. Moreover, we utilize the tolerance approach and sensitivity analysis for testing the necessary efficiency.Finally, based on the new conditions, a procedure to obtain some necessarily efficient and strictly possibly efficient solutions to multiobjective problems with interval objective functions is suggested.

A study of interval optimization problems

We study optimization problems with interval objective functions. We focus on set-type solution notions defined using the Kulisch–Miranker order between intervals. We obtain bounds for the asymptotic cones of level, colevel and solution sets that allow us to deduce coercivity properties and coercive existence results. Finally, we obtain various noncoercive existence results. Our results are easy to check since they are given in terms of the asymptotic cone of the constraint set and the asymptotic functions of the end point functions. This work extends, unifies and sheds new light on the theory of these problems.

Robust topology optimization for multi-material structures under interval uncertainty

In this paper, we propose an efficient method to design robust multi-material structures under interval loading uncertainty. The objective of this study is to minimize the structural compliance of linear elastic structures. First, the loading uncertainty can be decomposed into two unit forces in the horizontal and vertical directions based on the orthogonal decomposition, which separates the uncertainty into the calculation coefficients of structural compliance that are not related to the finite element analysis. In this manner, the time-consuming procedure, namely, the nested double-loop optimization, can be avoided. Second, the uncertainty problem can be transformed into an augmented deterministic problem by means of uniform sampling, which exploits the coefficients related to interval variables. Finally, an efficient sensitivity analysis method is explicitly developed. Thus, the robust topology optimization (RTO) problem considering interval uncertainty can be solved by combining orthogonal decomposition with uniform sampling (ODUS). In order to eliminate the influence of numerical units when comparing the optimal results to deterministic and RTO solutions, the relative uncertainty related to interval objective function is employed to characterize the structural robustness. Several multi-material structure optimization cases are provided to demonstrate the feasibility and efficiency of the proposed method, where the magnitude uncertainty, directional uncertainty, and combined uncertainty are investigated.

A New Method to Solve Fuzzy Interval Flexible Linear Programming Using a Multi-Objective Approach

ABSTRACT Recently fuzzy interval flexible linear programs have attracted many interests. These models are an extension of the classical linear programming which deal with crisp parameters. However, in most of the real-world applications, the nature of the parameters of the decision-making problems are generally imprecise. Such uncertainties can lead to increased complexities in the related optimisation efforts. Simply ignoring these uncertainties is considered undesired as it may result in inferior or wrong decisions. Therefore, inexact linear programming methods are desired under uncertainty. In this paper, we concentrate a fuzzy flexible linear programming model with flexible constraints and the interval objective function and then propose a new solving approach based on solving an associated multi-objective model. Finally, a numerical example is included to illustrate the mentioned solving process.

A novel approach based on preference-based index for interval bilevel linear programming problem

This paper proposes a new methodology for solving the interval bilevel linear programming problem in which all coefficients of both objective functions and constraints are considered as interval numbers. In order to keep as much uncertainty of the original constraint region as possible, the original problem is first converted into an interval bilevel programming problem with interval coefficients in both objective functions only through normal variation of interval number and chance-constrained programming. With the consideration of different preferences of different decision makers, the concept of the preference level that the interval objective function is preferred to a target interval is defined based on the preference-based index. Then a preference-based deterministic bilevel programming problem is constructed in terms of the preference level and the order relation preceq_{mw}. Furthermore, the concept of a preference δ-optimal solution is given. Subsequently, the constructed deterministic nonlinear bilevel problem is solved with the help of estimation of distribution algorithm. Finally, several numerical examples are provided to demonstrate the effectiveness of the proposed approach.

Models for solid transportation problems in logistics using particle swarm optimisation algorithm and genetic algorithm

Transportation policy seeks to improve agency freight and cargo management and enhance sustainable, efficient and effective transportation operations. In this paper, four new fuzzy fixed charge solid transportation problems (FFCSTP) are formulated to maximise the total profit and minimise the total cost. The interval objective function is approximated to an intervalvalued function, i.e., transformed to a single objective using weighted sum method and weighted multiplication method. The fuzzy constraints are converted to its equivalent deterministic form using different interval order relations. Genetic algorithm (GA) and particle swarm optimisation (PSO) algorithm are used to obtain the optimal transportation schedule for the proposed solid transportation problem. During the evaluation of the models, in one case, limitation on the transported amounts is imposed and in other case, no such limitation is used. The models are illustrated with numerical examples and the optimum results of the models are compared.

Models for solid transportation problems in logistics using particle swarm optimisation algorithm and genetic algorithm

Transportation policy seeks to improve agency freight and cargo management and enhance sustainable, efficient and effective transportation operations. In this paper, four new fuzzy fixed charge solid transportation problems (FFCSTP) are formulated to maximise the total profit and minimise the total cost. The interval objective function is approximated to an intervalvalued function, i.e., transformed to a single objective using weighted sum method and weighted multiplication method. The fuzzy constraints are converted to its equivalent deterministic form using different interval order relations. Genetic algorithm (GA) and particle swarm optimisation (PSO) algorithm are used to obtain the optimal transportation schedule for the proposed solid transportation problem. During the evaluation of the models, in one case, limitation on the transported amounts is imposed and in other case, no such limitation is used. The models are illustrated with numerical examples and the optimum results of the models are compared.

A production inventory model with price discounted fuzzy demand using an interval compared hybrid algorithm

Abstract An economic production quantity (EPQ) model in an imprecise environment is proposed, where the production rate, planning horizon and demand coefficients are fuzzy in nature. At the beginning of each cycle a price discount is offered for a period to boost the demand. During this period, demand increases with time depending upon the amount of discount. Here, demand also depends on the unit selling price. After withdrawal of the price discount, demand depends only on the unit selling price. The governing differential equation for the model is obviously fuzzy in nature as the production rate and demand are fuzzy. For this reason, the model is formulated using a fuzzy differential equation and the α -cut of the total profit from the planning horizon is obtained using fuzzy Riemann integration. To optimize the interval objective function, using a fuzzy preference relation on intervals and a fuzzy possibility/necessity measure, a hybrid algorithm with varying population size is developed by combining the features of particle swarm optimization (PSO) and a genetic algorithm (GA). This algorithm is named Interval Compared Hybrid Particle Swarm-Genetic Algorithm (ICHPSGA) and is used to find an optimal decision for the decision maker (DM) in different cases of the model. To test the efficiency of the algorithm, it is compared with two other established algorithms namely PSO and PSGA. Numerical experiments are performed to illustrate the model and some interesting observations are made.

Robust topology optimization for structures under interval uncertainty

This paper proposes a new non-probabilistic robust topology optimization approach for structures under interval uncertainty, as a complementarity of the probabilistic robust topology optimization methods. Firstly, to avoid the nested double-loop optimization procedure that is time consuming in computations, the interval arithmetic is introduced to estimate the bounds of the interval objective function and formulate the design problem under the worst scenario. Secondly, a type of non-intrusive method using the Chebyshev interval inclusion function is established to implement the interval arithmetic. Finally, a new sensitivity analysis method is developed to evaluate the design sensitivities for objective functions like structural mean compliance with respect to interval uncertainty. It can overcome the difficulty due to non-differentiability of intervals and enable the direct application of gradient-based optimization algorithms, e.g. the Method of Moving Asymptotes (MMA), to the interval uncertain topology optimization problems. Several examples are used to demonstrate the effectiveness of the proposed method.

An approach for solving a fuzzy bilevel programming problem through nearest interval approximation approach and KKT optimality conditions

In this paper, we consider a kind of bilevel linear programming problem where the coefficients of both objective functions are fuzzy numbers. In order to deal with such a problem, the original problem can be approximated by an interval bilevel programming problem in terms of the nearest interval approximation of a fuzzy number. Based on the Karush–Kuhn–Tucker (KKT) optimality conditions for the optimization problem with an interval-valued objective function, the interval bilevel programming problem can be converted into a single-level programming problem with an interval-value objective function. To minimize the interval objective function, the order relations of interval numbers are used to transform the uncertain single-objective optimization into a multi-objective optimization solved by global criteria method (GCM). Finally, illustrative numerical examples are provided to demonstrate the feasibility of the proposed approach.

Weighted sum of maximum regrets in an interval MOLP problem

AbstractThis paper presents a multiobjective linear programming problem with interval objective function coefficients. Considering the concept of maximum regret, the weighted sum problem of maximum regrets is introduced and its properties are investigated. It is proved that an optimal solution of the weighted sum problem of maximum regrets is at least possibly weakly efficient. Further, the circumstances under which the optimal solution is necessarily efficient (necessarily weakly efficient or possibly efficient) are discussed. Moreover, using a relaxation procedure, an algorithm is proposed, which for a given set of weights finds one feasible solution that minimizes the weighted sum of maximum regrets. A numerical example is provided to illustrate the proposed algorithm.