Multi-level Programming Problems Research Articles

On the Quasiconcave Multilevel Programming Problems

Multilevel programming appears in many decision-making situations. Investigation of the main properties of quasiconcave multilevel programming (QCMP) problems, to date, is limited to bilevel programming (only two levels). In this paper, first, we present an extension of the properties of quasiconcave bilevel programming (QCBP) problems for the case when three levels exist. Then, by induction on n (the number of levels), we prove the existence of an extreme point of the polyhedral constraint region that solves the QCMP problem under given conditions. Ultimately, a number of numerical examples are illustrated to verify the results.

Solving the general fully neutrosophic multi-level multiobjective linear programming problems

Solving the general fully neutrosophic multi-level multiobjective linear programming problems

Robust model predictive control with embedded multi-scenario closed-loop prediction

Robust MPC seeks to mitigate the effects of uncertainty which can lead to suboptimal MPC performance. Previous work on robust MPC includes a stochastic multi-scenario approach to simulate multiple plant realizations and create a control scheme to optimize a performance metric based on the plant scenarios. This paper seeks to combine a scenario-based approach with embedded closed-loop prediction under future MPC control action by directly incorporating MPC subproblems into the overall robust MPC formulation. This allows the current MPC to predict closed-loop MPC responses to a range of uncertain future plant realizations. The resulting multilevel programming problem is solved by reformulating the inner MPC optimization subproblems as algebraic constraints corresponding to their first-order optimality conditions, resulting in a single level mathematical program with complementarity constraints (MPCC). The performance of the robust MPC scheme is evaluated against a standard MPC formulation in linear and nonlinear case studies.

Interactive multilevel programming approaches in neutrosophic environments

Multilevel programming is a mathematical programming problem with hierarchical structure. A typical feature of multilevel programming is that the upper level exhibits a priority over the lower level. However, the solutions obtained by most existing programming methods either violate this rule or ignore the participants’ desire for a win–win outcome. The objective of this study is to propose new multilevel programming approaches for obtaining desirable solutions. First, three types of membership functions in neutrosophic set are defined to comprehensively describe fuzzy cognition of decision makers. Then, considering dissimilar intentions of experts, three different interactive approaches are proposed to solve multilevel programming problems. To demonstrate the feasibility of the proposed approaches, a case of pricing decision-making of data products is investigated and the impacts of four key parameters are discussed. Finally, several numerical examples are studied by using the proposed approaches and other existing methods. Two evaluation indexes, the equilibrium coefficient and distance measure, are utilized to appraise the performance of the developed programming methods. The results demonstrate that the proposed approaches can obtain sound solutions which obey the rule of multilevel programming, realize the mutual benefits of participants, and can provide guidelines for the pricing of satellite image data products.

Parametric approach to quadratically constrained multi-level multi-objective quadratic fractional programming

The paper proposed a method to study and obtain a set of Pareto optimal solutions or a set of representative solutions to a quadratically constrained multi-level multiobjective quadratic fractional programming problem. This problem involves several objectives to be fulfilled at multi levels under a common set of quadratic constraints. Initially, we used parametric approach to convert the fractional programming model to an equivalent non-fractional programming model by allocating a parametric vector to each fractional objective. Then, $$\varepsilon$$ -constraint method is used to convert this multiobjective programming model into an equivalent model with single objective. The solution of every previous level is followed by the next level in succession to find a solution which is suitable to each level decision maker. An algorithm and numerical example are also presented at the end of the paper to validate the proposed methodology for the Model.

(2003-5760) Building fuzzy approach with linearization technique for fully rough multi-objective multi-level linear fractional programming problem

This paper presents a suitable solution procedure to solve the fully rough multi-objective multi-level linear fractional programming (FRMMFP) problem. First, an extension of interval method is presented to deal with roughness of the stated problem. Then, an iterative technique is proposed for linearization of fractional objectives. Finally, a modification of fuzzy approach is provided in the environment of the fully rough to solve the linear model. An example is provided for understanding the solution procedure of the proposed method.

Novel metaheuristic based on multiverse theory for optimization problems in emerging systems.

Finding an optimal solution for emerging cyber physical systems (CPS) for better efficiency and robustness is one of the major issues. Meta-heuristic is emerging as a promising field of study for solving various optimization problems applicable to different CPS systems. In this paper, we propose a new meta-heuristic algorithm based on Multiverse Theory, named MVA, that can solve NP-hard optimization problems such as non-linear and multi-level programming problems as well as applied optimization problems for CPS systems. MVA algorithm inspires the creation of the next population to be very close to the solution of initial population, which mimics the nature of parallel worlds in multiverse theory. Additionally, MVA distributes the solutions in the feasible region similarly to the nature of big bangs. To illustrate the effectiveness of the proposed algorithm, a set of test problems is implemented and measured in terms of feasibility, efficiency of their solutions and the number of iterations taken in finding the optimum solution. Numerical results obtained from extensive simulations have shown that the proposed algorithm outperforms the state-of-the-art approaches while solving the optimization problems with large feasible regions.

A Novel Approach for Solving a Fully Rough Multi-Level Quadratic Programming Problem and Its Application

The most widely used actions and decisions of the real-world tasks frequently appear as hierarchical systems. To deal with these systems, the multi-level programming problem presents the most flourished technique. However, practical situations involve some the impreciseness regarding some decisions and performances; RST provides a vital role by considering the lower and upper bounds of any aspect of uncertain decision. By preserving the advantages of it, in the present study, solving fully rough multi-level quadratic programming problems over the variables, parameters of the objective functions, and the constraints such as rough intervals are focused on. The proposed approach incorporates the interval method, slice-sum method, Frank and Wolfe algorithm, and the decomposition algorithm to reach optimal values as rough intervals. The proposed is validated by an illustrative example, and also environmental-economic power dispatch is investigated as a real application. Finally, the proposed approach is capable of handling the fully rough multi-level quadratic programming models.

A Decomposition Algorithm for Solving Multi-Level Large-Scale Linear Programming Problems With Neutrosophic Parameters in the Constrains

A Decomposition Algorithm for Solving Multi-Level Large-Scale Linear Programming Problems With Neutrosophic Parameters in the Constrains

Volcano eruption algorithm for solving optimization problems

Meta-heuristic algorithms have been proposed to solve several optimization problems in different research areas due to their unique attractive features. Traditionally, heuristic approaches are designed separately for discrete and continuous problems. This paper leverages the meta-heuristic algorithm for solving NP-hard problems in both continuous and discrete optimization fields, such as nonlinear and multi-level programming problems through extensive simulations of volcano eruption process. In particular, a new optimization solution named volcano eruption algorithm is proposed in this paper, which is inspired from the nature of volcano eruption. The feasibility and efficiency of the algorithm are evaluated using numerical results obtained through several test problems reported in the state-of-the-art literature. Based on the solutions and number of required iterations, we observed that the proposed meta-heuristic algorithm performs remarkably well to solve NP-hard problem. Furthermore, the proposed algorithm is applied to solve some large-size benchmarking LP and Internet of vehicles problems efficiently.

On multi-level quadratic fractional programming problem with modified fuzzy goal programming approach

The present paper addresses an extended algorithm for the solution of multi-level quadratic fractional programming problem(ML-QFPP) based on fuzzy goal programming (FGP) approach. In this algorithm, suitable linear and nonlinear membership functions for the fuzzily described numerator and denominator of the quadratic objective functions of all levels as well as the control vectors of higher levels are respectively defined using individual optimal solutions. Then usual fuzzy goal programming approach is applied for the achievement of highest degree of each of the membership goal by minimising the negative deviational variables. The proposed algorithm is an extension of modified FGP approach for ML-QFPPs and a simple algorithm to obtain compromise optimal solution of ML-QFPPs with all major types of nonlinear membership functions. Comparative analysis over the variation in the types of membership functions is also carried out with numerical example to show suitability of different membership functions in the proposed algorithm.

On multi-level quadratic fractional programming problem with modified fuzzy goal programming approach

The present paper addresses an extended algorithm for the solution of multi-level quadratic fractional programming problem(ML-QFPP) based on fuzzy goal programming (FGP) approach. In this algorithm, suitable linear and nonlinear membership functions for the fuzzily described numerator and denominator of the quadratic objective functions of all levels as well as the control vectors of higher levels are respectively defined using individual optimal solutions. Then usual fuzzy goal programming approach is applied for the achievement of highest degree of each of the membership goal by minimising the negative deviational variables. The proposed algorithm is an extension of modified FGP approach for ML-QFPPs and a simple algorithm to obtain compromise optimal solution of ML-QFPPs with all major types of nonlinear membership functions. Comparative analysis over the variation in the types of membership functions is also carried out with numerical example to show suitability of different membership functions in the proposed algorithm.

On multi-level multi-objective linear fractional programming problem with interval parameters

This paper develops a method to solve multi-level multi-objective linear fractional programming problem (ML-MOLFPP) with interval parameters as the coefficients of decision variables and the constants involved in both the objectives and constraints. The objectives at each level are transformed into interval-valued fractional functions and approximated by intervals of linear functions using variable transformation and Taylor series expansion. Interval analysis and weighting sum method with analytic hierarchy process (AHP), are used to determine the non-dominated solutions at each level from which the aspiration values of the controlled decision variables are ascertained and linear fuzzy membership functions are constructed for all the objectives. Two multi-objective linear problems are equivalently formulated for the ML-MOLFPP with interval parameters and fuzzy goal programming is used to compute the optimal lower and upper bounds of all the objective values. A numerical example is solved to demonstrate the proposed solution approach.

Neutrosophic goal programming strategy for multi-level multi-objective linear programming problem

Neutrosophic set theory plays an important role in dealing with the impreciseness and inconsistency in data encountered in solving real life problems. This article aims to present a novel goal programming based strategy which will be helpful to solve Multi-Level Multi-Objective Linear Programming Problem (MLMOLPP) with parameters as neutrosophic numbers (NNs). Difficulty in decision making arises due to the presence of multiple decision makers (DMs) and impreciseness in information. Here each level DM has multiple linear objective functions with parameters considered as NNs which are represented in the form $$c + dI$$ , where c and d are considered real numbers and the symbol I denotes indeterminacy. The constraints are also linear with the parameters as NNs. Firstly the NNs are changed into intervals and the problem turns into a multi-level multi-objective linear programming problem considering interval parameters. Then interval programming technique is employed to obtain the target interval of each objective function. In order to avoid decision deadlock which may arise in hierarchical (multi-level) problem, a possible relaxation is imposed by each level DM on the decision variables under his/her control. Finally a goal programming strategy is presented to solve the MLMOLPP with interval parameters. The method presented in this paper facilitates to solve MLMOLPP with multiple conflicting objectives in an uncertain environment represented through NNs of the form $$c + dI$$ , where indeterminacy I plays a pivotal role. Lastly, a mathematical example is solved to show the novelty and applicability of the developed strategy.

A modified fuzzy approach for the fully fuzzy multi-objective and multi-level integer quadratic programming problems based on a decomposition technique

A modified fuzzy approach for the fully fuzzy multi-objective and multi-level integer quadratic programming problems based on a decomposition technique

Adjustable robust optimization through multi-parametric programming

Adjustable robust optimization (ARO) involves recourse decisions (i.e. reactive actions after the realization of the uncertainty, ‘wait-and-see’) as functions of the uncertainty, typically posed in a two-stage stochastic setting. Solving the general ARO problems is challenging, therefore ways to reduce the computational effort have been proposed, with the most popular being the affine decision rules, where ‘wait-and-see’ decisions are approximated as affine adjustments of the uncertainty. In this work we propose a novel method for the derivation of generalized affine decision rules for linear mixed-integer ARO problems through multi-parametric programming, that lead to the exact and global solution of the ARO problem. The problem is treated as a multi-level programming problem and it is then solved using a novel algorithm for the exact and global solution of multi-level mixed-integer linear programming problems. The main idea behind the proposed approach is to solve the lower optimization level of the ARO problem parametrically, by considering ‘here-and-now’ variables and uncertainties as parameters. This will result in a set of affine decision rules for the ‘wait-and-see’ variables as a function of ‘here-and-now’ variables and uncertainties for their entire feasible space. A set of illustrative numerical examples are provided to demonstrate the potential of the proposed novel approach.

Interactive programming approach for solving multi-level multi-objective linear programming problem

Interactive programming approach for solving multi-level multi-objective linear programming problem

A decentralized multi-level decision making model for solid transportation problem with uncertainty

In this paper, a multi-level model for the solid transportation problem having uncertain variables is presented. Multi-level programming deals with the situation where more than one decision maker is available to model decentralized planning problem in any hierarchical system. In the model presented in this paper, all the parameters involved viz. transportation costs, supply availabilities, demands and conveyance capacities are considered as uncertain parameters. As the problem is based on uncertainty theory, the idea of minimizing (or maximizing) the expectation of objective functions under the set of chance constraints is followed to formulate the problem. A crisp model equivalent to the multi-level uncertain solid transportation problem is also given. A fuzzy based solution approach for solving multi-level programming problems is discussed to solve the presented model. Further, a numerical example of a three-level uncertain solid transportation problem with two conveyance options is given in order to understand the model more clearly.

A Novel Cognitively Inspired State Transition Algorithm for Solving the Linear Bi-Level Programming Problem

Linear bi-level programming (LBLP) is a useful tool for modeling decentralized decision-making problems. It has two-level (upper-level and lower-level) objectives. Many studies have shown that the LBLP problem is NP-hard, meaning it is difficult to find a global solution in polynomial time. In this paper, we present a novel cognitively inspired computing method based on the state transition algorithm (STA) to obtain an approximate optimal solution for the LBLP problem in polynomial time. The proposed method is applied to a supply chain model that fits the definition of an LBLP problem. The experimental results indicate that the proposed method is more efficient in terms of solution accuracy through a comparison to other metaheuristic-based methods using four problems from the literature in addition to the supply chain distribution model. In this study, a cognitively inspired STA-based method was proposed for the LBLP problem. In the future, we expect to extent the proposed method for linear multi-level programming problems.

Presentation and Solving Non-Linear Quad-Level Programming Problem Utilizing a Heuristic Approach Based on Taylor Theorem

The multi-level programming problems are attractive for many researchers because of their application in several areas such as economic, traffic, finance, management, transportation, information technology, engineering and so on. It has been proven that even the general bi-level programming problem is an NP-hard problem, so the multi-level problems are practical and complicated problems therefore solving these problems would be significant. The literature shows several algorithms to solve different forms of the bi-level programming problems (BLPP).Not only there is no any algorithm for solving quad-level programming problem, but also it has not been studied by any researcher. The most important part of this paper is presentation and studying of a new model of non-linear multi-level problems.Then we attempt to develop an effective approach based on Taylor theorem for solving the non-linear quad-level programming problem. In this approach, by using aproposedsmoothing method the quad-level programming problem is converted to a linear single problem. Finally, the single level problem is solved using the algorithm based on Taylor algorithm. The presented approach achieves an efficient and feasible solution in an appropriate time which has been evaluated by solving test problems.