Multi-level Programming Problems Research Articles

Multi-level linear programming subject to addition-min fuzzy relation inequalities with application in Peer-to-Peer file sharing system

Multi-level linear programming problem subject to addition-min fuzzy relation inequalities is introduced to characterize a kind of optimization models in BitTorrent-like Peer-to-Peer file sharing systems. Based on some theorems, which contribute to the resolution of the proposed problem, we develop a novel algorithm to find the unique optimal solution. A practical application example is presented to illustrate the feasibility and efficiency of the algorithm.

A branch-and-bound multi-parametric programming approach for non-convex multilevel optimization with polyhedral constraints

In this paper we develop a general but smooth global optimization strategy for nonlinear multilevel programming problems with polyhedral constraints. At each decision level successive convex relaxations are applied over the non-convex terms in combination with a multi-parametric programming approach. The proposed algorithm reaches the approximate global optimum in a finite number of steps through the successive subdivision of the optimization variables that contribute to the non-convexity of the problem and partitioning of the parameter space. The method is implemented and tested for a variety of bilevel, trilevel and fifth level problems which have non-convexity formulation at their inner levels.

Multilevel Programming Problems with Fuzzy Parameters: A Fuzzy Goal Programming Approach

paper presents fuzzy goal programming approach for solving multilevel programming problems with fuzzy parameters. The proposed approach is based on  -cut and fuzzy goal programming. In the proposed approach, the tolerance membership functions for the fuzzily described objective functions are defined by determining individual best solution of the objective function of every decision maker. Since the objectives of the level decision makers are potentially conflicting in nature, decision deadlock may arise due to the dissatisfaction of the solution of upper level decision makers. Sometimes upper level decision makers insist to work more than stipulated working hours or overtime duty in order to meet the heavy demand of the market arising for festivals or emergency reasons. In order to survive in the open competitive market, the relaxations of lower level decision makers are very crucial for the upper level decision makers and for the organization. So in the proposed model relaxation of decision for each level decision maker is considered. The relaxation of decision is performed by providing preference bounds on the decision variables for avoiding decision deadlock. Then three fuzzy goal programming models for multilevel programming are formulated. In general, the fuzzy goal programming models offer different solutions. In order to find the best compromise solution Euclidean function is used. An illustrative numerical example is provided to demonstrate the efficiency of the

Modified FGP approach for multi-level multi objective linear fractional programming problems

In this paper, we present a new modified method for solving multi-level multi objective linear fractional programming problems (ML-MOLFPPs) based on fuzzy goal programming (FGP) approach with some modifications in the algorithm suggested by Baky (2010) [18] which dealt with multi-level multi objective linear programming problem (ML-MOLPP). In proposed modified approach, numerator and denominator function of each objective at each level are individually transformed into fuzzy goals and their aspiration levels are determined using individual best solutions. Different linear membership functions are defined for numerator and denominator function of each objective function. Then highest degree of each of these membership goals is achieved by minimising the sum of negative deviational variables. The proposed algorithm simplifies the ML-MOLFPP by eliminating solution preferences by the decision makers at each level, thereby avoiding difficulties associate with multi-level programming problems and decision deadlock situations. The aim of this paper is to present simple and efficient technique to obtain compromise optimal solution of ML-MOLFP problems. Numerical examples are illustrated in order to support the proposed modified FGP technique.

Chance Constrained Multi-Level Linear Programming Problem

ABSTRACT In the paper, we present chance constrained multi-level linear programming problem. The right hand parameters and the coefficients of the constraints are considered as the random variables of known distribution function and the chance constraints are transformed into equivalent deterministic constraints. Membership function for each level objective function is constructed subject to the equivalent deterministic constraints. In the multi-level decision making situation, lower level decision makers may not be satisfied with the decision of higher level decision maker. To avoid this problem, each level decision maker provides relaxation in his/ her decision. Three FGP models are adopted to get the membership goals. Euclidean distance function is used to select the best FGP model offering the most satisfactory solution. Two numerical examples are solved to demonstrate the proposed approach. General Terms Decision Making, Linear Programming, Optimization.

Fuzzy Goal Programming Approach to Solve Linear Multilevel Programming Problems using Genetic Algorithm

paper introduces a priority based fuzzy goal programming (FGP) method for modelling and solving multilevel programming problem (MLPP) through genetic algorithm (GA). In model formulation, the individual best solution of objectives of each of the decision makers (DMs) is determined by using the GA method for fuzzy description of the objectives. Then, tolerance membership functions of the defined fuzzy goals are constructed for measuring the degree of satisfaction of goal achievement and there by degree of optimality of the decision vectors controlled by the higher level DMs. In the executable FGP model, minimization of the under-deviational variables of the defined membership goals with highest membership value (unity) as the aspiration levels of them on the basis of pre- emptive priority is taken into consideration in the decision making context. In the solution process, sensitivity analysis with variations of priority structure of model goals is performed and then Euclidean distance function is used to identify the appropriate priority structure under which the most satisfactory decision can be reached in the decision making horizon. In the proposed GA scheme, roulette-wheel selection scheme, single point crossover and uniform mutation are adopted in the decision search process with regard to reach a satisfactory solution in the proposed hierarchical decision system. The effective use of the proposed approach is illustrated through a numerical example. Performance comparisons are also made to highlight the superiority of the proposed approach over the approaches studied previously.

An interactive intuitionistic fuzzy method for multilevel linear programming problems

In this paper, we propose an interactive method for solving the multilevel linear programming problems based on the intuitionistic fuzzy set theory. Firstly, the membership function and the non-membership function are introduced to describe the uncertainty of the decision makers. Secondly, a satisfactory solution is derived by updating the minimum satisfactory degrees with considerations of the overall satisfactory balance among all levels. In addition, the steps of the proposed method are given in this paper. Finally, numerical examples illustrate the feasibility of this method.

Taylor Series Approach for Solving Multi-level Large Scale Fractional Programming Problem with Stochastic Parameters in Constraints

This paper presents a solution algorithm to solve a multi-level large scale fractional programming problem with individual chance constraints (CH-MLLSFP). We assume that there is randomness in the right-hand side of the constraints only and that the random variables are normally distributed. The basic idea in treating (CH-MLLSFP) is to convert the probabilistic nature of this problem into a deterministic multi-level large scale fractional programming problem (MLLSFPP). A solution of multi-level large scale fractional programming problem is presented using aTaylor series to avoid the complexity of fractional nature. An illustrative example is discussed to demonstrate the correctness of the proposed solution method.

Combining a Continuous Search Algorithm with a Discrete Search Algorithm for Solving Non-linear Bi-level Programming Problem

The multi-level programming problems, have received much interest from researchers because of their application in several areas such as economic, traffic, finance, management, transportation and so on. Among these, the bi-level programming problem (BLPP) is an appropriate tool to model these real problems. It has been proven that the general BLPP is an NP-hard problem, so it is a practical and complicated problem therefore solving this problem would be significant. However the literature shows several algorithms to solve different forms of the bi-level programming problems (BLPP), but there is no any hybrid approach of combining of two meta-heuristic algorithms. In this paper, the authors combine particle swarm optimization (PSO), which is a continuous approach, with a proposed modified genetic algorithm (MGA), which is a discrete algorithm, using a heuristic function and constructing an effective hybrid approaches (PSOMGA). Using the Karush-KuhnOriginal Research Article Hosseini and Kamalabadi; JSRR, 6(7): 549-559, 2015; Article no.JSRR.2015.180 550 Tucker conditions the BLPP is converted to a non-smooth single level problem, and then it is smoothed by a new heuristic method for using PSOMGA. The smoothed problem is solved using PSOMGA which is a fast approximate method for solving the non-linear BLPP. The presented approach achieves an efficient and feasible solution in an appropriate time, as justified by comparison with test problems.

Multi-level linear programming problem involving some multi-choice parameters

In a multi-level decision making problem the decision is taken jointly by several decision makers who are at different levels. In this paper, we consider a multi-level linear programming problem where some of the cost coefficients of the objective functions and some of the right hand side parameters of the constraints are multi-choice type. The aim of this paper is to establish a suitable solution procedure to solve the stated multi-level programming problem. To tackle the multi-choice parameters of the multi-level programming problem, we use some interpolating polynomials and transform the stated problem into a multi-level mixed integer non-linear programming problem. Then we apply fuzzy programming method to solve the transformed multi-level programming problem. One numerical example is included to illustrate the solution procedure of the multi-level linear programming problem which has some multi-choice parameters.

A fuzzy approach with required minimum decision tolerances for multi-level multi-objective decision-making problems

This paper investigates multi-level multi-objective linear programming problems in fuzzy environments, where each decision level has one decision-maker (DM) with a vector of decision variables and possibly more than one objective function. The objective functions of each DM and the decision variables of higher-level DMs are characterized by membership functions, which are considered as fuzzy goals. In related studies, the tolerances used to define membership functions of higher-level DMs' decision variables are usually subjectively determined. This may lead to infeasible solutions. This paper focuses on the determination of required minimum tolerances for higher-level DMs' decision variables to ensure feasibility in the solution process. A fuzzy goal programming model is established to optimally satisfy all defined fuzzy goals by minimizing satisfaction deviations under the constraint of the leader-follower relationship. A numerical example is provided for demonstrating the proposed approaches. It is shown that existing approaches may not have feasible solutions if the required minimum tolerances are not adopted.

A two-phase fuzzy approach for solving multi-level decision-making problems

This paper develops a two-phase fuzzy goal programming (FGP) approach for multi-level linear programming (MLLP) problems in an uncertain environment. A numerical MLLP model is established based on a confidence level. In the first phase, an FGP model is used to find a solution that reaches the overall satisfaction of all decision-makers (DMs), ensuring that the fuzzy goal of each higher-level DM is more satisfied than those of lower-level DMs. Since the higher-level DM has the direct authority to manage the subordinate DM, an adjustment scheme is provided in the second phase for each higher-level DM, who has the opportunity to increase or decrease the satisfaction degree of their fuzzy goal by changing the relative satisfaction of the lower-level DM compared to that of their higher-level DM. A fuzzy variable of the relative satisfaction containing a set of linguistic terms is provided for these adjustments. The adjustment processes are carried out sequentially, from the top to bottom in the hierarchical decision structure. The proposed FGP model in the second phase considers these sequential adjustments. A numerical example and comparisons with existing methods are used to demonstrate the applicability and performance of the proposed approach.

Modified FGP approach and MATLAB program for solving multi-level linear fractional programming problems

In this paper, we present modified fuzzy goal programming (FGP) approach and generalized MATLAB program for solving multi-level linear fractional programming problems (ML-LFPPs) based on with some major modifications in earlier FGP algorithms. In proposed modified FGP approach, solution preferences by the decision makers at each level are not considered and fuzzy goal for the decision vectors is defined using individual best solutions. The proposed modified algorithm as well as MATLAB program simplifies the earlier algorithm on ML-LFPP by eliminating solution preferences by the decision makers at each level, thereby avoiding difficulties associate with multi-level programming problems and decision deadlock situation. The proposed modified technique is simple, efficient and requires less computational efforts in comparison of earlier FGP techniques. Also, the proposed coding of generalized MATLAB program based on this modified approach for solving ML-LFPPs is the unique programming tool toward dealing with such complex mathematical problems with MATLAB. This software based program is useful and user can directly obtain compromise optimal solution of ML-LFPPs with it. The aim of this paper is to present modified FGP technique and generalized MATLAB program to obtain compromise optimal solution of ML-LFP problems in simple and efficient manner. A comparative analysis is also carried out with numerical example in order to show efficiency of proposed modified approach and to demonstrate functionality of MATLAB program.

Fuzzy-based interactive method for solution of bi- and multi-level programming problems

In this paper, a fuzzy-based interactive method for bi-level and multi-level, linear and non-linear, programming problems is proposed. In this approach product operator is used for aggregation of different fuzzy goals and MI-LXPM, a genetic algorithm, is used as optimisation technique. Performance of the proposed method is tested on a set of five examples taken from literature.

A new modified randomly iterated search and statistical competency (RISC) approach for solving a nonlinear controlling model in three-level supply chain

This paper proposes a nonlinear controlling model in three-level supply chain. The proposed model consists of a manufacturer, a warehouse and two retailers. As nonlinear multi level programming problems are much more difficult to solve, the proposed model was converted to two nonlinear bilevel programming problems in order to make the model easier both to solve and to describe. The first model consists of warehouse's objective function at its first level and the manufacturer at the second level. In the second model, retailers are the leader and warehouse is the follower. Bilevel programming has been investigated to be NP-hard problem. Numerous algorithms have been developed so far for solving bilevel programming problem; however, algorithm proposed in this paper is easier than other algorithms for solving this type of problems. In this paper, we proposed a modified randomly iterated search and statistical competency approach in genetic algorithm to provide precise and reliable optimal solutions for manufacturer, warehouse and retailers' value in situations with no empirical observations and we presented confidence interval as solution.

On solving multi-level multi objective linear programming problems through fuzzy goal programming approach

In this paper, we propose an alternate technique based on fuzzy goal programming approach for solving multi-level multi objective linear programming problem (ML-MOLPP) which is simpler and requires less computational works than that of proposed algorithm by Baky, I. A. (Applied Mathematical Modelling, 34(2010), 2377–2387). In formulation of FGP model each objective functions at each level are transformed into fuzzy goals. Suitable membership function for every fuzzily described transformed objective functions at each level as well as the control vectors of each level decision makers are defined by determining individual optimal solution of each objective function at each of the decision making level. Then FGP approach is used for achieving highest degree of each of these membership goals by minimizing the sum of negative deviational variables. To avoid decision deadlock, solution preferences by the decision makers at each level are not taken into account of proposed FGP technique. The aim of this paper is to present simple technique to obtain compromise optimal solution of ML-MOLP problems. A comparative analysis based on numerical examples is also carried out to show similarity between two solution methodologies.

Mathematical solution of multilevel fractional programming problem with fuzzy goal programming approach

In this paper, we show a procedure for solving multilevel fractional programming problems in a large hierarchical decentralized organization using fuzzy goal programming approach. In the proposed method, the tolerance membership functions for the fuzzily described numerator and denominator part of the objective functions of all levels as well as the control vectors of the higher level decision makers are respectively defined by determining individual optimal solutions of each of the level decision makers. A possible relaxation of the higher level decision is considered for avoiding decision deadlock due to the conflicting nature of objective functions. Then, fuzzy goal programming approach is used for achieving the highest degree of each of the membership goal by minimizing negative deviational variables. We also provide sensitivity analysis with variation of tolerance values on decision vectors to show how the solution is sensitive to the change of tolerance values with the help of a numerical example.

Fuzzy Goal Programming Procedures for Multi-Level Multi-Objective Linear Fractional Programming Problems

Fuzzy Goal Programming Procedures for Multi-Level Multi-Objective Linear Fractional Programming Problems

Investment Planning for Electric Power Systems Under Terrorist Threat

Access to electric power is critical to societal welfare. In this paper, we analyze the interaction between a defender and a terrorist who threatens the operation of an electric power system. The defender wants to find a strategic defense to minimize the consequences of an attack. Both parties have limited budgets and behave in their own self-interest. The problem is formulated as a multi-level mixed-integer programming problem. A Tabu Search with an embedded greedy algorithm for the attack problem is implemented to find the optimum defense strategy. We apply the algorithm to a 24-bus network for a combination of four different defense budgets, three attack budgets, and three assumptions as to how the terrorists craft their attacks.

Interactive fuzzy programming for multi-level programming problems: a review

Decision-making problems in decentralised organisations are often modelled as Stackelberg games, and they are formulated as two-level mathematical programming problems with two decision-makers. If they do not have any motivation to cooperate mutually and behave rationally, outcomes of the problems can be explained by Stackelberg equilibrium which is not always Pareto optimal. From computational aspects, it is known that solving two-level programming problems is NP-hard even if the objective functions and the constraint functions are linear. In contrast, if the decision-makers can select strategies cooperatively, the most important aspect is to derive a Pareto optimal solution favourable to the decision-makers. As a method of this line of approach interactive fuzzy programming has been developed, taking into account fuzziness of human judgements. In this paper, after reviewing the development of solution methods for two- and multi-level programming problems, we focus on cooperative decision-making in decentralised organisations and give interactive fuzzy programming for two-level linear programming problems, which provides satisfactory solutions in accordance with the preference of the decision-makers. Moreover, we present extensions of interactive fuzzy programming for two-level linear programming problems under multi-objective environments and under uncertainty.