Multi-level Programming Problems Research Articles

A solution method for stochastic multilevel programming problems. A systematic sampling evolutionary approach

Stochastic multilevel programming is a mathematical programming problem with some given number of hierarchical levels of decentralized decision makers and having some kind of randomness properties in the problem definition. The introduction of some randomness property in its hierarchical structure makes stochastic multilevel problems computationally challenging and expensive. In this article, a systematic sampling evolutionary method is adapted to solve the problem. The solution procedure is based on realization of the random variables and systematic partitioning of each hierarchical level’s decision space for searching an optimal reaction. The search goes sequentially upwards starting from the bottom up through the top hierarchical level problem. The existence of solution and convergence of the solution procedure is shown. The solution procedure is implemented and tested on some selected deterministic test problems from literature. Moreover, the proposed algorithm can be used to solve stochastic multilevel programming problems with additional complexity in their problem definition.

An improved solution for fully neutrosophic multi-level linear programming problem based on Laplace transform

An improved solution for fully neutrosophic multi-level linear programming problem based on Laplace transform

Fully intuitionistic fuzzy multi-level linear fractional programming problem

In many real situations, it is frequently difficult to accurately determine the membership and non-membership degrees related to an element of the set with complete satisfaction because of the ambiguity in the input data. In instances like these, intuitionistic fuzzy (IF) numbers are crucial. We present a simple approach in this paper to solve the fully intuitionistic fuzzy multi-level linear fractional programming (FIFMLLFP) problem. By applying the new suggested approach to the problem under consideration, each level of the FIFMLLFP problem is converted into five crisp linear (MLMOLFP) problems, where each crisp problem has an additional bounded variables constraint and the upper problems' optimization variables are treated as parameters. This is done by using an iterative technique for linearizing fractional objectives. We converted the MLMOLFP problem into a multilevel multiobjective linear programming (MLMOLP) problem, in addition to the ε-constraint method that is used to reduce MLMOLP problem to a single objective linear programming problem. The method is demonstrated step-by-step with a numerical example.

Three different optimization techniques for solving the fully rough interval multi-level linear programming problem

Due to the importance of the multi-level fully rough interval linear programming (MLFRILP) problem to address a wide range of management and optimization challenges in practical applications, such as policymaking, supply chain management, energy management, and so on, few researchers have specifically discussed this point. This paper presents an easy and systematic roadmap of studies of the currently available literature on rough multi-level programming problems and improvements related to group procedures in seven basic categories for future researchers and also introduces the concept of multi-level fully rough interval optimization. We start remodeling the problem into its sixteen crisp linear programming LP problems using the interval method and slice sum method. All crisp LPs can be reduced to four crisp LPs. In addition, three different optimization techniques were used to solve the complex multi-level linear programming issues. A numerical example is also provided to further clarify each strategy. Finally, we have a comparison of the methods used for solving the MLFRILP problem.

On neutrosophic multi-level multi-objective linear programming problem with application in transportation problem

In this research, we use the harmonic mean technique to present an interactive strategy for addressing neutrosophic multi-level multi-objective linear programming (NMMLP) problems. The coefficients of the objective functions of level decision makers and constraints are represented by neutrosophic numbers. By using the interval programming technique, the NMMLP problem is transformed into two crisp MMLP problems, one of these problems is an MMLP problem with all of its coefficients being upper approximations of neutrosophic numbers, while the other is an MMLP problem with all of its coefficients being lower approximations of neutrosophic numbers. The harmonic mean method is then used to combine the many objectives of each crisp problem into a single objective. Then, a preferred solution for NMMLP problems is obtained by solving the single-objective linear programming problem. An application of our research problem is how to determine the optimality the cost of multi-objective transportation problem with neutrosophic environment. To demonstrate the proposed strategies, numerical examples are solved.

Branch-and-cut solution approach for multilevel mixed integer linear programming problems

A multilevel programming problem is an optimization problem that involves multiple decision makers, whose decisions are made in a sequential (or hierarchical) order. If all objective functions and constraints are linear and some decision variables in any level are restricted to take on integral or discrete values, then the problem is called a multilevel mixed integer linear programming problem (ML-MILP). Such problems are known to have disconnected feasible regions (called inducible regions), making the task of constructing an optimal solution challenging. Therefore, existing solution approaches are limited to some strict assumptions in the model formulations and lack universality. This paper presents a branch-and-cut (B&C) algorithm for the global solution of such problems with any finite number of hierarchical levels, and containing both continuous and discrete variables at each level of the decision-making hierarchy. Finite convergence of the proposed algorithm to a global solution is established. Numerical examples are used to illustrate the detailed procedure and to demonstrate the performance of the algorithm. Additionally, the computational performance of the proposed method is studied by comparing it with existing method through some selected numerical examples.

An adaptive method to solve multilevel multiobjective linear programming problems

This paper is a follow-up to a previous work where we defined and generated the set of all possible compromises of multilevel multiobjective linear programming problems (ML-MOLPP). We introduce a new algorithm to solve ML-MOLPP in which the adaptive method of linear programming is nested. First, we start by generating the set of all possible compromises (set of all non-dominated solutions). After that, an algorithm based on the adaptive method of linear programming is developed to select the best compromise among all the possible settlements achieved. This method will allow us to transform the initial multilevel problem into an ML-MOLPP with bonded variables. Then, apply the adaptive method which is the most efficient to solve all the multiobjective linear programming problems involved in the resolution process instead of the simplex method. Finally, all the construction stages are carefully checked and illustrated with a numerical example.

Multi-level multi-objective linear fractional programming problem: a solution approach

Multi-level multi-objective linear fractional programming problem: a solution approach

Multi-level multi-objective linear fractional programming problem: A solution approach

Multi-level multi-objective linear fractional programming problem: A solution approach

Uncertain multilevel programming with application to omni-channel vehicle routing problem.

Multilevel programming is widely applied to solve decentralized decision-making problems. In practice, indeterminacies are presented in these problems due to volatile factors or emergencies. As a type of indeterminacy, uncertainty is introduced in multilevel programming. For resolving multilevel programming problems with uncertain parameters, this paper constructs the uncertain expected value multilevel programming model and chance-constrained multilevel programming. Then, these models are converted to their equivalent forms. Moreover, the Stackelberg-Nash equilibrium solutions are obtained by using a genetic algorithm. Finally, these models are applied to the omni-channel vehicle routing problem, and a numerical experiment is given. The numerical experiment shows that the established models can optimize the distribution efficiency by coordinating the interests of decision-makers.

Solving Multi-Objective Multilevel Programming problems using two-phase Intuitionistic Fuzzy Goal Programming method

This paper presents a two-phase intuitionistic fuzzy goal programming (two-phase IFGP) algorithm to solve Multi-Objective Multilevel Programming (MO-MLP) problems. The coefficient of each objective and constraint function is assumed to be triangular intuitionistic fuzzy parameters and the crisp MO-MLP problems are obtained using the accuracy function method. To avoid decision lock, the top levels set tolerance limits for decision variables to control the lower levels. The problem is modeled in the intuitionistic fuzzy environment using membership and non-membership functions for each objective function at all levels and decision variables controlled by the top levels. Then, we proposed an IFGP algorithm to achieve the highest degree of each membership and non-membership goal by minimizing unwanted deviational variables and generating compensatory solutions for all decision-makers at all levels. Moreover, in the proposed approach, two-phase IFGP is modeled to yield a compromise solution that satisfies both the MN-Pareto optimal solution and the Pareto optimal solution at each level. Also, verification of the proposed method is discussed with numerical examples.

The set of all the possible compromises of a multi-level multi-objective linear programming problem

In this paper, a new approach is developed to solve multi-level multi-objective linear programming problems (ML-MOLPP), by providing the set of all the compromises without taking into account the hierarchy of the problem. A simple criterion is developed for testing whether a given feasible solution is a possible compromise or not. We describe a method used to generate the set of all the compromises, based on the P.L. Yu and M. Zeleny’s method. Theoretical results, numerical example and flow diagram are reported.

Solving Dual-Channel Supply Chain Pricing Strategy Problem with Multi-Level Programming Based on Improved Simplified Swarm Optimization

With the evolution of the Internet and the introduction of third-party platforms, a diversified supply chain has gradually emerged. In contrast to the traditional single sales channel, companies can also increase their revenue by selling through multiple channels, such as dual-channel sales: adding a sales channel for direct sales through online third-party platforms. However, due to the complexity of the supply chain structure, previous studies have rarely discussed and analyzed the capital-constrained dual-channel supply chain model, which is more relevant to the actual situation. To solve more complex and realistic supply chain decision problems, this paper uses the concept of game theory to describe the pricing negotiation procedures among the capital-constrained manufacturers and other parties in the dual-channel supply chain by applying the Stackelberg game theory to describe the supply chain structure as a hierarchical multi-level mathematical model to solve the optimal pricing strategy for different financing options to achieve the common benefit of the supply chain. In this study, we propose a Multi-level Improved Simplified Swarm Optimization (MLiSSO) method, which uses the improved, simplified swarm optimization (iSSO) for the Multi-level Programming Problem (MLPP). It is applied to this pricing strategy model of the supply chain and experiments with three related MLPPs in the past studies to verify the effectiveness of the method. The results show that the MLiSSO algorithm is effective, qualitative, and stable and can be used to solve the pricing strategy problem for supply chain models; furthermore, the algorithm can also be applied to other MLPPs.

On solving the multilevel rough interval linear programming problem

In real-world problems, the parameters of optimization problems are uncertain. A class of multilevel linear programming (MLLP) with uncertainty problem models cannot be determined exactly. Hence, in this paper, we are concerned with studying the uncertainty of MLLP problems. The main motivation of this paper is to obtain the solution to a multilevel rough interval linear programming (MLRILP) problem. To obtain that, we start turning the problem into its competent crisp equivalent using the interval method. Moreover, we rely on three methods to address the problem of multiple levels. First, by applying the constraint method in which upper levels give satisfactory solutions that are reasonable in rank order to the lower levels, second, by an interactive approach that uses the satisfaction test function, and third, by the fuzzy approach that is based on the concept of the tolerance membership function. A numerical example is given for illustration and to examine the validity of the approach. An application to deduce the optimality for the cost of the solid MLLP transportation problem in rough interval environment is presented.

Fuzzy harmonic mean technique for solving fully fuzzy multilevel multiobjective linear programming problems

This paper illustrates how the fuzzy harmonic mean technique can efficiently solve fully fuzzy multilevel multiobjective linear programming (FFMMLP) problems. First, at each level, the FFMMLP problem can be converted into three crisp multiobjective linear programming (CMLP) problems using the crisp linear technique. Then, the fuzzy harmonic mean technique is utilized to aggregate each crisp problem's multiobjective into a single objective. Second, the ensuing final, single-objective problem is constructed using the harmonic mean for each level. Finally, it is solved to obtain a fuzzy compromise solution for the FFMMLP problem in general. Two examples are given to obtain the application of the proposed method. One example is applying the proposed approach to a multilevel multiobjective production planning model for a supply chain under a fully fuzzy environment.

Alternate solution approach for ML-MOLFPP problems

This article presents an alternate solution approach for multi-level multi objective linear fractional programming problems (ML-MOLFPPs) with modifications in the modified fuzzy goal programming (FGP) approach suggested by Lachhwani [1]. In the proposed alternate technique, fractional objective functions at every level are individually considered and converted into individual membership function instead of treating numerator and denominator part of objective function separately for formulation of membership function. The proposed alternate approach for obtaining compromise optimal solution for ML-MOLFPPs is comparatively less computational and efficient. This is demonstrated with the use of proposed technique over numerical example and comparative analysis with earlier method.

Fuzzy Approach for Bi-Level Programming Problems

Bi-level programming problem (BLPP) can make the final decision result is a mutually coordinated scheme among all layers, which can solve complex practical problems well. This paper summarizes and describes the methods to solve the bi-level programming problem, and proposes a fuzzy method to solve the BLPP by using the concepts of membership function and multiobjective optimization to solve the bilevel programming problem, and gives a numerical example to solve it. It also points out that developing a method to solve fuzzy problems and new concepts to solve multi-level programming problems are practical directions for future research.

Optimizing Painting Sequence Scheduling Based on Adaptive Partheno-Genetic Algorithm

In this paper, we formulate and solve a novel real-life large-scale automotive parts paint shop scheduling problem, which contains color arrangement restrictions, part arrangement restrictions, bracket restrictions, and multi-objectives. Based on these restrictions, we construct exact constraints and two objective functions to form a large-scale multi-objective mixed-integer linear programming problem. To reduce this scheduling problem’s complexity, we converted the multi-objective model into a multi-level objective programming problem by combining the rule-based scheduling algorithm and the adaptive Partheno-Genetic algorithm. The rule-based scheduling algorithm is adopted to optimize color changes horizontally and bracket replacements vertically. The adaptive Partheno-Genetic algorithm is designed to optimize production based on the rule-based scheduling algorithm. Finally, we apply the model to the actual optimization problem that contained 829,684 variables and 137,319 constraints, and solved this problem by Python. The proposed method solves the optimal solution, consuming 575 s.

A simplex-based branch-and-cut method for solving integer tri-level programming problems

Optimization problems involving three decision makers at three hierarchical decision levels are referred to as tri-level decision making problems. In particular in the case where all objective functions and constraints are linear, and all involved variables are required to take integer values is known as integer tri-level programming problems (T-ILP). It has been known that the general T-ILP is an NP-hard problem. So far there are very few attempts in the literature that finds an exact global solution for integer programming hierarchical problems beyond two levels. This paper proposes a novel approach based on a branch-and-cut (B\(\&\)C) framework for solving T-ILP. The convergence of the algorithm is proved. Numerical examples are used to demonstrate the performance of the proposed algorithm. Finally, the results of this study show that the proposed algorithm is promising and more efficient to solve T-ILP. Moreover, it has been indicated in the remark that the proposed algorithm can be modified and used to solve discrete multilevel programming problems of any number of levels.

Solving multi-level multiobjective fractional programming problem with rough interval parameter in neutrosophic environment

In this study, a novel algorithm is developed to solve the multi-level multiobjective fractional programming problems, using the idea of a neutrosophic fuzzy set. The co-efficients in each objective functions is assumed to be rough intervals. Furthermore, the objective functions are transformed into two sub-problems based on lower and upper approximation intervals. The marginal evaluation of pre-determined neutrosophic fuzzy goals for all objective functions at each level is achieved by different membership functions, such as truth, indeterminacy/neutral, and falsity degrees in neutrosophic uncertainty. In addition, the neutrosophic fuzzy goal programming algorithm is proposed to attain the highest degrees of each marginal evaluation goals by reducing their deviational variables and consequently obtain the optimal solution for all the decision-makers at all levels. To verify and validate the proposed neutrosophic fuzzy goal programming techniques, a numerical example is adressed in a hierarchical decision-making environment along with the conclusions.