Multi-level Programming Problems Research Articles

Modeling supply chain operations as multi-level programming problems and their parametric programming based computation methodology

This paper proposes a modeling methodology for supply chain operations with a focus on the relationships of supply chain entities. Supply chain operation problems are mathematically formulated into a multilevel programming problem. A multiparametric programming-based computation methodology is proposed to compute the solution of the problems. Numerical examples are presented to illustrate the proposed modeling and computational methodology with some remarks.

A fuzzy algorithm for multilevel programming problems

This paper presents a Fuzzy Mathematical Programming (FMP) approach for solving Multilevel Decentralized Programming Problems (ML(D)PPs). The higher level Decision Maker(DM) makes the decision first to provide preferred values of the decision variables under control and the follower reacts by optimizing the objective function conditioned on the higher level DM’s decision. A simple method is employed to find a satisfactory solution under fuzziness with linear fractional objectives overcoming the selection of membership functions and tolerance values and terminates in a finite number of steps. The methodology proposes small number of iterations and aims at reducing the feasible space of a decision variable at each level of the hierarchical system until a satisfactory solution is obtained at the last level. The Stackelberg strategy is employed as a solution concept when decision problems are modeled as two-level programming problems, whether there is a cooperative relationship between the decision makers or not. Illustrative numerical example is provided to demonstrate the feasibility of the proposed method.

An algorithm for non-linear multi-level integer programming problems

In this paper, an algorithm is proposed to solve a tri-level integer programming problem in which the objective function for the first level is an indefinite quadratic, the second one is linear and the third one is linear fractional. The feasible space of the decision variable is reduced at each level until a satisfactory point is obtained at the last level. The higher level decision-maker reduces the feasible space for the lower level decision maker to search for his optimum. A satisfactory solution of the bilevel decentralised programming problem can also be obtained by the method proposed above. This method is illustrated with the help of examples.

Solving multi-level multi-objective linear programming problems through fuzzy goal programming approach

In this paper, two new algorithms are presented to solve multi-level multi-objective linear programming (ML-MOLP) problems through the fuzzy goal programming (FGP) approach. The membership functions for the defined fuzzy goals of all objective functions at all levels are developed in the model formulation of the problem; so also are the membership functions for vectors of fuzzy goals of the decision variables, controlled by decision makers at the top levels. Then the fuzzy goal programming approach is used to achieve the highest degree of each of the membership goals by minimizing their deviational variables and thereby obtain the most satisfactory solution for all decision makers. The first suggested algorithm groups the membership functions for the defined fuzzy goals of the objective functions at all levels and the decision variables for each level except the lower level of the multi-level problem. The second proposed algorithm lexicographically solves MOLP problems of the ML-MOLP problem by taking into consideration the decisions of the MOLP problems for the upper levels. An illustrative numerical example is given to demonstrate the algorithms.

An enumerative algorithm for non-linear multi-level integer programming problem

In this paper a multilevel programming problem, that is, three level programming problem is considered. It involves three optimization problems where the constraint region of the first level problem is implicitly determined by two other optimization problems. The objective function of the first level is indefinite quadratic, the second one is linear and the third one is linear fractional. The feasible region is a convex polyhedron. Considering the relationship between feasible solutions to the problem and bases of the coefficient sub-matrix associated to the variables of the third level, an enumerative algorithm is proposed, which finds an optimum solution to the given problem. It is illustrated with the help of an example. .

Agent-Based Evolutionary Game-Theoretic Framework for Optimizing Investment Plans of Competitive Port Facilities

Abstract The current paper deals with a special case of the Network Design Problem (NDP) that arises when multiple authorities are controlling certain parts of a network and in particular when these are engaged into a deliberative environment, forming a competitive market for transportation services. This alteration departs significantly from the classical paradigm of the NDP extending the classical single leader-multiple followers Stackelberg game-theoretic formulation to its complete form of multiple leaders-multiple followers Competitive NDP (CNDP). In order to address the particular problem setup, a multilevel vector-optimization programming problem is formulated, which is tackled by a proposed novel Artificial Intelligence (AI)-based decentralized evolutionary optimization algorithm. The application of this framework is done over a freight network that is composed of multiple transportation means, and in particular over a complete transportation chain formed by shippers, maritime and land carriers and port authorities, where the latter are competing for profits. The market of maritime facilities is modeled as an n-person non-cooperative game among port authorities who control the attractiveness of their terminal facilities. By taking the above interdependencies into consideration, the estimation of the equilibrium point of the above formulation is made by creating a game-theoretic platform where Intelligent Agents (corresponding to port authorities) are evolving their strategies towards equilibrium points. Results from the application of the proposed framework into a realistic part of the East Mediterranean freight network are provided and discussed.

Demand responsive urban public transport system design: Methodology and application

In this paper, we present a methodology for solving the Public Transport Network Design Problem (PTNDP) and describe its application in the context of the Design Study developed in order to propose a new structure for the transit system of the city of Santiago, Chile. Firstly, we briefly define the PTNDP as a multilevel programming problem and discuss the solution method implemented. Then, the application of this methodology to the Santiago transit system is presented, and the main results obtained are analyzed. The new restructured system, based on a hierarchy of specialized services that complement and coordinate their operations and using an integrated fare scheme, is compared with an optimized version (optimal frequencies) of the current one, a set of direct services, mainly based on the operation of independent itineraries, without fare integration. The most important conclusions are the following: (a) the private operating costs and the social costs of the restructured system, using higher standard buses, are considerably lower than the costs of the current system; (b) these cost reductions allow government authorities to introduce an important number of modernizing measures without subsidies and fare increases.

An algorithm for multi-level programming problem using goal programming

This paper presents an algorithm for solving a multi-level programming problem using a linear pre-emptive goal programming model. The higher level decision maker (DM) provides the preferred values of the decision variables under his control and the target value of his objective function to the next level DM to formulate a goal programming problem equivalent to the given multi-level programming problem. It is illustrated with the help of an example of a tri-level programming problem.

Interactive balance space approach for solving multi-level multi-objective programming problems

This paper studies a multi-level multi-objective decision-making (ML-MODM) problems with linear or non-linear constraints. The objective functions at each level are non-linear functions, which are to be maximized or minimized. This paper presents a three-level multi-objective decision-making (TL-MODM) model and an interactive algorithm for solving such a model. The algorithm simplifies three-level multi-objective decision-making problems by transforming them into separate multi-objective decision making problems at each level, thereby avoiding the difficulty associated with non-convex mathematical programming. Our algorithm is an extension of the work of Shi and Xia [X. Shi, H. Xia, Interactive bi-level multi-objective decision making, Journal of the Operational Research Society 48 (1997) 943–949], which dealt with interactive bi-level multi-objective decision-making problems, with some modifications in assigning satisfactoriness to each objective function in all the levels of the TL-MODM problem. Also, we solve each separate multi-objective decision making problem of the TL-MODM problem by the balance space approach. A new formula is introduced to interconnect the satisfactoriness and the proportions of deviation needed to reflect the relative importance of each objective function. Thus, we have the proportions of deviation including satisfactoriness. In addition, we present new definitions for the satisfactoriness and the preferred solution in view of singular-level multi-objective decision making problems that corresponds to the η-optimal solution of the balance space approach. Also, new definitions for the feasible solution and the preferred solution ( η-optimal point) of the TL-MODM problem are presented. An illustrative numerical example is given to demonstrate the algorithm.

Fuzzy goal programming approach to multilevel programming problems

The purpose of this paper is to propose a procedure for solving multilevel programming problems in a large hierarchical decentralized organization through linear fuzzy goal programming approach. Here, the tolerance membership functions for the fuzzily described objectives of all levels as well as the control vectors of the higher level decision makers are defined by determining individual optimal solution of each of the level decision makers. Since the objectives are potentially conflicting in nature, a possible relaxation of the higher level decision is considered for avoiding decision deadlock. Then fuzzy goal programming approach is used for achieving highest degree of each of the membership goals by minimizing negative deviational variables. Sensitivity analysis with variation of tolerance values on decision vectors is performed to present how the solution is sensitive to the change of tolerance values. The efficiency of our concept is ascertained by comparing results with other fuzzy programming approaches.

Interactive compensatory fuzzy programming for decentralized multi-level linear programming (DMLLP) problems

This paper presents interactive compensatory fuzzy programming for decentralized multi-level linear programming (DMLLP) problems. By adjusting the cooperative decision making process between the different levels and also between the decision makers of the same level; our aim is to obtain a preferred compensatory compromise Pareto-optimal solution for DMLLP. For this, the weights of objectives at each level are assigned by the next upper level decision maker (DM) by using analytic hierarchy process (AHP) or any other weighting methods. The weight of any objective for whole system is equal to the product of the weights on the path tying it to the top decision maker DM 0 . Using these weights, equivalence is established such that the satisfactory levels of all objectives are proportional to their own weights. Werners’ compensatory “fuzzy and’’ operator is offered to solve DMLLP problem. The most important idea to be emphasized is that equivalence is established such that the satisfactory levels of all objectives are proportional to their own weights. Thanks to this equivalence, DMLLP problem has been transformed to the multi-objective linear programming (MOLP) problem at level 0, the equivalence is reflected to the compensatory model within the constraints, and the equivalence also enables all DMs to obtain proportional satisfactions with their weights as much as possible. So, in our compensatory model, a reduction on equivalent satisfactory level of one DM can be compensated for by an increase in the equivalent satisfactory level of another DM. Furthermore, being developed a finite interactive iterative procedure with maximum interaction step, a set of compensatory solutions which are also Pareto-optimal is obtained, depending on compensation parameter γ . Giving a theorem, we will show that the solutions generated by Werners’ compensatory “fuzzy and’’ operator do guarantee Pareto-optimality for our DMLLP problem. And comparing it with some other computational efficient compensatory fuzzy aggregation operators we will conclude that this operator is more appropriate for DMLLP. Illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed interactive fuzzy compensatory method for DMLLP.

On the Non-Linear Multilevel Programming Problems

In this paper a multilevel programming problem viz. three level programming problem (TPP) is considered. It involves three optimization problems where the constraint region of the first level problem is implicitly determined by two other optimization problems. The objective function of the first level is indefinite quadratic, the second one is linear and the third one is linear fractional. The feasible region is a convex polyhedron. At first it is shown that minimizing the indefinite quadratic programming problem is equivalent to minimizing a quasiconcave function over a feasible region comprised of the faces of the convex polyhedron. Also, the second objective function being linear and the third being linear fractional, the optimal solution occurs at an extreme point. It is illustrated with the help of an example.

Fuzzy approach to multilevel knapsack problems

This study proposes a fuzzy approach for solving the multiobjective and multilevel knapsack problems (KPs). The problem was first formulated as a multilevel programming problem with multiple decision makers (DMs). Then the degree of satisfaction of each DM was established and represented by their individual membership functions. The recursive formulation of dynamic programming was used to solve the decisions of the interrelated stages. The overall satisfaction of the decision was obtained through this stage-wise operation on the hierarchical structure. Capacity allocation was developed and a step-by-step solution procedure was illustrated. A detailed comparison between multiobjective and multilevel KPs was also carried out. Finally, the possible use of turnpike theorem in KPs was scrutinized in the fuzzy domain.

Optimality Conditions and Geometric Properties of a Linear Multilevel Programming Problem with Dominated Objective Functions

In this paper, a model of a linear multilevel programming problem with dominated objective functions (LMPPD(l)) is proposed, where multiple reactions of the lower levels do not lead to any uncertainty in the upper-level decision making. Under the assumption that the constrained set is nonempty and bounded, a necessary optimality condition is obtained. Two types of geometric properties of the solution sets are studied. It is demonstrated that the feasible set of LMPPD(l) is neither necessarily composed of faces of the constrained set nor necessarily connected. These properties are different from the existing theoretical results for linear multilevel programming problems.

A neural network approach to multiobjective and multilevel programming problems

This study aims at utilizing the dynamic behavior of artificial neural networks (ANNs) to solve multiobjective programming (MOP) and multilevel programming (MLP) problems. The traditional and non-traditional approaches to the MLP are first classified into five categories. Then, based on the approach proposed by Hopfield and Tank [1], the optimization problem is converted into a system of nonlinear differential equations through the use of an energy function and Lagrange multipliers. Finally, the procedure is extended to MOP and MLP problems. To solve the resulting differential equations, a steepest descent search technique is used. This proposed nontraditional algorithm is efficient for solving complex problems, and is especially useful for implementation on a large-scale VLSI, in which the MOP and MLP problems can be solved on a real time basis. To illustrate the approach, several numerical examples are solved and compared.

Linear multi-level programming problems with random variables

In this paper we study hierarchical systems with stochastic data: be it the cost coefficients, right-hand-side vector, coefficient matrix or any of their combinations. The problem is converted to its equivalent deterministic form using any known stochastic approach. This deterministic model (in case of a linear hierarchy) may be linear or nonlinear depending on which parameter in the problem is random. Finally the linear nonlinear deterministic model is solved using fuzzy technique [9] via linear membership function. We have also shown the application of the deterministic models to find a solution using numerical examples.

A linear programming approach for linear multi-level programming problems

In an earlier paper, we proposed a modified fuzzy programming method to handle higher level multi-level decentralized programming problems (ML(D)PPs). Here we present a simple and practical method to solve the same. This method overcomes the subjectivity inherent in choosing the tolerance values and the membership functions. We consider a linear ML(D)PP and apply linear programming (LP) for the optimization of the system in a supervised search procedure, supervised by the higher level decision maker (DM). The higher level DM provides the preferred values of the decision variables under his control to enable the lower level DM to search for his optimum in a narrower feasible space. The basic idea is to reduce the feasible space of a decision variable at each level until a satisfactory point is sought at the last level.

KKT transformation approach for multi-objective multi-level linear programming problems

The earlier Karush–Kuhn–Tucker (KKT) transformation method has been applied to multi-level decentralized programming problems (ML(D)PPs) when the decision variable set was divided into subsets where each decision maker (DM) of the system controlled only a particular subset but had no control over any decision variables of some other subset. In this paper we give the mathematical formulation and corresponding development of ML(D)PPs by KKT transformation when DMs have absolute control over certain decision variables but some variables may be shared and hence controlled by two or more DMs.

Fuzzy programming approach to multi-level programming problems

Multi-level programming is characterized as mathematical programming to solve decentralized planning problems. We have considered a multi-level linear programming problem and applied fuzzy mathematical programming (FMP) approach to obtain the solution of the system. We have suggested FMP method for the minimization of the objectives using linear membership functions. FMP is a supervised search procedure (supervised by the higher level decision maker (DM)). The higher level DM provides the preferred values of decision variables under his control (to enable the lower level DM to search for his optimum in a wider feasible space) and the bounds of his objective function (to direct the lower level DM to search for his solutions in the right direction).

An interactive approach for integrated multilevel systems in a fuzzy environment

This study proposes a total solution of an interactive approach for integrated multilevel systems or multilevel programming problems (MLPPs) in a fuzzy environment. Simulating the actual decision-making process of the hierarchical structure of an organization, MLPP is a practical and useful approach to decentralized planning problems. Because of the complexity of the problems, there are no traditional techniques efficient enough to obtain the numerical solution of a reasonable size problem. Hence, Shih et al. [1] propose a fuzzy approach for MLPPs, to simplify the complex structure, which is proven to be feasible and efficient. The imprecise MLPPs will be involved when the coefficients of MLPPs cannot be estimated exactly. Because of such a complicated situation in the real world, we will take advantage of an interactive technique to improve the flexibility and robustness of its decision through progressive articulation of decision information from decision makers (DMs). Roughly speaking, there are two interactive procedures for imprecise MLPPs: inside loop and outside loop. The former is for the preference of the DMs, represented by fuzzy membership functions; the latter for the imprecision of coefficients, described by possibility distributions or cut-off values. Special considerations will be given to the compensatory operator, positive and negative ideal solutions, risk attitude, and ε-constraints. In the final section, linear-programming type and network-flow type of imprecise MLPPs will be solved separately as an integrated multilevel system.