Interval Linear Programming Problems Research Articles

Some new results on rough interval linear programming problems and their application to scheduling and fixed-charge transportation problems

This paper focuses on linear programming problems in a rough interval environment. By introducing four linear programming problems, an attempt is being made to propose some results on optimal value of a linear programming problem with rough interval parameters. To obtain optimal solutions of a linear programming problem with rough interval data, constraints of the four proposed linear problems are applied. In this regard, firstly, the largest and the smallest feasible spaces for a linear constraint set with rough interval coefficients and parameters are introduced. Then, a rough interval for optimal value of such problems is obtained. Further, an upper approximation interval and a lower approximation interval as the optimal solutions of linear programming problems with rough interval parameters are achieved. Moreover, two solution concepts, surely and possibly solutions, are defined. Some numerical examples demonstrate the validity of the results. In particular, a scheduling problem and a fixed-charge transportation problem (FCTP) under rough interval uncertainty are investigated.

On the new solution to interval linear fractional programming problems

On the new solution to interval linear fractional programming problems

A NOVEL METHOD FOR SOLVING FULLY FUZZY SOLID TRANSPORTATIONS PROBLEMS

In this paper, we proposes a new method for solving solid transportation problem under uncertainty environments. The fully fuzzy solid transportation problem has been formulated. To reduce the model into crisp equivalent, we have used existing method for approximation of fuzzy numbers by interval numbers and its arithmatics. A simplex method and existing method for solving Interval Linear Programming problems are used for solving solid transportation problem with fuzzy parameters and decision variables. Furthermore, for illustration, some numerical examples are used to demonstrate the correctness and usefulness of the proposed method. The proposed algorithm is flexible, easy and reasonable.

A study on the solution of interval linear fractional programming problem

Interval linear fractional programming problem (ILFPP) approaches uncertain-ties in real-world systems such as business, manufacturing, finance, and eco-nomics. In this study, we propose solving the interval linear fractional pro-gramming (ILFP) problem using interval arithmetic. Further, to construct the problem, a suitable variable transformation is used to form an equivalent ILP problem, and a new algorithm is depicted to obtain the optimal solution with-out converting the problem into its conventional form. This paper compares the range, solutions, and approaches of ILFP with fuzzy linear fractional pro-gramming (FLFP) in solving real-world optimization problems. The illustrated numerical examples show a better range of interval solutions on practical appli-cations of ILFPs and uncertain parameters.

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.

Parametric approach for multi-objective enhanced interval linear fractional programming problem

The design (decision) variables in the presented article of a multi-objective interval fractional optimization problem based on a linear function are assumed to take the form of a closed interval using the concept of the parametric form of an interval. The original problem is initially changed into equivalent multi-objective interval linear programming with the design variables as closed intervals. Further, it is made free from interval uncertainty by changing into a classical single-objective problem using the weighted-sum method. The solutions of the model are theoretically justified by its existence. Finally, a numerical example and a case study on the agricultural planting structure optimization problem with hypothetical data are presented to support the recommended technique for the model.

Fully hesitant fuzzy linear programming with hesitant fuzzy numbers

In this paper, we introduce a new approach to solve a fully hesitant fuzzy linear programming (FHFLP) problem with hesitant fuzzy numbers (HFNs) as parameters. Using an (α,k)-cut for the HFNs, we convert this problem into some interval linear programming (ILP) problems, and solve these problems through one of the available algorithms to solve the ILP problems. Then through statistical regression analysis, we get k fuzzy numbers as the final approximate solutions, and we have proved that they are well defined. Finally, using the hesitant fuzzy arithmetic, we obtain a hesitant fuzzy value of the objective function. An example is introduced to clarify the solution process of this method, and a comparison between the optimal solutions through the proposed method to an FHFLP problem and its converted form to a fully fuzzy linear programming problem has been done. The outcomes indicate that the obtained solutions for decision variables and objective function are reasonable.

GA/PD: a genetic algorithm based on problem decomposition for solving interval linear bilevel programming problems

For dealing with uncertain optimization problems, interval programming is a common model that can provide a risk range for decision makers between the best and the worst optimal solutions. Among all interval programming problems, the interval bilevel programming problem has always known as one of the hardest problems to solve. In this work, an efficient solution method is developed for general interval linear bilevel programs in which all coefficients and right-hand vectors are given by intervals. Firstly, a decomposition scheme is adopted to transform the original problem into two simplified subproblems in which only the follower's problem involves interval coefficients. Secondly, a genetic algorithm, one of the more effective evolutionary algorithms, is designed to search these intervals, and linear program optimality conditions are utilized to evaluate each individual in populations. Finally, by comparing the fitness values according to pre-determined rules, the best and the worst optimal solutions to the interval linear bilevel programming problem can be achieved. Simulation results show that the proposed algorithm is efficient and can obtain larger intervals of the optimal values on some computational examples than those in the literature.

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.

Iterative algorithms for determining optimal solution set of interval linear fractional programming problem

Determining the optimal solution (OS) set of interval linear fractional programming (ILFP) models is generally an NP-hard problem. Few methods have been proposed in this field which have only been able to obtain the optimal value of the objective function. Thus, there is a need for an appropriate method to determine the OS set of the ILFP model. In this paper, we introduce three algorithms to obtain the OS of ILFP. In the first and second algorithms, using the definition of strong and weak feasible solutions, the objective function of ILFP has been transformed to a linear objective function on the largest feasible region (LFR) and we obtain the OS of ILFP. These two algorithms, only introduce one point as the feasible OS. Since ILFP is an interval model, we seek an algorithm, where for the first time a solution set is obtained as the OS set by solving two sub-models. Hence, we transform the ILFP model into two pessimistic and optimistic sub-models, as one is in the smallest feasible region (SFR) and the other on the LFR. We add constraints to the optimistic model to ensure that the OS set is feasible. Then, we introduce pessimistic and modified optimistic model (PMOM) algorithm. In this algorithm, each PMOM is solved separately. The OSs obtained from these two models give the OS set so that this OS set is feasible. Note that the union of feasible OSs obtained from the proposed algorithms will be a more complete feasible OS set.

NEW IMPROVED METHOD FOR SOLVING THE FUZZY LINEAR PROGRAMMING PROBLEMS WITH VARIABLES GIVEN AS FUZZY NUMBERS

The present paper aims to propose an alternative solution approach in obtaining the fuzzy optimal solution to a fuzzy linear programming problem with variables given as fuzzy numbers with minimum uncertainty. In this paper, the fuzzy linear programming problems with variables given as fuzzy numbers is transformed into equivalent interval linear programming problems with variables given as interval numbers. The solutions to these interval linear programming problems with variables given as interval numbers are then obtained with the help of linear programming technique. A set of six random numerical examples has been solved using the proposed approach.

A Mathematical Model for Solving the Linear Programming Problems Involving Trapezoidal Fuzzy Numbers via Interval Linear Programming Problems

We define linear programming problems involving trapezoidal fuzzy numbers (LPTra) as the way of linear programming problems involving interval numbers (LPIn). We will discuss the solution concepts of primal and dual linear programming problems involving trapezoidal fuzzy numbers (LPTra) by converting them into two linear programming problems involving interval numbers (LPIn). By introducing new arithmetic operations between interval numbers and fuzzy numbers, we will check that both primal and dual problems have optimal solutions and the two optimal values are equal. Also, both optimal solutions obey the strong duality theorem and complementary slackness theorem. Furthermore, for illustration, some numerical examples are used to demonstrate the correctness and usefulness of the proposed method. The proposed algorithm is flexible, easy, and reasonable.

A Simplified Novel Technique for Solving Linear Programming Problems with Triangular Fuzzy Variables via Interval Linear Programming Problems

This study proposes a novel technique for solving Linear Programming Problems with triangular fuzzy variables. A modified version of the well-known simplex method and the Existing Method for Solving Interval Linear Programming problems are used for solving linear programming problems with triangular fuzzy variables. Furthermore, for illustration, some numerical examples and one real problem are used to demonstrate the correctness and usefulness of the proposed method. The proposed algorithm is flexible, easy, and reasonable.

Solution types of two-sided interval linear system and their application on interval linear programming problems

Two-sided interval linear systems contain variables on both sides of equations or inequalities. We propose six types of solutions (weak, strong, tolerance, control, left-localized and right-localized solutions) to the system of equations together with four types of solutions (weak, strong, tolerance and control solutions) to the system of inequalities. We prove the necessary and sufficient conditions for checking their solvabilities in the form of systems of linear inequalities depending on the sign of variables. As a result, an interval linear programming problem with nonnegative variables and two-sided interval linear constraints can be solved by a standard linear programming approach.

Interval linear fractional programming: optimal value range of the objective function

In the real world, some problems can be modelled by linear fractional programming with uncertain data as an interval. Therefore, some methods have been proposed for solving interval linear fractional programming (ILFP) problems. In this research, we propose two new methods for solving ILFP problems. In each method, we use two sub-models to obtain the range of the objective function. In the first method, we introduce two sub-models in which the objective functions are non-linear and the two sub-models have the largest and smallest feasible regions; therefore, the optimal value range of the objective function has been obtained. In the second method, two sub-models have been proposed in which the objective functions are linear and the optimal value of the objective function lies in the range obtained from the first method. We use our approaches to maximize the ratio of the facilities optimal allocation to the non-return fund in a bank.

Solving interval linear programming problem using generalized interval lu decomposition method

In this paper, we propose a method for solving interval linear programming problem by using generalized interval LU decomposition method based on Kaucher arithmetic for generalized interval numbers. Numerical example is also been provided.

A modified two-step method for solving interval linear programming problems

In this paper, we propose a new method for solving interval linear programming (ILP) problems. For solving the ILP problems, two important items should be considered: feasibility (i.e., solutions satisfy all constraints) and optimality (i.e., solutions are optimal for at least a characteristic model). In some methods, a part of the solution space is infeasible (i.e., it violates any constraints) such as the best and worst cases method (BWC) proposed by Tong in 1994 and two-step method (TSM) proposed by Huang et al. in 1995. In some methods, the solution space is completely feasible, but is not completely optimal (i.e., some points of the solution space are not optimal) such as modified ILP method (MILP) proposed by Zhou et al. in 2009 and improved TSM (ITSM) proposed by Wang and Huang in 2014. Firstly, basis stability for the ILP problems is reviewed. Secondly, the solving methods are analysed from the point of view of the feasibility and optimality conditions. Later, a new method which modifies the TSM by using the basis stability approach is presented. This method gives a solution space that is not only completely feasible, but also completely optimal.

A modified two-step method for solving interval linear programming problems

In this paper, we propose a new method for solving interval linear programming (ILP) problems. For solving the ILP problems, two important items should be considered: feasibility (i.e., solutions satisfy all constraints) and optimality (i.e., solutions are optimal for at least a characteristic model). In some methods, a part of the solution space is infeasible (i.e., it violates any constraints) such as the best and worst cases method (BWC) proposed by Tong in 1994 and two-step method (TSM) proposed by Huang et al. in 1995. In some methods, the solution space is completely feasible, but is not completely optimal (i.e., some points of the solution space are not optimal) such as modified ILP method (MILP) proposed by Zhou et al. in 2009 and improved TSM (ITSM) proposed by Wang and Huang in 2014. Firstly, basis stability for the ILP problems is reviewed. Secondly, the solving methods are analysed from the point of view of the feasibility and optimality conditions. Later, a new method which modifies the TSM by using the basis stability approach is presented. This method gives a solution space that is not only completely feasible, but also completely optimal.

Bi-level Linear Programming Problem with Neutrosophic Numbers

The paper presents a novel strategy for solving bi-level linear programming problem based on goal programming inneutrosophic numbers environment. Bi-level linear programming problem comprises of two levels namely upper or first leveland lower or second level with one objective at each level. The objective function of each level decision maker and the systemconstraints are considered as linear functions with neutrosophic numbers of the form [p + q I], where p, q are real numbers andI represents indeterminacy. In the decision making situation, we convert neutrosophic numbers into interval numbers and theoriginal problem transforms into bi-level interval linear programming problem. Using interval programming technique, the target interval of the objective function of each level is identified and the goal achieving function is developed. Since, the objectives of upper and lower level decision makers are generally conflicting in nature, a possible relaxation on the decision variables under the control of each level is taken into account for avoiding decision deadlock. Then, three novel goal programmingmodels are presented in neutrosophic numbers environment. Finally, a numerical problem is solved to demonstrate the feasibility, applicability and novelty of the proposed strategy.

Solving interval linear programming problems with equality constraints using extended interval enclosure solutions

This paper focuses on solving systems of interval linear equations and interval linear programming in a computationally efficient way. Since the computational complexity of most interval enclosure numerical methods is often prohibitive, a procedure to obtain a relaxation of the interval enclosure solution that is computationally tractable is proposed. We show that our approach unifies the four standard interval solutions—the weak, strong, control and tolerance solutions. The interval linear system methods require $$n\cdot 2^{n}$$ linear solutions. However, in the case of linear programming problems, we show that this requires just two optimization problem of the size of the problem itself. Numerical examples illustrate our results.