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Article
Publication date: 5 May 2020

Davood Darvishi Salookolaei and Seyed Hadi Nasseri

For extending the common definitions and concepts of grey system theory to the optimization subject, a dual problem is proposed for the primal grey linear programming problem.

Abstract

Purpose

For extending the common definitions and concepts of grey system theory to the optimization subject, a dual problem is proposed for the primal grey linear programming problem.

Design/methodology/approach

The authors discuss the solution concepts of primal and dual of grey linear programming problems without converting them to classical linear programming problems. A numerical example is provided to illustrate the theory developed.

Findings

By using arithmetic operations between interval grey numbers, the authors prove the complementary slackness theorem for grey linear programming problem and the associated dual problem.

Originality/value

Complementary slackness theorem for grey linear programming is first presented and proven. After that, a dual simplex method in grey environment is introduced and then some useful concepts are presented.

Details

Grey Systems: Theory and Application, vol. 10 no. 2
Type: Research Article
ISSN: 2043-9377

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Article
Publication date: 17 June 2020

Davood Darvishi, Sifeng Liu and Jeffrey Yi-Lin Forrest

The purpose of this paper is to survey and express the advantages and disadvantages of the existing approaches for solving grey linear programming in decision-making problems.

Abstract

Purpose

The purpose of this paper is to survey and express the advantages and disadvantages of the existing approaches for solving grey linear programming in decision-making problems.

Design/methodology/approach

After presenting the concepts of grey systems and grey numbers, this paper surveys existing approaches for solving grey linear programming problems and applications. Also, methods and approaches for solving grey linear programming are classified, and its advantages and disadvantages are expressed.

Findings

The progress of grey programming has been expressed from past to present. The main methods for solving the grey linear programming problem can be categorized as Best-Worst model, Confidence degree, Whitening parameters, Prediction model, Positioned solution, Genetic algorithm, Covered solution, Multi-objective, Simplex and dual theory methods. This survey investigates the developments of various solving grey programming methods and its applications.

Originality/value

Different methods for solving grey linear programming problems are presented, where each of them has disadvantages and advantages in providing results of grey linear programming problems. This study attempted to review papers published during 35 years (1985–2020) about grey linear programming solving and applications. The review also helps clarify the important advantages, disadvantages and distinctions between different approaches and algorithms such as weakness of solving linear programming with grey numbers in constraints, inappropriate results with the lower bound is greater than upper bound, out of feasible region solutions and so on.

Details

Grey Systems: Theory and Application, vol. 11 no. 1
Type: Research Article
ISSN: 2043-9377

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Article
Publication date: 28 January 2014

Qiao-Xing Li, Sifeng Liu and Nai-Ang Wang

This paper attempts to establish the general formula for computing the inverse of grey matrix, and the results are applied to solve grey linear programming. The inverse of…

Abstract

Purpose

This paper attempts to establish the general formula for computing the inverse of grey matrix, and the results are applied to solve grey linear programming. The inverse of a grey matrix and grey linear programming plays an important role in establishing a grey computational system.

Design/methodology/approach

Starting from the fact that missing information often appears in complex systems, and therefore that true values of elements are uncertain when the authors construct a matrix, as well as calculate its inverse. However, the authors can get their ranges, which are called the number-covered sets, by using grey computational rules. How to get the matrix-covered set of inverse grey matrix became a typical approach. In this paper, grey linear programming was explained in detail, for the point of grey meaning and the methodology to calculate the inverse grey matrix can successfully solve grey linear programming.

Findings

The results show that the ranges of grey value of inverse grey matrix and grey linear programming can be obtained by using the computational rules.

Practical implications

Because the matrix and the linear programming have been widely used in many fields such as system controlling, economic analysis and social management, and the missing information is a general phenomenon for complex systems, grey matrix and grey linear programming may have great potential application in real world. The methodology realizes the feasibility to control the complex system under uncertain situations.

Originality/value

The paper successfully obtained the ranges of uncertain inverse matrix and linear programming by using grey system theory, when the elements of matrix and the coefficients of linear programming are intervals and the results enrich the contents of grey mathematics.

Details

Grey Systems: Theory and Application, vol. 4 no. 1
Type: Research Article
ISSN: 2043-9377

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Article
Publication date: 3 April 2018

Davood Darvishi Salookolaei, Sifeng Liu and Sayed Hadi Nasseri

The purpose of this paper is to discuss the animal diet problem in grey environment which is adapted to the real situations. In particular, a new approach to solve these…

Abstract

Purpose

The purpose of this paper is to discuss the animal diet problem in grey environment which is adapted to the real situations. In particular, a new approach to solve these problems is proposed.

Design/methodology/approach

With the objective to produce the least-cost diet, in the traditional model for optimizing the diet problem, the price of foods, the nutrients requirements and the necessity of foods requirement have been considered as grey interval numbers. Grey linear programming approach has been employed to solve the grey diet problem. Grey linear programming with flexibility in selection of the coefficients can be more effective for solving the diet problems. In this research, only the positioned method has been used. The grey diet model is solved by using GAMS software based on the positioned method.

Findings

The main contribution of this work is to introduce a new model in the practical case that is concerned with diet problem under a kind of uncertainty environment and furthermore, proposing a novel method to solve the formulated problem. In this way, using a grey model and applying all restrictions, the least cost for one kilogram of total mixed ration was 6,893-10,163 Rials, and at this level, cow’s nutrient requirement was met. Based on the numerical examination, which was done on the real case, the achieved results have showed that the uncertainty of foods requirement and nutrients requirements had slight effect on the animal budget diet.

Originality/value

This problem must be viewed from another perspective because of the uncertainty regarding the amount of nutrients per unit of foods and the diversity of animals’ daily needs to receive them. In particular, a new method to optimize the fully mixed diet of lactating cows in early lactation that are readily available in the northeast of Iran in uncertainty environment has been proposed.

Details

Grey Systems: Theory and Application, vol. 8 no. 2
Type: Research Article
ISSN: 2043-9377

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Article
Publication date: 17 August 2012

Seyed Hossein Razavi Hajiagha, Hadi Akrami and Shide Sadat Hashemi

The purpose of this paper is to extend an approach to solve linear programming problems with grey data and variables, based on a developed multi‐objective programming approach.

Abstract

Purpose

The purpose of this paper is to extend an approach to solve linear programming problems with grey data and variables, based on a developed multi‐objective programming approach.

Design/methodology/approach

The proposed approach to generally solve the grey linear programming problems is based on the notion of order relation between interval grey numbers. This notion is applied to cascade the grey objective function to a bi‐objective problem based on the objective function of the original problem. The same approach is taken to transform grey constraints to a set of corresponding linear constraints. Finally, the obtained multi‐objective model can be solved by any existing methods in the literature.

Findings

One of the shortcomings of previous approaches to solve grey linear programming problems was that they required the grey coefficients of objective function to be both side negative or positive. The approach proposed here does not have such a requirement and guarantees the feasibility of solutions.

Originality/value

A different approach is developed in the paper that can be used to solve grey linear programming problems in general form. The method relaxes the limitation of existing approaches.

Details

Grey Systems: Theory and Application, vol. 2 no. 2
Type: Research Article
ISSN: 2043-9377

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Article
Publication date: 1 February 2004

Zhijie Chen, Qile Chen, Weizhen Chen and Yinao Wang

This paper describes the use of grey system theory in mathematical programming problems. In particular, the linear programming problem, which is one of the most widely…

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Abstract

This paper describes the use of grey system theory in mathematical programming problems. In particular, the linear programming problem, which is one of the most widely used mathematical programming problems, with grey interval and grey forecasting are developed. The adaptability of both these linear programming problems is rather satisfactory.

Details

Kybernetes, vol. 33 no. 2
Type: Research Article
ISSN: 0368-492X

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Article
Publication date: 5 February 2018

Amin Mahmoudi, Mohammad Reza Feylizadeh and Davood Darvishi

The purpose of this paper is to examine the shortcomings and problems associated with the method proposed by Razavi Hajiagha et al. (2012).

Abstract

Purpose

The purpose of this paper is to examine the shortcomings and problems associated with the method proposed by Razavi Hajiagha et al. (2012).

Design/methodology/approach

A multi-objective approach is proposed to solve the grey linear programming problems. In this method, the grey linear problem is converted into a multi-objective problem and then solved.

Findings

According to the numerical example presented in the study by Razavi Hajiagha et al. (2012), this method does not have a correct solution because the solution does not satisfy the constraints and the upper bounds of the variables are equal or less than their lower bound.

Originality/value

In recent years, various methods have been proposed for solving grey linear programming problems. Razavi Hajiagha et al. (2012) proposed a multi-objective approach to solve grey linear programming problems, but this method does not have a correct solution and using this method in other researches studies can reduce the value of the grey system theory.

Details

Grey Systems: Theory and Application, vol. 8 no. 1
Type: Research Article
ISSN: 2043-9377

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Article
Publication date: 28 May 2021

Zainab Asim, Syed Aqib Aqib Jalil, Shakeel Javaid and Syed Mohd Muneeb

This paper aims to develop a grey decentralized bi-level multi-objective programming (MOP) model. A solution approach is also proposed for the given model. A production…

Abstract

Purpose

This paper aims to develop a grey decentralized bi-level multi-objective programming (MOP) model. A solution approach is also proposed for the given model. A production and transportation plan for a closed loop supply chain network under an uncertain environment and different scenarios is also developed.

Design/methodology/approach

In this paper, we combined grey linear programming (GLP) and fuzzy set theory to present a solution approach for the problem. The proposed model first solves the given problem using GLP. Membership functions for the decision variables under the control of the leader and for the goals are created. These membership functions are then used to generate the final solutions.

Findings

This paper provides insight for fomenting the decision-making process while providing a more flexible approach in uncertain logistics problems. The deviations of the final solution from the individual best solutions of the two levels are very little. These deviations can further be reduced by adjusting the tolerances associated with the decision variables under the control of the leader.

Practical implications

The proposed approach uses the concept of membership functions of linear form, and thus, requires less computational efforts while providing effective results. Most of the organizations exhibit decentralized decision-making under the presence of uncertainties. Therefore, the present study is helpful in dealing with such scenarios.

Originality/value

This is the first time, formulation of a decentralized bi-level multi-objective model under a grey environment is carried out as per the best knowledge of the authors. A solution approach is developed for bi-level MOP under grey uncertainty.

Details

Journal of Modelling in Management, vol. 16 no. 3
Type: Research Article
ISSN: 1746-5664

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Article
Publication date: 23 May 2018

Amin Mahmoudi and Mohammad Reza Feylizadeh

The purpose of this paper is to examine projects crashing based on the factors including cost, time, quality, risk and the law of diminishing returns.

Abstract

Purpose

The purpose of this paper is to examine projects crashing based on the factors including cost, time, quality, risk and the law of diminishing returns.

Design/methodology/approach

The paper first investigated effective factors on project crashing then proposed a grey linear programming model. In the proposed grey linear programming model, the costs of quality of works that include the cost of conformance and non-conformance of deliverables in the project were studied. The results are presented for considering the existing uncertainties using positioned programming under the sensitivity analysis table and graphs.

Findings

The lack of consideration of project risks will reduce the project success probability in future. The proposed model reduces the existing uncertainties to a significant extent by covering the project risks completely. Based on the law of diminishing returns, after a certain point technically known as saturation point, the increase of resources does not lead to the reduction of time and may even have negative impacts. Finding the saturation point for each activity prevents the excessive allocation of resources that can lead to reduction of productivity.

Practical implications

The main duty of each project manager is finishing the project in the framework of the determined objectives. In most of the cases, after the preparation of the initial project schedule by the project team, it is seen that there is a need for the time reduction. This study has used a grey linear programming model for optimum crashing of project activities. In order to make the model more realistic and applicable, the authors endeavoured to consider most of the factors that are involved in doing a project.

Originality/value

In the present study, to the best of the authors’ knowledge the factors of time, cost, quality, risk and the law of diminishing returns are simultaneously considered in project crashing for the first time and the grey theory was used for considering the uncertainties of project parameters. Also, “the law of diminishing returns” has not been considered during crashing in the studies conducted so far.

Details

Grey Systems: Theory and Application, vol. 8 no. 3
Type: Research Article
ISSN: 2043-9377

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Article
Publication date: 25 May 2018

Amin Mahmoudi, Mohammad Reza Feylizadeh, Davood Darvishi and Sifeng Liu

The purpose of this paper is to propose a method for solving multi-objective linear programming (MOLP) with interval coefficients using positioned programming and…

Abstract

Purpose

The purpose of this paper is to propose a method for solving multi-objective linear programming (MOLP) with interval coefficients using positioned programming and interactive fuzzy programming approaches.

Design/methodology/approach

In the proposed algorithm, first, lower and upper bounds of each objective function in its feasible region will be determined. Afterwards using fuzzy approach, considering a membership function for each objective function and finally using grey linear programming, the solution for this problem will be obtained.

Findings

According to the presented example, in this paper, the proposed method is both simple in use and suitable for solving different problems. In the numerical example mentioned in this paper, the proposed method provides an acceptable solution for such problems.

Practical implications

As in most real-world situations, the coefficients of decision models are not known and exact. In this paper, the authors consider the model of MOLP with interval data, since one of the solutions to cover uncertainty is using interval theory.

Originality/value

Based on using grey theory and interactive fuzzy programming approaches, an appropriate method has been presented for solving MOLP problems with interval coefficients. The proposed method, against the complex methods, has less effort and offers acceptable solutions.

Details

Grey Systems: Theory and Application, vol. 8 no. 3
Type: Research Article
ISSN: 2043-9377

Keywords

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