Grey Linear Programming Research Articles

Land use structure optimization in Los Angeles County based on grey linear programming

In the rapidly developing modern world, population growth and urbanization have driven the prioritization of industrial settlements over the natural environment in land use. Besides losing ecological services, many inherent problems emerge in cities due to inefficient planning. This study presents an application of a grey linear program to maximize the socio-economic and environmental benefits of the land use structure in Los Angeles County. A technique in operations research, linear programming fits the purpose of this study since it optimizes a function concerning constraints. For each of the fourteen land types in this research, a monetary value is assigned to its economic output and environmental services, and the scalar sum is the goal to be maximized. The constraints consist of inequalities built from historical trends, developmental needs, and targets. On the other hand, based on past data, the grey model employs differential equations to predict the value of several parameters for the linear program, such as GDP output. The results are calculated by the Scipy library in a Python program and improve the quantified benefit by 8.26%, with an increase in residential, commercial, and industrial land and a slower decrease in natural landscape. Environmental initiatives and improvements in LA Countys livability are necessary to fulfill long-term values.

A bi-level multi objective programming approach to solve grey problems: an application to closed loop supply chain network

PurposeThis 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/approachIn 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.FindingsThis 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 implicationsThe 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/valueThis 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.

A Comparative Study between GLP and GBWM

When decision-makers’ judgments are uncertain, they often express their opinions using grey linguistic variables. Once used, the data often retains its grey nature throughout all subsequent decision-making iterations. Multicriteria decision-making (MCDM) is a tool used when making complicated decisions and in circumstances where several criteria require evaluation to choose the most desirable option. Grey data serves as the basis for several MCDM methods. This paper compares two MCDM methods, Grey-Linear-Programming (GLP) and Grey-Best-Worst-Method (GBWM), in terms of the weights of decision criteria and their rankings. Moreover, Grey-The Technique for Order of Preference by Similarity to Ideal Solution (GTOPSIS) was used to rank the weights of the two methods. Study findings demonstrated that GBWM requires more mathematical calculations than GLP, based on linear programming's classic simplex method. On the other hand, when GTOPSIS follows GLP, the alternative rank does not change compared to when GTOPSIS followed GBWM. For the applications used in this comparison, GLP procedure is considered simpler than GBWM procedure.

Enhancement of sustainable agricultural production system by integrated natural resources management framework under climatic and operational uncertainty

Constant decline of freshwater reserve poses a major challenge for coastal agricultural systems. The present study focuses on the integrated land and water resources management framework under operational and climatic uncertainties to achieve sustainable agricultural production in a tropical coastal river basin. To capture the operational and climatic uncertainties, Grey Linear Programming (GLP) and Acceptability Constrained Linear Programming (ACLP) optimization models were developed. GLP can consider both climatic and operational uncertainties. However, ACLP can accommodate operational uncertainty only. Groundwater model was coupled within the optimization models (GLP and ACLP) by a grey response matrix. The SWAT estimated recharge rates were utilized in the MODFLOW model as the groundwater flux. The calibrated (R2 = 0.80) and validated (R2 = 0.84) groundwater flow simulation model was utilized to generate the grey response matrix. The eight plausible scenarios were tested for GLP (Scenarios I, II, III, IV, V, VI, VII, VIII) and ACLP (Scenarios Ia, IIa, IIIa, Iva, Va, VIa, VIIa, VIIIa) with groundwater table fluctuation (GWTF) method and grey response matrix as drawdown constraint. A reduction in net annual return was observed in Scenarios III and IV (with grey response matrix) in comparison to Scenarios I and II (with groundwater table fluctuation) under any change in crop yield. Decrease was more prominent for Rabi (winter) season compared to Kharif (monsoon) season. Futuristic Scenarios (V, VI, VII, VIII) considering crop yield reduction showed a possible change required in cropping pattern (specifically rice area reduction in Kharif Season). ACLP model results showed similar trends. Results suggest that the GWTF method overestimates the groundwater availability compared to the physically based simulation model. Obtained land and water resources management plans were cross-validated for the safe yield limit using MODFLOW simulation model.

Sensitivity analysis of grey linear programming for optimisation problems

Sensitivity analysis of parameters is usually more important than the optimal solution when it comes to linear programming. Nevertheless, in the analysis of traditional sensitivities for a coefficient, a range of changes is found to maintain the optimal solution. These changes can be functional constraints in the coefficients, such as good values or technical coefficients, of the objective function. When real-world problems are highly inaccurate due to limited data and limited information, the method of grey systems is used to perform the needed optimisation. Several algorithms for solving grey linear programming have been developed to entertain involved inaccuracies in the model parameters; these methods are complex and require much computational time. In this paper, the sensitivity of a series of grey linear programming problems is analysed by using the definitions and operators of grey numbers. Also, uncertainties in parameters are preserved in the solutions obtained from the sensitivity analysis. To evaluate the efficiency and importance of the developed method, an applied numerical example is solved.

Optimization of Sustainable Land Use Management in Water Source Area Using Water Quality Dynamic Monitoring Model.

Land use management is the primary source of resource planning, and the management part of the sustainable ecosystem of water and soil resources is an important evidence for the sustainable development of the economic and social system. This is guided by the concept of sustainable development, and on the basis of the accumulation of relevant research practices and outcomes at home and abroad, water and land based systems are a research object and study the status of water and soil resource utilization, the state of water and soil coupling, and the supply and demand status of water resources. A balance analysis was carried out, and the gray linear programming model was used to optimize the allocation of land resources using the water quality dynamic monitoring model, which achieved the best coupling of water and soil resources and the greatest benefit. In this paper, aiming at the two types of problems in comprehensive water quality evaluation, namely, aiming at indifference and spatiotemporal changes, this article explores a powerful calculation method based on variable identification models and compiles a GIS geostatistical model (it is a computer-based tool that can draw and analyze ground objects; event GIS technology integrates seamless visual effects between map and local analysis services and general data processing services) to perform spatial analysis and visual expression of the evaluation results, in-depth analysis of the connotation, and theory and optimal allocation model of land resources optimal allocation. On the basis of the conceptual framework of the best share of land sources, the theories that should follow in the best share of land sources are discussed, and the available models and their characteristics are analyzed and compared. Experimental results show that, in the data provided by the analysis of water supply and demand balance at the annual spring system site by constructing an energy monitoring model, the water supply conditions of different water sources are rough, but the data of this study shows that the water shortage rate has reached 25%. In addition, the article explains the setting variables for the optimal allocation of land resources in water sources and compares and analyzes the optimization and planning of land resources in water sources.

Land use optimization by integrating GLP and CLUE-S model to control land degradation risk in mountainous area of Southwest China

Large-scale artificial plantations in mountainous areas of Southwest China have changed land use status and aggravated land degradation risk (LDR). This study taking Menglian County as an example, optimizes land use in 2025 to reduce the regional LDR, by integrating Grey Linear Programming (GLP) and CLUE-S model. Results showed that: The high-risk and medium-risk levels are main LDR types in Menglian County, accounting for 56.36% of total area. The regions with high LDR consistent with the distribution of concentrated garden land and cultivated land. The regions with low LDR consistent with the forestland. While the distribution of medium-risk regions relates to small plots garden land and cultivated land. In the optimization results, the LDR reduced 461.80, 168.95 and 34.23 in three schemes respectively, comparing to 2015. Thereinto, the strict-demand scheme has good applicability and guidance for study area relatively, in which the LDR is reduced while ensuring sustainable development. After spatial allocation, garden land, cultivated land, forestland and construction land tend to be centralized. It is effective for solving the optimal problem of mountainous land resource by integrating GLP and CLUE-S. The methods and results can provide a scientific reference for controlling LDR in mountainous area in Southwest China.

Grey linear programming: a survey on solving approaches and applications

PurposeThe 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/approachAfter 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.FindingsThe 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/valueDifferent 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.

A dual simplex method for grey linear programming problems based on duality results

PurposeFor 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/approachThe 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.FindingsBy 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/valueComplementary 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.

Optimization of Land Reuse Structure in Coal Mining Subsided Areas Considering Regional Economic Development: A Case Study in Pei County, China

Land subsidence, which has caused large-scale settlement loss and farmland degradation, was regarded as the main constraint for land reclamation in the High Groundwater Coal Basins (HGCBs) in the eastern China plain. Both coal mining and agricultural production are important for regional development in this area. In general, the land reclamation direction in this area is greatly affected by the adequacy of filling materials and the land use demand of regional economic development. Taking seven coal mining subsided areas in Pei county, located in the eastern China plain, for example, this study proposed an integrated model (including the Limit Condition model, Logistic Regression model, Grey Linear Programming model and the conversion of land use and its effects at small regional extent (CLUE-S) model) to simulate and optimize the post-mining land use structure to meet the economic development needs of Pei county. Then, the post-mining land use structure under different scenarios, which were set based on the subsidence depth, were compared to explore the optimal collapse depth for separating the damaged land into the filling area and non-filling area. The landscape structure, ecological benefits, engineering quantity and reclaimed farmland area were used to compare the reclaimed land use structure of different scenarios. The results showed that the integrated model could efficiently simulate the reclaimed land use structure to meet the land demand for regional development. The optimal collapse depth for separating the damaged area into the filling area and non-filling area was 2.6 m. Currently, the reclaimed land use structure not only needs a low quantity of filling material, but also obtains a good landscape structure and elevated ecosystem services value. Furthermore, the reclaimed urban land was mainly distributed around Pei town, and the reclaimed farmland was mostly distributed in the area between Pei town and Weishan lake, which were consistent with the pattern of urbanization. The study provides valuable information for future land use plans for Pei county and will contribute to the methods of post-mining land reclamation in other HGCBs.

Application of Center and Width Concepts to Solving Grey Linear Programming

Linear programming is the optimization of a linear function under constraints of linear equations or inequalities. Since much of the information in the real world is unknown, linear programming does not produce acceptable results due to the need for accurate information and data in many real-world decisions. The grey system is one of the most efficient approaches to dealing with uncertainty and incomplete data. Accordingly, the present study is an attempt to propose a novel method for solving grey linear programming problems. The linear programming with grey information methods presented to date have drawbacks, such as the weakness of solving linear programming with grey information in constraints, inappropriate lower bound results than high bound, a high volume of operations, and high complexity. To address these drawbacks, the present study proposes a novel method of linear programming with grey parameters. As opposed to the other existing methods, which include several steps, the proposed method in this study is much smaller and has only five simple steps. It is easy to work and applicable to any grey linear programming problem.

A grey production planning model on a ready-mixed concrete plant

ABSTRACTThe parameters of decision models are not clearly known and they are not exact for most real-life problems. Therefore, this article presents a model of linear programming with grey interval numbers for right-hand-side parameters under an uncertainty case. The purpose of this article is to compare the results of a fuzzy linear programming model and those of four proposed grey linear programming models and to find effective solutions for alternative production planning. The results produced a more effective plan to use as an industrial solution according to a fuzzy linear programming model. The objective function value is about 3% more effective than with the fuzzy linear programming method, and the customer satisfaction of total demand is about 25% more effective for the number of served customers.

Solving Linear Programming Problems with Grey Data

A Grey Linear Programming problem differs from an ordinary one to the fact that the coefficients of its objective function and / or the technological coefficients and constants of its constraints are grey instead of real numbers. In this work a new method is developed for solving such kind of problems by the whitenization of the grey numbers involved and the solution of the obtained in this way ordinary Linear Programming problem with a standard method. The values of the decision variables in the optimal solution may then be converted to grey numbers to facilitate a vague expression of it, but this must be strictly checked to avoid non creditable such expressions. Examples are also presented to illustrate the applicability of our method in real life applications.

Grey-fuzzy solution for multi-objective linear programming with interval coefficients

PurposeThe 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/approachIn 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.FindingsAccording 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 implicationsAs 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/valueBased 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.

A grey mathematical model for crashing of projects by considering time, cost, quality, risk and law of diminishing returns

PurposeThe 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/approachThe 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.FindingsThe 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 implicationsThe 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/valueIn 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.

Sustainable resource allocation for power generation: The role of big data in enabling interindustry architectural innovation

Economic, social and environmental requirements make planning for a sustainable electricity generation mix a demanding endeavour. Technological innovation offers a range of renewable generation and energy management options which require fine tuning and accurate control to be successful, which calls for the use of large-scale, detailed datasets. In this paper, we focus on the UK and use Multi-Criteria Decision Making (MCDM) to evaluate electricity generation options against technical, environmental and social criteria. Data incompleteness and redundancy, usual in large-scale datasets, as well as expert opinion ambiguity are dealt with using a comprehensive grey TOPSIS model. We used evaluation scores to develop a multi-objective optimization model to maximize the technical, environmental and social utility of the electricity generation mix and to enable a larger role for innovative technologies. Demand uncertainty was handled with an interval range and we developed our problem with multi-objective grey linear programming (MOGLP). Solving the mathematical model provided us with the electricity generation mix for every 5 min of the period under study. Our results indicate that nuclear and renewable energy options, specifically wind, solar, and hydro, but not biomass energy, perform better against all criteria indicating that interindustry architectural innovation in the power generation mix is key to sustainable UK electricity production and supply.

A new approach in animal diet using grey system theory

PurposeThe 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/approachWith 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.FindingsThe 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/valueThis 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.

A note on “a multi-objective programming approach to solve grey linear programming”

PurposeThe purpose of this paper is to examine the shortcomings and problems associated with the method proposed by Razavi Hajiagha et al. (2012).Design/methodology/approachA 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.FindingsAccording 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/valueIn 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.

Integrated grey relational analysis and multi objective grey linear programming for sustainable electricity generation planning

Sustainable energy generation is a key feature in sustainable development and among various sources of energy electricity due to some unique characteristics seems particularly important. Optimising electricity generation mix is a highly complex task and requires consideration of numerous conflicting criteria. To deal with uncertainty of experts’ opinions, inaccuracy of the available data and including more factors, some of which are difficult to quantify, in particular for environmental and social criteria, we applied grey relational analysis (GRA) with grey linguistic, and grey interval values to obtain the rank of each system. Then the obtained ranking were used as coefficients for a multi objective decision making problem, aimed to minimize the cost, import dependencies and emissions as well as maximizing the share of generation sources with better ranking. Due to existence of interval variables multi objective grey linear programming (MOGLP) method was used to solve the problem. Our results for the UK as a case study suggest increased role for all low carbon energy technologies and sharp reduction in the use of coal and oil. We argue that the integrated GRA–MOGLP approach provides an effective tool for the evaluation and optimisation of complex sustainable electricity generation planning. It is particularly promising in dealing with uncertainty and imprecisions, which reflect real-life scenarios in planning processes.

A Single Machine Capacitated Production Planning Problem Under Uncertainty: A Grey Linear Programming Approach

The production planning is an important problem in most of manufacturing systems in practice. Unlike many researches existing in literature, this problem encounters with great uncertainties in parameters and input data. In this paper, a single machine capacitated production planning problem is considered and a linear programming formulation is presented. The production costs are assumed to be uncertain parameters. To handle the uncertainties in the model, the grey systems theory is employed and the concept of grey numbers is incorporated into an optimization framework. In such systems, the uncertain parameters with unknown distributions can be handled by grey numbers. The grey linear programming (GLP) is a development of the classical linear programming which allows uncertainty to be effectively communicated into the optimization process. Finally, the uncertain problem is transformed into a GLP, and is solved by two linear deterministic sub-models.