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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.

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.

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.

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.

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.

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.

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.

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.

The paper introduces a resource-based linear programming model for resource optimization in small innovative enterprises (SIE).

The model is grounded on resource-based view on the firm and dynamic capabilities approach. Linear programming technique is used to provide the actual framework to the resource-based model.

The paper introduces a new resource-based linear programming model for resource optimization in small innovative enterprises. The conceptual model is grounded on resource-based view (RBV) and dynamic capabilities strategy. The RVB of firm and firm strategy is based on the concept of economic rent. Linear programming technique is used to provide the actual framework to the resource-based model. In developing the versatility concept, study suggests a distinct sight regarding resource fungibility. Study classifies resources into multipliable, rentable and expendable resources to increases adequacy of the model. The developed model includes both tangible and intangible assets such as human capital. The survival rate of SIE in the early stages of life cycle is very low due to the competition among SIEs. In this regard, the greatest advancement of the developed resource-based linear programming model is its simplicity and versatility which is much desirable for the SIE especially in their initial stages of the life cycle. Kelliher and Reinl (2009) argued that micro firms have unique advantage over bigger firms in following term: rate of learning or redeployment of strategy in micro firms is faster than the rate of change in their environment. One very significant feature of the developed resource-based linear programming model is that mathematically the proposed model could easily be transformed into mixed integer or stochastic linear programming models to meet the time variant requirement of small firms especially when it expands its operation.

The survival rate of SIE in the early stages of life cycle is very low due to the competition among SIEs. In this regard, the greatest advancement of the developed resource-based linear programming model is its simplicity and versatility which is much desirable for the SIE especially in their initial stages of the life cycle. Kelliher and Reinl (2009) argued that micro firms have unique advantage over bigger firms in following term: rate of learning or redeployment of strategy in micro firms is faster than the rate of change in their environment. One very significant feature of the developed resource-based linear programming model is that mathematically the proposed model could easily be transformed into mixed integer or stochastic linear programming models to meet the time variant requirement of small firms especially when it expands its operation.

One very significant contribution of the present study is that the study develops a new resource-based model for SIE especially for the SIE in the initial stages of the life cycle, to gain competitive advantages. Furthermore, the present study contributes to the existing literature in strategy at least in three senses as mentioned below: 1. further addition of SIE research based on the RBV and dynamic capabilities in the strategy literature 2. in developing the versatility concept, the study suggests a distinct sight regarding resource fungibility and it classifies resources into three categories as follows: multipliable, rentable and expendable resources to increases adequacy of the model. 3. Finally, the study introduces a new resource-based linear programming model for SIE resources allocation. To the best of author’s knowledge, no such similar model is introduced by any previous studies for small firm. The greatest advancement of the developed resource-based linear programming model is its simplicity and versatility.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Train re-scheduling remains a longstanding challenge in railway operation. To design high-quality timetable in fuzzy environment, the purpose of this paper is to study train re-scheduling problem under the fuzzy environment, in which the fuzzy coefficients of the constraint resources have the fuzzy boundaries.

Based on the improved fuzzy linear programming, the train re-scheduling model is constructed. Aiming at dealing with the fuzzy characteristics of the constraint coefficients value range boundaries, the description method of this kind of objective function is proposed and the solving approach is presented. The model has more adaptability to model a common train re-scheduling problem, in which some resources of the constraints are uncertain and have the characteristics of fuzziness and the boundaries of the resources are fuzzy.

Two numerical examples are carried out and it shows that the model proposed in this paper can describe the train re-scheduling problem precisely, dealing with the fuzzy boundaries of the fuzzy coefficients of the constraint resources. And the algorithm present is suitable to solve the problem. The approach proposed in this paper can be a reference for developers of railway dispatching system.

It is the first time to study train re-scheduling problem under the fuzzy environment, in which the fuzzy coefficients of the constraint resources have the fuzzy boundaries.

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.

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.

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.

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.

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.

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.

Negative effects of habitat isolation that arise from landscape fragmentation can be mitigated, by connecting natural areas through a network of habitat corridors. To increase the permeability of a given network, i.e. to decrease the resistance to animal movements through this network, often many developments can be made. The available financial resources being limited, the most effective developments must be chosen. This optimization problem, suggested in Finke and Sonnenschein, can be treated by heuristics and simulation approaches, but the method is heavy and the obtained solutions are sub‐optimal. The aim of the paper is to show that the problem can be efficiently solved to optimality by mathematical programming.

The moves of the individual in the network are modeled by an absorbing Markov chain and the development problem is formulated as a mixed‐integer quadratic program, then this program is linearized, and the best developments to make are determined by mixed‐integer linear programming.

First, the approach allows the development problem to be solved to optimality contrary to other methods. Second, the definition of the mathematical program is relatively simple, and its implementation is immediate by using standard, commercially available, software. Third, as it is well known with mixed‐integer linear programming formulation it is possible to add new constraints easily if they are linear (or can be linearized).

With a view to propose a simple and efficient tool to solve a difficult combinatorial optimization problem arising in the improvement of permeability across habitat networks, the approach has been tested on simulated habitat networks. The research does not include the study of some precise species movements in a real network.

The results provide a simple and efficient decision‐aid tool to try to improve the permeability of habitat networks.

The joint use of mathematical programming techniques and Markov chain theory is used to try to lessen the negative effects of landscape fragmentation.