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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 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 propose a scenario-based grey methodology using clustering and optimizing with imprecise and uncertain body size data in an emergency assembly point area to assign the people on a campus to reach the emergency assembly points under uncertain disaster times.

Grey clustering and a new grey p-median linear programming model are developed to determine which units to assign to the pre-determined assembly points for a main campus in case of a disaster. The models have two scenarios: 70 and 100% occurrence capacities of administrative and academic personnel and students.

In this study, the academic and administrative units have been assigned to determine five different emergency assembly points on the main campus by using the numbers of the academic and administrative personnel and student and distances of the units to the assembly point areas of each other. The alternative solutions are obtained effectively by evaluating capacity utilization rates in the scenarios.

It is often unclear when disasters can occur and therefore, a preliminary preparation time must be required to minimize the risk. In the case of natural, man-made (unnatural) or technological disasters, the people are required to defend themselves and move away from the disaster area as soon as possible in a proper direction. The proposed assignment model yields a final solution that effectively eliminates uncertainty regarding the selection of emergency assembly points for administrative and academic staff as well as students, in the event of disasters.

Grey clustering suggests an assignment plan and concurrently, an investigation is underway utilizing the grey p-median linear programming model. This investigation aims to optimize various scenarios and body sizes concerning emergency assembly areas. All campus users who are present at the disaster in units of the campus are getting uncertainty about which emergency assembly point to use, and with this study, the vital risks aim to be ultimately reduced with reasonable plans.

The purpose of this paper is to manage the demand uncertainty considered as lower and upper levels for a medium-scale industrial distribution planning problem in a biobjective inventory routing problem (IRP). In order to achieve this, the grey system theory is applied since no statistical distribution from the past data and incomplete information.

This study is investigated with optimizing the distribution plan, which involves 30 customers of 12 periods in a manufacturing company under demand uncertainty that is considered as lower and upper levels for a biobjective IRP with using grey demand parameters as a grey integer programming model. In the data set, there are also some missing demand values for the customers. So, the seven different grey models are developed to eliminat the effects on demand uncertainties in computational analysis using a piece of developed software considering the logistical performance indicators such as total deliveries, total cost, the total number of tours, distribution capacity, average remaining capacity and solution time.

Results show that comparing the grey models, the cost per unit and the maximum number of vehicle parameters are also calculated as the new key performance indicator, and then results were ranked and evaluated in detail. Another important finding is the demand uncertainties can be managed with a new approach via logistics performance indicators using alternative solutions.

The results enable logistics managers to understand the importance of demand uncertainties if more reliable decisions are wanted to make with obtaining a proper distribution plan for effective use of their expectations about the success factors in logistics management.

The study is the first in terms of the application of grey models in a biobjective IRP with using interval grey demand data. Successful implementation of the grey approaches allows obtaining a more reliable distribution plan. In addition, this paper also offers a new key performance indicator for the final decision.

This paper investigates a problem in a reverse logistics (RLs) network to decide whether to dispose of unsold goods in primary stores or re-commercialize them in outlet centers. By deducting the costs associated with each policy from its revenue, this study aims to maximize the profit from managing unsold goods.

A new mixed-integer linear programming model has been developed to address the problem, which considers the selling prices of products in primary and secondary stores and the costs of transportation, cross-docking and returning unwanted items. As a result of uncertain nature of the cost and time parameters, gray numbers are used to deal with it. In addition, an innovative uncertain solution approach for gray programming problems is presented that considers objective function satisfaction level as an indicator of optimism.

According to the results, higher costs, including transportation, cross-docking and return costs, make sending goods to outlet centers unprofitable and more goods are disposed of in primary stores. Prices in primary and secondary stores heavily influence the number of discarded goods. Higher prices in primary stores result in more disposed of goods, while higher prices in secondary stores result in fewer. As a result of the proposed method, the objective function satisfaction level can be viewed as a measure of optimism.

An integral contribution of this study is developing a new mixed-integer linear programming model for selecting the appropriate goods for re-commercialization and choosing the best outlet center based on the products' price and total profit. Another novelty of the proposed model is considering the matching percentage of boxes with secondary stores’ desired product lists and the probability of returning goods due to non-compliance with delivery dates. Moreover, a new uncertain solution approach is developed to solve mathematical programming problems with gray parameters.

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.

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.

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.

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.

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.

One important assumption in the conventional cooperative game theory is that payoffs are assumed to be deterministic. In terms of the players’ cognitive ability of the realistic world, this is a very strict assumption. The classical game theory can find no way out when a particular game circumstance involves uncertainty, such as limited knowledge, small sample, and inadequate information, the payoff values of the game can only be described with interval grey numbers. The paper aims to discuss these issues.

In this study the concept of N-person grey games is proposed in which payoffs are represented with interval grey numbers opposed to the classical game theory. A straight forward solution methodology is submitted compatible to grey numbers. Then, a currency war between anonymous countries is handled and modeled as an N-Person grey game. A generic currency war scenario is developed to follow the proposed solution procedure thoroughly.

Based on the outcomes of this work, the authors can say that N-person grey game is an expansion of the classical N-person game under uncertain grey information and can be applied in more complex and uncertain environments, such as those seen in complicated currency warfare.

This study combines the grey system theory with the classic N-person game theory and sets up the N-person grey game with grey payoff functions. Based on the grey number operating methods, the grey linear programming algorithm is established to calculate and distribute benefits to the players. In this respect this study has the feature of being the pioneer in the N-person grey game area.

The purpose of this paper is to present a mixed integer programming model for simple assembly line balancing problems (SALBP) with Type 1 when the annual demand and task durations are uncertain and encoded with grey numbers.

Grey theory and grey numbers are used for illustrating the uncertainty of parameters in an SALBP, where the objective is to minimize the total number of workstations. The paper proposes a 0-1 mathematical model for SALBP of Type 1 with grey demand and grey task durations.

The uncertainty of the demand and task durations are encoded with grey numbers and a well-known 0-1 mathematical model for SALBP of Type 1 is modified to find the minimum number of workstations in order to meet both the lower and upper bounds of the uncertain demand. The results obtained from the proposed mathematical model show a task-workstation assignment that does not distribute precedence relations among tasks and workstations and the sum of task durations in each single workstation is less than or equal to the grey cycle time.

The grey theory and grey numbers have not been previously used to identify uncertainties in assembly line balancing problems. Therefore, this study provides an important contribution to the literature.

The purpose of this paper is to attempt to realize a distribution network optimization in supply chain using grey systems theory for uncertain information.

There is much uncertain information in the distribution network optimization of supply chain, including fuzzy information, stochastic information and grey information, etc. Fuzzy information and stochastic information have been studied in supply chain, however grey information of the supply chain has not been covered. In the distribution problem of supply chain, grey demands are taken into account. Then, a mathematics model with grey demands has been constructed, and it can be transformed into a grey chance‐constrained programming model, grey simulation and a proposed hybrid particle swarm optimization are combined to resolve it. An example is also computed in the last part of the paper.

The results are convincing: not only that grey system theory can be used to deal with grey uncertain information about distribution of supply chain, but grey chance‐constrained programming, grey simulation and particle swarm optimization can be combined to resolve the grey model.

The method exposed in the paper can be used to deal with distribution problems with grey information in the supply chain, and network optimization results with a grey uncertain factor could be helpful for supply chain efficiency and practicability.

The paper succeeds in realising both a constructed model of the distribution of supply chain with grey demands and a solution algorithm of the grey mathematics model by using one of the newest developed theories: grey systems theory.

This paper presents a mixed-integer programming model for a single machine earliness/tardiness scheduling problem where the objective is to minimize total earliness/tardiness duration when the uncertainty of parameters such as processing times and due date is coded with grey numbers.

Grey theory and grey numbers are used for illustrating the uncertainty of parameters in processing times and common due date, where the objective is to minimize the total earliness/tardiness duration. The paper proposes a 0–1 mathematical model for the problem and an effective heuristic method for the problem by using expected processing times for ordering jobs.

The uncertainty of the processing times and common due date are encoded with grey numbers and a position-dependent mixed-integer mathematical programming model is proposed for the problem in order to minimize total grey earliness/tardiness duration of jobs having grey processing times and a common due date. By using expected processing times for ranking grey processing times, V-shaped property of the problem and an efficient heuristic method for the problem are proposed. Solutions obtained from the heuristic method show that the heuristic is effective. The experimental study also reveals that while differences between upper and lower bounds of grey processing times decrease, the proposed heuristic's performance decreases.

The grey theory and grey numbers have been rarely used as machine scheduling problems. Therefore, this study provides an important contribution to the literature.