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Logistics is the part of the supply chain (SC) that plans, executes and handles forward and reverse movement and storage of products, services and related information, in order to respond to customers' needs effectively and efficiently. The main concern for logistics is to ensure that the correct product is placed at the right time. This paper introduces a linear model of shipping focused on decision-making, which includes configuration of shipping network, choosing of transport means and transfer of individual customer shipments through a particular transport system.

In this study, authors try to address the problem of supply chain network (SCN) where the primary goal is to determine the appropriate order allocation of products from different sources to different destinations. They also seek to minimize total transportation cost and inventory cost by simultaneously determining optimal locations, flows and shipment composition. The formulated problem of getting optimal allocation turns out to be a problem of multi-objective programming, and it is solved by using the max-addition fuzzy goal programming approach, for obtaining optimal order allocation of products. Furthermore, the problem demand and supply parameters have been considered random in nature, and the maximum likelihood estimation approach has been used to assess the unknown probabilistic distribution parameters with a specified probability level (SPL).

A case study has also been applied for examining the effectiveness and applicability of the developed multi-objective model and the proposed solution methods. Results of this study are very relevant for the manufacturing sector in particular, for those facing logistics issues in SCN. It enables researchers and managers to cope with various types of uncertainty and logistics risks associated with SCN.

The principal contribution of the proposed model is the improved modelling of transportation and inventory, which are affected by different characteristics of SCN. To demonstrate computational information of the suggested methods and proposed model, a case illustration of SCN is provided. Also, environmentalism is increasingly becoming a significant global concern. Hence, the concept proposed could be extended to include environmental aspects as an objective function or constraint.

Efficient integration of logistical cost components, such as transportation costs, inventory costs, with mathematical programming models is an important open issue in logistics optimization. This study expands conventional facility location models to incorporate a range of logistic system elements such as transportation cost and different types of inventory cost, in a multi-product, multi-site network. The research is original and is focused on case studies of real life.

This paper proposes a fuzzy random multi-objective portfolio model with different entropy measures and designs a hybrid algorithm to solve the proposed model.

Because random uncertainty and fuzzy uncertainty are often combined in a real-world setting, the security returns are considered as fuzzy random numbers. In the model, the authors also consider the effects of different entropy measures, including Yager's entropy, Shannon's entropy and min-max entropy. During the process of solving the model, the authors use a ranking method to convert the expected return into a crisp number. To find the optimal solution efficiently, a fuzzy programming technique based on artificial bee colony (ABC) algorithm is also proposed.

(1) The return of optimal portfolio increases while the level of investor risk aversion increases. (2) The difference of the investment weights of the optimal portfolio obtained with Yager's entropy are much smaller than that of the min–max entropy. (3) The performance of the ABC algorithm on solving the proposed model is superior than other intelligent algorithms such as the genetic algorithm, differential evolution and particle swarm optimization.

To the best of the authors' knowledge, no effect has been made to consider a fuzzy random portfolio model with different entropy measures. Thus, the novelty of the research is constructing a fuzzy random multi-objective portfolio model with different entropy measures and designing a hybrid fuzzy programming-ABC algorithm to solve the proposed model.

The purpose of this paper is to develop a combined intuitionistic fuzzy analytic hierarchy process (IFAHP) and fuzzy multi-objective optimization approach to select suppliers and allocate the orders to them in the bottled water production context.

First, the primary weights of criteria associated with the supplier selection problem are calculated using the IFAHP technique. Then a fuzzy multi-objective optimization model is developed to allocate the appropriate amount of orders to each supplier.

The proposed methodology has been successfully implemented in the case of an Iranian food company in its bottled water factory. Results demonstrate our model is capable of practically handling the uncertainty in DMs’ preference that leads to effective and efficient supplier selection and order allocation decisions.

The authors develop a novel hybrid decision-making tool to tackle the uncertainty in decision-makers’ opinions with a demonstrated applicability and some promising outcomes in efficiently allocating the order quantity to suppliers in the area of bottled water production.

This paper aims to introduce a novel structure for the physical internet (PI)–enabled sustainable supplier selection and inventory management problem under uncertain environments.

To address hybrid uncertainty both in the objective function and constraints, a novel interactive hybrid multi-objective optimization solution approach combining Me-based fuzzy possibilistic programming and interval programming approaches is tailored.

Various numerical experiments are introduced to validate the feasibility of the established model and the proposed solution method.

Due to its interconnectedness, the PI has the opportunity to support firms in addressing sustainability challenges and reducing initial impact. The sustainable supplier selection and inventory management have become critical operational challenges in PI-enabled supply chain problems. This is the first attempt on this issue, which uses the presented novel interactive possibilistic programming method.

Globalization of markets and pace of technological change have caused the growing importance of paying attention to supplier selection problem. Therefore, this study aims to choose the best suppliers by providing a mathematical model for the supplier selection problem considering the green factors and stochastic parameters. This paper aims to propose a multi-objective model to identify optimal suppliers for a green supply chain network under uncertainty.

The objective of this model is to select suppliers considering total cost, total quality parts and total greenhouse gas emissions. Also, uncertainty is tackled by stochastic programming, and the multi-objective model is solved as a single-objective model by the LP-metric method.

Twelve numerical examples are provided, and a sensitivity analysis is conducted to demonstrate the effectiveness of the developed mathematical model. Results indicate that with increasing market numbers and final product numbers, the total objective function value and run time increase. In case that decision-makers are willing to deal with uncertainty with higher reliability, they should consider whole environmental conditions as input parameters. Therefore, when the number of scenarios increases, the total objective function value increases. Besides, the trade-off between cost function and other objective functions is studied. Also, the benefit of the stochastic programming approach is proved. To show the applicability of the proposed model, different modes are defined and compared with the proposed model, and the results demonstrate that the increasing use of recyclable parts and application of the recycling strategy yield more economic savings and less costs.

This paper aims to present a more comprehensive model based on real-world conditions for the supplier selection problem in green supply chain under uncertainty. In addition to economic issue, environmental issue is considered from different aspects such as selecting the environment-friendly suppliers, purchasing from them and taking the probability of defective finished products and goods from suppliers into account.

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 extend a methodology for solving multi‐objective linear programming (MOLP) problems, when the objective functions and constraints coefficients are stated as interval numbers.

The approach proposed in this paper for the considered problem is based on the maximization of the sum of membership degrees which are defined for each objective of multi objective problem. These membership degrees are constructed based on the deviation from optimal solutions of individual objectives. Then, the final model based on membership degrees is itself an interval linear programming which can be solved by current methods.

The efficiency of the solutions obtained by the proposed method is proved. It is shown that the obtained solution by the proposed method for an interval multi objective problem is Pareto optimal.

The proposed method can be used in modeling and analyzing of uncertain systems which are modeled in the context of multi objective problems and in which required information is ill defined.

The paper proposed a novel and well‐defined algorithm to solve the considered problem.

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.

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.

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.

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.

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.

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.

This study aims to deal with supplier selection problem. The supplier selection problem has significantly become attractive to researchers and practitioners in recent years. Many real-world supply chain problems are assumed as multiple objectives combinatorial optimization problems.

In this paper, the authors propose a multi-objective model with fuzzy parameters to select suppliers and allocate orders considering multiple periods, multiple resources, multiple products and two-echelon supply chain. The objective functions consist of total purchase costs, transportation, order and on-time delivery, coverage and the weights of suppliers. Distance-based partial and general coverage of suppliers makes the number of orders of products more realistic. In this model, the weights of suppliers are determined by fuzzy Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method, as a multi-criteria decision analysis method, in the objective function. Also, the authors consider the parameters related to delays as triangular fuzzy numbers.

A small-sized numerical example is provided to clearly show the proposed model. The exact epsilon constraint method is used to solve this given multi-objective combinatorial optimization problem. Subsequently, the sensitivity analysis is conducted to testify the proposed model. The obtained results demonstrate the validity of the proposed multiple objectives mixed integer mathematical programming model and the efficiency of the solution approach.

In real-life situations, supplier selection parameters are uncertain and incomplete. Hence, the fuzzy set theory is used to tackle uncertainty. In this paper, a multi-objective supplier selection problem is formulated taking into consideration the coverage of suppliers and suppliers’ weights. Integrating coverage of suppliers to select and allocate the order to them can be mentioned as the main contribution of this study. The proposed model considers the delay from suppliers as fuzzy parameters.