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This chapter presents the survey of selected linear and mixed integer programming multi-objective portfolio optimization. The definitions of selected percentile risk measures are presented. Some contrasts and similarities of the different types of portfolio formulations are drawn out. The survey of multi-criteria methods devoted to portfolio optimization such as weighting approach, lexicographic approach, and reference point method is also presented. This survey presents the nature of the multi-objective portfolio problems focuses on a compromise between the construction of objectives, constraints, and decision variables in a portfolio and the problem complexity of the implemented mathematical models. There is always a trade-off between computational time and the size of an input data, as well as the type of mathematical programming formulation with linear and/or mixed integer variables.

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.

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.

Explores the use of a multi‐objective programming approach as a method for supplier selection in a just‐in‐time (JIT) setting. Based on a case study, develops a model of JIT supplier selection which allows for simultaneous trade‐offs of price, delivery and quality criteria. The analysis occurs in a decision support system environment. A multi‐objective programming decision support system is seen as advantageous because such an environment allows for judgement in decision making while simultaneously trading off key supplier selection criteria. An additional flexibility of this model is that it allows a varying number of suppliers into the solution, and provides suggested volume allocation by supplier.

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 study aims to evaluate the performance of the most popular multi-objective programming scalarization methods in the literature for the aircraft sequencing and scheduling problem (ASSP). These methods are the weighted sum method, weighted goal programming, the ε-constraint method, the elastic constraint method, weighted Tchebycheff and augmented weighted Tchebycheff.

First, the ASSP for a single runway case was modeled using mixed-integer programming considering the safety and operational constraints and the objectives of the minimization of total delay and total flight time for a sample airport. The objectives were then combined by using the multi-objective programming scalarization methods and various expected times of arrival–departure samples were run for the mathematical models. Finally, the methods were evaluated in terms of the number of nondominated solutions, superior nondominated solution and the average solution time using the Measurement of Alternatives and Ranking according to Compromise Solution method, which is a popular multi-criteria decision-making method.

Augmented Weighted Tchebycheff was found to be the most effective approach to ASSP in terms of the evaluation criteria followed by Weighted Tchebycheff and then weighted sum method.

The methodology presented in this study could provide more efficient air traffic management in terminal maneuvering areas when multiple objectives need to be optimized.

Although there are studies including the comparison of several scalarization methods for other problems, the comparison of the methods for ASSP has not yet been handled in the literature. As there are several stakeholders in the air traffic system, ASSP includes several objectives, and as a result, this problem can benefit from analyses using this comparison.

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 examine the shortcomings and problems associated with the method proposed by Razavi Hajiagha et al. (2012).

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.

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.

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.

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.

The purpose of this paper is to study a transportation service procurement bid construction problem from a less than a full truckload perspective. It seeks to establish stochastic mixed integer programming to allow for the proper bundle of loads to be chosen based on price, which could improve the likelihood that carrier can earn its maximum utility.

The authors proposes a bi-level programming that integrates the bid selection and winner determination and a discrete particle swarm optimization (PSO) solution algorithm is then developed, and a numerical simulation is used to make model and algorithm analysis.

The algorithm comparison shows that although GA could find a little more Pareto solutions than PSO, it takes a longer time and the quality of these solutions is not dominant. The model analysis shows that compared with traditional approach, our model could promote the likelihood of winning bids and the decision effectiveness of the whole system because it considers the reaction of the shipper.

The highlights of this paper are considering the likelihood of winning the business and describing the conflicting and cooperative relationship between the carrier and the shipper by using a stochastic mixed integer programming, which has been rarely examined in previous research.