Multi-objective Programming Method Research Articles

Uncertain programming model for designing multi-objective reverse logistics networks

Given the contradiction between the rapid growth of products and the modest recovery rate of end-of-life products, there is a pressing need to understand the societal significance of establishing a reverse logistics network for end-of-life products. This research constructs an open-loop five-tier reverse logistic network model encompassing customers, centres for collection, disassembly and inspection, remanufacturing, and disposal. A multi-objective mixed-integer nonlinear programming model under uncertainty has been developed. Unlike previous research, this model accounts for surrounding residents' disutility of facilities while simultaneously minimizing economic costs and environmental impact. Besides, uncertainty theory is introduced in solving the proposed model. More specifically, the formulated model converts all uncertain variables into uncertain distributions by implementing the uncertain multi-objective programming method. Furthermore, a customised non-dominated sorting genetic algorithm III (NSGA-III) is proposed and is employed for the first time to address facility selection and recycling volume distribution within the network. The model is then validated using a real-life case study focusing on end-of-life vehicles in Changchun, China. This research could assist decision-makers in both governmental and private sectors in achieving a balanced approach to the interests of the economy, environment, and local communities comprehensively when designing reverse supply chains.

A new cutting plane method for lexicographic multi-objective integer linear programming

This work presents a new cutting plane method for lexicographic multi-objective integer linear programming (LMOILP). The method uses Grossone Methodology to reformulate a LMOILP problem into one having a single scalar objective function involving Grossone, as done in Cococcioni et al. (2018) (but in that case in the absence of the integer constraints). The problem, without the integer constraints, is solved using the GrossSimplex algorithm presented in Cococcioni et al. (2018) to find a candidate optimal solution. Here a novel cutting plane is introduced, named Gross-based Objective Function Cutting Plane whenever the optimal value of the Gross-scalar objective function is Gross-fractional. When it happens to be not Gross-fractional, cutting planes are generated using the Fractional Cutting Plane, derived from fractional components of the optimal solution. Moreover, by combining them, we proposed an algorithm that we called Gross-based Cutting Plane (GCP) method. It has been proved that it finds the optimal solution of a LMOILP problem and terminates after a finite number of iterations. To speed-up the GCP, at each iteration subsequent the first one, we re-use the optimal basis of the last continuous relaxation, by running the GrossDualSimplex algorithm. This is the well-known warm-start technique, which, however, needs specific attention due to the need to solve a linear problem having a right-hand side involving Grossone-based numbers. In the experimental part, we show the efficacy of the proposed approach.

Multi-Objective Mixed-Integer Convex Programming Method for the Siting and Sizing of UPFCs in a Large-Scale AC/DC Transmission System

Multi-Objective Mixed-Integer Convex Programming Method for the Siting and Sizing of UPFCs in a Large-Scale AC/DC Transmission System

Spatiotemporal pattern and multi-scenario simulation of ecological risk in mountainous cities: a case study in Chongqing, China.

Scientific descriptions and simulations of the ecological risks in mountainous areas can promote the sustainable use of land resources in these areas and improve the reliability of decision-making for ecological risk management. Taking Chongqing, China, as an example, we constructed a landscape ecological risk (LER) evaluation model based on land use data from 1995 to 2020 and analysed the spatiotemporal evolution characteristics of the LER pattern. Moreover, we coupled the patch-generating land use simulation (PLUS) model and multi-objective programming (MOP) method and input multiple scenarios (inertial development, ID; economic priority development, ED; ecological priority development, PD; and sustainable development, SD) to simulate the ecological risk pattern in 2030. The model coupling the "top-down" and "bottom-up" processes obtained optimal land use patterns in different contexts, and it was used to perform a spatially explicit examination of LER evolutionary trends in different contexts. The results showed that LER evolution in Chongqing has had obvious stage characteristics. The high-risk area decreased significantly under various constraints, including topographic, economic, and other constraints, and the distribution showed a trend of high in the west and low in the east. The LER spatial clustering characteristics were highly coupled with the risk level pattern. The ED scenario presented the most severe risk, the PD scenario presented a moderate risk, and the SD scenario balanced the land demand for economic and ecological development and had a better land use structure and LER compared with the other scenarios. The coupled model proposed in this study helps to obtain the optimal land use structure and mitigate ecological risks, thus providing a scientific basis for future urban development.

An ε-Constraint Method for Multiobjective Linear Programming in Intuitionistic Fuzzy Environment

Effective decision-making requires well-founded optimization models and algorithms tolerant of real-world uncertainties. In the mid-1980s, intuitionistic fuzzy set theory emerged as another mathematical framework to deal with the uncertainty of subjective judgments and made it possible to represent hesitancy in a decision-making problem. Nowadays, intuitionistic fuzzy multiobjective linear programming (IFMOLP) problems are a topic of extensive research, for which a considerable number of solution approaches are being developed. Among the available solution approaches, ranking function-based approaches stand out for their simplicity to transform these problems into conventional ones. However, these approaches do not always guarantee Pareto optimal solutions. In this study, the concepts of dominance and Pareto optimality are extended to the intuitionistic fuzzy case by using lexicographic criteria for ranking triangular intuitionistic fuzzy numbers (TIFNs). Furthermore, an intuitionistic fuzzy ε-constraint method is proposed to solve IFMOLP problems with TIFNs. The proposed method is illustrated by solving two intuitionistic fuzzy transportation problems addressed in two studies (S. Mahajan and S. K. Gupta’s, “On fully intuitionistic fuzzy multiobjective transportation problems using different membership functions,” Ann Oper Res, vol. 296, no. 1, pp. 211–241, 2021, and Ghosh et al.’s, “Multi-objective fully intuitionistic fuzzy fixed-charge solid transportation problem,” Complex Intell Syst, vol. 7, no. 2, pp. 1009–1023, 2021). Results show that, in contrast with Mahajan and Gupta’s and Ghosh et al.’s methods, the proposed method guarantees Pareto optimality and also makes it possible to obtain multiple solutions to the problems.

A Regional Water Resource Allocation Model Based on the Human–Water Harmony Theory in the Yellow River Basin

Considering the issues of water scarcity, water environment deterioration, and unreasonable allocation of water resources in the urban area of the Yellow River Basin, this paper introduces the human–water harmony theory to the allocation of regional water resources. Based on an analysis of the structural characteristics of the regional water resource system, the harmonious water resource allocation (HWRA) model—which includes three sub-systems (i.e., the water service system, ecological environmental system, and economic and social system)—is established. In addition, considering the uncertain factors in the HWRA model, the inexact fuzzy multi-objective programming (IFMOP) method is used to solve the model, aiming at achieving the minimum amount of regional water scarcity, the minimum amount of sewage discharge, and the maximum total economic benefit. A case study of water resource allocation of Binzhou, a city located in the Yellow River Basin, is conducted to validate the model. The model solution results show that the water resource system in Binzhou in 2025 and 2035 could be optimized after harmonious allocation, especially in terms of the water service and ecological environmental systems. Compared with the optimal water resource allocation (OWRA) model, the HWRA model has a more scientific water supply structure, and a smaller amount of sewage discharge. The HWRA model solves the variables using an interval number, so it can flexibly and scientifically reflect the decision-making process.

Evaluation of future wetland changes under optimal scenarios and land degradation neutrality analysis in the Guangdong-Hong Kong-Macao Greater Bay Area

Wetlands are one of the most productive ecosystems on Earth and are also focused on by the Sustainable Development Goals (SDGs). However, global wetlands have suffered from considerable degradation due to rapid urbanization and climate change. To support wetland protection and SDG reporting, we predicted future wetland changes and assessed land degradation neutrality (LDN) from 2020 to 2035 under four scenarios in the Guangdong-Hong Kong-Macao Greater Bay Area (GBA). A simulation model combining random forest (RF), CLUE-S and multi-objective programming (MOP) methods was developed to predict wetland patterns under the natural increase scenario (NIS), economic development scenario (EDS), ecological protection and restoration scenario (ERPS) and harmonious development scenario (HDS). The simulation results indicated that the integration of RF and CLUE-S achieved good simulation accuracy, with OA over 0.86 and kappa indices over 0.79. From 2020 to 2035, the mangrove, tidal flat and agricultural pond increased while the coastal shallow water decreased under all scenarios. The river decreased under NIS and EDS, while increased under ERPS and HDS. The Reservoir decreased under NIS, while increased under the remaining scenarios. Among scenarios, the EDS had the largest built-up land and agricultural pond, and the ERPS had the largest forest and grassland. The HDS was a coordinated scenario that balanced economic development and ecological protection. Its natural wetlands were almost equal to these of ERPS, and its built-up land and cropland were almost equal to these of EDS. Then, the land degradation and SDG 15.3.1 indicators were calculated to support the LDN target. From 2020 to 2035, the ERPS had a smallest gap of 705.51 km2 from the LDN target, following the HDS, EDS and NIS. The SDG 15.3.1 indicator was lowest under the ERPS, with a value of 0.85 %. Our study could offer strong support for urban sustainable development and SDGs reporting.

Application of Fuzzy Multi-Objective Programming to Regional Sewer System Planning

Planning of sewer systems typically involves limitations and problems, regardless of whether traditional planning methods or optimization models are used. Such problems include non-quantifiability, fuzzy objectives, and uncertainties in decision-making variables which are commonly applied in the planning of any process. Particularly, uncertainties have prevented the inclusion of these variables in models. Consequently, the theoretical optional solution of the mathematical models is not the true optimum solution to practical problems. In this study, to solve the above problems for regional sewer system planning, multi-objective programming (MOP), nonlinear programming, mixed-integer programming, and compromise fuzzy programming were used. The objectives of this study were two-fold: (1) determination of the necessary decision-making variables or parameters, such as the optimum number of plants, piping layout, size of the plant, and extent of treatment; (2) establishment of a framework and methodology for optimal planning for designing a regional sewer system, matching demanded targets with the lowest cost, which would achieve the aim of lower space and energy requirements as well as consumption and high treatment efficiency for the purpose of meeting effluent standards. The findings of this study revealed that individual regional sewage treatment plants could be merged to form a centralized system. Land acquisition was difficult; thus, reducing the number of plants was required. Therefore, the compromise-fuzzy-based MOP method could effectively be used to build a regional sewer system plan, and the amount of in-plant establishment reached its maximized value with a minimized cost.

Solving multi level multi-objective linear programming (ML MOLP) problems with fuzzy parameters (FPs)

Making numerous business decisions in complex decentralized organizations can be presented as a ML MOLP problem with FPs expressed as triangular fuzzy numbers. This paper presents a methodology that uses several multi-objective programming methods to solve this problem. The Iskander’s method was used for defuzzification of the objective functions and constraints of the problem and a multi-objective programming method based on the cooperation among decision makers was used to determine the aspired values of the variables controlled by the decision-makers and to obtain the preferred non-dominated solution of the entire problem. The efficiency of the proposed methodology was tested on an example of production planning in a complex decentralized company.

Optimal Design of Heat Dissipation Scheme of Submarine Data Center Based on Multi-objective Programming

Aiming at the heat dissipation scheme of submarine data center, firstly, the main constituent materials of submarine container are screened by using deviation matrix; Secondly, considering the budget conditions, the water temperature equation is established, and the placement water depth and container wall thickness are determined by multi-objective programming method; Finally, considering the influence of seasonal factors and tidal factors, the influence of temperature on container heat dissipation is estimated. The research shows that: it is more suitable to choose AISI 1080 Steel as the material for the submarine data center under the conditions of comprehensive consideration of budget, water pressure, water temperature and other factors.

A Project Scheduling Game Equilibrium Problem Based on Dynamic Resource Supply

In a resource-constrained project scheduling problem, most studies ignore that resource supply is a separate optimization problem, which is not in line with the actual situation. In this study, the project scheduling problem and the resource supply problem are regarded as a dynamic game system, with interactive influences and constraints. This study proposes a Stackelberg dynamic game model based on the engineering supply chain perspective. In this model, the inherent conflicts and complex interactions between the Multi-mode Resource-Constrained Project Scheduling Problem (MRCPSP) and the Multi-Period Supply Chain Problem (MPSCP) are studied to determine the optimal equilibrium strategy. A two-level multi-objective programming method is used to solve the problem. The MRCPSP is the upper-level planning used to optimize project scheduling and activity mode selection to minimize project cost and duration; MPSCP is a lower-level planning method that seeks to make resource transportation decisions at a lower cost. A two-layer hybrid algorithm, consisting of Genetic Algorithm (GA) and Particle Swarm Optimization (PSO), is proposed to determine the optimal equilibrium strategy. Finally, the applicability and effectiveness of the proposed optimization method are evaluated through a case study of a large hydropower construction project, and management suggestions for related departments are provided.

A Closure Jacking Force Calculation Algorithm for Curved Prestressed Concrete Continuous Rigid-Frame Bridges with Asymmetric Cantilevers and Piers

Concrete continuous rigid-frame bridges are the most commonly used to cross exciting highway and railway. Before the bridge closure, the closure jacking force (CJF) needs to be determined firstly, which plays an important role in mid-span deflection controlling and the whole bridge stress state. This study proposed a closure jacking force calculation algorithm for curved prestressed concrete continuous rigid-frame bridges with asymmetric cantilevers and piers. Firstly, the mechanics method was used to analyze the relationship between the CJF and normal stress of main-girder and the pier. In addition, an actual bridge was taken as a numerical simulation case to determine the influence of the deformation and the stress induced by concrete shrinkage and creep of the main-girder and the jacking force. Finally, in order to obtain the optimal stress state of bridge piers, the multi-objective linear programming method was proposed to calculate the CJF. The results show that curved prestressed concrete continuous rigid-frame bridges usually suffered longitudinal and transverse bending and torsion under the concrete shrinkage and creep of the main-girder and the CJF, and the CJF significantly affected the normal stress of the pier with a low anti-push rigidity. The normal stress of piers was obviously linear with the CJF during whole construction and operation state of bridges at the following three states: the jacking process, bridge completion, and ten years of bridge operation. The multi-objective linear programming algorithm was an effective way to determine the optimal CJF and can fully reflect both longitudinal and transverse bending of piers.

A hybrid multi-objective bi-level interactive fuzzy programming method for solving ECM-DWTA problem

Electronic countermeasure (ECM) has become one of the most significant factors in modern warfare, in the course of combat, the electronic jamming allocation tasks need to be flexibly adjusted with the change of combat stage, which puts forward higher requirements for the modeling and solution method of this kind of problems. To solve the ECM dynamic weapon target assignment (ECM-DWTA) problem, a hybrid multi-target bi-level programming model is established. The upper level takes the sum of the electronic jamming effects in the whole combat stage as an optimization objective, and locally optimizes the ECM weapon (ECM-WP) assignment scheme in each stage. The lower level takes the importance expectation value of the target subjected to interference and combat consumption as double optimization objectives to globally optimize the ECM-WP assignment scheme. Focus on solving this complex model, a hybrid multi-objective bi-level interactive fuzzy programming algorithm (HMOBIF) is proposed, in this method, exponential membership function is used to describe the satisfaction degree of each level. When solving the multi-objective optimization problem composed of membership functions in the upper and lower levels, we use the MOEA/D algorithm to obtain the Pareto Front (PF) solution set, and then each solution in PF is evaluated and selected by the TOPSIS multi-criteria evaluation method. This local and global interactive optimization process of bi-level model is actually the process of executing observation-orientation-decision-action loop in practical combat. According to the current example, we conduct numerical simulation on the parameters in the model and obtain the parameter values suitable for the model solution. The computational experiments on different scale ECM-DWTA problems show that HMOBIF method is superior to four bi-level programming algorithms in terms of performance index, and can better solve ECM-DWTA problems.

Investigating a multi-objective programming method with common weights for a two-stage efficiency evaluation

In this article, two stage data envelopment analysis (DEA) models are reviewed initially and then two methods of sum weighted and product in efficiency combination are presented. In this two models, the weight of intermediate data are presumed to be the same whereas the results of implementing these two models show different weights for intermediate data. Hence, in the second section, a multi-objective model for each stage, considering an ideal unit, is used which computes the intermediate data weight for two stages uniformly and the efficiency of stages are accordingly calculated. Furthermore, considering the homogeneous nature of input, intermediate and output data for all units, it seems allocating various weights for these factors, is not fair. Thus in Section 3, a model for evaluation of two stage process, considering a common weight for input, intermediate and output factors of different stages is presented and the stage and overall efficiency and inefficiency values for each decision making unit using common weight is calculated. Finally, the relationship of these two models with respect to previous models was investigated and the results were explained using some examples.

Sustainable supplier evaluation and transportation planning in multi-level supply chain networks using multi-attribute- and multi-objective decision making

Supplier selection and transportation planning are essential tasks in managing supply chain networks (SCNs). The choice of a supplier has a significant influence on the transportation planning of SCNs, as the transportation routes will change. A limited number of studies in the literature have addressed the supplier selection and transportation planning issues simultaneously. This study proposes a two-stage multi-criteria decision-making (MCDM) approach for sustainable supplier evaluation and transportation planning in multi-level SCNs. Initially, a supplier evaluation framework is proposed using the modified indifference threshold-based attribute ratio analysis (ITARA) and the performance calculation technique of the integrated multiple multi-attribute decision-making (PCIM-MADM) to determine the supplier’s performance index. Subsequently, a multi-level SCN model is established, and the sustainable supplier selection and transportation planning problem is solved by the fuzzy multi-objective linear programming (FMOLP) method. A set of real-world data provided by a smart audio manufacturing company has been used to demonstrate the proposed approach. Decision-makers understand the importance of criteria through the weights constructed by the modified ITARA. Moreover, the results of the PCIM-MADM can effectively determine the supplier’s ranking and provide improving strategies for underperforming suppliers. The augmented max–min technique of the FMOLP can produce a single optimal solution of the transportation allocation. The results show that our approach can effectively evaluate sustainable supplier performance and optimize transportation planning in SCNs.

Optimal wastewater allocation with the development of an SECA multi-criteria decision-making method

The aim of this study is to develop a multi-objective decision-making method, named Simultaneous Evaluation of Criteria and Alternatives (SECA) for optimal ranking of wastewater allocation alternatives. This multi-objective decision-making method is for the continuous environment, meaning that experts can choose the best solution among the solutions obtained. The SECA multi-objective non-linear programming method has three objective functions: (i) maximization of the overall performance of alternatives, (ii) minimization of the deviation of criterion weights from reference point based on the within-criteria variation information, and (iii) minimization of the deviation of criterion weights based on the between-criteria variation information as well as two limitations for the weights of criteria. First, 15 criteria from different fields, including economic, socio-cultural, technological, and environmental, and six alternatives were determined for reusing wastewater, including reuse in the industrial sector, for recreational consumption, supplying environmental demand, artificial aquifer recharge, agricultural irrigation, and landscape irrigation. Then, decision-making matrix was constructed and reference points were calculated. By programming the SECA code in Lingo software and considering different β values, different weights for criteria and alternatives were ranked. In the other words, determining the best value for β was the most important step, so different values for β from 0.1 to 10 were examined. Results showed that in the value of β equal to 4, the maximum value of the objective function was obtained as 0.6926. Therefore, it was the best value for β and the weight of the effects on water resources criterion was equal to 0.0957 and was the most important criterion, and the allocation alternative in the environmental sector with a score of 0.8575 was the best wastewater allocation alternative. Followed by the environmental sector alternative, landscape irrigation, industrial sector, artificial aquifer recharge, agricultural irrigation, and recreational consumption were placed. Therefore, water supply for the environmental sector by considering the effect of wastewater on water resources is the best alternative for optimal allocation of wastewater.

Assignment model with multi-objective linear programming for allocating choice ranking using recurrent neural network

Classic linear assignment method is a multi-criteria decision-making approach in which criteria are weighted and each rank is assigned to a choice. In this study, to abandon the requirement of calculating the weight of criteria and use decision attributes prioritizing and also to be able to assign a rank to more than one choice, a multi-objective linear programming (MOLP) method is suggested. The objective function of MOLP is defined for each attribute and MOLP is solved based on absolute priority and comprehensive criteria methods. For solving the linear programming problems we apply a recurrent neural network (RNN). Indeed, the Lyapunov stability of the proposed model is proved. Results of comparing the proposed method with TOPSIS, VICOR, and MORA methods which are the most common multi-criteria decision schemes show that the proposed approach is more compatible with these methods.

An Algorithm to Improve Accuracy of Flush Air Data Sensing

In the field of aviation industry, flush air data sensing (FADS) is an advanced sensor for hypersonic aircraft. However, since the sensor has not been used in engineering practice for a long time, there is still much room for improvement. The main purpose of this paper is to improve the measurement accuracy of FADS from the perspective of fuzzy theory. First, the aerodynamic model of FADS is established based on the knowledge of aerodynamics under supersonic and subsonic conditions. Further, to describe the uncertainty and randomness of redundant signals, normal cloud model, a significant concept in fuzzy theory, is employed. Meanwhile, to reduce the influence of abnormal data on the final measurement accuracy, 3En principle is adopted to preprocess the data. In the process of data fusion using the aggregation operate, a new method of calculating the objective weight vector is derived based on Lagrange multiplier method and simplex evolutionary method of multi-objective programming (MOP). To illustrate the feasibility and validity of the proposed method, two FADS systems with six measurement taps and thirteen taps are adopted, respectively. The experimental results show that for the FADS with six measurement taps, the proposed method can improve the measurement accuracy by 4.55% and reduce the dispersion of data by 61.38%. For the latter, the proposed method can improve the measurement accuracy by 3.43% and reduce the dispersion of data by 50.14%. To further demonstrate the superiority of the proposed method, a detailed comparison with other six widely used methods is carried out.

Optimal Land Use Structure for Sustainable Agricultural Development—A Case Study in Changsha County, South Central China

Environmental and social problems caused by overfertilization, excessive pesticides, and encroachment on farmland are increasingly serious in agricultural settings, especially in suburban agricultural areas and highly intensive agricultural areas. Hence, modern agriculture not only pursues economic benefits, but it also pays more attention to ecological functions and social stability. This paper proposes a set of methods which are designed to realize optimal agricultural benefits and sustainable development by scientifically adjusting the land use structure. Taking Changsha County in South Central China as a case study, this paper first built an index system and adopted the information entropy-TOPSIS method to assess the economic, social, and ecological benefits of agricultural land use. Next, a coupled coordination model and an obstacle model were chosen to diagnose those factors that remained as obstacles to achieving the sustainable and coordinated development of the benefits of agricultural land use. Finally, based on the analysis of the changes in the benefits and obstacles over time, socio-economic and ecological constraints were established, and the multi-objective linear programming method (MOLP) was used to determine the comprehensive benefits and optimal land use structure. The results indicate that: (1) The agricultural benefits were stably increasing from 0.20 in 1996 to 0.79 in 2016. (2) The economic benefit index is no longer the main obstacle, while the social benefit index, which includes components such as the food security index, has become the principal influencing factor. (3) The optimal land use structure and comprehensive benefits were presented by taking into consideration the economic development, environmental protection, and social needs. This study emphasizes economic development, but it also seeks coordinated development with comprehensive benefits. The results of the study could provide scientific recommendations for optimizing the agricultural land use spatial patterns and sustainable land use.

Addressing a sustainable supplier selection and order allocation problem by an integrated approach: a case of automobile manufacturing

ABSTRACT This study introduces a novel integration of language entropy weight method (LEWM) and multi-objective programming (MOP) method to address a sustainable supplier selection and order allocation problem. To show the applicability of the proposed method, a real case study of automobile manufacturing is considered. First of all, the indicators of the proposed problem are defined. Then, the LEWM is considered to select the suppliers. To meet the standards of the sustainable development goals and the global supply chains for a case of automobile manufacturing system, an optimization model is established. The proposed model has three objectives including the total cost and the carbon emissions to be minimized while the procurement value to be maximized. These objectives are based on the economic, environmental, and social impacts simultaneously to address the triple bottom lines of sustainability. In addition, the proposed method improves the relationship between the supply chain practitioners and the potential suppliers to achieve the product development capabilities and qualities to achieve the sustainable development goals and the global supply chain trend.