A two-stage evolutionary strategy based MOEA/D to multi-objective problems

Bao C., Xu L., Goodman E.D.

Maintaining a good balance between convergence and diversity in many-objective optimization is a key challenge for most Pareto dominance-based multi-objective evolutionary algorithms. In most existing multi-objective evolutionary algorithms, a certain fixed metric is used in the selection operation, no matter how far the solutions are from the Pareto front. Such a selection scheme directly affects the performance of the algorithm, such as its convergence, diversity or computational complexity. In this paper, we use a more structured metric, termed augmented penalty boundary intersection, which acts differently on each of the non-dominated fronts in the selection operation, to balance convergence and diversity in many-objective optimization problems. In diversity maintenance, we apply a distance-based selection scheme to each non-dominated front. The performance of our proposed algorithm is evaluated on a variety of benchmark problems with 3 to 15 objectives and compared with five state-of-the-art multi-objective evolutionary algorithms. The empirical results demonstrate that our proposed algorithm has highly competitive performance on almost all test instances considered. Furthermore, the combination of a special mate selection scheme and a clustering-based selection scheme considerably reduces the computational complexity compared to most state-of-the-art multi-objective evolutionary algorithms.

Sun Y., Xue B., Zhang M., Yen G.G.

Convergence and diversity are interdependently handled during the evolutionary process by most existing many-objective evolutionary algorithms (MaOEAs). In such a design, the degraded performance of one would deteriorate the other, and only solutions with both are able to improve the performance of MaOEAs. Unfortunately, it is not easy to constantly maintain a population of solutions with both convergence and diversity. In this paper, an MaOEA based on two independent stages is proposed for effectively solving many-objective optimization problems (MaOPs), where the convergence and diversity are addressed in two independent and sequential stages. To achieve this, we first propose a nondominated dynamic weight aggregation method by using a genetic algorithm, which is capable of finding the Pareto-optimal solutions for MaOPs with concave, convex, linear and even mixed Pareto front shapes, and then these solutions are employed to learn the Pareto-optimal subspace for the convergence. Afterward, the diversity is addressed by solving a set of single-objective optimization problems with reference lines within the learned Pareto-optimal subspace. To evaluate the performance of the proposed algorithm, a series of experiments are conducted against six state-of-the-art MaOEAs on benchmark test problems. The results show the significantly improved performance of the proposed algorithm over the peer competitors. In addition, the proposed algorithm can focus directly on a chosen part of the objective space if the preference area is known beforehand. Furthermore, the proposed algorithm can also be used to effectively find the nadir points.

Tian Y., Cheng R., Zhang X., Li M., Jin Y.

Diversity preservation plays an important role in the design of multi-objective evolutionary algorithms, but the diversity performance assessment of these algorithms remains challenging. To address this issue, this paper proposes a performance metric and a multi-objective test suite for the diversity assessment of multiobjective evolutionary algorithms. The proposed metric assesses both the evenness and spread of a solution set by projecting it to a lower-dimensional hypercube and calculating the "volume" of the projected solution set. The proposed test suite contains eight benchmark problems, which pose stiff challenges for existing algorithms to obtain a diverse solution set. Experimental studies demonstrate that the proposed metric can assess the diversity of a solution set more precisely than existing ones, and the proposed test suite can be used to effectively distinguish between algorithms with respect to their diversity performance.

Zhao F., Qin S., Zhang Y., Ma W., Zhang C., Song H.

The no-wait flow shop scheduling problem (NWFSP) plays an essential role in the manufacturing industry. Inspired by the overall process of biogeography theory, the standard biogeography-based optimization (BBO) was constructed with migration and mutation operators. In this paper, a hybrid biogeography-based optimization with variable neighborhood search (HBV) is implemented for solving the NWFSP with the makespan criterion. The modified NEH and the nearest neighbor mechanism are employed to generate a potential initial population. A hybrid migration operator, which combines the path relink technique and the block-based self-improvement strategy, is designed to accelerate the convergence speed of HBV. The iterated greedy (IG) algorithm is introduced into the mutation operator to obtain a promising solution in exploitation phase. A variable neighbor search strategy, which is based on the block neighborhood structure and the insert neighborhood structure, is designed to perform the local search around the current best solution in each generation. Furthermore, the global convergence performance of the HBV is analyzed with the Markov model. The computational results and comparisons with other state-of-art algorithms based on Taillard and VRF benchmark show that the efficiency and performance of HBV for solving NWFSP.

Li H., Deng J., Zhang Q., Sun J.

Complicated geometric shapes of Pareto fronts can cause difficulties for multiobjective evolutionary algorithms. To deal with these difficulties, efficient diversity strategies must be highly addressed in order to obtain a set of representative Pareto solutions. In decomposition-based multiobjective evolutionary algorithms, this is often done by optimizing multiple single objective subproblems defined by a set of weight vectors. For complicated Pareto fronts with extreme convexity, disconnection or degeneracy, however, it is nontrivial to set these weight vector properly. To overcome this shortcoming, we propose a new decomposition-based multiobjective evolutionary algorithm based on a hybrid weighting strategy, which optimizes both random subproblems and fixed subproblems. To maintain diversity of nondominated solutions stored in external population, a new archiving strategy based on adaptive Epsilon dominance is also suggested in our proposed algorithm. Our experimental results have showed that our proposed algorithm is superior to several other state-of-the-art multiobjective evolutionary algorithms on a set of benchmark multiobjective test problems with different challenging difficulties regarding the geometric shapes of Pareto fronts.

Li Z., Lin K., Nouioua M., Jiang S., Gu Y.

Maintaining a good balance between the convergence and the diversity is particularly crucial for the performance of the evolutionary algorithms (EAs). However, the traditional multi-objective evolutionary algorithms, which have shown their competitive performance with a variety of practical problems involving two or three objectives, face significant challenges in case of problems with more than three objectives, namely many-objective optimization problems (MaOPs). This paper proposes a dynamic convergence–diversity guided evolutionary algorithm, namely (DCDG-EA) for MaOPs by employing the decomposition technique. Besides, the objective space of MaOPs is divided into K subspaces by a set of uniformly distributed reference vectors. Each subspace has its own subpopulation and evolves in parallel with the other subspaces. In DCDG-EA, the balance between the convergence and the diversity is achieved through the convergence–diversity based operator selection (CDOS) strategy and convergence–diversity based individual selection (CDIS) strategy. In CDOS, for each operator of the set of operators, a selection probability is assigned which is related to its convergence and diversity capabilities. Based on the attributed selection probabilities, an appropriate operator is selected to generate the offsprings. Furthermore, CDIS is used which allows to greatly overcome the inefficiency of the Pareto dominance approaches. It updates each subpopulation by using two independent distance measures that represent the convergence and the control diversity, respectively. The experimental results on DTLZ and WFG benchmark problems with up to 15 objectives show that our algorithm is highly competitive comparing with the four state-of-the-art evolutionary algorithms in terms of convergence and diversity.

Qi Y., Li X., Yu J., Miao Q.

This paper proposes a novel decomposition method based on user-preference and developed a variation of the decomposition based multi-objective optimization algorithm (MOEA/D) targeting only solutions in a small region of the Pareto-front defined by the preference information supplied by the decision maker (DM). This is particularly advantageous for solving multi-objective optimization problems (MOPs) with more than 3 objectives, i.e., many-objective optimization problems (MaOPs). As the number of objectives increases, the ability of an EMO algorithm to approximate the entire Pareto front ( PF ) is rapidly diminishing. In this paper, we first propose a novel scalarizing function making use of a series of new reference points derived from a reference point specified by the DM in the preference model. Based on this scalarizing function, we then develop a user-preference-based EMO algorithm, namely R-MOEA/D. One key merit of R-MOEA/D is that it does not rely on an estimation of the ideal point, which may impact significantly the performances of state-of-the-art decomposition based EMO algorithms. Our experimental results on multi-objective and many-objective benchmark problems have shown that R-MOEA/D provides a more direct and efficient search towards the preferred PF region, resulting in competitive performances. In an interactive setting when the DM changes the reference point during optimization, R-MOEA/D has a faster response speed and performance than the compared algorithms, showing its robustness and adaptability to changes of the preference model. Furthermore, the effectiveness of R-MOEA/D is verified on a real-world problem of reservoir flood control operations.

Fan Z., Fang Y., Li W., Cai X., Wei C., Goodman E.

This paper proposes a novel constraint-handling mechanism named angle-based constrained dominance principle (ACDP) embedded in a decomposition-based multi-objective evolutionary algorithm (MOEA/D) to solve constrained multi-objective optimization problems (CMOPs). To maintain the diversity of the working population, ACDP utilizes the information of the angle of solutions to adjust the dominance relation of solutions during the evolutionary process. This paper uses 14 benchmark instances to evaluate the performance of the MOEA/D with ACDP (MOEA/D-ACDP). Additionally, an engineering optimization problem (which is I-beam optimization problem) is optimized. The proposed MOEA/D-ACDP, and four other decomposition-based CMOEAs, including C-MOEA/D, MOEA/D-CDP, MOEA/D-Epsilon and MOEA/D-SR are tested by the above benchmarks and the engineering application. The experimental results manifest that MOEA/D-ACDP is significantly better than the other four CMOEAs on these test instances and the real-world case, which indicates that ACDP is more effective for solving CMOPs.

Qiao J., Zhou H., Yang C., Yang S.

A multiobjective evolutionary algorithm based on decomposition (MOEA/D) decomposes a multiobjective optimization problem (MOP) into a number of scalar optimization subproblems and optimizes them in a collaborative manner. In MOEA/D, decomposition mechanisms are used to push the population to approach the Pareto optimal front (POF), while a set of uniformly distributed weight vectors are applied to maintain the diversity of the population. Penalty-based boundary intersection (PBI) is one of the approaches used frequently in decomposition. In PBI, the penalty factor plays a crucial role in balancing convergence and diversity. However, the traditional PBI approach adopts a fixed penalty value, which will significantly degrade the performance of MOEA/D on some MOPs with complicated POFs. This paper proposes an angle-based adaptive penalty (AAP) scheme for MOEA/D, called MOEA/D-AAP, which can dynamically adjust the penalty value for each weight vector during the evolutionary process. Six newly designed benchmark MOPs and an MOP in the wastewater treatment process are used to test the effectiveness of the proposed MOEA/D-AAP. Comparison experiments demonstrate that the AAP scheme can significantly improve the performance of MOEA/D.

Ho-Huu V., Hartjes S., Visser H.G., Curran R.

The multi-objective evolutionary algorithm based on decomposition (MOEA/D) has been recognized as a promising method for solving multi-objective optimization problems (MOPs), receiving a lot of attention from researchers in recent years. However, its performance in handling MOPs with complicated Pareto fronts (PFs) is still limited, especially for real-world applications whose PFs are often complex featuring, e.g., a long tail or a sharp peak. To deal with this problem, an improved MOEA/D (named iMOEA/D) that mainly focuses on bi-objective optimization problems (BOPs) is therefore proposed in this paper. To demonstrate the capabilities of iMOEA/D, it is applied to design optimization problems of truss structures. In iMOEA/D, the set of the weight vectors defined in MOEA/D is numbered and divided into two subsets: one set with odd-weight vectors and the other with even-weight vectors. Then, a two-phase search strategy based on the MOEA/D framework is proposed to optimize their corresponding populations. Furthermore, in order to enhance the total performance of iMOEA/D, some recent developments for MOEA/D, including an adaptive replacement strategy and a stopping criterion, are also incorporated. The reliability, efficiency and applicability of iMOEA/D are investigated through seven existing benchmark test functions with complex PFs and three optimal design problems of truss structures. The obtained results reveal that iMOEA/D generally outperforms MOEA/D and NSGA-II in both benchmark test functions and real-world applications.

Lin Q., Liu S., Zhu Q., Tang C., Song R., Chen J., Coello C.A., Wong K., Zhang J.

Recently, it was found that most multiobjective particle swarm optimizers (MOPSOs) perform poorly when tackling many-objective optimization problems (MaOPs). This is mainly because the loss of selection pressure that occurs when updating the swarm. The number of nondominated individuals is substantially increased and the diversity maintenance mechanisms in MOPSOs always guide the particles to explore sparse regions of the search space. This behavior results in the final solutions being distributed loosely in objective space, but far away from the true Pareto-optimal front. To avoid the above scenario, this paper presents a balanceable fitness estimation method and a novel velocity update equation, to compose a novel MOPSO (NMPSO), which is shown to be more effective to tackle MaOPs. Moreover, an evolutionary search is further run on the external archive in order to provide another search pattern for evolution. The DTLZ and WFG test suites with 4-10 objectives are used to assess the performance of NMPSO. Our experiments indicate that NMPSO has superior performance over four current MOPSOs, and over four competitive multiobjective evolutionary algorithms (SPEA2-SDE, NSGA-III, MOEA/DD, and SRA), when solving most of the test problems adopted.

Tian Y., Cheng R., Zhang X., Jin Y.

Over the last three decades, a large number of evolutionary algorithms have been developed for solving multi-objective optimization problems. However, there lacks an upto-date and comprehensive software platform for researchers to properly benchmark existing algorithms and for practitioners to apply selected algorithms to solve their real-world problems. The demand of such a common tool becomes even more urgent, when the source code of many proposed algorithms has not been made publicly available. To address these issues, we have developed a MATLAB platform for evolutionary multi-objective optimization in this paper, called PlatEMO, which includes more than 50 multiobjective evolutionary algorithms and more than 100 multi-objective test problems, along with several widely used performance indicators. With a user-friendly graphical user interface, PlatEMO enables users to easily compare several evolutionary algorithms at one time and collect statistical results in Excel or LaTeX files. More importantly, PlatEMO is completely open source, such that users are able to develop new algorithms on the basis of it. This paper introduces the main features of PlatEMO and illustrates how to use it for performing comparative experiments, embedding new algorithms, creating new test problems, and developing performance indicators. Source code of PlatEMO is now available at: http://bimk.ahu.edu.cn/index.php?s=/Index/Software/index.html.

Tanabe R., Ishibuchi H., Oyama A.

Recently, a large number of multi-objective evolutionary algorithms (MOEAs) for many-objective optimization problems have been proposed in the evolutionary computation community. However, an exhaustive benchmarking study has never been performed. As a result, the performance of the MOEAs has not been well understood yet. Moreover, in almost all previous studies, the performance of the MOEAs was evaluated based on nondominated solutions in the final population at the end of the search. Such traditional benchmarking methodology has several critical issues. In this paper, we exhaustively investigate the anytime performance of 21 MOEAs using an unbounded external archive (UEA), which stores all nondominated solutions found during the search process. Each MOEA is evaluated under two optimization scenarios called UEA and reduced UEA in addition to the standard final population scenario. These two scenarios are more practical in real-world applications than the final population scenario. Experimental results obtained under the two scenarios are significantly different from the previously reported results under the final population scenario. For example, results on the Walking Fish Group test problems with up to six objectives indicate that some recently proposed MOEAs are outperformed by some classical MOEAs. We also analyze the reason why some classical MOEAs work well under the UEA and the reduced UEA scenarios.

Wu M., Kwong S., Jia Y., Li K., Zhang Q.

By transforming a multi-objective optimization problem into a number of single-objective optimization problems and optimizing them simultaneously, decomposition-based evolutionary multi-objective optimization algorithms have attracted much attention in the field of multi-objective optimization. In decomposition-based algorithms, the population diversity is maintained using a set of predefined weight vectors, which are often evenly sampled on a unit simplex. However, when the Pareto front of the problem is not a hyperplane but more complex, the distribution of the final solution set will not be that uniform. In this paper, we propose an adaptive method to periodically regenerate the weight vectors for decomposition-based multi-objective algorithms according to the geometry of the estimated Pareto front. In particular, the Pareto front is estimated via Gaussian process regression. Thereafter, the weight vectors are reconstructed by sampling a set of points evenly distributed on the estimated Pareto front. Experimental studies on a set of multi-objective optimization problems with different Pareto front geometries verify the effectiveness of the proposed adaptive weights generation method.

Wang Z., Zhang Q., Li H., Ishibuchi H., Jiao L.

Decomposition based multiobjective evolutionary algorithms approximate the Pareto front of a multiobjective optimization problem by optimizing a set of subproblems in a collaborative manner. Often, each subproblem is associated with a direction vector and a reference point. The settings of these parameters have a very critical impact on convergence and diversity of the algorithm. Some work has been done to study how to set and adjust direction vectors to enhance algorithm performance for particular problems. In contrast, little effort has been made to study how to use reference points for controlling diversity in decomposition based algorithms. In this paper, we first study the impact of the reference point setting on selection in decomposition based algorithms. To balance the diversity and convergence, a new variant of the multiobjective evolutionary algorithm based on decomposition with both the ideal point and the nadir point is then proposed. This new variant also employs an improved global replacement strategy for performance enhancement. Comparison of our proposed algorithm with some other state-of-the-art algorithms is conducted on a set of multiobjective test problems. Experimental results show that our proposed algorithm is promising.

Li C., Deng L., Qiao L., Zhang L.

Zou D., Ma L., Li F., Ouyang H.

Yue Y., Zhao D., Zhou Y., Xu L., Tang Y., Peng H.

Chen H., Liu J., Shen G.Q., Feng Z.

Zhang P., Fei B., Bi J., Wang M., Zhao C., Zhang J.

Huang S., Wang C., Bian W.

With the improvement of people’s living standards, the issue of dietary health has received extensive attention. In order to simultaneously meet people’s demands for dietary preferences and nutritional balance, we have conducted research on the issue of personalized food recommendations. For this purpose, we have proposed a hybrid food recommendation model, which can provide users with scientific, reasonable, and personalized dietary advice. Firstly, the collaborative filtering (CF) algorithm is adopted to recommend foods to users; then, the improved Multi-Objective Evolutionary Algorithm Based on Decomposition (MOEA/D) is used to adjust the nutritional balance and symmetry of the recommended foods. In view of the existing problems in the current nutritional balance algorithm, such as slow convergence speed and insufficient local search ability, the autonomous optimization (AO) adjustment strategy, the self-adaptive adjustment strategy, and the two-sided mirror principle to optimize boundary strategy are introduced in the MOEA/D. According to the characteristics of the food nutrition regulation problem, an adaptive food regulation (AFR) adjustment strategy is designed to achieve more accurate nutritional regulation. Based on the above improvements, a food nutritional recommendation algorithm based on MOEA/D (FNR-MOEA/D) is proposed. Experiments show that compared with MOPSO, MOABC, and RVEA, FNR-MOEA/D performs more superiorly in solving the problem of nutritional balance in food recommendation.

Sun N., Chen Q.

Yu B., Bi X., Liu X., Sun H., Buysse J.

Zou D., Ma L., Li C.

In this paper, an enhanced NSGA-II (ENSGA-II) is presented for the combined heat and power dynamic economic emission dispatch (CHPDEED). ENSGA-II produces offspring individuals by a novel crossover of the Cauchy distribution with adaptive location and scale parameters. It can not only sufficiently explore and exploit the decision space but also remain a relatively high population diversity. Also, ENSGA-II adopts a modified crowding distance to penalize crowded individuals, seeking to evenly distribute the individuals in the objective space. In addition, a fast constraint repairing method (FCRM) is proposed to significantly reduce the constraint violations and guide infeasible individuals to move towards feasible zones rapidly. The proposed ENSGA-II is applied to a number of multi-objective optimization problems and compared with the other eight multi-objective evolutionary algorithms (MOEAs). ENSGA-II is found to produce better results on most problems, such as smaller inverse generational distances, larger hypervolumes, larger coverage rates and smaller spacings. Therefore, ENSGA-II can obtain a satisfactory Pareto set with broad spread, high diversity, strong convergence and good evenness, and it is an efficient alternative for CHPDEED.

Altiok M., Gündüz M.

AbstractWhen real-world engineering challenges are examined adequately, it becomes clear that multi-objective need to be optimized. Many engineering problems have been handled utilizing the decomposition-based optimization approach according to the literature. The performance of multi-objective evolutionary algorithms is highly dependent on the balance of convergence and diversity. Diversity and convergence are not appropriately balanced in the decomposition technique, as they are in many approaches, for real-world problems. A novel Multi-Objective Artificial Algae Algorithm based on Decomposition (MOAAA/D) is proposed in the paper to solve multi-objective structural problems. MOAAA/D is the first multi-objective algorithm that uses the decomposition-based method with the artificial algae algorithm. MOAAA/D, which successfully draws a graph on 24 benchmark functions within the area of two common metrics, also produced promising results in the structural design problem to which it was applied. To facilitate the design of the "rectangular reinforced concrete column" using MOAAA/D, a solution space was derived by optimizing the rebar ratio and the concrete quantity to be employed.

Zhou G., Xie Y., Lan H., Tian W., Buyya R., Wu K.

Optimizing multi-dimensional resource utilization is a critical research area in distributed computing, particularly in cloud computing, where various heterogeneous resources are integrated to offer a wide range of services. Addressing this issue necessitates the simultaneous consideration of multiple resource bottlenecks. This paper presents a new solution, called the Multi-Population Growth Genetic Algorithm (MPGGA), which consists of a central management unit responsible for executing information interaction and growth quota reallocation, and multiple population evolution executors to perform crossover and regeneration within each population. The proposed MPGGA combines elite sharing and priority support for the weaker population (ESPW), resulting in better convergence and optimality than other combinations of strategies. This outcome is corroborated by extensive ablation experiments on various strategies. Furthermore, the experimental results for minimizing the maximum utilization of resources in each dimension indicate that MPGGA-ESPW outperforms other popular algorithms, such as GHW-NSGA II (1.363x), GHW-MOEA/D (1.339x), NSGA II (1.948x), and MOEA/D (2.151x) in terms of convergence speed. For energy consumption-related optimization problems, the experimental results demonstrate that the adaptability of a single algorithm in MPGGA family is limited by the algorithm of growth route, while also showing that the MPGGA framework is flexible to allow various algorithms as its growth route to adapt to various scenarios.

Ren X., Li L.

Combined cooling, heating and power (CCHP) integrated energy systems can realize the gradient utilization of energy and the coordination and complementarity of multiple energy sources, which is the primary choice for solving the energy crisis and alleviating the environmental pressure. However, most current design and operational optimizations for CCHP are performed under rule-based operational strategies. This paper constructs a two-level alternating optimization framework for the planning-operation of the CCHP system with an embedded scheduling strategy. The upper-level solves for the optimal capacity configuration of the system to optimize energy, economic and environmental performance, and the lower-level solves for the operating capacity of the equipment system to optimize the operation and maintenance cost. In order to obtain the optimal configuration scheme and scheduling strategy for the system, a novel multi-objective weIghted meaN oF vectOrs (MOINFO)-mixed integer linear programming (MILP) method is proposed. The optimization results suggest that the hybrid CCHP for three buildings (market, office, and restaurant) under the two-level optimization model can achieve better economic, energy, and environmental performance than the rule-based single-level optimization model. Generally, for the market, office and restaurant, the two-Level model achieves maximum economic, energy and environmental performance improvements of 29.12%, 29.88%, and 42.13%; 38.32%, 19.56%, and 34.68%; and 47.37%, 30.93%, and 45.47%, respectively.

Luo X., Guo S., Du B., Luo X., Guo J.

This paper addresses a novel multi-skill resource-constrained project scheduling problem with flexible resource profiles (F-MSRCPSP), in which the resource allocation of each activity consists of a certain number of discrete resources and is allowed to be adjusted over its duration. The F-MSRCPSP aims, therefore, to determine the flexible resource profile of each activity to minimize the make-span and total cost simultaneously. Then, a hybrid multi-objective fruit fly optimization algorithm is proposed to handle the concerned problem. In the proposed algorithm, two flexible parallel and serial schedule generation schemes are introduced, aiming to schedule activities and adjust allocated resource combinations. Additionally, two heuristic strategies are proposed to effectively select suitable resource combinations for activities. Moreover, a series of operators has been developed, including the rejoining operator, empirical re-arrangement operator, and empirical re-selection operator. These operators aim to accelerate the convergence speed and enhance the exploration of the proposed algorithm. Finally, the orthogonal test is used to select the optimal parameter combination, and comparative experiments based on tests with different scales are conducted, along with a t-test. The experimental results demonstrate that MOFOA-HS is effective in solving the F-MSRCPSP.

Du Z., Xu W., Wang Z., Zhu X., Wang J., Wang H.

To enhance the forming quality of the forging and minimize the forging cost in the concave radial forging process, this article examines the influence of process parameters (radial reduction ∆h, rotation angle β, friction coefficient μ) on the forging process through numerical simulation. A multi-objective optimization method is employed to balance the objective functions (strain homogeneity E, forging load F). First, sample points for different combinations of process parameters were obtained using a central composite experimental design. Then, a mathematical model between the process parameters and the objective function was established using the response surface method, which underwent variance analysis and sensitivity analysis. Finally, the optimal process parameter combination was determined based on the NSGA-II algorithm and satisfaction function. The optimization results were verified by finite element simulations. The optimized process combination: ∆h = 0.25 mm, β = 21.68°, μ = 0.05. The corresponding E and F are 0.241367 and 577.029, respectively. Compared with the initial process, the standard deviation of the overall strain was reduced by 14.25%, and the forging load was reduced by 1.76%. The results indicate that the quality of the forgings was significantly improved while the forging cost was reduced to some extent.

Wang Y., Luo S., Fan J., Xu M., Wang H.

With the growing interest in win-win cooperation within the logistics industry recently, collaborative multicenter logistics networks have witnessed the establishment of numerous alliances aimed at achieving cost savings. However, alliance members may choose to withdraw from an alliance due to changes in market positioning strategies and short-term profit considerations. Such withdraws often lead to the disintegration of the alliance and result in financial losses for the remaining non-defaulting members. This study focuses on addressing the issues of compensation and profit allocation in a specific scenario known as the collaborative multicenter vehicle routing problem with time windows and defaulting members withdrawal (CMVRPTWDMW). The objective is to ensure alliance stability and fair profit distribution among its members. To achieve this, a tri-objective mathematical model is proposed to optimize the re-formed alliances by considering the total operating cost, number of vehicles, and service waiting time. Additionally, an improved minimum cost remaining saving method is introduced to accurately quantify the deserved profits of non-defaulting members. Compensation quantification and default cost allocation models are then developed based on this analysis. To solve the CMVRPTWDMW and obtain Pareto optimal solutions, a hybrid heuristic algorithm is designed. This algorithm combines a 3D k-means clustering algorithm for customer reassignment after the withdrawal of defaulting members, effectively reducing the complexity of the problem. Furthermore, the algorithm employs a Clarke-Wright saving method-based non-dominated sorting genetic algorithm-III (CW-NSGA-III) to search for optimal solutions. To enhance the algorithm’s search ability, an elite alteration strategy and a penalty-based boundary intersection approach are incorporated into CW-NSGA-III. Finally, a real-world case study conducted in Chongqing City, China, is presented to demonstrate the feasibility of the proposed models and solution algorithms. The study contributes valuable insights into the sustainable operation of urban logistics networks that may experience unpredictable defaults, providing guidance for the stability of logistics alliances in the transportation sector.