A simulated annealing approach to a bi-criteria sequencing problem in a two-stage supply chain

In this paper, a multi-objective simulated annealing (MOSA) solution approach is proposed to a bi-criteria sequencing problem to coordinate required set-ups between two successive stages of a supply chain in a flow shop pattern. Each production batch has two distinct attributes and a set-up occurs in each stage when the corresponding attribute of the two successive batches are different. There are two objectives including: minimizing total set-ups and minimizing the maximum number of set-ups between the two stages that are both NP-hard problems. The MOSA approach starts with an initial set of locally non-dominated solutions generated by an initializing heuristic. The set is then iteratively updated through the annealing process in search for true Pareto-optimal frontier until a stopping criterion is met. Performance of the proposed MOSA was evaluated using true Pareto-optimal solutions of small problems found via total enumeration. It was also compared against a lower bound in large problems. Comparative experiments show that the MOSA is robust in finding true Pareto-optimal solutions in small problems. It was also shown that MOSA is very well-performing in large problems and that it outperforms an existing multi-objective genetic algorithm (MOGA) in terms of quality of solutions.     

A multi-objective approach to facility layout problem by genetic search algorithm and Electre method

Classical approaches to layout design problem tend to maximise the efficiency of layout, measured by the handling cost related to the interdepartmental flow and to the distance among the departments. However, the actual problem involves several conflicting objectives hence requiring a multi-objective formulation. Multi-objective approaches, recently proposed, in most cases lead to the maximisation of a weighted sum of score functions. The poor practicability of such an approach is due to the difficulty of normalising these functions and of quantifying the weights. In this paper, this difficulty is overcome by approaching the problem in two subsequent steps: in the first step, the Pareto-optimal solutions are determined by employing a multi-objective constrained genetic algorithm and the subsequent selection of the optimal solution is carried out by means of the multi-criteria decision-making procedure Electre. This procedure allows the decision maker to express his preferences on the basis of the knowledge of candidate solution set. Quantitative (handling cost) and qualitative (adjacency and distance requests between departments) objectives are considered referring to a bay structure-based layout model, that allows to take into account also practical constraints such as the aspect ratio of departments. Results obtained confirm the effectiveness of the proposed procedure as a practicable support tool for layout designers.     

A non dominated ranking Multi Objective Genetic Algorithm and electre method for unequal area facility layout problems

The unequal area facility layout problem (UA-FLP) comprises a class of extremely difficult and widely applicable optimization problems arising in diverse areas and meeting the requirements for real-world applications. Genetic Algorithms (GAs) have recently proven their effectiveness in finding (sub) optimal solutions to many NP-hard problems such as UA-FLP. A main issue in such approach is related to the genetic encoding and to the evolutionary mechanism implemented, which must allow the efficient exploration of a wide solution space, preserving the feasibility of the solutions and ensuring the convergence towards the optimum. In addition, in realistic situations where several design issues must be taken into account, the layout problem falls in the broader framework of multi-objective optimization problems. To date, there are only a few multi-objective FLP approaches, and most of them employ over-simplified optimization techniques which eventually influence the quality of the solutions obtained and the performance of the optimization procedure. In this paper, this difficulty is overcome by approaching the problem in two subsequent steps: in the first step, the Pareto-optimal solutions are determined by employing Multi Objective Genetic Algorithm (MOGA) implementing four separate fitness functions within a Pareto evolutionary procedure, following the general structure of Non-dominated Ranking Genetic Algorithm (NRGA) and the subsequent selection of the optimal solution is carried out by means of the multi-criteria decision-making procedure Electre. This procedure allows the decision maker to express his preferences on the basis of the knowledge of candidate solution set. Quantitative and qualitative objectives are considered referring to the slicing-tree layout representation scheme. The numerical results obtained outperform previous referenced approaches, thus confirming the effectiveness of the procedure proposed.     

A memetic algorithm for bi-objective integrated forward/reverse logistics network design

Logistics network design is a major strategic issue due to its impact on the efficiency and responsiveness of the supply chain. This paper proposes a model for integrated logistics network design to avoid the sub-optimality caused by a separate, sequential design of forward and reverse logistics networks. First, a bi-objective mixed integer programming formulation is developed to minimize the total costs and maximize the responsiveness of a logistics network. To find the set of non-dominated solutions, an efficient multi-objective memetic algorithm is developed. The proposed solution algorithm uses a new dynamic search strategy by employing three different local searches. To assess the quality of the novel solution approach, the quality of its Pareto-optimal solutions is compared to those generated by an existing powerful multi-objective genetic algorithm from the recent literature and to exact solutions obtained by a commercial solver.     

基于幂变换的多目标进化算法MOEA?D权重设计方法

在多目标最优化问题中,如何求解一组均匀散布在前沿界面上的有效解具有重要意义.MOEA?D是最近出现的一种杰出的多目标进化算法,当前沿界面的形状是某种已知的类型时,MOEA?D使用高级分解的方法容易求出均匀散布在前沿界面上的有效解.然而,多目标优化问题的前沿界面的形状通常是未知的.为了使MOEA?D能求出一般多目标优化问题的均匀散布的有效解,利用幂函数对目标进行数学变换,使变换后的多目标优化问题的前沿界面在算法的进化过程中逐渐接近希望得到的形状,提出了一种求解一般的多目标优化问题的MOEA?D算法的权重设计方法,并且讨论了经过数学变换后前沿界面的保距性问题.采用建议的权重设计方法,MOEA?D更容易求出一般的多目标优化问题均匀散布的有效解.数值结果验证了算法的有效性.     

Introducing robustness in multi-objective optimization

In optimization studies including multi-objective optimization, the main focus is placed on finding the global optimum or global Pareto-optimal solutions, representing the best possible objective values. However, in practice, users may not always be interested in finding the so-called global best solutions, particularly when these solutions are quite sensitive to the variable perturbations which cannot be avoided in practice. In such cases, practitioners are interested in finding the robust solutions which are less sensitive to small perturbations in variables. Although robust optimization is dealt with in detail in single-objective evolutionary optimization studies, in this paper, we present two different robust multi-objective optimization procedures, where the emphasis is to find a robust frontier, instead of the global Pareto-optimal frontier in a problem. The first procedure is a straightforward extension of a technique used for single-objective optimization and the second procedure is a more practical approach enabling a user to set the extent of robustness desired in a problem. To demonstrate the differences between global and robust multi-objective optimization principles and the differences between the two robust optimization procedures suggested here, we develop a number of constrained and unconstrained test problems having two and three objectives and show simulation results using an evolutionary multi-objective optimization (EMO) algorithm. Finally, we also apply both robust optimization methodologies to an engineering design problem.     

A hybrid approach for multi-objective combinatorial optimisation problems in ship design and shipping

Numerous real-world problems relating to ship design and shipping are characterised by combinatorially explosive alternatives as well as multiple conflicting objectives and are denoted as multi-objective combinatorial optimisation (MOCO) problems. The main problem is that the solution space is very large and therefore the set of feasible solutions cannot be enumerated one by one. Current approaches to solve these problems are multi-objective metaheuristics techniques, which fall in two categories: population-based search and trajectory-based search. This paper gives an overall view for the MOCO problems in ship design and shipping where considerable emphasis is put on evolutionary computation and the evaluation of trade-off solutions. A two-stage hybrid approach is proposed for solving a particular MOCO problem in ship design, subdivision arrangement of a ROPAX vessel. In the first stage, a multi-objective genetic algorithm method is employed to approximate the set of pareto-optimal solutions through an evolutionary optimisation process. In the subsequent stage, a higher-level decision-making approach is adopted to rank these solutions from best to worst and to determine the best solution in a deterministic environment with a single decision maker.     

A steady-state genetic algorithm for multi-product supply chain network design

Supply chain network (SCN) design is to provide an optimal platform for efficient and effective supply chain management (SCM). The problem is often an important and strategic operations management problem in SCM. The design task involves the choice of facilities (plants and distribution centers (DCs)) to be opened and the distribution network design to satisfy the customer demand with minimum cost. This paper presents a solution procedure based on steady-state genetic algorithms (ssGA) with a new encoding structure for the design of a single-source, multi-product, multi-stage SCN. The effectiveness of the ssGA has been investigated by comparing its results with those obtained by CPLEX, Lagrangean heuristic, hyrid GA and simulated annealing on a set of SCN design problems with different sizes.     

基于Pareto最优的PID多目标优化设计

现有的PID优化方法往往难以同时兼顾系统对时域和频域性能的要求,针对这一缺陷,提出了一种PID多目标优化方法：将动态性能指标作为优化目标,频域性能指标作为约束条件,采用基于Pareto最优的多目标优化算法对其求解。该算法采用新的拥挤距离计算方法,引入双重精英机制,进化效率高,得到的Pareto最优解集多样性好,决策者可根据当前工作需求从中选择最终的满意解。仿真结果证明了本文方法的有效性。     

一类新型动态多目标鲁棒进化优化方法

传统动态多目标优化问题（Dynamic multi-objective optimization problems,DMOPs）的求解方法,通常需要在新环境下,通过重新激发寻优过程,获得适应该环境的Pareto最优解.这可能导致较高的计算代价和资源成本,甚至无法在有限时间内执行该优化解.由此,提出一类寻找动态鲁棒Pareto最优解集的进化优化方法.动态鲁棒Pareto解集是指某一时刻下的Pareto较优解可以以一定稳定性阈值,逼近未来多个连续动态环境下的真实前沿,从而直接作为这些环境下的Pareto解集,以减小计算代价.为合理度量Pareto解的环境适应性,给出了时间鲁棒性和性能鲁棒性定义,并将其转化为两类鲁棒优化模型.引入基于分解的多目标进化优化方法和无惩罚约束处理方法,构建了动态多目标分解鲁棒进化优化方法.特别是基于移动平均预测模型实现了未来动态环境下适应值的多维时间序列预测.基于提出的两类新型性能评价测度,针对8个典型动态测试函数的仿真实验,结果表明该方法得到满足决策者精度要求,且具有较长平均生存时间的动态鲁棒Pareto最优解.     

Multi-objective optimization of stochastic disassembly line balancing with station paralleling

One of the major activities performed in product recovery is disassembly. Disassembly line is the most suitable setting to disassemble a product. Therefore, designing and balancing efficient disassembly systems are important to optimize the product recovery process. In this study, we deal with multi-objective optimization of a stochastic disassembly line balancing problem (DLBP) with station paralleling and propose a new genetic algorithm (GA) for solving this multi-objective optimization problem. The line balance and design costs objectives are simultaneously optimized by using an AND/OR Graph (AOG) of the product. The proposed GA is designed to generate Pareto-optimal solutions considering two different fitness evaluation approaches, repair algorithms and a diversification strategy. It is tested on 96 test problems that were generated using the benchmark problem generation scheme for problems defined on AOG as developed in literature. In addition, to validate the performance of the algorithm, a goal programming approach and a heuristic approach are presented and their results are compared with those obtained by using GA. Computational results show that GA can be considered as an effective and efficient solution algorithm for solving stochastic DLBP with station paralleling in terms of the solution quality and CPU time.     

Uncertain location–allocation decisions for a bi-objective two-stage supply chain network design problem with environmental impacts

In the cases that the historical data of an uncertain event is not available, belief degree-based uncertainty theory is a useful tool to reflect such uncertainty. This study focuses on uncertain bi-objective supply chain network design problem with cost and environmental impacts under uncertainty. As such network may be designed for the first time in a geographical region, this problem is modelled by the concepts of belief degree-based uncertainty theory. This article is almost the first study on belief degree-based uncertain supply chain network design problem with environmental impacts. Two approaches such as expected value model and chance-constrained model are applied to convert the proposed uncertain problem to its crisp form. The obtained crisp forms are solved by some multi-objective optimization approaches of the literature such as TH, Niroomand, MMNV. A deep computational study with several test problems are performed to study the performance of the crisp models and the solution approaches. According to the results, the obtained crisp formulations are highly sensitive to the changes in the value of the cost parameters. On the other hand, Niroomand and MMNV solution approaches perform better than other solution approaches from the solution quality point of view.     

基于正交设计的多目标演化算法

提出一种基于正交设计的多目标演化算法以求解多目标优化问题(MOPs)．它的特点在于：(1)用基于正交数组的均匀搜索代替经典EA的随机性搜索,既保证了解分布的均匀性,又保证了收敛的快速性;(2)用统计优化方法繁殖后代,不仅提高了解的精度,而且加快了收敛速度;(3)实验结果表明,对于双目标的MOPs,新算法在解集分布的均匀性、多样性与解精确性及算法收敛速度等方面均优于SPEA;(4)用于求解一个带约束多目标优化工程设计问题,它得到了最好的结果——Pareto最优解,在此之前,此问题的Pareto最优解是未知的．     

A new efficient encoding/decoding procedure for the design of a supply chain network with genetic algorithms

Supply chain network (SCN) design is a strategic issue which aims at selecting the best combination of a set of facilities to achieve an efficient and effective management of the supply chain. This paper presents an innovative encoding–decoding procedure embedded within a genetic algorithm (GA) to minimize the total logistic cost resulting from the transportation of goods and the location and opening of the facilities in a single product three-stage supply chain network. The new procedure allows a proper demand allocation procedure to be run which avoids the decoding of unfeasible distribution flows at the stage of the supply chain transporting products from plants to distribution centers. A numerical study on a benchmark of problems demonstrates the statistical outperformance of the proposed approach vs. others currently available in literature in terms of total supply chain logistic cost saving and reduction of the required computation burden to achieve an optimal design.     

Preference-based evolutionary multi-objective optimization in ship weather routing

In evolutionary multi-objective optimization (EMO) the aim is to find a set of Pareto-optimal solutions. Such approach may be applied to multiple real-life problems, including weather routing (WR) of ships. The route should be optimal in terms of passage time, fuel consumption and safety of crew and cargo while taking into account dynamically changing weather conditions. Additionally it must not violate any navigational constraints (neither static nor dynamic). Since the resulting non-dominated solutions might be numerous, some user support must be provided to enable the decision maker (DM) selecting a single “best” solution. Commonly, multi-criteria decision making methods (MCDM) are utilized to achieve this goal with DM’s preferences defined a posteriori. Another approach is to apply DM’s preferences into the very process of finding Pareto-optimal solutions, which is referred to as preference-based EMO. Here the Pareto-set is limited to those solutions, which are compliant with the pre-configured user preferences. The paper presents a new tradeoff-based EMO approach utilizing configurable weight intervals assigned to all objectives. The proposed method has been applied to ship WR problem and compared with a popular reference point method: r-dominance. Presented results prove applicability and competitiveness of the proposed method to solving multi-objective WR problem.     

Design for Six Sigma through collaborative multiobjective optimization

This paper presents a global methodology for designing product for Six Sigma. First, we combine a feasibility-modeling technique with an interactive multiobjective algorithm taking into account the decision maker’s preferences (IMOP) to generate several Pareto-optimal solutions that maintain a probability of constraint satisfaction. These solutions are called reliable Pareto-optimal solutions.The solutions found by the algorithm fulfill as much as possible the decision makers’ requirements. Second, we develop a procedure for choosing a solution for implementation from among the reliable Pareto-optimal solutions generated by the algorithm. This procedure is based on the robust design and philosophy of Six Sigma. Finally, the critical characteristics are identified to help the managers develop the manufacturing system and its related control plans in order to achieve quality products. The proposed methodology is applied to vehicle crash-worthiness design optimization for side impact with structural weight and front door velocity under side impact as objectives.     

A novel nondominated sorting simplified swarm optimization for multi-stage capacitated facility location problems with multiple quantitative and qualitative objectives

Capacitated facility location problems (CFLPs) arise in the practical application of many supply chain networks that select a set of suppliers, plants, distribution centers, and customers. In general, the goal of CFLPs is to consider multiple critical performances that involve quantitative and qualitative factors, such as cost, transportation time, inventory, profit, and customer satisfaction, to obtain various perspectives from decision makers in most real-world applications. CFLP becomes increasingly complex and challenging when decision makers simultaneously consider both factors; however, offering comprehensive decisions is important. In this study, a novel solution based on simplified swarm optimization (SSO) and a nondominated sorting technique is proposed to provide Pareto-optimal solutions for enhancing search efficiency and solution quality. To yield feasible solutions, three repairer mechanisms, namely, random repair, cost-based, and utility-based mechanisms, are proposed to enhance the search efficiency and diversity of each population. A fuzzy analytic hierarchy process is used to calculate the weight of qualitative objectives. To evaluate the efficiency and effectiveness of the proposed algorithm, extensive experiments are conducted on benchmark and newly generated instances of the four stages of CFLPs. Then, results are compared with those of the nondominated sorting genetic algorithm-II, multi-objective SSO, and multi-objective particle swarm optimization reported from the literature. The computational results demonstrate that the proposed algorithm is highly competitive and performs well in terms of solution quality and computational time. The Pareto set in the investigated type of facility location problems leads to solutions that may better support decision-making.     

Bi-objective integrating sustainable order allocation and sustainable supply chain network strategic design with stochastic demand using a novel robust hybrid multi-objective metaheuristic

Sustainability has been considered as a growing concern in supply chain network design (SCND) and in the order allocation problem (OAP). Accordingly, there still exists a gap in the quantitative modeling of sustainable SCND that consists of OAP. In this article, we cover this gap through simultaneously considering the sustainable OAP in the sustainable SCND as a strategic decision. The proposed supply chain network is composed of five echelons including suppliers classified in different classes, plants, distribution centers that dispatch products via two different ways, direct shipment, and cross-docks, to satisfy stochastic demand received from a set of retailers. The problem has been mathematically formulated as a multi-objective optimization model that aims at minimizing the total costs and environmental effect of integrating SCND and OAP, simultaneously. To tackle the addressed problem, a novel multi-objective hybrid approach called MOHEV with two strategies for its best particle selection procedure (BPSP), minimum distance, and crowding distance is proposed. MOHEV is constructed through hybridization of two multi-objective algorithms, namely the adapted multi-objective electromagnetism mechanism algorithm (AMOEMA) and adapted multi-objective variable neighborhood search (AMOVNS). According to achieved results, MOHEV achieves better solutions compared with the others, and also crowding distance method for BPSP outperforms minimum distance. Finally, a case study for an automobile industry is used to demonstrate the applicability of the approach.     

Conservative multi-objective optimization considering design robustness and tolerance: a quality engineering design approach

Methods of multi-objective optimization are proposed to account for tolerance of design variable and variation in problem parameter. The post-optimization effort is initiated from deterministic Pareto-optimal solutions that were obtained from NSGA-II. The successive process to determine search directions and step sizes toward conservative multi-objective solutions was conducted by design of experiments to determine the worst design that had the highest constraint violation. The signal-to-noise (S/N) ratio was also employed to represent the robustness of constrained objective functions under parameter variation. Structural optimization was explored to accommodate both design tolerance and parameter variation and further apply S/N ratio in conservative multi-objective optimization.     

A random search heuristic for a multi-objective production planning

Real world production planning is involved in optimizing different objectives while considering a spectrum of parameters, decision variables, and constraints of the corresponding cases. This comes from the fact that production managers desire to utilize from an ideal production plan by considering a number of objectives over a set of technological constraints. This paper presents a new multi-objective production planning model which is proved to be NP-Complete. So a random search heuristic is proposed to explore the feasible solution space with the hope of finding the best solution in a reasonable time while extracting a set of Pareto-optimal solutions. Then each Pareto-optimal solution is considered as an alternative production plan in the hand of production manager. Both the modeling and the solution processes are carried out for a real world problem and the results are reported briefly. Also, performance of the proposed problem-specific heuristic is verified by comparing it with a multi-objective genetic algorithm on a set randomly generated test data.