Multi-objective Integer Programming Research Articles

An efficient branch‐and‐bound algorithm to optimize a function over a nondominated set

AbstractThis study introduces an algorithm based on the branch‐and‐bound approach for optimizing a main function over the nondominated set of a multiobjective integer programming (MOIP) problem. Initially, is optimized within the feasible solution set of the MOIP. A new efficiency test combining Benson's test with is then developed using an auxiliary optimization program. This program provides both an efficient solution and a lower bound for . Moreover, this solution is the best one for when compared to its alternative solutions for MOIP. Subsequently, efficient cuts are incorporated into the criteria space to eliminate dominated points. Furthermore, the algorithm is tailored to handle scenarios where the objective involves optimizing a linear combination of multiobjective programming criteria over the nondominated set. The study concludes by showcasing the superior performance of the proposed two algorithms through comparison with existing approaches on well‐known problem instances from the literature.

A network design of a canola-to-biodiesel supply chain by considering the water-energy-food-land-ecosystem-undesirable climate change nexus

Global warming, adverse environmental consequences, and the rapid depletion of oil resources are some of the critical challenges that humanity presently confronts due to the utilization of fossil fuels. Replacing fossil fuels with renewable sources of energy such as biofuel is crucial for a clean environment. In addition to its numerous advantages, the production of biofuels has detrimental effects on natural resources as well. Water, energy, food, and land are all required to produce food and energy, which are inextricably linked as the essential inputs for agriculture and fuel production. Undesirable climate change and its impacts on ecosystems are also consequences of this process. This study offers a two-phase approach to designing a canola-to-biodiesel supply chain. The first phase is concerned with finding appropriate sites for canola cultivation in accordance with climatic, social, economic and design criteria. To achieve an ideal solution, the fuzzy best-worst method andthe fuzzy technique of order preference similarity to the ideal solution are simultaneously used. In the second phase, a new multi-objective integer programming model that involves a water-energy-food-land-ecosystem-undesirable climate change nexus is introduced. The suggested model enhances crop harvest from desirable agricultural areas, maximizes the profit, preserves ecosystem values, mitigates undesirable climate changes, and reduces water and energy consumption. An interactive fuzzy method is used to solve the model, and the efficiency of the proposed model is evaluated through a real case study for Iran. Sensitivity analyses are also performed on the key parameters. According to the findings, production activities are connected with the increase of network costs, adverse changes in climate, and increased use of energy. Thus, using high-yield catalysts, increasing the production scale, complying with environmental regulations, receiving government subsidies, investing in research to improve production technologies, and identifying cost-saving strategies can minimize harmful environmental effects and reduce biodiesel production costs, thereby boosting the profit. Furthermore, given Iran’s water scarcity and reduced rainfall, increasing the canola yield per hectare might lead to less land use and water consumption for irrigation. Utilizing specialized machinery, enhancing farmers’ skills in canola cultivation, supplying seeds on time, planting promptly, providing optimal nutrition, and promoting pest management are all effective methods to boost canola yields. The insights provided in this study can help managers in the canola-based biodiesel production sector make better decisions.

On the strength of Lagrangian duality in multiobjective integer programming

AbstractThis paper investigates the potential of Lagrangian relaxations to generate quality bounds on non-dominated images of multiobjective integer programs (MOIPs). Under some conditions on the relaxed constraints, we show that a set of Lagrangian relaxations can provide bounds that coincide with every bound generated by the convex hull relaxation. We also provide a guarantee of the relative quality of the Lagrangian bound at unsupported solutions. These results imply that, if the relaxed feasible region is bounded, some Lagrangian bounds will be strictly better than some convex hull bounds. We demonstrate that there exist Lagrangian multipliers which are sparse, satisfy a complementary slackness property, and generate tight relaxations at supported solutions. However, if all constraints are dualized, a relaxation can never be tight at an unsupported solution. These results characterize the strength of the Lagrangian dual at efficient solutions of an MOIP.

A simple, efficient and versatile objective space algorithm for multiobjective integer programming

In the last years a multitude of algorithms have been proposed to solve multiobjective integer programming problems. However, only few authors offer open-source implementations. On the other hand, new methods are typically compared to code that is publicly available, even if this code is known to be outperformed. In this paper, we aim to overcome this problem by proposing a new state-of-the-art algorithm with an open-source implementation in C++. The underlying method falls into the class of objective space methods, i.e., it decomposes the overall problem into a series of scalarized subproblems that can be solved with efficient single-objective IP-solvers. It keeps the number of required subproblems small by avoiding redundancies, and it can be combined with different scalarizations that all lead to comparably simple subproblems. Our algorithm bases on previous results but combines them in a new way. Numerical experiments with up to ten objectives validate that the method is efficient and that it scales well to higher dimensional problems.

A joint planning strategy for EV charging system towards net-zero transportation electrification

Electric vehicles (EVs) are pivotal in the progression toward sustainable transportation. However, a growing body of recent studies has challenged the often-assumed notion that electric vehicles are often considered to be net-zero emissions. These studies suggest that electric vehicles could potentially generate substantial emissions due to their reliance on fossil fuels to generate electricity, especially when receiving fast charging services. This phenomenon has been mitigated with the use of renewable energy sources in the power system. However, due to the intermittent nature of renewable energy sources, additional requirements have arisen. To address this situation, this paper presents an interdisciplinary study of collaborative fast-charging stations (FCS) and distributed renewable energy planning based on the interactions between traffic and power networks in order to increase the adoption rate of electric vehicles and reduce emissions caused by traffic and power systems. The locations and sizing of fast-charging stations, distributed photovoltaic generation, and renewable energy power supply for electric vehicles are determined in the proposed strategy through multi-objective integer programming. The proposed planning strategy considers heterogeneous electric vehicle driving range constraints, the operation security of the power distribution system, the reliability and expansion of the current power distribution, as well as different types of air pollutant classifications resulting from the charging behaviors. To guarantee the feasibility and accuracy of the planning results, an interactive algorithm is applied to solve the multi-objective integer model. Several cases are conducted to validate the proposed approach. The results demonstrate that the proposed planning strategy has good performance in the planning of fast-charging stations and distributed renewable energy integration.

A Multi-Objective Optimization Method for Maritime Search and Rescue Resource Allocation: An Application to the South China Sea

To effectively address the increase in maritime accidents and the challenges posed by the trend toward larger ships for maritime safety, it is crucial to rationally allocate the limited maritime search and rescue (MSAR) resources and enhance accident response capabilities. We present a comprehensive method for allocating MSAR resources, aiming to improve the overall efficiency of MSAR operations. First, we use long short-term memory to predict the number of future accidents and employ the K-medoids algorithm to identify the accident black spots in the studied area. Next, we analyze the multi-constraint conditions in the MSAR resource allocation process. A multi-objective integer programming model is constructed to minimize the response time and allocation cost. Finally, we use the non-dominated sorting genetic algorithm II (DNSGA-II) with Deb’s rules to solve the model, and we propose a multi-attribute decision optimization-based method for MSAR resource allocation. We found that the DNSGA-II exhibits better convergence and generates higher-quality solutions compared to the NSGA-II, particle swarm optimization (PSO), and enhanced particle swarm optimization (EPSO) algorithms. Compared with the existing MSAR resource emergency response system, the optimized scheme reduces the response time and allocation cost by 11.32% and 6.15%, respectively. The proposed method can offer decision makers new insights when formulating MSAR resource allocation plans.

Multi-product disassembly line balancing optimization method for high disassembly profit and low energy consumption with noise pollution constraints

Remanufacturing attains much attention from both industrial and academic sectors due to its beneficial roles in energy conservation and environment protection, where disassembly is a crucial part. To reach the comprehensive sustainability and global optimization of disassembling multiple end-of-life products, this paper suggests a multi-objective multi-product disassembly line balancing problem with considering disassembly profit, energy consumption and noise pollution simultaneously. According to the natures of the problem under consideration, a multi-objective integer programming model is constructed. Its goals are to reach maximum disassembly profit and realize minimum energy consumption while observing noise pollution requirements and resource constraints. Accordingly, a multi-objective group teaching optimization algorithm is specially devised. In it, rank and crowding distance methods are employed to partition the population into two groups. Moreover, precedence preserving crossover and mutation methods are severally used on the two groups to realize the teacher phase. Furthermore, to achieve the student phase, an adaptive local search method is applied to refine solutions in an external archive, and thus its exploitation ability is enhanced. By executing contrast experiments between the devised approach and its powerful competitors on a set of real-world test instances, the experimental results validate that it has highly-adaptive and well-superior performance in tackling the problem of concern.

A simultaneous time and fuel minimization robust possibilistic multiobjective programming approach for truck-sharing scheduling in container terminals under uncertainty

The issue of integrated scheduling and sequencing operation of unloading and loading equipment in container ports has been one of the most important issues concerning time efficiency. In addition, with the emergence of green harbor concepts, the inclusion of criteria for minimizing energy consumption, fuel and emission reduction are among the other issues that have been noticed by planners in the field of energy efficiency. Furthermore, due to the complexity and scope of activities of a container terminal, uncertainty in operational parameters such as transportation time, time of readiness and entry of work into the system and the velocity of the transportation fleet are inevitable in this operational environment. Therefore, this research with the aim of sharing trucks among loading and unloading equipment, proposes a robust multi-objective integer programming model for the synchronized scheduling of truck operations with other handling equipment to decrease the fuel consumption of trucks and the flow time of containers, considering the uncertainty in operational parameters as fuzzy numbers. To find the Pareto solutions for this model, the ε-Constraint technique is employed. Finally, the performance of the model in deterministic and uncertain modes is evaluated, compared and analyzed employing the inputs gathered from Shahid Rajaei port. The findings demonstrate that using this model will result in a substantial decrease in both fuel consumption and flow time of containers in comparison to the current procedure. Additionally, results will demonstrate the extent to which the terminal's fuel and time consumption will increase under conditions of uncertainty in operational parameters when the optimal plans derived from the robust model are implemented.

Relaxations and duality for multiobjective integer programming

Relaxations and duality for multiobjective integer programming

A multi-objective joint optimisation method for simultaneous part family formation and configuration design in delayed reconfigurable manufacturing system (D-RMS)

ABSTRACT In the era of Industry 4.0, the demand fluctuation has become fiercer due to the characteristics of diversification, customisation, and uncertainty. Reconfigurability of manufacturing systems has been proven to be a useful and necessary feature when it comes to handling demand uncertainty. This feature can be achieved through the implementation of reconfigurable manufacturing system (RMS) and delayed reconfigurable manufacturing system (D-RMS). D-RMS is a subclass of RMS that focuses primarily on improving the convertibility of the manufacturing system. The two main phases involved in implementing D-RMS are part family formation and configuration design. Therefore, we proposed a multi-objective joint optimisation method of part family formation and configuration design according to the philosophy of D-RMS. Firstly, we develop a multi-objective joint optimisation model that takes into account investment cost, reconfiguration cost, similarity coefficient, and delayed reconfiguration to optimise the part family and configuration of D-RMS simultaneously. Three types of machine tools namely dedicated machine tools, flexible machine tools, and reconfigurable machine tools are considered in the optimisation model. Secondly, the non-dominated sorting genetic algorithm-III (NSGA-III) is adopted to solve the proposed multi-objective integer programming problem. Finally, numerical experiments are presented to demonstrate the effectiveness of the proposed multi-objective joint optimisation method.

Layout Optimization of Construction Waste Recycling Facilities for Development of New Urban Areas from Centralized and Decentralized Processing Collaboration Perspective

The generation, treatment and safe disposal of construction waste have become major problems in new urban areas of China. And the existing research has not been carried out based on the recycling characteristics of scattered, low efficiency and large negative externalities of construction waste processing in the development of new areas. This research studies the optimal location of construction waste centralized and decentralized processing facilities based on the characteristics of the generation end and the demand side of new urban areas. Firstly, grey relational analysis method is used to determine the location of centralized processing facilities. Then, a multi-objective integer programming model is built by comprehensively considering cost, side effect on society and carbon emissions during transportation, and the optimal location and dynamic adjustment of decentralized processing facilities are determined. Finally, taking the development of a new area in China as an example, the network of construction waste recycling facilities is established to verify the effectiveness of this model. The research results show that the layout strategy formulated with this method can maintain the dynamic optimal layout of the system from the perspective of the whole region, and realize the utilization of construction waste with low cost, low transport carbon emission and low social negative impact. The analysis method system established in this paper is universal, which can provide reference for the relevant government departments to plan the construction waste recycling facility network of the area, and provide guidance for the investment decisionmaking and operation management of related enterprises.

Parallel and Collaborative Passenger Flow Control of Urban Rail Transit Under Comprehensive Emergency Situation

The artificial cooperative control methods are used to fully utilize the train capacity and reduce the passenger delays by coordinating the number of entering and boarding passengers at each station, but it ignores the potential risk associated with the aggregation attributes of passengers in the station under emergency and the unfairness caused by the imbalance of travel opportunities for different origin–destination (OD) passengers. A new parallel and cooperative control model under comprehensive emergency situation is constructed from the perspective of guaranteeing travel safety, improving travel fairness, and increasing travel efficiency to generate passenger flow control strategies. The model aims to maximize passenger turnovers, minimize passenger delays and reduce the difference in the ratio of stranded passengers at stations. The complex multi-objective integer programming problem involving nonlinear equation constraints is difficult to solve numerically. A dynamic programming model taking train sequence as stage variable is then constructed as well as an NSGA-II algorithm with multiple improved strategies (MIS-NSGA-II) is proposed to solve the model. Numerical experiments on a line of Beijing Rail Transit show that the MIS-NSGA-II outperforms the traditional NSGA-II, and that increasing the overload coefficient of the train and the tolerance coefficient of each area in the station can further alleviate passenger congestion and promote the operation efficiency. Compared with other control strategies mentioned in this study, the proposed control model from a comprehensively perspective, which can ensure the travel safety and promote the travel efficiency while forcefully decongesting the bottleneck areas and improving the fairness of passenger travel.

Novo rešenje problema višekriterijumskog celobrojnog programiranja pomoću višekriterijumske optimizacije zasnovane na verovatnoći

Introduction/purpose: In this paper, a new solution for solving a multiobjective integer programming problem with probabilistic multi - objective optimization is formulated. Furthermore, discretization by means of the good lattice point and sequential optimization are employed for a successive simplifying treatment and deep optimization. Methods: In probabilistic multi - objective optimization, a new concept of preferable probability has been introduced to describe the preference degree of each performance utility of a candidate; each performance utility of a candidate contributes a partial preferable probability and the product of all partial preferable probabilities deduces the total preferable probability of a candidate; the total preferable probability thus transfers a multi-objective problem into a single-objective one. Discretization by means of the good lattice point is employed to conduct discrete sampling for a continuous objective function and sequential optimization is used to perform deep optimization. At first, the requirements of integers in the treatment could be given up so as to simply conduct above procedures. Finally, the optimal solutions of the input variables must be rounded to the nearest integers. Results: This new scheme is used to deal with two production problems, i.e., maximizing profit while minimizing pollution and determining a purchasing plan for spending as little money as possible while getting as large amount of raw materials as possible. Promising results are obtained for the above two problems from the viewpoint of the probability theory for simultaneous optimization of multiple objectives. Conclusion: This method properly considers simultaneous optimization of multiple objectives in multi-objective integer programming, which naturally reflects the essence of multi-objective programming, and opens a new way of solving multi-objective problems.

Optimizing the ground intra-city express delivery network: An integrated multiple centrality assessment, multi-criteria decision-making, and multi-objective integer programming model

Optimization of an intra-city express delivery network from three to two levels is of great interest to suppliers and customers for reducing costs and improving service efficiency. One feasible solution is to identify critical nodes in the three-level network and upgrade them as transshipment facilities in the two-level one. However, traditional optimization models seldom combine empirical business data, composite metrics, and objective evaluation rules. We proposed an approach integrating empirical data, multi-criteria decision-making methods based on the real-world application of the SF Express Chengdu branch. We also developed a mathematical optimization model using statistical and operations management techniques combined with logistics expertise for a location decision. First, the appropriateness of each service point as a candidate transshipment facility is evaluated from internal and external perspectives by applying multiple centrality assessment from complex network theory and fuzzy Technique for Order Preference by Similarity to an Ideal Solution, respectively. Second, 16 candidate transshipment facilities are selected by combining these two ways. Then, a multi-objective integer programming model is built to obtain the optimal number, locations of transshipment facilities, and the corresponding service points covered by each transshipment facility. Using this multi-methodologic approach, we show that the optimized two-level network is economically feasible and simply applicable, with the total cost and average delivery time reduced by 18.41% and 6 h, respectively. This article is of practical significance and provides an important reference for optimizing ground express service networks for other large cities.

Finding preferred solutions under weighted Tchebycheff preference functions for multi-objective integer programs

Many interactive approaches in multi-objective optimization assume the existence of an underlying preference function that represents the preferences of a decision maker (DM). In this paper, we develop the theory and an exact algorithm that guarantees finding the most preferred solution of a DM whose preferences are consistent with a Tchebycheff function for multi-objective integer programs. The algorithm occasionally presents pairs of solutions to the DM and asks which one is preferred. It utilizes the preference information together with the properties of the Tchebycheff function to generate solutions that are candidates to be the most preferred solution. We test the performance of the algorithm on a set of three and four-objective combinatorial optimization problems.

Order dispatch optimization with considering flexible one-to-three matching strategy under hybrid ride-hailing service modes

ABSTRACT In practice, the ride-hailing platform generally provides two products: express services and ridesplitting services, and driver resources are usually shared across these two products on the supply side, i.e., hybrid ride-hailing mode. This study tackles the order dispatch problem with considering flexible one-to-three matching strategy under hybrid modes. In this problem, if a driver serves express requests, only one order can be matched with; if a driver serves ridesplitting requests, this driver can be matched with at most three orders simultaneously or sequentially. Considering one-to-three matching, en-route matching, and multiple passengers per order, this problem is formulated as a multi-objective integer programming, a hierarchical approach is adopted to address it. Using the ride-hailing trip data, we examine characteristics of this problem, such as the impact of service priority and the benefits of one-to-three matching. Finally, some beneficial management implications are proposed to improve the operation of the ride-hailing service.

Exact Methods for Multi-Objective Integer Nonlinear Programming

Multi-objective integer nonlinear programming (MOINLP) problems are multi-objective integer programming problems with at least one nonlinear objective function or constraint. To date, MOINLP problem has not been exactly solved. Although traditional ε-constraint method can be used to solve MOINLP problem, the obtained solution may not be Pareto-optimal. To overcome this shortcoming, a basic ε-constraint method (BEM) is proposed to solve MOINLP problem exactly. However, the time complexity of BEM is as high as O ( ( p − 1 ) M 2 N ) , where p, M, and N are the numbers of objectives, Pareto-optimal solutions, and feasible solutions, respectively. For this reason, an improved BEM (IBEM) is developed whose time complexity is O ( M 2 N ) . That is, the time complexity of IBEM for solving MOINLP problem is equal to solving the single-objective one. Finally, to avoid using all the feasible solutions (N) in obtaining each Pareto-optimal solution, three methods to eliminate dominated solutions effectively are used before performing IBEM. The test results illustrate that our method can not only solve MOINLP problem exactly but also has high efficiency.

A Pareto-based discrete particle swarm optimization for parallel casting workshop scheduling problem with fuzzy processing time

The parallel casting workshop scheduling problem is essentially a type of unrelated parallel machine scheduling (UPMS) problem with fuzzy order processing time. In this paper, a tri-objective parallel casting workshop optimization model is investigated to minimize the order advance/delay penalty, makespan, and workload imbalance, and a formula of the solution scale with the number of orders and workshops is derived. In line with the multi-objective integer programming model, a new Pareto-based discrete particle swarm optimization (PDPSO) is proposed. In this algorithm, the particles are encoded with integers. A novel two-phase heuristic is developed for integer-encoded particle updates. The first phase determines the order assignment through a weighting vote, and the second phase sorts the orders according to their dominance in each workshop. Every particle’s update is simultaneously guided by the best compromise solution selected from the personal and global Pareto solution. Meanwhile, the Pareto simulated annealing strategy is introduced in PDPSO to enhance its exploration ability. Multiple simulations under different scales are tested to validate the effectiveness of our proposed model and algorithm. The results show that the proposed PDPSO outperforms the other two multi-objective algorithms and has large advantages in finding the Pareto frontier (PF) with an acceptable solution number and solving speed, especially for small and medium-sized instances.

A Bifactor Approximation Algorithm for Cloudlet Placement in Edge Computing

Emerging applications with low-latency requirements such as real-time analytics, immersive media applications, and intelligent virtual assistants have rendered Edge Computing as a critical computing infrastructure. Existing studies have explored the cloudlet placement problem in a homogeneous scenario with different goals such as latency minimization, load balancing, energy efficiency, and placement cost minimization. However, placing cloudlets in a highly heterogeneous deployment scenario considering the next-generation 5G networks and IoT applications is still an open challenge. The novel requirements of these applications indicate that there is still a gap in ensuring low-latency service guarantees when deploying cloudlets. Furthermore, deploying cloudlets in a cost-effective manner and ensuring full coverage for all users in edge computing are other critical conflicting issues. In this article, we address these issues by designing a bifactor approximation algorithm to solve the heterogeneous cloudlet placement problem to guarantee a bounded latency and placement cost, while fully mapping user applications to appropriate cloudlets. We first formulate the problem as a multi-objective integer programming model and show that it is a computationally NP-hard problem. We then propose a bifactor approximation algorithm, ACP, to tackle its intractability. We investigate the effectiveness of ACP by performing extensive theoretical analysis and experiments on multiple deployment scenarios based on New York City OpenData. We prove that ACP provides a (2,4)-approximation ratio for the latency and the placement cost. The experimental results show that ACP obtains near-optimal results in a polynomial running time making it suitable for both short-term and long-term cloudlet placement in heterogeneous deployment scenarios.

Pigeon-inspired fuzzy multi-objective task allocation of unmanned aerial vehicles for multi-target tracking

In this paper, a pigeon-inspired fuzzy multi-objective optimization algorithm is proposed for task allocation of multiple unmanned aerial vehicles tracking multiple ground targets in urban environment. Firstly, a multi-objective integer programming of task allocation, involving minimum total flight distance, best task allocation balance and minimum completion time, is established. Secondly, fuzzy two-phase optimization based on the relaxed order of desirable satisfactory degrees is proposed to formulate mixed integer programming regarding the linguistic importance preference of objectives. Then, an adaptive pigeon-inspired algorithm combined with auction mechanism is proposed to solve the optimization model. The position of pigeon is defined as the bidding price given by unmanned aerial vehicle for target. To satisfy the constraints and avoid existence of inferior pigeons, the auction mechanism is designed to decode the pigeon position into a feasible task allocation scheme. Finally, by comparing with the conventional particle swarm optimization, simulations validate the effectiveness and efficiency of the proposed method.