Bi-objective Integer Programming Model Research Articles

A modified bi-objective NSGA-II approach to sustainability in reconfiguration planning of dynamic cellular manufacturing systems

Manufacturing plant layouts are developed to facilitate optimal process flow. Modern manufacturing systems must meet present production demands and be adaptable to changes in process flow in the future. Dynamic Cellular Manufacturing Systems (DCMS) increase the flexibility of layouts by reconfiguring cell structure and equipment distribution, to effectively adjust part routings for optimal process flow. Frequent reconfiguring of plant layout may not always be feasible or economical, however, when new product releases are planned, reconfiguring the plant layout to optimise the work-flow may be extremely beneficial. This paper presents a Non-dominated Sorting Genetic Algorithm (NSGA-II) approach to solving a DCMS problem in a sustainable, and responsible manner. A bi-objective integer programming model was developed over multiple planning horizons with fluctuating product demands. This model aims to achieve sustainability by reducing the cost of production, mitigating the environmental impact of production, and minimise negative social impacts on labourers that work in such environments. A penalty function approach was used to enforce the model constraints during optimisation. This study details trade-offs between the economic factors of a DCMS, the environmental implications of reconfiguring such a system, and the social impacts of reconfigurations on the workforce.

Optimizing Rack Locations in the Mobile-Rack Picking System: A Method of Integrating Rack Heat and Relevance

The flexible movement of racks in the mobile-rack picking system (MRPS) significantly improves the picking efficiency of e-commerce orders with the characteristics of “one order multi–items” and creates a challenging problem of how to place racks in the warehouse. This is because the placement of each rack in the MRPS directly influences the distance that racks need to be moved during order picking, which in turn affects the order picking efficiency. To handle the rack location optimization problem (RLOP), this work introduces a novel idea and methodology, taking into account the heat degree and the relevance degree of racks, to enhance the efficiency of rack placements in the MRPS. Specifically, a two-stage solution strategy is implemented. In stage 1, an integer programming model (Model 1) is developed to determine the heat and relevance degree of racks, and it can be solved quickly by the Gurobi. Stage 2 entails developing a bi-objective integer programming model (Model 2) with the objective to minimize the travel distances of robots in both heavy load and no-load conditions, using the rack heat and relevance degree as inputs. In light of the challenge of decision coupling and the vast solution space in stage 2, we innovatively propose two lower bounds by slacking off the distance between storage locations. A matheuristic algorithm based on Benders decomposition (MABBD) is designed, which utilizes Benders-related rules to reconstruct Model 2, introduces an enhanced cut and an improved optimal cut with RLOP characteristics, and designs the warm start strategy and the master variable fixed strategy. Given the substantial size of real-life problems, the Memetic algorithm (MA) is specifically devised to address them. Instances of varying sizes are also employed to validate the science and efficacy of the model and algorithm.

An integrated integer programming and forecasting model with geographic information systems for spray operations of West Nile disease insecticides

Public health agencies and medical organizations have increased their relentless efforts to control the outbreaks of mosquitoes causing West Nile disease epidemics with the re-emergence of deadly disease in the United States. Such measures include a large-scale spray operation of insecticides that can counter the population growth of mosquitoes. This study proposes a business analytics tool that helps determine the optimal number and location of airfields housing insecticide-spraying aircraft, the type (mode) and the number of insecticide-spraying airplanes, and their flight patterns. This study also identifies factors that can enhance and hinder the efficiency of insecticide spray operations by combining a bi-objective integer programming model with a geographic information system and forecasting models within a business analytics framework for the first time.

A New Product Configuration Model for Low Product Cost and Carbon-Neutral Expenditure

In the background of global carbon-neutral requirements, enterprises need to control carbon emissions in the process of product lifecycles in order to gain market competitive advantages. Previous product configuration studies, mostly focused on minimizing carbon dioxide emissions, have ignored the issue of carbon-neutral costs. This study quantifies the product costs borne by enterprises and the carbon-neutral cost borne by the government, respectively. A carbon-neutral cost model for suppliers, enterprises, customers, and recycling plants in the whole life cycle of products was constructed. The whole life cycle carbon emissions and the unit carbon removal costs were taken into account in the carbon-neutral cost model. By minimizing product and carbon-neutral costs, a bi-objective integer programming model was constructed. The NSGA-II algorithm was introduced to solve the Pareto front of the model. The feasibility and effectiveness of this method were then illustrated through a case study and results comparison. It showed that, compared with the scheme of carbon emissions reduction, the optimization scheme with carbon-neutral costs as the object had a significant change. Integrating carbon-neutral costs into product development activities was effective in reducing the enterprise’s product cost and the government’s financial expenditure on carbon removal simultaneously. The proposed model could provide a win–win product configuration scheme for the government and enterprises.

An integrated Multi-Criteria Decision Making and Multi-Choice Conic Goal Programming approach for customer evaluation and manager assignment

Companies adopt a customer-centric approach to maintain their competitive position or dominate the market. In this context, managers are pivotal in navigating the competitive markets in accomplishing these goals. Setting targets and conducting performance evaluations are vital for managers’ long-term success. This study aims to optimise customer evaluation and manager assignment in the defence industry. An integrated approach utilising the Analytical Hierarchy Process (AHP) and the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is employed to conduct a comprehensive market analysis and conduct this evaluation. A bi-objective integer programming model is formulated to maximise​ total integrated scores and potential returns. The efficient number of visits is determined by combining Multi-Criteria Decision Making (MCDM) and Multi-Choice Conic Goal Programming (MCCGP). This study offers an objective alternative to traditional survey-based evaluations by proposing a novel decision support system for Goal-Based Performance Appraisals (GBPA) of managers. The proposed approach yields performance values that are considerably more reliable than questionnaire results. Moreover, the proposed decision support system can effectively benefit employees across diverse sectors, extending beyond the confines of the defence industry.

Landmark selection and path planning for unmanned vehicles with position error corrections

Unmanned vehicles have been widely used for sensing, surveillance, surveying, communication, transportation, and many other purposes. Navigation is always an essential operational function in the course of their movements. In the case of the absence of Global Positioning System (GPS) signals, the positioning need of an unmanned vehicle is generally empowered by its onboard Inertial Navigation System (INS). An inherent deficiency of INS, however, is that position errors inevitably occur and will gradually accumulate over time. A feasible approach to correcting errors is to use the fixed reference points with known coordinates in the cruising environment, which we call landmarks, to calibrate the vehicle’s position when it passes by. The vehicle must choose a path along which a sufficient number of landmarks can serve the position calibration requirement. In such a context, this paper introduces a joint landmark selection and path planning problem for unmanned vehicles navigating in two-dimensional (2D) and three-dimensional (3D) environments, in which the vehicles’ position drifting errors need to be rectified through communicating with preset landmarks. To characterize and tackle the problem, a set of new types of subpaths are defined and a bi-objective integer programming model using this building block is formulated for simultaneously minimizing the landmark deployment cost and the vehicle cruise distance. An efficient dynamic programming algorithm, which features a multi-criterion label-correcting form, is developed for searching for the Pareto-optimal solution set of landmark and path selections. Numerical and computational analyses are conducted in both 2D and 3D scenarios, in which a pair of origin and destination places are typically hundreds of kilometers apart and 4 datasets of different scales including 500, 1,000, 1,500, and 2,000 candidate landmarks are randomly generated. A sensitivity analysis reveals the impact of different position drifting increments caused by external environment interferences, varying correction abilities, and heterogeneous landmark deployment costs on the solution behavior. The influence of the number of candidate landmarks, position error limit, and error increment rate on computing time is also evaluated. All these test results conclude that the algorithm is capable of finding in seconds the entire set of Pareto-optimal landmarks and paths in real-world applications.

Multi-depot vehicle scheduling with multiple vehicle types on overlapped bus routes

In China, overlapped routes from multiple bus lines are widespread. Some buses are overcrowded while others have few passengers. The goal of this study is to reduce onboard passenger crowding without wasting bus capacity. This paper examines an extended multiple-depot vehicle scheduling problem (VSP) with multiple vehicle types. First, a bi-objective integer programming model was created. The trade-off between operating cost and passenger travel cost was established and the crowding cost due to different vehicle sizes was especially considered. Then, the capacity restraint incremental assignment algorithm and a vehicle scheduling method with multiple vehicle types were developed to find Pareto-optimal solutions that considered vehicle capacity constraints. Finally, the proposed methodology was validated using a real-world transit network in Beijing, China. The optimization results from about 1.18 million bus card swiping records show that our model and solution algorithm can effectively reduce the passenger crowding cost and avoid the waste of transport capacity.

Facility planning and schedule design in the pandemic: Eliminating contacts at construction workplace

The construction industry has been severely affected by the COVID-19 pandemic and the associated restrictions on person-to-person contacts issued by the government. A construction site usually has a high number of workers working at the same time; therefore, the question of how to ensure their safety during the pandemic—that is, how to protect them from getting infected—has become an urgent problem. In this study, we propose a bi-objective integer programming model to establish the optimal schedule plan under COVID-19 regulations. We develop a solution method and conduct numerical experiments to solve and validate our model. The optimal schedule plan can avoid contacts between workers of different groups while minimizing the total costs of complying with government policy. Our proposed model can be applied in practice to help project managers establish a reasonable and cost-effective schedule plan. This study contributes to reducing the operating costs of contractors and protecting the health of construction workers.

A TWO-STAGE MATHEMATICAL PROGRAMMING MODEL FOR DISTRIBUTED PHOTOVOLTAIC PROJECT PORTFOLIO SELECTION WITH INCOMPLETE PREFERENCE INFORMATION

With the rapid growth of the solar photovoltaic (PV) market, many distributed PV power projects are introduced to the market. Selecting a rational project investment portfolio is a complex and challenging task for energy enterprises as both financial and non-financial factors of projects are needed to be considered under limited information and resources. This study presents a two-stage hybrid multi-attribute decision-making and integer programming model for distributed PV project portfolio selection. In Stage I, a multiple attribute group decision-making method based on mathematical programming is used to evaluate the non-financial value of projects under incomplete preference information. Compensative weighted averaging operators with an adjustable parameter are utilized to capture the subjective attitudinal character of an expert in the aggregation process. Then, a rank acceptability index is developed to measure each project’s group support degree in non-financial dimension. In Stage II, a bi-objective integer programming model is constructed to optimize project portfolios, which considers both financial and non-financial values of projects under resource, carbon emission and other strategic constraints. The applicability and effectivity of the proposed approach are demonstrated by a case study of a distributed PV project portfolio selection.

Bi-objective operation optimization in multi-shuttle automated storage and retrieval systems to reduce travel time and energy consumption

The automated warehouses currently in widespread use maintain highly efficient operations at the cost of consuming a large amount of energy. This study first considers the problem of operation optimization in multi-shuttle automated storage and retrieval systems (AS/RSs) to reduce both time and energy consumption. A time and energy bi-objective integer programming model is proposed to jointly optimize the storage/retrieval (S/R) location assignment and S/R scheduling for a multi-shuttle AS/RS. The adaptive variable neighbourhood search algorithm and Lagrangian relaxation algorithm are developed to tackle large-size instances. The results demonstrate that the bi-objective model is effective at finding the compromise solution considering travel time and energy consumption simultaneously. The decrease in travel time is obtained at the expense of increasing energy consumption and vice versa. The study provides valuable insights for warehouse managers to operate a time-efficient and energy-efficient automated warehouse.

Two-Echelon Multidepot Logistics Network Design with Resource Sharing

Resource sharing within a logistics network offers an effective way to solve problems resulting from inefficient and costly operations of individual logistics facilities. However, the existing analysis of resource sharing and profit allocation is still limited. Therefore, this study aims to model resource sharing in two-echelon delivery and pickup logistics networks to improve the overall efficiency and decrease the total network operating cost. A bi-objective integer programming model is first proposed for two-echelon collaborative multidepot pickup and delivery problems with time windows (2E-CMDPDTW) to seek the minimization of operating costs and number of vehicles. Integrating a customer clustering algorithm, a greedy algorithm, and an improved nondominated sorting genetic algorithm-II (Im-NSGA-II), a hybrid method is then designed to handle the 2E-CMDPDTW model. The customer clustering and the greedy algorithms are employed to generate locally optimized initial solutions to accelerate the calculating velocity and guarantee the diversity of feasible solutions. The Im-NSGA-II combines the order crossover operation and the polynomial mutation process to find the optimal solution of the 2E-CMDPDTW. The comparative results show that the proposed hybrid method outperforms the NSGA-II and the multiobjective genetic algorithm. Furthermore, a Shapley value method is used for allocating total profits of established alliances and finding an optimal coalition sequence of the logistics facilities joining alliances based on the strictly monotonic path strategy. Finally, a case study of 2E-CMDPDTW in Chongqing China is conducted to validate the feasibility. Results indicate that this study contributes to long-term partnerships between logistics facilities within multi-echelon logistics networks in practice and contributes to the long-term sustainability of urban logistics pickup and delivery networks’ development.

Green and sustainable tunnel maintenance activities scheduling under uncertainty

The maintenance of tunnel infrastructure is fundamental to the reliability, safety, and efficiency of tunnel operations. However, environmental deterioration has dramatically changed maintenance scheduling of tunnel infrastructure. Therefore, the trade-off between maintenance cost and environmental impact is crucial when formulating maintenance schedules. This paper extends the preventive maintenance scheduling problem (PMSP) from three perspectives: social impact, environmental impact, and unexpected maintenance over the entire planning horizon. We then propose a stochastic bi-objective integer programming model to minimize the total cost and CO2 emissions of tunnel maintenance over the entire planning horizon. The model is applied to a case study developed for the Dalian Road Tunnel in Shanghai. A scenario-based method is adopted to account for uncertain failures. A hybrid algorithm using particle swarm optimization (PSO) and genetic algorithm (GA) is proposed to solve the model in realistic large-scale environments. Extensive numerical experiments are performed to verify the effectiveness of the proposed model and the efficiency of the proposed algorithm. Some meaningful management implications are revealed based on the experimental results.

Maintenance Decision-Making of an Urban Rail Transit System in a Regionalized Network-Wide Perspective

The networked operation of Urban Rail Transit (URT) brings the new challenge of network-wide maintenance. This research focuses on the URT Network-Wide Maintenance Decision-Making Problem (URT-NMDP), including regionalized maintenance network design and maintenance resource allocation. In this work, we proposed a bi-objective integer programming model that integrates the characteristics of set coverage and P-median models, resulting in the regionalized maintenance network design model. Some critical factors are considered in the model, such as the importance of node, the maximum failure response time, and maintenance guarantee rules. We designed a NSGA-II based algorithm to solve the model. Moreover, due to the uncertainty of failures in the URT network, we developed the method of allocating maintenance resources based on Monte Carlo simulation to strengthen the reliability of the regionalized maintenance network. With the model and algorithm presented in this work, we obtained Pareto optimal solutions of URT-NMDP, i.e., URT network-wide maintenance planning schemes, which include the number and location of maintenance points, the allocation of demand points, and the amount of maintenance units. Finally, a real-world case is studied to evaluate the operating performance of these schemes for verifying the method in our paper. The results of the case study demonstrate that the reasonable and tested-in-practice maximum failure response time is the precondition for the efficient URT maintenance network. The maintenance scheme considered the weighted importance of node shows the optimal performance, with the shortest overall maintenance path and the minimum average failure response time and investment cost on maintenance resources.

Berth Allocation and Quay Crane Assignment for the Trade-off Between Service Efficiency and Operating Cost Considering Carbon Emission Taxation

To sustain the development of maritime transportation, “green ports,” which operate with a good balance between environmental impact and economic interests, have been the focus of port operators and government agencies and are required to look into energy saving and emission reduction initiatives. One such reduction strategy proposed by the International Maritime Organization suggested imposing a carbon emission tax on ports as a long-term solution to reduce carbon emissions, but this would definitely increase the operating cost of ports. Quay cranes (QCs), as one type of handling equipment, play an important role in the service efficiency and carbon emission of ports. Therefore, this paper makes an effort to explore the study of the integrated berth allocation and QC assignment problem with the consideration of carbon emission taxation. This problem is formulated as a biobjective integer programming model, aimed at minimizing the total completion delay of all tasks and the total operating costs for all QCs. Finally, numerical experiments are performed to assess the applicability of the proposed models and evaluate the efficiency of the developed solution algorithm.

A Cost and Passenger Responsible Optimization Method for the Operation Plan of Additional High-Speed Trains in a Peak Period

In the peak period of a railway system, operators typically add additional trains to provide increased capacity to satisfy the increasing passenger demand. The paper proposes a new optimization framework for designing the operation plan, which includes the number of additional trains, train type, stop plan, and timetable, for additional trains in a peak period. A space-time network representation is used to obtain a feasible primary operation plan by finding a set of feasible space-time paths in the space-time network. Considering simultaneously the passenger demand and the trains’ total travel times, we formulate a biobjective integer programming model for generating a cost and passenger responsible primary operation plan. A set of loading capacity constraints are formulated in the model to guarantee a suitable loading capacity for each station’s passenger demand and better service for passengers. The CPLEX solver is used to solve the proposed model and to generate the optimal operation plan. Two sets of numerical experiments are conducted on a small-scale rail corridor and on the Wuhan-Guangzhou rail corridor to evaluate the performance of the proposed method. The results of the experiments show that the primary operation plan can be obtained within an acceptable computation time.

Battery-electric transit vehicle scheduling with optimal number of stationary chargers

Because of zero emissions and other social and economic benefits, electric vehicles (EVs) are currently being introduced in more and more transit agencies around the world. One of the most challenging tasks involves efficiently scheduling a set of EVs considering the limited driving range and charging requirement constraints. This study examines the battery-electric transit vehicle scheduling problem (BET-VSP) with stationary battery chargers installed at transit terminal stations. Two equivalent versions of mathematical formulations of the problem are provided. The first formulation is based on the deficit function theory, and the second formulation is an equivalent bi-objective integer programming model. The first objective of the math-programming optimization is to minimize the total number of EVs required, while the second objective is to minimize the total number of battery chargers required. To solve this bi-objective BET-VSP, two solution methods are developed. First, a lexicographic method-based two-stage construction-and-optimization solution procedure is proposed. Second, an adjusted max-flow solution method is developed. Three numerical examples are used as an expository device to illustrate the solution methods, together with a real-life case study in Singapore. The results demonstrate that the proposed math-programming models and solution methods are effective and have the potential to be applied in solving large-scale real-world BET-VSPs.

Research on the module configuration of complex products considering the evolution of the product family

The evolution of the product family is the essential driving force for the development of a complex product. Only customer satisfaction is emphasized in the traditional module configuration methods, which is not beneficial for product family evolution that is due to non-customer factors such as the emergence of new technology. In this study, the intuitionistic fuzzy number is employed to quantify the degree of correlation between each module and configuration targets, namely customer satisfaction and the degree of evolution of the product family, respectively. The bi-objective integer programming model is constructed by maximizing the degree of customer satisfaction and product family evolution. An improved Pareto ant colony optimization (P-ACO) is designed to solve this model and subsequently the Pareto frontier is obtained. The radar chart is adopted to represent the performance of each configuration scheme in the Pareto frontier. The feasibility and effectiveness of the proposed method are expounded by a case study and result comparison, showing that this method can provide a more competitive product configuration scheme to customers in the future market.

Route Optimization Model for Hazardous Materials Transportation Considering Traffic Restrictions and Delivery Time Windows

The government's strict traffic control and the requirement of enterprise logistics efficiency make the hazardous materials transportation process more complex. The challenge of existing hazardous materials transportation is to determine vehicle routes that minimize risks based on cost considerations of customer needs and road closures. According to the objective of minimizing risk and cost, a bi-objective integer programming model with joint constraints of road closure time window and delivery time window is constructed and solved by adaptive weight genetic algorithm. The case study of practical transportation network shows that the model can reasonably plan the node closure time of each route and the departure time of vehicles, and obtain a variety of optimal combinations of transportation schemes, so as to obtain the optimal strategies with different preferences for decision-making and provide decision support for the optimization of hazardous materials transportation route.

A bi-objective integer programming model for locating garbage accumulation points: a case study

Enhancing efficiency in Municipal Solid Waste (MSW) management is crucial for local governments, which are generally in charge of collection, since this activity explains a large proportion of their budgetary expenses. The incorporation of decision support tools can contribute to improve the MSW system, specially by reducing the required investment of funds. This article proposes a mathematical formulation, based on integer programming, to determine the location of garbage accumulation points while minimizing the expenses of the system, i.e., the installment cost of bins and the required number of visits the collection vehicle which is related with the routing cost of the collection. The model was tested in some scenarios of an important Argentinian city that stills has a door-to-door system, including instances with unsorted waste, which is the current situation of the city, and also instances with source classified waste. Although the scenarios with classified waste evidenced to be more challenging for the proposed resolution approach, a set of solutions was provided in all scenarios. These solutions can be used as a starting point for migrating from the current door-to-door system to a community bins system.

A New Efficient Algorithm for Hazardous Material Transportation Problem via Lane Reservation

Hazardous material transportation is well-known for its high potential risk. Minimizing the transportation risk is an important issue for hazardous material transportation. This paper focuses on a novel algorithm for the hazardous material transportation problem via lane reservation, whose goal is to obtain a best compromise between the impact on normal traffic due to lane reservation and the transportation risk. Firstly, a bi-objective integer programming model for the considered problem is formulated and transformed into a series of single objective models by\varepsilon-constraint method. For the transformed single objective models, a cut-and-solve and cutting plane combined method is proposed to reduce the computational time. The performance of the proposed algorithm is evaluated by an instance using a real network topology and randomly generated instances. Computational results demonstrate that the cut-and-solve and cutting plane combined method runs faster than direct use of software package CPLEX.