Bi-objective Mixed Integer Linear Programming Research Articles

A new approach for reliability modeling in green closed-loop supply chain design under post-pandemic conditions: A case study

Climate change, pandemics, and economic crises have created complex challenges for supply chains. Managing such situations requires the development of reliable decision-making frameworks. In this paper, a multi-level, multi-product, and multi-period closed-loop supply chain is studied with environmental considerations. A bi-objective mixed-integer linear programming model is presented for facility location, flow allocation, and transportation mode determination. The objectives of the model are to minimize the total cost and maximize the reliability of suppliers to meet the needs of factories. In the area of reliability engineering, a new approach is defined for modeling the probability of supplier availability considering catastrophic failures caused by pandemics, economic sanctions, and other failure modes. Furthermore, the decision-maker can handle the emission of greenhouse gases by an upper-bound constraint. In order to face the simultaneous uncertainty of demand and the maximum CO2 emission allowed, a scenario-based two-stage stochastic programming approach is proposed. The improved version of the augmented ε-constraint method, known as AUGMECON2, is used to solve the proposed model. The efficiency of the model and the proposed solution approach are investigated through a real-world case study of a battery manufacturing company in Iran.

An efficient bi-objective optimization approach for the optimal design of omnichannel supply chain distribution networks with sustainability

ABSTRACT This article studies a bi-objective omnichannel supply chain network design (SCND) problem, while simultaneously minimizing the overall supply chain (SC) cost and associated environmental emissions. A novel bi-objective mixed integer linear programming model is formulated, then an efficient optimization approach is proposed for the problem. Experimental studies on a case study and 210 random testing instances were conducted to evaluate the performance of the proposed model and approach. For the case study, 10 Pareto-optimal solutions were obtained within 1.9 s. For the random testing instances, the average number of Pareto-optimal solutions and computational time were 10.19 and 18.85 s, respectively. The overall results show that the proposed method is efficient and outperforms the adapted non-dominated sorting genetic algorithm II on test instances. Computational results indicate that this approach could assist decision makers in making good decisions, with considerable SC cost saving and carbon emission reductions in different situations.

A scenario-based game theory integrating with a location-allocation-routing problem in a pre- and post-disaster humanitarian logistics network under uncertainty

Purpose Pre-positioning and distributing relief items are important parts of disaster management as it simultaneously considers activities from both pre- and post-disaster stages. This study aims to address this problem with a novel mathematical model. Design/methodology/approach In this research, a bi-objective mixed-integer linear programming model is developed to tackle pre-positioning and distributing relief items, and it is formulated as an integrated location-allocation-routing problem with uncertain parameters. The humanitarian supply chain consists of relief facilities (RFs) and demand points (DPs). Perishable and imperishable relief commodities (RCs), different types of vehicles, different transportation modes, a time window for delivering perishable commodities and the occurrence of unmet demand are considered. A scenario-based game theory is applied for purchasing RCs from different suppliers and an integrated best-worst method-technique for order of preference by similarity to ideal solution technique is implemented to determine the importance of DPs. The proposed model is used to solve several random test problems for verification, and to validate the model, Iran’s flood in 2019 is investigated as a case study for which useful managerial insights are provided. Findings Managers can effectively adjust their preferences towards response time and total cost of the network and use sensitivity analysis results in their decisions. Originality/value The model locates RFs, allocates DPs to RFs in the pre-disaster stage, and determines the routing of RCs from RFs to DPs in the post-disaster stage with respect to minimizing total costs and response time of the humanitarian logistics network.

Efficient Humanitarian Logistics: Multi-Commodity Location–Inventory Model Incorporating Demand Probability and Consumption Coefficients

Background: A logistics network plan could be a major key issue due to its effect on supply chain effectiveness and responsiveness. This study aims to investigate the inventory location in the humanitarian logistics response stage using a three-level logistics network to integrate location–allocation problems such as warehouse location and shelter allocation to each facility, and then determine the inventory level in each warehouse. Methods: In this research, the center and its distribution, as well as the reduction in service-level costs due to inventory deficit, have been considered to increase the level of shelter services. In order to investigate the network, in this study, bi-objective mixed-integer linear programming (BOMILP) is presented. Results: The first objective is to reduce location costs and inventory costs that take into account probable demand, consumption factors, and transportation costs, and the second objective is to raise the level of services offered to victims in the model. The software programs GAMS win32, 25.1.2 and MATLAB have been utilized with numerical examples in various dimensions. Conclusions: To maximize the efficiency and quality of the service, first, the model was numerically solved, and then the location where the most commodities could be transported at the lowest possible cost was identified.

Logic-based Benders decomposition for bi-objective parallel machine selection and job scheduling with release dates and resource consumption

This work addresses a new bi-objective parallel machine selection and job scheduling problem with release dates and resource consumption. It consists in optimally selecting subcontractors (machines) from a set of geographically dispersed locations and scheduling the orders (jobs) to the selected subcontractors for processing while meeting the order release dates and resource consumption restrictions, so as to simultaneously minimize the maximum completion time, i.e., the makespan, and the total cost including machine usage cost and resource consumption cost. The problem is first formulated into a bi-objective mixed-integer linear program based on linear ordering (LO-MILP), and then valid inequalities are explored based on property analysis. To solve it, an ɛ-constraint method based on LO-MILP (ɛ-LO-MILP) is first proposed. To more efficiently solve it, we also develop a tailored logic-based Benders decomposition combined with ɛ-constraint method (ɛ-LBBD) where a novel method to obtain a tight lower bound of the identical parallel machine with machine-dependent release dates and some problem-specific cuts are proposed. Numerical experiments on an illustrative example are conducted to show the applicability of the model and algorithm and intuitively reveal the trade-off between production efficiency and cost. Experimental results on 200 instances with up to 100 orders demonstrate that ɛ-LBBD can reduce the computation time by about 28.92% compared to ɛ-LO-MILP and yields better Pareto solutions than ɛ-LO-MILP and the well-known non-dominated sorted genetic algorithm II do.

Grid-based artificial bee colony algorithm for multi-objective job shop scheduling with manual loading and unloading tasks

This paper investigates a multi-objective job shop scheduling problem with manual loading and unloading tasks (MOJSSPLU) aiming to minimize makespan and total workload simultaneously. Taking human factors into consideration achieves a better balance between production efficiency and resource utilization. To address this problem, we first establish a bi-objective mixed-integer linear programming (MILP) model. Afterward, we combine the artificial bee colony algorithm with the grid technique to develop a grid-based artificial bee colony algorithm (GABC) for solving MOJSSPLU. The algorithm uses a decomposition approach with the earliest-shortest dispatching rule to reduce the complexity of MOJSSPLU. A grid coordinate system is constructed to divide the objective space into cells, facilitating individuals’ location and evaluation in the decision space. Moreover, we focus on improving the cooperation between the employed and onlooker bees and design a mathematical formula based on priority weights to generate onlooker bees to maximize information utilization and improve the exploration and exploitation capabilities of the algorithm. To ensure that the most promising individuals are retained and protected from being eliminated during the evolutionary, we propose an elitist non-dominated solutions reservation strategy for improving the diversity of the population. We further propose a dynamic adjustment of the grid-division method that focuses on improving the exploitation capacity of the algorithm in the objective space to enhance the diversity and convergence of the population. To test the performance of the GABC algorithm, we compare it to the MILP approach and other algorithms using modified benchmark instances. Furthermore, we conduct the Friedman and Wilcoxon nonparametric tests to validate the comparison results among the different algorithms. The computational results demonstrate that the algorithm GABC is efficient.

Optimizing worker productivity and the exposure to hand-arm vibration: a skill-based job rotation model

ABSTRACT Productivity and concerns regarding the well-being of workers exposed to vibrations stand as significant topics within labor-intensive sectors. In particular, this study contributes to the existing research by analyzing the problem with linkages among worker skill level, production rates, and vibration exposure. A bi-objective mixed integer linear programming model was employed to optimize both productivity and the exposure to hand-arm vibration in the manufacturing workplace. A sensitivity analysis was carried out to examine the impact of key parameters on the trade-off between productivity and vibration exposure. The results demonstrate the model’s effectiveness in determining the best job rotation schedules by achieving optimal productivity and vibration exposure for low and medium problem sizes. Moreover, the numerical case study points out that strengthening the workforce by adding more proficient skilled workers can maintain a good level of productivity with a decreased likelihood of excessive vibration exposure.

Avoiding sudden maritime risk: A new variable speed route planning model by integrating Spatio-temporal dimension

Variable speed route planning is important for ships to avoid large-scale and sudden risk. However, in the face of massive amounts of grid data, current algorithms are still unable to establish a mathematical model to effectively avoid such risk in a variable speed manner. Therefore, this paper proposes a variable speed maritime route planning model integrating spatial and temporal considerations. First, in the temporal dimension, a bi-objective mixed-integer linear programming model is developed considering the minimization of risk and residual action power, and selection and mutation operators in the genetic algorithm are improved to solve the programming model. Second, a segment parallel A*(SPA*) method is proposed to improve spatial planning capacity. SPA* successfully reduces runtime by designing a parallel computing structure and adaptive heuristic function to simultaneously handle several segments of the path considering variable speed and reduce the expansion of useless grids. Three rounds of simulation experiments are conducted on small and large datasets with sudden and continuous risk scenarios in the vicinity of the South China Sea and Indonesian Throughflow, respectively. These experiments involve comparative analyses of our algorithm with the uniform speed scheme and with three conventional algorithms. The results show that in the face of both sudden and continuous risks, the path of our model outperforms the uniform speed scheme in all cases, takes an average of only 13.11 s to plan a route on the large dataset and is more compliant with the norms of the International Maritime Organization (IMO). Numerically, it is approximately 6 times and 7 times lower than the best performing of the remaining algorithms in terms of runtime and the rate of violating IMO.

Timetable Rescheduling Using Skip-Stop Strategy for Sustainable Urban Rail Transit

Unanticipated events inevitably occur in daily urban rail transit operations, disturbing the scheduled timetable. Despite the mild delay, the busy operation system probably tends to worsen a larger disturbance and even lead to a knock-on disruption if no rescheduling is timely carried out. We propose a bi-objective mixed-integer linear programming model (MILP) that employs the skip-stop operation strategy to eliminate unscheduled delays. This model addresses two distinct, yet interconnected objectives. Firstly, it aims to minimize the difference between the plan and the actual operation. Secondly, it strives to minimize the number of left-behind passengers. In order to resolve this MILP problem, we devised a Pareto-based genetic algorithm (GA). Based on the case study, we certify the superior effectiveness with comparisons to the whale optimization algorithm and the epsilon constraint method. The outcomes affirm that our model has the potential to reduce the total delay time of the line by 44.52% at most compared with the traditional all-stop running adjustment model. The optimal scheme saved 6.08% of the total costs based on a trade-off between operators’ interests and passenger satisfaction.

Planning for pharma supply chain under uncertainty considering inventory optimization

In this paper, a bi-objective mixed-integer linear programming model is constructed to manage the pharmaceutical supply chain of a hospital. The proposed model aims to concurrently reduce the overall cost of obtaining drugs from several vendors and choose the best suitable source. The suggested model takes into account supplier distance, inventory management, and multi-product and multi-period. The major assumptions of the proposed model are product storage for future periods of decreased demand and supplier capacity. The results indicate that the ideal approach can minimize hospital supply and pharmaceutical planning expenses. The Best-Worst and TOPSIS methods determine which pharmaceutical supplier should be selected for future orders. The suggested model identifies human resource capability as an essential factor that might significantly affect the system’s total cost. The results of applying the model and the sensitivity analysis validate the efficacy and validity of the suggested mathematical model and solution strategy.

A robust chance-constrained programming approach for a bi-objective pre-emptive multi-mode resource-constrained project scheduling problem with time crashing

The presented study proposes a novel bi-objective mixed integer linear programming (MILP) framework for the multi-mode resource-constrained project scheduling problem (MRCPSP) with pre-emptive and non-preemptive activities splitting under uncertain conditions. Minimising the project makespan and resource costs are the considered objectives. Renewable and non-renewable resources along with different modes are taken into account for activities implementation. Additionally, some activities can be crashed by consuming additional renewable and non-renewable resources. Model uncertainty is efficiently addressed by utilising a fuzzy chance constrained programming (CPP) method as well as extending two robust possibilistic programming models. The capability of the presented mathematical framework is validated using problem instances from PSPLIB (j10, j14, j20, and j30) and MMLIB (MM50 and MM100). Finally, a detailed computational comparison is presented to assess the performance of the two robust possibilistic programming models.

A new fuzzy bi-objective mixed-integer linear programming for designing a medical waste management network in the Coronavirus epidemic: a case study

ABSTRACT The COVID-19 pandemic has increased medical assistance and service demand, necessitating the development of a comprehensive model for managing medical waste networks. Despite extensive research, there is a lack of models examining infectious and hazardous waste simultaneously. This paper presents a multi-layered bi-objective mixed-integer linear program model, dividing infectious and non-infectious waste into different categories. The model consists of several stages for safe recycling of medical waste. Fuzzy programming and an augmented-constraint approach are employed to tackle uncertainty in parameters. A real case study from Tehran is examined, and numerical and computational results are provided to prove the model’s efficiency. The results indicate that insufficient disposal centers may require increasing their capacity.

Integrated optimization of rolling stock allocation and train timetables for urban rail transit networks: A benders decomposition approach

We investigate the integrated optimization of rolling stock allocation and train timetables (RATT) in an urban rail transit network with multiple connected lines and rolling stock depots. Different from most existing research on single-line cases, we consider that the rail manager plans to allocate a certain fleet of rolling stock to depots to implement the operational timetables of multiple lines, where each depot can serve more than one line but has limited capacity. By means of the construction of a space–time network, we formulate RATT into a bi-objective mixed-integer linear programming (MILP) model. The model can simultaneously generate the rolling stock allocation plan and train schedules for the considered lines, with the aim of optimizing both the investment cost of rolling stock and the service quality of passengers. Due to the computational complexity of large-scale real-world instances, we develop a Benders decomposition-based solution algorithm, which decomposes the MILP into a rolling stock assignment problem and a set of independent subproblems, i.e., a train scheduling subproblem for each line. As the subproblems also contain integral variables, we further prove that a large portion of the subproblems, after proper reformulation, has the integrality property and can be solved rapidly via linear relaxation. We test our integrated approach and solution algorithms on real-world instances from the Beijing rail transit network. The results show that our solution approach can obtain near optimal solutions, while the commercial solvers cannot even return feasible solutions on real-world instances. Compared to the existing plan in the Beijing metro, our RATT approach can improve the service quality by 17.6%, on average, using the same fleet size of rolling stock.

A simultaneous facility location and vehicle routing problem with recyclable express packaging consideration for sustainable city logistics

Environmental pollution from express packaging waste is a crucial issue for the development of sustainable city logistics. The use of recyclable express packaging can reduce resource consumption and pollution, but it also leads to additional recycling time and costs. To manage this problem, this paper considers the joint optimization of express parcel delivery and recyclable express packaging recycling. The optimization problem is defined as a simultaneous location and vehicle routing problem with simultaneous pickup and delivery. To exploit better potential to balance vehicle working time and operating costs, a bi-objective mixed integer linear programming model is developed. Numerical experiments show that the proposed method provides a more feasible allocation for facility location and route planning. The cost savings from REP recycling account for 45.36% of the operating costs, which means that the packaging costs are saved and the burden of express packaging waste on the urban environment is correspondingly alleviated. The joint optimization of delivery and recycling is helpful to promote the forward and backward connection of the industrial chain and supply chain, which can provide a reference for recycling network planning in the large-scale application of recyclable express packaging and contribute to sustainable city logistics.

Optimization models for cloud seeding network design and operations

Water shortage has become a significant challenge in many regions, affecting sectors such as agriculture, energy, industry, and the economy. Based on this subject matter and considering cloud seeding as an emerged water achieving promising technique, this paper provides an effective and innovative optimization approach for cloud seeding. This approach involves location analysis and operational planning, which offers significant potential for addressing water shortage challenges. The proposed approach considers atmospheric conditions, functional properties of facilities, seeding method variants, and cloud seeding system vulnerabilities as strategic programming for integrated facility network design. Additionally, operational planning, including flight routing and ground-based facility scheduling during rainfall storms, is considered to manipulate storms according to atmospheric conditions and systemic/facility operational constraints. To this end, the paper proposes a multi-objective mixed-integer linear programming model for facility location, which uses an innovative and application depended point-to-polygon covering algorithm, with the principal objective of maximization of spatial coverage. Moreover, a bi-objective mixed-integer linear programming model for seeding operations is developed based on the optimally located facilities in the strategic model, which uses an innovative line-to-polygon covering algorithm. The first objective of this model is to maximize precipitation augmentation, while the second objective of the both models is to minimize total system-wide costs. Finally, the applications of the proposed models and the managerial insights are also illustrated in a real case study domain.

A Bi-Objective Mixed-Integer Linear Programming Model for a Sustainable Agro-Food Supply Chain with Product Perishability and Environmental Considerations

Background: Agro-food supply chains possess specific characteristics due to the diverse nature of products involved and contribute to all three pillars of sustainability, making the optimal design of a sustainable agro-food supply chain a complex problem. Therefore, efficient models incorporating the unique characteristics of such chains are essential for making optimal supply chain decisions and achieving economically and environmentally sustainable agro-food supply chains that contribute to global food security. Methods: This article presents a multi-objective mixed-integer linear programing model that integrates agricultural-related strategic decisions into the tactical design of an agro-food supply chain. The model considers transportation, inventory, processing, demand fulfilment, and waste disposal decisions. It also accounts for seasonality and perishability, ensuring a comprehensive approach to sustainability. The model aims to maximize the total generated profits across the supply chain while simultaneously minimizing CO2 emissions as a measure of environmental impact. Results: By implementing the model on a sugar beet supply chain in the Netherlands, strategic crop rotation farm schedules for the crop rotation cycle and the optimum supply network decisions are obtained. Furthermore, different objectives are analyzed and the Pareto-efficient frontier is investigated to analyze the underlying trade-offs. Additionally, the model serves as a decision support tool for managers facilitating informed investment decisions in technologies that prolong product shelf life while maintaining profitability. Conclusions: The proposed multi-objective model offers a valuable framework for designing economically and environmentally sustainable agro-food supply chains. By aligning with sustainability goals and providing decision support, this research contributes to enhancing global food security and promoting sustainable resource utilization.

Designing reverse logistics network for healthcare waste management considering epidemic disruptions under uncertainty

Population growth and recent disruptions caused by COVID-19 and many other man-made or natural disasters all around the world have considerably increased the demand for medical services, which has led to a rise in medical waste generation. The improper management of these wastes can result in a serious threat to living organisms and the environment. Designing a reverse logistics network using mathematical programming tools is an efficient and effective way to manage healthcare waste. In this regard, this paper formulates a bi-objective mixed-integer linear programming model for designing a reverse logistics network to manage healthcare waste under uncertainty and epidemic disruptions. The concept of epidemic disruptions is employed to determine the amount of waste generated in network facilities; and a Monte Carlo-based simulation approach is used for this end. The proposed model minimizes total costs and population risk, simultaneously. A fuzzy goal programming method is developed to deal with the uncertainty of the model. A simulation algorithm is developed using probabilistic distribution functions for generating data with different sizes; and then used for the evaluation of the proposed model. Finally, the efficiency of the proposed model and solution approach is confirmed using the sensitivity analysis process on the objective functions’ coefficients.

The balanced [formula omitted]-median problem with unitary demand

We consider a bi-objective variant of the p-median problem where p facilities must be located to serve a set of n customers with unitary demand. The considered objectives are: minimizing the average traveled distance between customers and facilities, and balancing the number of allocated customers per facility. We denote the latter by customer allocation inequity and measure it as the mean absolute deviation of the number of customers assigned to each median. We formulate this new problem as a bi-objective mixed-integer linear program ,and use a weighted sum method to generate a representative set of Pareto optimal solutions. Considering the single-objective subproblem solved by the weighted sum method, we develop a primal–dual algorithm that handles large-scale instances by combining a Lagrangian relaxation heuristic within a variable neighborhood search metaheuristic. This algorithm relies on the solution of a tailored minimum cost flow problem for the case where the locations of the facilities are known. We evaluate the proposed formulation and algorithm on test instances from the literature. After demonstrating the effectiveness of the developed algorithm, we test it on a series of large instances derived from an industrial application of districting for last-mile delivery. We analyze the trade-off between the assignment cost and customer allocation inequity, and evaluate the quality of the solutions by comparing them with those attained through alternative inequity measures.

A Robust Optimization Model for Multi-Period Railway Network Design Problem Considering Economic Aspects and Environmental Impact

The railway network design problem is one of the critical issues in the transportation sector due to its significance and variety of necessary applications. The major issue of this field relates to the decision of whether to increase the railways’ capacity or construct a new route to meet demand. Although the budget is a great concern of the managers for making such a decision, environmental factors should be necessarily included in the decision-making process. Therefore, this research proposes a novel robust bi-objective mixed-integer linear programming (MILP) model to simultaneously minimize the total cost and environmental impact under uncertain conditions and within a given time horizon. The proposed problem addresses strategic and operational decisions through railway project selection and product flow determination. To deal with the bi-objectiveness of the model and tackle the complexity of the problem, a nondominated sorting genetic algorithm (NSGA-II) is employed. The proposed NSGA-II could reach near-optimal Pareto solutions in a reasonable solution time and showed a reliable performance for being employed in large-sized instances. It also indicates that the proposed NSGA-II can be utilized for solving large-sized samples in a very short time.

Energy-efficient job shop scheduling problem with transport resources considering speed adjustable resources

This work extends the energy-efficient job shop scheduling problem with transport resources by considering speed adjustable resources of two types, namely: the machines where the jobs are processed on and the vehicles that transport the jobs around the shop-floor. Therefore, the problem being considered involves determining, simultaneously, the processing speed of each production operation, the sequence of the production operations for each machine, the allocation of the transport tasks to vehicles, the travelling speed of each task for the empty and for the loaded legs, and the sequence of the transport tasks for each vehicle. Among the possible solutions, we are interested in those providing trade-offs between makespan and total energy consumption (Pareto solutions). To that end, we develop and solve a bi-objective mixed-integer linear programming model. In addition, due to problem complexity we also propose a multi-objective biased random key genetic algorithm that simultaneously evolves several populations. The computational experiments performed have show it to be effective and efficient, even in the presence of larger problem instances. Finally, we provide extensive time and energy trade-off analysis (Pareto front) to infer the advantages of considering speed adjustable machines and speed adjustable vehicles and provide general insights for the managers dealing with such a complex problem.