Bi-objective Mixed-integer Linear Programming Model Research Articles

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.

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.

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.

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.

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.

Optimization of Synchronized Scheduling for Dual-Source Trolleybus Network

With the development of new energy technologies, dual-source trolleybuses are widely used in public transportation systems. A synergistic schedule can further improve the service quality of vehicles and optimize the passenger travel experience. This study investigated the synchronized scheduling of dual-source trolleybus networks to reduce the transfer time of passengers and the fleet size problem to reduce the operating cost of enterprises. A bi-objective mixed-integer linear programming model was developed to maximize the total synchronizations and minimize the total fleet size of dual-source trolleybus lines. A two-stage algorithm was designed to obtain multiple sets of Pareto effective solutions. Meanwhile, to demonstrate the effectiveness of the proposed method, the results of numerical examples solved by the two-stage algorithm were compared with those of a genetic algorithm (GA) and a nondominated sorting genetic algorithm (NSGA-II). A real-world case study based on the Beijing dual-source trolleybus network was studied to validate the proposed model. The results show that the model can obtain synchronized schedules. The optimization of the number of synchronizations and fleet size is obvious. The model and algorithm can be applied to a large dual-source trolleybus network.

A location-inventory-routing problem to design a circular closed-loop supply chain network with carbon tax policy for achieving circular economy: An augmented epsilon-constraint approach

The emergence of the circular economy concept has pushed industry owners towards the green supply chain and waste reduction. The closed-loop supply chain originates from the concept of the circular economy and its purpose is to increase efficiency and profitability by reducing waste and energy consumption. Hence, in this research, an integrated bi-objective mixed-integer linear programming model is developed with the aim of optimizing both operational and strategic decisions in a closed-loop supply chain network. The proposed model benefits the location-inventory-routing problem to structure the network, and it applies a carbon tax policy and vehicle scheduling problem to reduce emissions and vehicle waiting time, respectively. Considering problems such as split-delivery, storage possibility, shortage, supplier selection, order allocation, heterogenous vehicles, and uncertain demand has led to the development of a comprehensive model. A stochastic scenario-based approach is used to deal with demand uncertainty, and an augmented epsilon-constraint method is employed to solve the proposed bi-objective model. The applicability of the proposed model and the effectiveness of the bi-objective solution approach for achieving circular economy are examined through its implementation in a cable and wire industry in Iran.

A Bi-objective optimization model for the last train timetabling problem

In cities where the urban rail transit (URT) systems do not provide 24-h services, passengers may not be able to reach their destinations if the last train services have closed by the time they arrive at the transfer stations. This paper aims to seek a well-coordinated last train timetable that can transport as many passengers as possible to their destinations (referred to as reachable passengers) and also transport those passengers who cannot reach their destinations (referred to as unreachable passengers) to the stations as close as possible to their destinations. A bi-objective mixed-integer linear programming (MILP) model is developed to maximize the number of reachable passengers and minimize the total remaining travel distance of all passengers. The augmented ε-constraint method is applied to generate all Pareto optimal solutions of the bi-objective MILP model. Numerical experiments were implemented in the Chengdu URT network. Results indicate that compared to the current-in-use timetable, the optimized timetable by our methods significantly increased the number of reachable passengers and meanwhile reduced the average remaining travel distance of unreachable passengers. In addition, we discussed two possible strategies to improve passengers’ destination reachability, which are encouraging passengers to arrive early at their origin stations, and optimizing the timetable of last trains and non-last trains at the same time.

A sustainable reverse supply chain network design considering inventory-location-routing problem and cross-docking scheduling under uncertainty

Designing an integrated supply chain network (SCN) by considering economic, social, and environmental aspects will increase its sustainability. This paper proposes a bi-objective mixed-integer linear programming model to design a sustainable closed-loop SCN considering location-inventory-routing problem, cross-docking scheduling, vehicles failure, inflation, routes reliability, impact of load on fuel consumption, discount on purchases from suppliers, supplier selection, and order allocation under uncertainty. The proposed bi-objective model minimises the total costs and maximises the created jobs, simultaneously. A scenario-based approach is used to cope with demand uncertainty while a fuzzy multi-objective solution approach is employed to convert the bi-objective model into a single-objective one. Finally, the performance of the proposed bi-objective mathematical model is evaluated using data from a wire and cable industry in Iran.

Medical waste management during coronavirus disease 2019 (COVID-19) outbreak: A mathematical programming model

Municipal solid waste (MSW) directly impacts community health and environmental degradation; therefore, the management of MSW is crucial. Medical waste is a specific type of MSW which is generally divided into two categories: infectious and non-infectious. Wastes generated by coronavirus disease 2019 (COVID-19) are classified among infectious medical wastes; moreover, these wastes are hazardous because they threaten the environment and living organisms if they are not appropriately managed. This paper develops a bi-objective mixed-integer linear programming model for medical waste management during the COVID-19 outbreak. The proposed model minimizes the total costs and risks, simultaneously, of the population’s exposure to pollution. This paper considers some realistic assumptions for the first time, including location-routing problem, time window-based green vehicle routing problem, vehicles scheduling, vehicles failure, split delivery, population risk, and load-dependent fuel consumption to manage both infectious and non-infectious medical waste. We apply a fuzzy goal programming approach for solving the proposed bi-objective model, and the efficiency of the proposed model and solution approach is assessed using data related to 13 nodes of medical waste production in a location west of Tehran.

An integrated mathematical model of dynamic production and maintenance planning in pumped-storage hydroelectricity

PurposePumped-storage hydroelectricity (PSH) is considered as an effective method to moderate the difference in demand and supply of electricity. This study aims to understanding of the high capacity of energy production, storage and permanent exploitation has been the prominent feature of pumped-storage hydroelectricity.Design/methodology/approachIn this paper, the optimization of energy production and maintenance costs in one of the large Iranian PSH has been discussed. Hence, a mathematical model mixed integer nonlinear programming developed in this area. Minimizing the difference in supply and demand in the energy production network to multiple energies has been exploited to optimal attainment scheme. To evaluate the model, exact solution CPLEX and to solve the proposed programming model, the efficient metaheuristics are utilized by the tuned parameters achieved from the Taguchi approach. Further analysis of the parameters of the problem is conducted to verify the model behavior in various test problems.FindingsThe results of this paper have shown that the meta-heuristic algorithm has been done in a suitable time, despite the approximation of the optimal answer, and the consequences of research indicate that the model proposed in the studied power plant is applicable.Originality/valueIn pumped-storage hydroelectricity plants, one of the main challenges in energy production issues is the development of production, maintenance and repair scheduling concepts that improves plant efficiency. To evaluate the mathematical model presented, exact solution CPLEX and to solve the proposed bi-objective mixed-integer linear programming model, set of efficient metaheuristics are used. Therefore, according to the level of optimization performed in the case study, it has caused the improvement of planning by 7%–12% and effective optimization processes.

Green reverse supply chain network design considering location-routing-inventory decisions with simultaneous pickup and delivery.

In this work, a bi-objective mixed-integer linear programming model is formulated for the first time to design a green supply chain while considering both forward and reverse flows, location-inventory-routing problem with simultaneous pickup and delivery, scheduling of vehicles, and time window. This model is presented to manage the production and distribution of perishable products and aims to minimize total costs and lost demands simultaneously. To solve the proposed bi-objective model under uncertainty, an approach based on fuzzy theory is developed. The performance of the proposed model and solution approach is evaluated using data from a bread production and distribution industry in Alborz province. The results showed that Hashtgerd center should be set up as the production and separation center and suppliers 1 and 2 (i.e., Seifabad and Mahdasht) should be selected to purchase raw materials when designing this network. The results obtained from the model's implementation in the real world and the sensitivity analysis of coefficients of objective functions indicate the desired efficiency and effectiveness of the proposed model and solution approach.

A location-inventory-routing model for green supply chains with low-carbon emissions under uncertainty.

The location-inventory-routing modeling is an integrated and comprehensive approach to the interconnected location planning, inventory management, and vehicle routing problems in supply chain management. Supplier selection and order allocation are critical operational and strategic decisions in green supply chains. Green supply chain management is an environmental approach to sourcing and production that considers sustainability in every supply chain stage. In this study, a novel bi-objective mixed-integer linear programming model is formulated to solve the location-inventory-routing problems in green supply chains with low-carbon emissions under uncertainty. The proposed model is used for supplier selection and order allocation by considering the location priorities, heterogeneous vehicle routing, storage needs, uncertain demand, and backorder shortage. The formulated bi-objective model is solved with a weighted fuzzy multi-objective solution approach coupled with a novel intelligent simulation algorithm to ensure the feasibility of the solution space. We generate and solve different-sized problems to demonstrate the applicability and efficacy of the proposed model.

Suppliers’ coalition strategy for green-Resilient supply chain network design

ABSTRACT Risk mitigation strategies have an environmental effect besides their economic benefits. The current research investigates designing a green-resilient supply chain network under the risk of disruptions. A new risk mitigation strategy namely “coalition” is applied between suppliers in a bi-objective mixed-integer linear programming model to minimize total cost and carbon emissions, simultaneously. Along with the coalition strategy between suppliers, the multi-sourcing strategy in the downstream supply chain is applied for retailers. The performance of the coalition strategy is compared with the three reputable risk mitigation strategies in both single-source and multi-source retailers. Results revealed that the average total cost and total carbon emissions of the green-resilient model have been reduced by 14% and 3.6% compared to the non-resilient model, respectively. Besides, multiple-sourcing for retailers has no improvement in the non-resilient model. On the contrary, the economic and environmental performance of the resilient model varies for the different number of sources.

Integrated Production and Distribution Scheduling in Mobile Facilities

Many supply chains lack flexibility and adaptability in today's competitive market, resulting in customer dissatisfaction, backorders, and several extra costs for the business. Additionally, the inability to quickly meet the customer's demands and the unnecessary transportation costs is also one of the significant challenges faced by the fixed facilities' supply chain. To address these challenges, this study analyzed the mobile facilities supply chain and the production, distribution, and delivery of goods conducted by trucks based on customer preferences. This study proposes a bi-objective mixed-integer linear programming model to ensure the mobile facilities' routing and manufacturing schedules are optimized to meet the customer's needs. Furthermore, this model minimizes production and distribution costs in the shortest amount of time. An exact decomposition algorithm based on Benders decomposition is used to find high-quality solutions in a reasonable amount of time to tackle the problem efficiently. We present several acceleration strategies for increasing the convergence rate of Benders' decomposition algorithm, including Pareto optimality cut and warm-up start. The warm-up start acceleration strategy itself is a meta-heuristic based on particle swarm optimization (PSO). Using the Benders decomposition, we demonstrate the superior accuracy of our solution methodology for large-scale cases with 10 kinds of products ordered by 30 customers using 10 mobile facilities.

Eco-friendly multi-skilled worker assignment and assembly line balancing problem

Workforce assignment and energy consumption impact greatly on the manufacturing performance. In this work, we study a multi-skilled worker assignment and assembly line balancing problem with the consideration of energy consumption. The problem consists of scheduling products and assigning workers to workstations appropriately under a given cycle time. Two objectives are minimized simultaneously, i.e., (1) the total costs including the processing cost and the fixed cost induced by employing workers, and (2) the energy consumption. A bi-objective mixed-integer linear programming model is formulated and an ϵ-constraint method is adopted to obtain the Pareto front for small-scale problems. For solving large-size problems, a processing time and energy consumption sorted-first rule (PT-EC SFR), a multi-objective genetic algorithm (NSGA-II) and a multi-objective simulated annealing method (MOSA) are developed. Numerical experiments are conducted and computational results show that the designed PT-EC SFR outperforms the other two algorithms in terms of computational time and quality of solutions.