Bi-objective Mixed-integer Linear Programming Model Research Articles

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.

A new bi-objective vehicle routing-scheduling problem with cross-docking: Mathematical model and algorithms

This paper addresses a vehicle routing problem with cross-docking (VRPCD) that considers truck scheduling, splitting pickup and delivery orders with time-windows at supplier and retailer locations, while optimizing two conflicting objectives (i.e., cost efficiency and responsiveness). The objectives are to minimize the total operational cost and the sum of the maximum earliness and tardiness. A new bi-objective mixed-integer linear programming model is presented and a multi-objective meta-heuristic evolutionary algorithm is proposed for solving the problem. Numerical results indicate the effectiveness of our proposed algorithm comparing with two multi-objective meta-heuristic algorithms (i.e., non-dominated sorting genetic algorithm (NSGA-II) and Pareto archived evolution strategy (PAES)). Also, we report findings from a hypothetical case study in a retail chain in Houston, Texas.

Integrated resource leveling and material procurement with variable execution intensities

This paper addresses the concurrent planning of resource leveling and material ordering, aiming to minimize the completion time of the project and the total cost of project execution. The latter is comprised of material ordering, purchase and holding costs as well as the cost of fluctuations in resource utilization. All activities of the project are performed regarding variable execution intensities, which means that the effort spent to execute an activity is considered to vary over time. A bi-objective mixed-integer linear programming model is proposed to deal with the above-mentioned time-cost trade-off problem. The proposed model determines the start time of activities, the fraction of executing activities in each time period as well as the quantity and timing of orders for materials, while satisfying precedence relationships among activities. The problem belongs to the class of NP-hard problems. Hence, a genetic algorithm as well as two multi-objective meta-heuristic algorithms, namely non-dominated sorting genetic algorithm II (NSGA-II) and multi-objective vibration damping optimization (MOVDO) algorithm, calibrated by Taguchi method, are utilized to solve 30 different sized test problems. Then, four performance metrics are used to assess the performance of both NSGA-II and MOVDO algorithms. Results show that MOVDO outperforms NSGA-II in the small-, medium- and large-sized test problems. Moreover, a sensitivity analysis on different-sized test problems, using MOVDO, shows that when the durations of activities are considered to be flexible, the Pareto front is significantly improved in terms of objective values, and the number of Pareto-optimal solutions (NPS) increases. This helps the project manager(s) make decisions on the schedule and the execution intensity of activities and material procurement in a more flexible manner.

A chance constrained fuzzy goal programming approach for perishable pharmaceutical supply chain network design

In this paper, a bi-objective mixed-integer linear programming model is formulated for designing a perishable pharmaceutical supply chain network under demand uncertainty. The objectives of the proposed model are to simultaneously minimize the total cost of the network and lost demand amount. The proposed model is multi-product and multi-period and includes simultaneous facilities location, vehicle routing, and inventory management; hence, it is considered an operational-strategic model. Procurement discounts, the lifetime of products, storing products for future periods, lost demand, and soft and hard time windows are the main assumptions of the proposed model. A novel hybrid approach, based on fuzzy theory, chance constrained programming, and goal programming approach, is developed for solving the proposed bi-objective model. The validity of the proposed model and developed solution approach is evaluated using data from Avonex, a prefilled syringe distribution chain serving 11 health centers in Tehran. The proposed model indicates that some lost sales exist, and to overcome the lost sales, the case company needs to invest a little more in addition to the initial investment of around 2 billion tomans. The results obtained from implementing the model and the sensitivity analysis, using real-world data, confirm the efficiency and validity of the proposed mathematical model and solution approach.

Hybrid optimization procedures applying for two-echelon urban underground logistics network planning: A case study of Beijing

Motivated by urban land scarcity and surging demand, the underground space has become a frontier domain for future city logistics industry. The Underground Logistics System (ULS) is a sustainable option to overcome traffic congestion while improving freight transport efficiency. This study presents a comprehensive layout design and facility planning method for the two-echelon urban ULS network with hub-and-spoke structure. The underground network is organized by the primary and secondary ULS nodes and tunnels which have strong alignment with existing logistics nodes such as distribution centers and major demand terminals. A bi-objective mixed-integer linear programming model considering minimal costs and maximal system utilization was established to formulate the location-allocation of ULS nodes, tunnel layout and flow routing in the network. Given the inherent problem complexity, a series of simplification techniques and hybrid optimization procedures integrating both exact and meta-heuristic algorithms were developed to obtain high quality solutions. Finally, a real-world simulation was carried out in Beijing, China to demonstrate the applicability of proposed method. Results show that our procedures can effectively obtain the best configurations of ULS network under various scenarios and has better convergence efficiency and optimality performance. Moreover, a group of horizontal tests and sensitivity analysis reveals further insights regarding ULS network decision-making and layout strategies.

Synchronizing Last Trains of Urban Rail Transit System to Better Serve Passengers from Late Night Trains of High-Speed Railway Lines

Synchronizing last trains of Urban Rail Transit (URT) system is important for passenger transportation, especially for travelers from other modes, such as high-speed railway (HSR) trains in late night, to reach their destinations. This paper develops a bi-objective mixed-integer linear programming model for the problem of Last-Train Timetable Synchronization (LTTS) of URT system in late night hours. The bi-objective optimization model maximizes the number of passengers who can arrive at their destinations successfully, and minimizes the total operation ending time for last trains of URT system. This model can deliver both a last-train transfer scheme solution (i.e. which transfer connections to keep at which stations) and a last-train timetable of URT system (i.e. arrival and departure times for last trains at stations) at the same time. Transfer walking time is considered different for passengers, and suggestions for passengers to adopt certain transfer connections are dynamically dependent on the last-train timetable and passenger assignment. The Epsilon-constraint method is used to find the Pareto optimal solutions for the proposed bi-objective model. Numerical tests based on a sample network demonstrate the validity of the model on solving the LTTS problem. To evaluate the effectiveness and efficiency of the bi-objective model in practice, real-world experiments are conducted based on Beijing-Shanghai high-speed railway (HSR-BS) line, Beijing-Tianjin inter-city high-speed railway (HSR-BT) line and Beijing urban rail transit (URT) system. We also present a set of experiments to evaluate the benefits of considering different Transfer Walking Time (TWT) of passengers in the model, compared to models with constant and identical TWT for all passengers.

Designing a resilient-green closed loop supply chain network for perishable products by considering disruption in both supply chain and power networks

The effects of disruption in both supply chain and its infrastructures like power network have not been addressed in previous works, simultaneously. Besides, disruption risks have environmental effects besides their economic effects. For this purpose, this paper studies a resilient-green closed loop supply chain design problem considering the perishable nature of products and disruption risks in both supply chain and power networks. To cope with disruptions five different risk mitigation strategies are employed. We have developed an integrated bi-objective mixed-integer linear programming model to minimize the expected total network cost and the expected total amount of carbon emissions of both networks. The results demonstrate that applying strategy of integrating interdependent networks along with four other resilient strategies, especially “intermediate facility” and “lateral transshipment” strategies, will improve the performance of integrated network significantly. Also, with increasing the lifetime of products, the performance of the proposed model will be improved.

A green delivery-pickup problem for home hemodialysis machines; sharing economy in distributing scarce resources

In this paper, we address a green delivery-pickup problem for Home Hemodialysis Machines (HHMs) categorized as scarce commodities. The system supplies the HHMs either from the central depot of the company or from the individual owners. Based on the sharing economy concept, the individuals who own the HHM devices can involve in this home health care system and share them with others through the fleet of the company to make money. After delivery of portable HHM devices to the clients (patients), they will be collected, disinfected and reallocated to fulfill the demands of the other customers. Moreover, respecting the environmental concerns, the vehicles’ fuel consumption and consequently the GHG emissions are realistically assumed as a function of the vehicles’ load, such that the company and especially the individual owners contribute to reducing GHG emissions, in addition to the primary economic motivations. Current research provides a bi-objective mixed-integer linear programming model which seeks minimizing total system cost and total carbon emissions. In order to solve the problem, Torabi and Hassini’s (TH) technique is applied and then a multi-objective meta-heuristic algorithm, self-learning non-dominated sorting genetic algorithm (SNSGA-II), is developed for medium- and large-sized problems. Finally, the application of the problem is investigated by a real case study from the healthcare sector. Numerical analyses indicate that the proposed green sharing-enabled model has a meaningful impact on both operational-level logistics determinations as well as the environmental important attainment indicators. As notable savings are guaranteed in terms of total system cost and emission, the proposed model has a great potential to provide the item sharing activities with a proper sustainable solution.

Logic-based benders decomposition algorithm for contamination detection problem in water networks

To prevent the serious hazards caused by the intrusion of contaminants into the water distribution network, equipping the network with monitoring sensors is necessary. In this regard, the identification criterion is an important and recently addressed issue indicating that the sensors should be located so that in the case of intrusion of contamination, not only at least one sensor responds, but also it is possible to identify the source of contamination intrusion, as well. This paper addresses the sensor location problem with the identification criterion assuming that a limited budget is available for the sensor placement, and the aim is to minimize the number of vulnerable nodes having the same alarm pattern. First, the problem is formulated as a bi-objective mixed-integer linear programming model, assuming that the objective functions are ordered based on a given prioritization. Then, by utilizing the underlying problem structure, an exact logic-based Benders decomposition algorithm is presented. Computational results over moderate and large-sized instances confirm the efficiency of the proposed algorithm.

An integrated production-distribution planning with a routing problem and transportation cost discount in a supply chain

PurposeThe motivation behind this research refers to the significant role of integration of production-distribution plans in effective performance of supply chain networks under fierce competition of today’s global marketplace. In this regard, this paper aims to deal with an integrated production-distribution planning problem in deterministic, multi-product and multi-echelon supply chain network. The bi-objective mixed-integer linear programming model is constructed to minimize not only the total transportation costs but also the total delivery time of supply chain, subject to satisfying retailer demands and capacity constraints where quantity discount on transportation costs, fixed cost associated with transportation vehicles usage and routing decisions have been included in the model.Design/methodology/approachAs the proposed mathematical model is NP-hard and that finding an optimum solution in polynomial time is not reasonable, two multi-objective meta-heuristic algorithms, namely, non-dominated sorting genetic algorithm II (NSGAII) and multi-objective imperialist competitive algorithm (MOICA) are designed to obtain near optimal solutions for real-sized problems in reasonable computational times. The Taguchi method is then used to adjust the parameters of the developed algorithms. Finally, the applicability of the proposed model and the performance of the solution methodologies in comparison with each other are demonstrated for a set of randomly generated problem instances.FindingsThe practicality and applicability of the proposed model and the efficiency and efficacy of the developed solution methodologies were illustrated through a set of randomly generated real-sized problem instances. Result. In terms of two measures, the objective function value and the computational time were required to get solutions.Originality/valueThe main contribution of the present work was addressing an integrated production-distribution planning problem in a broader view, by proposing a closer to reality mathematical formulation which considers some real-world constraints simultaneously and accompanied by efficient multi-objective meta-heuristic algorithms to provide effective solutions for practical problem sizes.

Enhanced multi-directional local search for the bi-objective heterogeneous vehicle routing problem with multiple driving ranges

The transportation sector accounts for a significant amount of greenhouse gas emissions. To mitigate this problem, electric vehicles have been widely recommended as green vehicles with lower emissions. However, the driving range of electric vehicles is limited due to their battery capacity. In this paper, a bi-objective mixed-integer linear programming model is proposed to minimise total costs (fixed plus variable) as well as CO2 emissions caused by the vehicles used in the fleet for a Heterogeneous Vehicle Routing Problem with Multiple Loading Capacities and Driving Ranges (HeVRPMD). To solve the proposed model, an enhanced variant of Multi-Directional Local Search (EMDLS) is developed to approximate the Pareto frontier. The proposed method employs a Large Neighbourhood Search (LNS) framework to find efficient solutions and update the approximated Pareto frontier at each iteration. The LNS algorithm makes use of three routing-oriented destroy operators and a construction heuristic based on a multi-round approach. The performance of EMDLS is compared to MDLS, an Improved MDLS (IMDLS), non-dominated sorting genetic algorithm II (NSGAII), non-dominated sorting genetic algorithm III (NSGAIII), and the weighting and epsilon-constraint methods. Extensive experiments have been conducted using a set of instances generated from the Capacitated Vehicle Routing Problem benchmark tests in the literature. In addition, real data is utilised to estimate fixed and variable costs, CO2 emissions, capacity, and the driving range of each type of vehicle. The results show the effectiveness of the proposed method to find high-quality non-dominated solutions.

Green permutation flowshop scheduling problem with sequence-dependent setup times: a case study

Increasing global energy consumption, large variations in its cost and the environmental degradation effects are good reasons for the manufacturing industries to become greener. Green shop floor scheduling is increasingly becoming a vital factor in the sustainable manufacturing. In this paper, a green permutation flowshop scheduling problem with sequence-dependent setup times is studied. Two objectives are considered including minimisation of makespan as a measure of service level and minimisation of total energy consumption as a measure of environmental sustainability. We extend a bi-objective mixed-integer linear programming model to formulate the stated problem. We develop a constructive heuristic algorithm to solve the model. The constructive heuristic algorithm includes iterated greedy (CHIG) and local search (CHLS) algorithms. We develop an efficient energy-saving method which decreases energy consumption, on average, by about 15%. To evaluate the effectiveness of the constructive heuristic algorithm, we compare it with the famous augmented ϵ-constraint method using various small-sized and large-sized problems. The results confirm that the heuristic algorithm obtains high-quality non-dominated solutions in comparison with the augmented ϵ-constraint method. Also, they show that the CHIG outperforms the CHLS. Finally, this paper follows a case-study, with in-depth analysis of the model and the constructive heuristic algorithm.

Bi-Objective Mixed-Integer Linear Programming Model for High-Level Planning of Biochar-Based Carbon Management Networks

Biochar is the stable, carbon-rich solid co-product of thermochemical biomass conversion. It has recently gained considerable interest, driven by the need to mitigate climate change through carbon sequestration in soil. During pyrolysis, much of the carbon in biomass is transformed into recalcitrant form, so that biochar applied to soils results in storage of carbon for hundreds of years and simultaneous improvement of soil fertility. Biochar can display a wide range of properties such as pH, cation exchange capacity (CEC) and elemental composition due to the assortment of raw materials and reaction conditions. The suitability of soil to biochar application depends on these properties. This has led to the “designer biochar” concept, wherein biochar could be tailored with relevant properties to address specific soil quality improvements. This aspect and the promising results of biochar application to soil can be potentially optimized through biochar-based carbon management networks (CMN) with the aid of mathematical programming. In this work, a bi-objective mixed-integer linear programming (MILP) model is developed for the high-level planning of biochar-based CMN. Additional parameters, variables and constraints are given to account for the incompatibilities of biochar sources and sinks. An illustrative case study is presented here to demonstrate the applicability of the developed model.

A bi-objective MILP model for blocking hybrid flexible flow shop scheduling problem: robust possibilistic programming approach

ABSTRACTOne of the important issues in scheduling due to the frequent use of it in manufacturing industries and factories is the hybrid flow shop (HFS) scheduling problem. In this paper, a bi-objective mixed integer linear programming (MILP) model for the problem is presented in which blocking constraint is also considered. The first objective function tries to minimize the makespan and the second one tries to minimize the total costs of machine allocation at each stage. In fact, in this model, the number and the type of machines at each stage are determined by the model according to the processing and setup times and cost of machines. Because most issues in the real world are uncertain, in this study, processing times, sequence-dependent setup times, and costs are considered as uncertain parameters. The robust possibilistic programming (RPP) approach is used to cope with the uncertainty. In this paper, the realistic and the hard worst-case robust approaches are used. The realistic and the soft worst-case robust models became the same because we are only concerned about the robustness of the makespan. Comparing the results between fuzzy and robust fuzzy models shows that the realistic model is more suitable than fuzzy and hard worst-case models in terms of mean and standard deviation.

Bi-criteria single-machine batch scheduling with machine on/off switching under time-of-use tariffs

The industrial sector is the largest consumer of the world’s total energy and most of its consumption is in form of electricity. In recent years, to strengthen the peak load regulation capability, time-of-use (TOU) pricing has been implemented in many countries to encourage consumers to shift their use from peak to mid- and off-peak periods such that their energy bills can be reduced. In this paper, we study a new single-machine batch scheduling problem with machine on/off switching under TOU tariffs, which aims to simultaneously minimize total electricity cost and makespan. For the problem, we first develop a bi-objective mixed-integer linear programming model. Based on optimal batch rule analysis, an improved model is further provided which greatly reduces Pareto optimal solution search space. To efficiently solve large-size problems, we propose a heuristic based ε-constraint method. The results from extensive computational experiments confirm the effectiveness and efficiency of the proposed model and the algorithm.

The Bi-Objective Location-Routing Problem based on Simultaneous Pickup and Delivery with Soft Time Window

The location-routing problem is the most significant and yet new research field in location problems that considers simultaneously vehicle routing problem features with original one for achieving high-quality integrated distribution systems in beside of the global optimum. Simultaneous pickup and delivery based on time windows are the two main characteristics of logistic management that have been used separately in most of the location routing problem in spite of their various real-life application with together. Furthermore, distribution manager always trying to create a distributed system layout along with the lowest total system cost and enhancing service levels for providing all customers satisfaction. Accordingly, in the current paper is considered the mentioned gap, that is to say the bi-objective capacitated location-routing problem based on simultaneous pickup and delivery with soft time window and multi depots (BOCLRPSPDSTW). For achieving the main goal, bi-objective mixed-integer linear programming model for BOCLRPSPDSTW, on the one hand minimizing summation of all problem costs and on the other hand, for meeting customer service level minimizing maximum summation of delivery times and service times are addressed. To solve the presented model, NSGAII and NRGA are proposed and at last efficiency of the anticipated solutions are depicted by testing them in a data set.

Bi-objective scheduling on uniform parallel machines considering electricity cost

ABSTRACTThis article investigates a bi-objective scheduling problem on uniform parallel machines considering electricity cost under time-dependent or time-of-use electricity tariffs, where electricity price changes with the hours within a day. The aim is to minimize simultaneously the total electricity cost and the number of machines actually used. A bi-objective mixed-integer linear programming model is first formulated for the problem. An insertion algorithm is then proposed for the single-objective scheduling problem of minimizing the total electricity cost for a given number of machines. To obtain the whole Pareto front of the problem, an iterative search framework is developed based on the proposed insertion algorithm. Computational results on real-life and randomly generated instances demonstrate that the proposed approach is quite efficient and can find high-quality Pareto fronts for large-size problems with up to 5000 jobs.

Multiobjective Optimization for Multiperiod Reverse Logistics Network Design

In recent years, the ever-rising return streams for repair service have forced the electronics manufacturers to expand their reverse logistics capacities. However, most existing papers on the reverse logistics network design neglected the time sensitivity of the return flows. Moreover, most of these investigations were primarily concerned with the single objective problems of either minimizing the total cost or maximizing the profit. In this paper, we propose a biobjective mixed-integer linear programming model for the multiperiod design problem of a reverse logistics network for repair service. A multiperiod setting is taken into account to make the reverse logistics network flexible to accommodate the gradual changes in the capacity of the facilities and the network configuration. To solve the NP-hard problem with biobjective, we develop a hybrid evolutionary algorithm that combines nondominated sorting genetic algorithm II (NSGA-II) with a local search method. We compare the hybrid evolutionary algorithm with NSGA-II and ϵ-constraint method using numerical examples. The comparison results indicate that the hybrid evolutionary algorithm outperforms the NSGA-II in most cases. The ϵ-constraint method performs best for the small instances, but it cannot solve large instances within reasonable time. Finally, an extensive parametric analysis is conducted and several managerial insights are derived.