Bi-objective Mixed Integer Linear Programming Research Articles

Electric Vehicle Tour Planning Considering Range Anxiety

In this study, the tour planning problem for electric vehicles is investigated. We aim to derive the optimal route and thus, to maximize profitability and minimize range anxiety within the time horizon. To solve this problem, a bi-objective mixed integer model is proposed. Specifically, we first introduced the reliability of route planning and quantified it as a cost with specific functions. The nonlinear model was then converted into a bi-objective mixed integer linear program, and an interactive branch and bound algorithm was adopted. Numerical experiments conducted on different networks have shown that the model that considers range anxiety offers more effective solutions. This means that our model is able to plan the routes with high reliability and low risk of profit loss and accidents.

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.

Ensemble of metaheuristics for energy-efficient hybrid flowshops: Makespan versus total energy consumption

Abstract Due to its practical relevance, the hybrid flowshop scheduling problem (HFSP) has been widely studied in the literature with the objectives related to production efficiency. However, studies regarding energy consumption and environmental effects have rather been limited. This paper addresses the trade-off between makespan and total energy consumption in hybrid flowshops, where machines can operate at varying speed levels. A bi-objective mixed-integer linear programming (MILP) model and a bi-objective constraint programming (CP) model are proposed for the problem employing speed scaling. Since the objectives of minimizing makespan and total energy consumption are conflicting with each other, the augmented epsilon (e)-constraint approach is used for obtaining the Pareto-optimal solutions. While close approximations for the Pareto-optimal frontier are obtained for small-sized instances, sets of non-dominated solutions are obtained for large instances by solving the MILP and CP models under a time limit. As the problem is NP-hard, two variants of the iterated greedy algorithm, a variable block insertion heuristic and four variants of ensemble of metaheuristic algorithms are also proposed, as well as a novel constructive heuristic. The performances of the proposed seven bi-objective metaheuristics are compared with each other as well as the MILP and CP solutions on a set of well-known HFSP benchmarks in terms of cardinality, closeness, and diversity of the solutions. Initially, the performances of the algorithms are tested on small-sized instances with respect to the Pareto-optimal solutions. Then, it is shown that the proposed algorithms are very effective for solving large instances in terms of both solution quality and CPU time.

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.

Dynamic Gaming Case of the R-Interdiction Median Problem with Fortification and an MILP-Based Solution Approach

This paper studies the cyclic dynamic gaming case of the r-interdiction median problem with fortification (CDGC-RIMF), which is important for strengthening a facility’s reliability and invulnerability under various possible attacks. We formulated the CDGC-RIMF as a bi-objective mixed-integer linear programming (MILP) model with two opposing goals to minimize/maximize the loss from both the designer (leader) and attacker (follower) sides. The first goal was to identify the most cost-effective plan to build and fortify the facility considering minimum loss, whereas the attacker followed the designer to seek the most destructive way of attacking to cause maximum loss. We found that the two sides could not reach a static equilibrium with a single pair of confrontational plans in an ordinary case, but were able to reach a dynamically cyclic equilibrium when the plan involved multiple pairs. The proposed bi-objective model aimed to discover the optimal cyclic plans for both sides to reach a dynamic equilibrium. To solve this problem, we first started from the designer’s side with a design and fortification plan, and then the attacker was able to generate their worst attack plan based on that design. After that, the designer changed their plan again based on the attacker’s plan in order to minimize loss, and the attacker correspondingly modified their plan to achieve maximum loss. This game looped until, finally, a cyclic equilibrium was reached. This equilibrium was deemed to be optimal for both sides because there was always more loss for either side if they left the equilibrium first. This game falls into the subgame of a perfect Nash equilibrium—a kind of complete game. The proposed bi-objective model was directly solved by the CPLEX solver to achieve optimal solutions for small-sized problems and near-optimal feasible solutions for larger-sized problems. Furthermore, for large-scale problems, we developed a heuristic algorithm that implemented dynamic iterative partial optimization alongside MILP (DIPO-MILP), which showed better performance compared with the CPLEX solver when solving large-scale problems.

Green supply chain network design considering inventory-location-routing problem: a fuzzy solution approach

The growing rate of population and technological advancement have led to an increase in natural resource consumption, which has caused irreparable damage to the environment. The implementation of green supply chain management is one of the most effective ways to deal with environmental degradation. Therefore, in this paper, a bi-objective mixed integer linear programming model is developed to design a green supply chain network. In the proposed model, the possibility of customer storage, being faced with shortage, locating of distribution centres, green vehicle routing problem, split delivery, multi-depot vehicle routing problem (VRP), capacitated VRP, and uncertainty in demands will be considered. The aim of the proposed model is to minimise the total cost and total shortages simultaneously and, therefore, a fuzzy solution approach is applied for this purpose. The results of implementing this model in a production chain of automotive parts in Iran indicate the exact and efficient performance of the proposed model.

A robust bi-objective programming approach to environmental closed-loop supply chain network design under uncertainty

Imposition of strict environmental protection acts and the imperative need of the best possible allocation of resources have given birth to the concept of low carbon logistics. Environmental laws force the manufacturers to extend their existing supply chains to form a closed-loop supply chain (CLSC) through the setup of an efficient recovery system. In this paper, a multi-objective robust optimisation model is proposed for the design of CLSC network under uncertainty. First, a deterministic bi-objective mixed integer linear programming (BOMILP) model is developed for designing a CLSC network. Then, the robust counterpart of the proposed BOMILP is presented to cope with the real-world uncertainty. The first objective aims to maximise the total profit generated in the CLSC network and the second objective minimises the environmental pollution of the CLSC network. The proposed bi-objective model is solved using a multi-choice goal programming (MCGP) approach.

Green Supply Chain Network Design Considering Inventory- Location- Routing Problem: A Fuzzy Solution Approach

The growing rate of population and technological advancement have led to an increase in natural resource consumption, which has caused irreparable damage to the environment. The implementation of green supply chain management is one of the most effective ways to deal with environmental degradation. Therefore, in this paper, a bi-objective mixed integer linear programming model is developed to design a green supply chain network. In the proposed model, the possibility of customer storage, being faced with shortage, locating of distribution centres, green vehicle routing problem, split delivery, multi-depot vehicle routing problem (VRP), capacitated VRP, and uncertainty in demands will be considered. The aim of the proposed model is to minimise the total cost and total shortages simultaneously and, therefore, a fuzzy solution approach is applied for this purpose. The results of implementing this model in a production chain of automotive parts in Iran indicate the exact and efficient performance of the proposed model.

DESIGNING A LOGISTIC NETWORK FOR HOSPITAL WASTE MANAGEMENT: A BENDERS DECOMPOSITION ALGORITHM

Healthcare wastes are produced from all medical and therapeutic activities in hospitals and healthcare centres. Around 15-20% of these wastes are classified as infectious wastes that could be hazardous. Therefore, an effective approach is required for handling costs and environmental issues. In this paper, a bi-objective Mixed-Integer Linear Programming (MILP) model is presented for logistics of infectious and non-infectious wastes. The proposed bi-objective model aims to minimize network costs and the risk of exposure to contamination. In this respect, a multi-stage network consisting of hospitals, recycling centres, treatment centres, disposal centres, mainly covering the location-routing problem, is considered. To deal with computational complexity of the proposed model, a Benders Decomposition Algorithm (BDA) has been employed. The researchers were faced with two important questions: (1) whether the proposed model can be applied to real-world scenarios or not, and (2) how efficient is the proposed algorithm in solving standard test problems and real-world cases. To answer these questions, the outputs of the BDA have been compared with the optimal solutions provided by the CPLEX software. The results imply that the BDA has been able to achieve optimal solutions in less computation times. Moreover, the proposed model and algorithm have been applied to a real-world study in Alborz Province of Iran. The outputs demonstrate that the proposed model and algorithm can yield applicable solutions for the case study which have been approved by experts. The proposed model can be extended by considering compatibility of wastes in storage.

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.

Route and speed optimization for liner ships under emission control policies

Pollutants such as nitrogen oxides (NOx), sulfur dioxide (SO2), and particulate matters (PM) generated by shipping industry are increasing in recent years. In order to control the ship emission pollution, the International Maritime Organization (IMO) has established the Emission Control Areas (ECAs). In the fierce competition of the shipping market, liner shipping companies are looking for strategies to maintain their core competencies under the emission control policy. To achieve this goal, this paper first proposes a bi-objective mixed integer linear programming model, aiming to optimize sailing routes and speeds within and outside the ECA while minimizing the total fuel cost and SO2 emissions. Then, a new algorithm is developed to solve the proposed model by combining the two-stage iterative algorithm and fuzzy logic method based on ∊-constraint. Finally, this paper compares and analyzes the navigation plan of a real sailing route considering and not considering the effects of ECA. Some experiments are conducted to analyze the effects of fuel cost, decision makers, and ECA boundaries on the total fuel cost and SO2 emissions. The results indicate that the proposed model and algorithm can contribute to save fuel cost and reduce SO2 emissions under the ECA policy and provide different Pareto optimal solutions. Thus, the effectiveness of the model and the efficiency of the algorithm are validated.

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.

Bioethanol supply chain network design considering land characteristics

Biomass is becoming an increasingly widespread source of energy. Yet land, as one of the most important resources in biomass production, is surprisingly understudied in the literature of biomass supply chain planning. This study proposes a novel framework that combines the literature of bioethanol supply chain design with agricultural land planning to simultaneously address optimal supply chain planning and sustainable land use in a bioethanol supply chain. A bi-objective mixed-integer linear programming (MILP) model is proposed to formulate the optimal design and planning of a bioethanol supply chain network considering competition of food and biomass feedstock over the available croplands. The proposed model is capable of making strategic decisions (i.e. locations and capacities of facilities, sourcing and allocation of biomass feedstocks to biorefineries), along with some tactical decisions (i.e. land planning, inventory and production of both biomass feedstock and bioethanol). The model incorporates the two objectives of minimum cost and maximum suitability of crops with their assigned croplands. A novel integration of the FAO framework, the best-worst multi-criteria decision-making method, PROMETHEE II and GIS is used to determine the suitability of available croplands according to the croplands’ soil and topographical characteristics. The performance of the proposed model is demonstrated through a multi-feedstock bioethanol supply chain in Fars province, Iran. It is concluded that the proposed integrated land planning-network design framework outperforms hierarchical approaches in which network design and land planning problems are solved separately in a sequential manner. Also, the case study shows that conditional on implementing second generation bioethanol production, Fars province has the potential to satisfy three percent of the fuel demand for transportation in the country.

Scheduling of periodic services to customers in dispersed locations from heterogeneous multi-agent companies considering uncertainty: A real case study

The scheduling problem of periodic services from service providers to customers located in different places and need different services. The service centers are also located in different positions, each of which has limited number of teams with the capability of performing one or some services. The goal is to simultaneously minimize ‘total service costs’ and ‘total earliness/tardiness’ in providing services to customers. Providing an optimal maintenance schedule is a big challenge in those companies with dispersed supply centers. In this paper, a novel bi-objective mixed integer linear programming model along with augmented epsilon constraint method is presented to exactly solve this problem. Then, a bi-objective meta-heuristic technique based on genetic algorithm is proposed and its performance in solving large-scaled problems is assessed. The uncertain parameters are faced through robust possibilistic programming approach to diminish the risk of decision making. Finally, the performance of the proposed model and solution approaches are evaluated through a real case study in maintenance scheduling of CNG stations equipment in Iran.

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.

A New Exact Algorithm to Optimize a Linear Function over the Set of Efficient Solutions for Biobjective Mixed Integer Linear Programs

We present the first (criterion space search) algorithm for optimizing a linear function over the set of efficient solutions of biobjective mixed integer linear programs. The proposed algorithm is developed based on the triangle splitting method [Boland N, Charkhgard H, Savelsbergh M (2015) A criterion space search algorithm for biobjective mixed integer programming: The triangle splitting method. INFORMS J. Comput. 27(4):597–618.], which can find a full representation of the nondominated frontier of any biobjective mixed integer linear program. The proposed algorithm is easy to implement and converges quickly to an optimal solution. An extensive computational study shows the efficacy of the algorithm. We numerically show that the proposed algorithm can be used to quickly generate a provably high-quality approximate solution because it maintains a lower and an upper bound on the optimal value of the linear function at any point in time.

Collaborative energy management in a micro-grid by multi-objective mathematical programming

In micro-grids, distributed energy generation based on renewable sources allows reducing the fossil fuel emissions. In order to manage the limited availability of renewable sources and to meet users’ requirements, a proper scheduling of both tasks and storage activities is needed. Moreover, the difficulty of storing energy on a large scale represents an opportunity for the micro-grid actors to collaborate for better balancing energy supply and demand. Collaboration may be achieved through the application of suitable energy management policies tackling the specific consumers’ needs. We address the Energy Management Problem in a micro-grid with collaboration among users. Specifically, a building is considered where each apartment can use both Shared and Local energy resources, the former managed by a Building Manager (BM). The problem is formulated through Bi-objective Mixed Integer Linear Programming, maximizing both the BM and the apartments’ profits and solved through the Augmented ϵ-Constraint approach. Numerical results, obtained on a set of realistic scenarios, are compared to those of the Weighted Sum method and evaluated according to proper performance metrics. The approach flexibility, in terms of selected management policies, is also discussed. Finally, the superiority of the collaborative paradigm versus the non-collaborative one is experimentally proven.

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.

Stable maintenance tasks scheduling: A bi-objective robust optimization model

This study aims to introduce an original concept entitled “stable maintenance tasks scheduling”. It is a useful contribution to the literature as it incorporates Robust Optimization (RO) concept into the maintenance and repair tasks scheduling problem. Developing a novel robust Bi-Objective Mixed-Integer Linear Programming (BOMILP) model to support the framework, where the maintenance tasks duration is uncertain, makes the contribution of the paper novel and unique. The main objective is to optimally schedule the maintenance and repair tasks in a minimum cost, while guaranteeing the stability of the scheduling. To provide more realistic scheduling, planning horizon is divided into some pre-purchased time buckets. The goal is to schedule the tasks in the available time buckets corresponding to a set of plannedmaintenance stopperiods and preventive maintenance actions. To solve the problem, different Multi-Objective Decision-Making (MODM) techniques are applied to the proposed model. The best solution technique is then selected corresponding to such criteria as total cost, total float, and run time, by employing Displaced Ideal Solution (DIS) method considering three different-sized examples. It is revealed that Weighted Goal Programming (WGP) can generate high-quality solutions. Results further show the superiority of the proposed framework in maintenance tasks scheduling taking into account their nominal frequency. Besides, a negligible amount of objective functions variations, despite the uncertainty of the tasks duration, is indicative of achieving the main goal of the research in achieving the stability of the scheduling. However, task duration is revealed as the most sensitive parameter to the objective functions.

On single-source capacitated facility location with cost and fairness objectives

We consider a location problem where a planner has to decide where to open facilities which have to be reached by customers at their own cost. The planner has two objectives: cost minimization and customer satisfaction. We argue that performance and fairness, as structural components of customer satisfaction, are both captured by an equity measure called conditional β-mean. We thus formulate the Fair Single-Source Capacitated Facility Location (F-SSCFL) problem, where the cost minimization objective is paired with the conditional β-mean minimization objective. The resulting formulation is a bi-objective mixed-integer linear program. A weighted sum method is developed to generate a small representative set of efficient solutions to the F-SSCFL problem, and a Benders decomposition approach is implemented to handle large-scale instances. On small/medium-scale instances, we analyse the trade-off between cost and conditional β-mean, showing that the model may be used as a managerial tool to balance direct costs and customer service. We also compare the quality of the solutions obtained with those obtained using alternative equity measures. On large-scale instances, we test Benders decomposition, showing that it is able to find good quality solutions in a relatively short computing time.