Bi-objective Mixed Integer Linear Programming Research Articles

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.

Order Assignment and Scheduling for Personal Protective Equipment Production During the Outbreak of Epidemics

This paper investigates a new multi-objective order assignment and scheduling problem for personal protective equipment (PPE) production and distribution during the outbreak of epidemics like COVID-19. The objective is to simultaneously minimize the total cost and maximize the PPE supply timeliness. For the problem, we first develop a bi-objective mixed-integer linear program (MILP). Then an <inline-formula> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula>-constraint combined with logic-based Benders decomposition method is proposed based on some explored properties. We then extend the proposed model to handle dynamics and randomness. In particular, we design a predictive reactive rescheduling approach to address random order arrivals and manufacturer disruptions. Computational experiments on a real case from China and 100 randomly generated instances are conducted. Results show that the proposed algorithm significantly outperforms an adapted <inline-formula> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula>-constraint method combined with the proposed MILP and the widely used non-dominated sorting genetic <xref ref-type="algorithm" rid="alg2">algorithm II</xref> (NSGA-II) in obtaining high-quality Pareto solutions. <i>Note to Practitioners</i>&#x2014;The unprecedented outbreak of COVID-19 and its rapid spread caught numerous national and local governments unprepared. Healthcare systems faced a vital scarcity of PPEs. The urgency of producing and delivering PPEs increases as the number of infected cases rapidly increases. A key challenge in response to the epidemic is effectively and efficiently matching the demands and needs. Performing practical and efficient order assignment and scheduling for PPE production during the COVID-19 outbreak is critical to curbing the COVID-19 pandemic. This work first proposes a bi-objective mixed-integer linear program for optimal order assignment and scheduling for PPE production. The aim is to achieve an economical and timely PPE production and supply. A novel method that combines the <inline-formula> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula>-constraint framework and the logic-based Benders decomposition is proposed to yield high-quality Pareto solutions for practical-sized problems. Computational results indicate that the proposed approaches are practical and feasible, which can help decision-makers to perform acceptable order assignment and scheduling decisions.

An Integrated Production and Distribution Planning Model in Shrimp Agroindustry Supply Chain

This study discusses the integration model of production and distribution planning in the shrimp agroindustry supply chain, consisting of four echelons: shrimp suppliers, shrimp agroindustry, logistics provider companies, and buyers. The shrimp agroindustry supply chain is an essential part of the supply chain of processed product food, which transforms raw shrimp into various processed shrimp frozen products. One form of collaboration between supply chain actors is the integration of production and distribution planning activities. A model is developed to determine the flow of goods from each echelon, the number of processed shrimp products in the agroindustry, and the supplies of processed shrimp products. The bi-objective mixed-integer linear programming is proposed to describe the characteristics of the problem to minimize the total supply chain costs and maximize service level. Non-dominated sorting genetic algorithm II (NSGA-II) is designed to solve the shrimp agroindustry supply chain problem. The sample problem from the shrimp agroindustry in East Java, Indonesia, is applied to exhibit an algorithm’s efficiency. The result shows the best solution for the total supply chain is 1.75 trillion, and the service level is 1,502,264.5.

A NOVEL BI-OBJECTIVE MODEL FOR A MULTI-PERIOD MULTI-PRODUCT CLOSED-LOOP SUPPLY CHAIN

Closed-loop supply chain (CLSC) is a kind of supply chain which contains forward and backward flows of commodities within a logistics network. In the decision-making process of CLSC, locational, inventory control and transportation issues are addressed to deal with strategic, tactical and operational decisions. This paper utilizes a novel bi-objective mixed-integer linear programming (MILP) model to formulate a multi-period multi-product CLSC design problem considering aggregate cost minimization and service level maximization at the same time. To tackle the bi-objectiveness of the model, goal attainment method (GAM) is applied which is then executed by Gurobi Python API to test the applicability of the suggested model for three different scales (small, medium and large). It is demonstrated that the proposed methodology can find the optimal solutions for different problems in a maximum of 500 seconds. Finally, a set of sensitivity analyses is carried out on the main parameters in order to test the behaviors of the objective functions and suggest managerial insights as well as decision aids. The results reveal that the model is highly dependent on the demand parameter, that is, an increase in demand is closely related to an increase in the aggregate cost and a simultaneous downward trend in the service level.

Bi-objective optimization for a multi-period COVID-19 vaccination planning problem

This work investigates a new multi-period vaccination planning problem that simultaneously optimizes the total travel distance of vaccination recipients (service level) and the operational cost. An optimal plan determines, for each period, which vaccination sites to open, how many vaccination stations to launch at each site, how to assign recipients from different locations to opened sites, and the replenishment quantity of each site. We formulate this new problem as a bi-objective mixed-integer linear program (MILP). We first propose a weighted-sum and an ϵ-constraint methods, which rely on solving many single-objective MILPs and thus lose efficiency for practical-sized instances. To this end, we further develop a tailored genetic algorithm where an improved assignment strategy and a new dynamic programming method are designed to obtain good feasible solutions. Results from a case study indicate that our methods reduce the operational cost and the total travel distance by up to 9.3% and 36.6%, respectively. Managerial implications suggest enlarging the service capacity of vaccination sites can help improve the performance of the vaccination program. The enhanced performance of our heuristic is due to the newly proposed assignment strategy and dynamic programming method. Our findings demonstrate that vaccination programs during pandemics can significantly benefit from formal methods, drastically improving service levels and decreasing operational costs.

A robust multi-objective model for managing the distribution of perishable products within a green closed-loop supply chain

&lt;p style='text-indent:20px;'&gt;The required processes of supply chain management include optimal strategic, tactical, and operational decisions, all of which have important economic and environmental effects. In this regard, efficient supply chain planning for the production and distribution of perishable productsis of particular importance due to its leading role in the human food pyramid. One of the main challenges facing this chain is the time when products and goods are delivered to the customers and customer satisfaction will increase through this.In this research, a bi-objective mixed-integer linear programming (MILP)model is proposedto design a multi-level, multi-period, multi-product closed-loop supply chain (CLSC) for timely production and distribution of perishable products, taking into account the uncertainty of demand. To face the model uncertainty, the robust optimization (RO) method is utilized. Moreover, to solve and validate the bi-objective model in small-size problems, the &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \epsilon $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-constraint method (EC) is presented. On the other hand, a Non-dominated Sorting Genetic Algorithm (NSGA-II) is developed for solving large-size problems. First, the deterministic and robust models are compared by applying the suggested solutions methods in a small-size problem, and then, the proposed solution methods are compared in large-size problems in terms of different well-known metrics. According to the comparison, the proposed model has an acceptable performance in providing the optimal solutions and the proposed algorithm obtains efficient solutions.Finally, managerial insights are proposed using sensitivity analysis of important parameters of the problem.&lt;/p&gt;

Global and local supply chain sourcing design: cost and delivery reliability comparison in unimodal and intermodal transportation

The design of the sourcing strategy is key to gain a competitive advantage. In recent times, supplier failure is one of the most crucial risks of supply chains. The purpose of this study is to investigate the cost of local and global sources of the supply chain (SC) with the comparison of delivery reliability. The approach of this study applied data from one of the largest paper and board industries in Asia. We used a bi-objective mixed-integer linear programming (BOMILP) model for the formulation of the problem. Pareto optimal solutions are generated by implementing the multi-objective genetic algorithm (MOGA) to facilitate decision-makers to trade-off between cost and delivery reliability. The results reveal significant savings of 138.51 million Pkr per year by contracting with local suppliers (Pakistan) as compared to global suppliers from Turkey, China and Vietnam. However, the delivery reliability of global suppliers is higher than local suppliers. Further, this study can be developed by investigating the performance, wastage, and quality of the supplies provided by local and global suppliers.

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.

A multi-objective integrated procurement, production, and distribution problem of supply chain network under fuzziness uncertainties

In this paper, we devoted a design under uncertainty of a four-echelon supply chain network including multiple suppliers, multiple plants, multiple distributors and multiple customers. The proposed model is a bi-objective mixed integer linear programming which considers several constraints and aims to minimize the total costs including the procurement, production, storage and distribution costs as well as to maximize on-time deliveries (OTD). To bring the model closer to real-world planning problems, the objective function coefficients (e.g. procurement cost, production cost, inventory holding and transport costs) and other parameters (e.g., demand, production capacity and safety stock level), are all considered triangular fuzzy numbers. Besides, a hybrid mathematical model-based on credibility approach is constructed for the problem, i.e., expected value and chance constrained models. Moreover, to build the crisp equivalent model, we use different property of the credibility measure. The resulted crisp equivalent model is a bi-objective mixed integer linear programs (BOMILP). To transform this crisp BOMILP into a single objective mixed integer linear programs (MILP) model, we apply three different aggregation functions. Finally, numerical results are reported for a real case study to demonstrate the efficiency and applicability of the proposed model.

Vehicle Routing Problem in Relief Supply under a Crisis Condition considering Blood Types

Despite the advances achieved in Medical Sciences, no substitute has been found for blood as a vital factor. Therefore, preparing sufficient and healthy blood in crisis conditions is a challenge that health systems encounter. Along with examining the conducted investigations in this field, the main contribution of current research is to develop a biobjective Mixed-Integer Linear Programming (MILP) model for relief supply under crisis condition. For this purpose, this paper proposes a model for routing of bus blood receiver under crisis conditions considering different blood groups. Besides, hours of unnecessary travel by bloodmobiles (buses) between each blood station (BS) and the crisis-stricken city for dispatching the collected blood is prevented thanks to considering a helicopter. The mentioned model has two objectives: maximizing the amount of blood collected by bloodmobiles and minimizing the arrival time of the blood receiver buses and a helicopter to a crisis-stricken city after the collected blood is used up. The model is coded by CPLEX software, and the results obtained from solving the model indicate that, without considering a helicopter, the demand is not supplied within the critical period after crisis. Given that blood cannot be artificially produced, its primary resource is blood donors. Concerning the importance of this issue under crisis conditions, this research investigates the relief vehicles’ routing problem, including bus and helicopter, in a crisis considering supply and transfer of different blood groups to a crisis-stricken city for maximum relief supply and blood transfer within the shortest period.

Branch-and-Bound for Biobjective Mixed-Integer Linear Programming

We present a generic branch-and-bound algorithm for finding all the Pareto solutions of a biobjective mixed-integer linear program. The main contributions are new algorithms for obtaining dual bounds at a node, checking node fathoming, presolve, and duality gap measurement. Our branch-and-bound is predominantly a decision space search method because the branching is performed on the decision variables, akin to single objective problems, although we also sometimes split gaps and branch in the objective space. The various algorithms are implemented using a data structure for storing Pareto sets. Computational experiments are carried out on literature instances and on a new set of instances that we generate using a benchmark library (MIPLIB2017) for single objective problems. We also perform comparisons against the triangle splitting method from literature, which is an objective space search algorithm. Summary of Contribution: Biobjective mixed-integer optimization problems have two linear objectives and a mixed-integer feasible region. Such problems have many applications in operations research, because many real-world optimization problems naturally comprise two conflicting objectives to optimize or can be approximated in such a manner and are even harder than single objective mixed-integer programs. Solving them exactly requires the computation of all the nondominated solutions in the objective space, whereas some applications may also require finding at least one solution in the decision space corresponding to each nondominated solution. This paper provides an exact algorithm for solving these problems using the branch-and-bound method, which works predominantly in the decision space. Of the many ingredients of this algorithm, some parts are direct extensions of the single-objective version, but the main parts are newly designed algorithms to handle the distinct challenges of optimizing over two objectives. The goal of this study is to improve solution quality and speed and show that decision-space algorithms perform comparably to, and sometimes better than, algorithms that work mainly in the objective-space.

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.

A JIT part supply scheduling strategy with electric transport device between central receiving store and supermarkets in the automobile industry considering energy

Sustainable scheduling has been attracting arousing attention of modern manufacturers, and energy efficiency becomes a critical issue related to sustainability. This paper aims at providing an effective solution method for a real-world energy-efficient part supply scheduling problem (EPSSP) between the central receiving store and supermarkets with electric transport device on purpose of coordinating production and transportation in the automobile industry. A bi-objective mixed integer linear programming (MILP) model is formulated to jointly optimize the total energy consumption and a just-in-time (JIT) metric. An epsilon–constraint approach is presented to obtain the optimal solutions in small-scale problems. Owing to EPSSP's N P-hard nature, an indicator-based bi-objective teaching-learning-based optimization (I-BTLBO) algorithm is proposed by combining the binary quality indicator and teaching-learning-based optimization (TLBO) metaheuristic. I-BTLBO adopts a novel solution codification embedded with an adjustment technique to accommodate EPSSP's characteristic. For infeasible solutions, penalty function is applied to deal with the solution that violate vehicle capacities. A variable neighborhood search-based local optimizer is introduced to enhance I-BTLBO's convergence ability. In addition, an external population (EP) is employed to preserve non-dominated solutions' diversity and to guide the evolution process based on the Pareto dominance and preference indicators. Experimental studies demonstrate that the developed novel approach can effectively and efficiently solve EPSSP problem at different scales, and a case study is carried out to give instructive significance on real-world production. • A bi-objective MILP is formulated to minimize energy consumption and stocks. • Introduce an epsilon–constraint approach to obtain small-scale optimal solutions. • Propose an I-BTLBO algorithm to solve EPSSP. • Develop a novel solution codification with adjustment technique. • Develop a variable neighborhood search-based local optimizer.

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.

A new bi-objective green medicine supply chain network design under fuzzy environment: Hybrid metaheuristic algorithms

This paper studies the design of the green medicine supply chain network under uncertainty, which integrates allocation, location, production, distribution, routing, inventory, and purchasing problems. The main contribution of the current paper is to design a fuzzy bi-objective Mixed-Integer Linear Programming (MILP) model for a multi-period, three-echelon, multi-product, and multi-modal transportation green medicine supply chain network (GMSCN). Additionally, the main aim of this network is to consider the environmental impacts related to the establishment of pharmacies and hospitals, by focusing on the reduction of greenhouse gases and the control of environmental pollutants. Therefore, to cope with uncertain parameters, fuzzy programming is utilized to examine uncertainty parameters. To solve the GMSCN model, meta-heuristic algorithms are used, including social engineering optimization, improved kill herd, improved social spider optimization, and hybrid whale optimization and simulated annealing. In this regard, two new hybrid algorithms called hybrid Firefly Algorithm and Simulated Annealing (HFFA-SA) and Hybrid Firefly Algorithm and Social Engineering Optimization (HFFA-SEO) to solve the proposed model for the first time are developed. In order to show the applicability of our paper and the lack of benchmark functions in the literature, a set of simulated data in two sizes including small- and large-sized problems is provided. Finally, the results of the analysis and the designed problems indicate that the GMSCN model and developed solution approaches are promising.

An Integrated Bi-Objective Mathematical Model for Minimizing Take-off Delay and Passenger Dissatisfaction

As air transportation has increased in recent years, it is necessary for airport planners to optimally manage aircraft ground traffic on stands, taxiways and runways in order to minimize flight delay and passenger dissatisfaction. A closer look at the literature in this area indicates that most studies have merely focused on one of these resources which in a macroscopic level may result in aircrafts’ collision and ground traffic at the airport. In this paper, a new bi-objective Mixed-Integer Linear Programming (MILP) model is developed to help airport management to integrate Gate Assignment Problem (GAP) and Runway Scheduling Problem (RSP) considering taxiing operation for departing flights. The proposed model aims to help airport planners to 1) minimize any deviation from preferred schedule and 2) minimize transit passengers’ walking distance. Due to the complexity of the research problem, a Normalized Weighted Sum Method (NWSM) is applied to solve small-sized problems and two meta-heuristics, namely NSGA-II and MOGWO, are used for large-scale instances to generate Pareto optimal solutions. The performance of these algorithms is assessed by well-known coverage and convergence measures. Based on the most criteria, the results indicate that MOGWO outperforms NSGA-II.

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 Novel Matheuristic based on bi-level optimization for the multi-Objective design of hydrogen supply chains

This work introduces an efficient tool for the design of sustainable hydrogen supply chains (HSCs), considering both economic and environmental concerns, through an appropriate multi-objective strategy. The original problem, being formulated as a bi-objective mixed-integer linear programming (MILP) problem, takes into consideration the availability of different energy sources, the installation and operation of hydrogen facilities of different sizes and technologies, and the transportation of hydrogen from production units to storage facilities. The area of study is divided into grids which have a specific hydrogen demand that evolves over time, thus a multi-period model of the HSC is considered. In order to overcome the computational burden associated to the solution of large size instances of the resulting problem, we proposed a solution strategy consisting of a hybrid algorithm. The original problem is reformulated into a bi-level optimization problem: the upper level (discrete problem) consists of finding the optimal location for production plants and storage facilities, whereas the lower level (continuous problem) minimizes their corresponding costs associated to transportation and facility operation. A multi-objective evolutionary algorithm is employed for the solution of the bi-objective upper level, whereas the bi-objective lower level is decomposed using a scalarizing function, which is then solved using a linear programming solver. The proposed methodology is validated through the comparison of the true Pareto fronts given by CPLEX with ε-constraint method, for six increasing size instances. Numerical results prove that the proposed hybrid approach produces an accurate approximation of the Pareto-optimal fronts, more efficiently than the exact 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.

Assessing the benefits of horizontal collaboration using an integrated planning model for two-echelon energy efficiency-oriented logistics networks design

The transition to modern and ecologically oriented industries requires the application of new approaches and advanced models of logistics management. In this context, horizontal collaboration has recently become a promising and emerging practice for companies intending to improve their freight transportation efficiency and achieve environmental sustainability. This strategy is usually involved in the independent planning of decision-making problems with economic objectives. However, integrated optimisation of planning decisions with environmental considerations is nowadays considered as one of the major challenges to enhance the competitiveness of logistics networks. In this paper, the integrated planning problem of location, inventory and routing to design two-echelon green logistics networks is studied in collaborative and non-collaborative scenarios. The main purpose of this study is to assess the benefits of horizontal collaboration between suppliers in integrated freight transportation optimization by proposing a new decision support tool. A bi-objective Mixed Integer Linear Programming is proposed and adapted to address both distribution strategies. First, a resolution of the several scenarios with a single objective and a comparison of the results are performed. Second, a multi-objective optimization using an aggregation approach is conducted to find a good trade-off between the economic objectives and environmental ones. The experimental results show that horizontal collaboration among suppliers may decrease the total aggregated cost and emissions in the logistics system.