Bi-objective Mixed Integer Linear Programming Research Articles

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

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula>-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.</p>

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.

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.

A bi-objective model for determining an optimal warehouse capacity, and product allocation in a green multi-product, multi-period distribution network

Nowadays, the growing trend of environmental concerns is significantly higher than in the past while the business also focuses on the supply chain more to serve the customer demand using the lowest resources possible which leads to two main decisions which are capacity, and product allocation. Therefore, this research aims at designing a multi-product, multi-period, and multi-echelon supply chain network with factories, internal warehouses, external warehouses, and customers while also trying to lower the environmental effect of the supply chain through gas emission. For this problem, large data of products and 20 periods of planning horizon of a real-world polymer industry are considered. The product deliveries use direct shipment from both internal and external warehouses. We develop bi-objective mixed-integer linear programming to find the most optimal product allocation and capacity while minimizing the total logistic costs which include warehouse and transportation cost and total emission through transportation. The problem is solved by the min-max approach through a mixed integer linear programming model using CPLEX software. After we get the result, we compare it with the single-objective model’s result to determine the trade-off between the total logistic cost and emission gas. Our base case result shows a better overall satisfaction level among all the models.

A coordinated scheduling problem for the supply chain in a flexible job shop machine environment

In this study, a new coordinated scheduling problem is proposed for the multi-stage supply chain network. A multi-product and multi-period supply chain structure has been developed, including a factory, warehouses, and customers. Furthermore, the flexible job shop scheduling problem is integrated into the manufacturing part of the supply chain network to make the structure more comprehensive. In the proposed problem, each product includes a sequence of operations and is processed on a set of multi-functional machines at the factory to produce the final product. Final products are delivered to the warehouses to meet customers’ demands. If the demands of customers are not fulfilled, the shortage in the form of backorder may occur at any period. The problem is expressed as a bi-objective mixed-integer linear programming (MILP) model. The first objective function is to minimize the total supply chain costs. On the other hand, the second objective function aims to minimize the makespan in all periods. A numerical example is presented to evaluate the performance of the proposed MILP model. Five multi-objective decision-making (MODM) methods, namely weighted sum, goal programming, goal attainment, LP metric, and max–min, are used to provide different alternative solutions to the decision-makers. The performance of the methods is evaluated according to both objective function values and CPU time criteria. In order to select the best solution technique, the displaced ideal solution method is applied. The results reveal that the weighted sum method is the best among all MODM methods.

A green closed-loop supply chain network: a bi-objective mixed integer linear programming model

The increasing level of customer awareness and application of environmental laws by governments, on the one hand, and the increasing number of competitors, on the other hand, has obliged industry owners to include green activities in the design of the supply chain network. Green activities include any type of action that reduces environmental degradation. Hence, this study seeks to develop a bi-objective, mixed integer linear programming (MILP) model for designing a green supply chain network. In the proposed model, the minimisation of costs and detrimental environmental effects are discussed. LP metric method is used to solve the bi-objective model and the proposed model is run by using simulated data in small and medium sizes in GAMS software. Finally, the model sensitivity analysis is measured in order to evaluate the validity and performance of the proposed model by using demand-driven scenarios. The results of this model indicate the effectiveness of the proposed model.

A green closed-loop supply chain network: a bi-objective mixed integer linear programming model

The increasing level of customer awareness and application of environmental laws by governments, on the one hand, and the increasing number of competitors, on the other hand, has obliged industry owners to include green activities in the design of the supply chain network. Green activities include any type of action that reduces environmental degradation. Hence, this study seeks to develop a bi-objective, mixed integer linear programming (MILP) model for designing a green supply chain network. In the proposed model, the minimisation of costs and detrimental environmental effects are discussed. LP metric method is used to solve the bi-objective model and the proposed model is run by using simulated data in small and medium sizes in GAMS software. Finally, the model sensitivity analysis is measured in order to evaluate the validity and performance of the proposed model by using demand-driven scenarios. The results of this model indicate the effectiveness of the proposed model.