Bi-objective Mixed Integer Linear Programming Research Articles

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

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

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.

Integrated Production and Distribution Scheduling in Mobile Facilities

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

Improved model and efficient method for bi-objective closed-loop food supply chain problem with returnable transport items

ABSTRACT Closed-loop supply chains (CLSC) for food products mostly focus on the recovery of the residual value of food itself. Food packaging that plays an important role in food supply chain has rarely been studied. This paper aims to investigate an integrated multi-period closed-loop food supply chain planning problem with returnable transport items (RTIs) in which the total profit and the environmental impact are simultaneously considered. For the problem, an improved bi-objective mixed-integer linear program is formulated with obtained valid inequalities. Especially, the model with valid inequalities can reduce nearly 50% of the average computation time compared with the initial one. To solve the problem, a kernel-search heuristic based ε-constraint method is developed to obtain an approximate Pareto front. Finally, a fuzzy logic-based technique is adapted to help decision makers select a preferred solution according to his/her preference. A real case study from a slaughterhouse illustrates that the proposed method can improve the company’s current strategy for a 7-day planning. Computational results of randomly generated instances demonstrate that the proposed method outperforms the exact ε-constraint method in terms of computational time while providing good approximation.

Minimal covering unrestricted location of obnoxious facilities: bi-objective formulation and a case study

This paper addresses the problem of locating obnoxious facilities aiming to mitigate the adverse effects of such facilities by minimizing the total covered demand and reducing the harmful effects of multiple-coverage by placing the facilities far from each other. Unlike the classical approaches, it is assumed that the facilities can be located not only on the nodes but also on the network’s edges. Additionally, demands are not restricted to reside on the nodes but are distributed along the edges. In such a condition, the problem of locating obnoxious facilities is much closer to the real-world. A bi-objective mixed-integer linear programming formulation is developed for this novel problem. This bi-objective problem is solved using the e-constraint method and NSGA-II. The e-constraint method can be used to solve the small and medium-sized problems optimally in a reasonable time. As a realistic example, this approach is implemented in a case study in Isfahan for a particular urban planning problem. The case is locating obnoxious solid waste disposal facilities that should be located as far away as possible and simultaneously cover the least demands. Large scale problems cannot be solved efficiently using the e-constraint method. However, the numerical analysis showed the efficiency and effectiveness of the NSGA-II approach for these problems. Finally, sensitivity analysis is applied to evaluate the effect of changes in coverage distance and the number of facilities on the conflicting objective functions.

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

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

Data-driven predictive maintenance policy based on multi-objective optimization approaches for the component repairing problem

In systems with many components that are required to be constantly active, such as refineries, predicting the components that will break in a time interval after a stoppage may significantly increase their reliability. However, predicting the set of components to be repaired is a challenging task, especially when several conditions (e.g. breakage probability, repair time and cost) have to be considered simultaneously. A data-driven predictive maintenance policy is proposed for maximizing the system reliability and minimizing the maximum repair time, considering both budget and human resources constraints. Therefore, a data-driven algorithm is designed for extracting component breakage probabilities. Then, two bi-objective optimization approaches are proposed for determining the set of components to repair. The former is based on the formulation of a bi-objective mixed integer linear programming model solved through the AUGMEnted ε-CONstraint (AUGMECON) method. The latter implements a bi-objective large neighbourhood search, outperforming the first approach.

Heuristic algorithms for the bi-objective hierarchical multimodal hub location problem in cargo delivery systems

In this paper, we introduce an extended version of hub location problem, called bi-objective hierarchical multimodal hub location problem to simultaneously minimize the overall system-wide costs and the maximum delivery time. This problem is distinct from the classic hub location problem in designing a hierarchical multimodal hub-and-spoke network involving multiple transportation modes, multi-class hubs and corresponding layers. Combining cost and time dimensions, we first propose a bi-objective mixed-integer linear programming to model this problem formally with diverse flow balance constraints. We then show that the proposed model can be efficiently solved by a reformulation approach based on the ε-constraint method for only small instances. Hence, we develop two heuristics, a variable neighborhood search algorithm and an improved non-dominated sorting genetic algorithm-II to obtain high-quality Pareto solutions for realistic-sized instances. We further illustrate the application of the proposed model to provide decision support for cargo delivery systems. Finally, we conduct extensive numerical experiments based on Turkish network to demonstrate the superiority of the proposed solution methods compared to the standard non-dominated sorting genetic algorithm-II. The statistical results confirm the efficacy of the developed heuristic algorithms by adopting the Wilcoxon test.

A robust bi-objective mathematical model for disaster rescue units allocation and scheduling with learning effect

This paper proposes a novel bi-objective mixed-integer linear programming (MILP) model for allocating and scheduling disaster rescue units considering the learning effect. When a natural phenomenon (e.g., earthquake or flood) occurs, the presented decision support model is expected to help decision-makers of emergency relief centers to provide efficient planning for rescue units to minimize the total weighted completion time of rescue operations, as well as the total delay in rescue operations. The problem has some features in common with unrelated parallel machine scheduling (UPMS) problem and traveling salesman problem (TSP). To deal with the inherent uncertainty and bi-objective nature of the problem, an uncertainty-set based robust optimization technique and multi-choice goal programming (MCGP) with utility functions are applied. To demonstrate the applicability of the proposed model, a real case study in Mazandaran province in Iran is presented. The computational results confirm the high complexity of the problem and the significant impacts of the uncertainty on the solution. Moreover, the analytical results provide useful insights to decision-makers for disastrous situations.

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

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

A new dynamic two-stage mathematical programming model under uncertainty for project evaluation and selection

Project portfolio evaluation and selection is a complex task involving an exhaustive assessment of competing projects with interdependencies and synergies based on multiple and often conflicting criteria. The additional factor of uncertainty further complicates this complex task. This study proposes a two-stage hybrid multi-criteria decision making and mixed-integer linear programming for evaluating and selecting projects with interdependencies under uncertainty. In Stage I, we use the fuzzy technique for order of preference by similarity to ideal solution (TOPSIS) to evaluate the alternative projects under uncertainty. In Stage II, we formulate a bi-objective mixed-integer linear program to optimize profit and qualitative values for each portfolio by considering project synergies, human resources capabilities, and employee training opportunities under different scenarios. The proposed model produces portfolios with quantitative and qualitative values for each scenario under consideration. We demonstrate and validate the applicability and efficacy of the proposed approach through a real-world case study in the cybersecurity industry.

A stochastic network design problem for hazardous waste management

Hazardous waste management is of paramount importance due to the potential threats posed to the environment and local residents. The design of a hazardous waste management system involves several important decisions, i.e., the determination of the locations and sizes of treatment, recycling and disposal facilities, and organizing the transportation of hazardous waste among different facilities. In this paper, we proposed a novel stochastic bi-objective mixed integer linear program (MILP) to support these decisions in order to reduce the population exposure to risk while simultaneously maintaining a high cost efficiency of the transportation and treatment of hazardous waste. Moreover, considering the inherent uncertainty within the planning horizon, the cost, demand and affected population are defined as stochastic parameters. A sample average approximation based goal programming (SAA-GP) approach is used to solve the mathematical model. The proposed model and solution method are validated through numerical experiments whose results show that uncertainty may not only affect the objective value but also lead to different strategic decisions in the network design of a hazardous waste management system. In this regard, the strategic decisions obtained by the stochastic model is more robust to the change of external environment. Finally, the model is applied in a real-world case study of healthcare waste management in Wuhan, China, in order to show its applicability.

Multi-objective optimization for the reliable pollution-routing problem with cross-dock selection using Pareto-based algorithms

Cross-docking practice plays an important role in improving the efficiency of distribution networks, especially, for optimizing supply chain operations. Moreover, transportation route planning, controlling the Greenhouse Gas (GHG) emissions and customer satisfaction constitute the major parts of the supply chain that need to be taken into account integratedly within a common framework. For this purpose, this paper tries to introduce the reliable Pollution-Routing Problem with Cross-dock Selection (PRP-CDS) where the products are processed and transported through at least one cross-dock. To formulate the problem, a Bi-Objective Mixed-Integer Linear Programming (BOMILP) model is developed, where the first objective is to minimize total cost including pollution and routing costs and the second is to maximize supply reliability. Accordingly, sustainable development of the supply chain is addressed. Due to the high complexity of the problem, two well-known meta-heuristic algorithms including Multi-Objective Simulated-annealing Algorithm (MOSA) and Non-dominated Sorting Genetic Algorithm II (NSGA-II) are designed to provide efficient Pareto solutions. Furthermore, the ε-constraint method is applied to the model to test its applicability in small-sized problems. The efficiency of the suggested solution techniques is evaluated using different measures and a statistical test. To validate the performance of the proposed methodology, a real case study problem is conducted using the sensitivity analysis of demand parameter. Based on the main findings of the study, it is concluded that the solution techniques can yield high-quality solutions and NSGA-II is considered as the most efficient solution tool, the optimal route planning of the case study problem in delivery and pick-up phases is attained using the best-found Pareto solution and the highest change in the objective function occurs for the total cost value by applying a 20% increase in the demand parameter.

Integrated resource leveling and material procurement with variable execution intensities

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

Planning for Medical Emergency Transportation Vehicles during Natural Disasters

One of the main critical steps that should be taken during natural disasters is the assignment and distribution of resources among affected people. In such situations, this can save many lives. Determining the demands for critical items (i.e., the number of injured people) is very important. Accordingly, a number of casualties and injured people have to be known during a disaster. Obtaining an acceptable estimation of the number of casualties adds to the complexity of the problem. In this paper, a location-routing problem is discussed for urgent therapeutic services during disasters. The problem is formulated as a bi-objective Mixed-Integer Linear Programming (MILP) model. The objectives are to concurrently minimize the time of offering relief items to the affected people and minimize the total costs. The costs include those related to locations and transportation means (e.g., ambulances and helicopters) that are used to carry medical personnel and patients. To address the bi-objectiveness and verify the efficiency and applicability of the proposed model, the e-constraint method is employed to solve several randomly-generated problems with CLEPX solver in GAMS. The obtained results include the objective functions, the number of the required facility, and the trade-offs between objectives. Then, the parameter of demands (i.e., number of casualties), which has the most important role, is examined using a sensitivity analysis and the managerial insights are discussed.

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

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

Fuzzy Mathematical Programming and Self-Adaptive Artificial Fish Swarm Algorithm for Just-in-Time Energy-Aware Flow Shop Scheduling Problem With Outsourcing Option

Flow shop scheduling (FSS) problem constitutes a major part of production planning in every manufacturing organization. It aims at determining the optimal sequence of processing jobs on available machines within a given customer order. In this article, a novel biobjective mixed-integer linear programming (MILP) model is proposed for FSS with an outsourcing option and just-in-time delivery in order to simultaneously minimize the total cost of the production system and total energy consumption. Each job is considered to be either scheduled in-house or to be outsourced to one of the possible subcontractors. To efficiently solve the problem, a hybrid technique is proposed based on an interactive fuzzy solution technique and a self-adaptive artificial fish swarm algorithm (SAAFSA). The proposed model is treated as a single objective MILP using a multiobjective fuzzy mathematical programming technique based on the ϵ-constraint, and SAAFSA is then applied to provide Pareto optimal solutions. The obtained results demonstrate the usefulness of the suggested methodology and high efficiency of the algorithm in comparison with CPLEX solver in different problem instances. Finally, a sensitivity analysis is implemented on the main parameters to study the behavior of the objectives according to the real-world conditions.