Mixed Integer Nonlinear Programming Model Research Articles

Multi-period fourth-party logistics network design from the viability perspective: a collaborative hyper-heuristic embedded with double-layer Q-learning algorithm

In the ‘new normal’ setting of a mega-crisis, the viability becomes the driving force for the fourth party logistics (4PL) network design. In this paper, the viability is characterised in terms of agility, resilience and survival sustainability as the response to changes in demand, disruption and survivability. The fortification and recovery strategies are considered in possible disruptions at transfer centres and third-party logistics providers. A novel mixed integer non-linear programming model is proposed to obtain the multi-period 4PL network solution with minimum total cost under viability constraints. Considering the NP-hard characteristic of problem and the non-convex of proposed model, the hyper-heuristic algorithm is designed. To take advantage of both global optimality seeking and local search ability, a collaborative hyper-heuristic embedded with double-layer Q-learning (CHHDLQL) algorithm is proposed. The effectiveness and efficiency of the proposed algorithm is demonstrated by the promising numerical results. By stress-testing the existing network, appropriate adjustments to fortification and recovery strategies can effectively cope with changes in demand and disruption. Furthermore, the impact of 4PL strategy, fortification and recovery strategies, and viability constraints are investigated. The demand satisfaction, network resilience and capacity can be improved by adjusting agility, resilience and survival sustainability to influence different component network costs. Highlights A novel multi-period 4PL network model under viability constraints is presented. The CHHDLQL algorithm for construction and perturbation coevolution is proposed. The stress tests apply to the 4PL network, including demand, disruption and 3PL. 4PL improves demand satisfaction, resilience and capacity by adjusting viability.

First train timetabling and passenger transfer routing problems in urban rail transit networks

This paper incorporates the shortest passenger travel paths into the first train timetabling problem in urban rail transit networks. We first propose a path-based first train timetabling problem, which is formulated as a mixed-integer nonlinear programming model to reduce the travel time for passengers. To solve this model, we present a tailored branch-and-bound combined with the timetable-based Dijkstra algorithm. To further enhance the efficiency of the exact algorithm, we design a rolling time heuristic algorithm as a substitute for the branch-and-bound algorithm. Then, we proved that the rolling time heuristic combined with the timetable-based Dijkstra algorithm achieves 97.86% and 99.39% reductions in CPU time while sacrificing only 0.007% and 0.057% of optimality gap by two numerical experiments. Finally, the results based on two test networks with different structures demonstrate that the path-based first train timetabling model effectively improves passenger travel efficiency. Specifically, the travel times are reduced by 744 mins and 761 mins, while the transfer waiting times are reduced by 37.87% and 58.91% compared to the existing timetables. Moreover, the results also demonstrate that loop line networks will increase the difficulty of timetabling and lead to many passengers taking detours. To further enhance the efficiency of commuting for passengers who take the first train, URT operators may consider relaxing the limitations on the departure time of the depot stations.

Simultaneous Optimization of Work and Heat Exchange Networks

This paper introduces a simultaneous optimization approach to synthesizing work and heat exchange networks (WHENs). The proposed work and heat integration (WHI) superstructure enables different thermodynamic paths of pressure and temperature-changing streams. The superstructure is connected to a heat exchanger network (HEN) superstructure, enabling the heat integration of hot and cold streams identified within the WHI superstructure. A two-step solution strategy is proposed, consisting of initialization and design steps. In the first step, a thermodynamic path model based on the WHI superstructure is combined with a model for simultaneous optimization and heat integration. This nonlinear programming (NLP) model aims to minimize operating expenditures and provide an initial solution for the second optimization step. In addition, hot and cold streams are identified, enabling additional model reduction. In the second step of the proposed solution approach, a thermodynamic path model is combined with the modified HEN model to minimize the network’s total annualized cost (TAC). The proposed mixed integer nonlinear programming (MINLP) model is validated by several examples, exploring the impact of the equipment costing and annualization factor on the optimal network design. The results from these case studies clearly indicate that the new synthesis approach proposed in this paper produces solutions that are consistently similar to or better than the designs presented in the literature using other methodologies.

Large-scale optimization of nonconvex MINLP refinery scheduling

Modeling and optimization of large-scale refinery scheduling problems is challenging because of their complexity and size. Herein, we propose a mathematical model to represent such problems more accurately and realistically, and a state-of-the-art optimization framework for its solution. The framework leverages the use of mathematical optimization and algorithmic methods by combining modeling approaches (process design, model decompositions), solving strategies (rescheduling, heuristics), and machine learning regression (surrogate models). An industrial-size refinery scheduling problem (2 blenders, 4 feed tanks, distillation network with 5 towers, processing network with FCC, hydrotreaters, debutanizers, superfractionator, catalytic reformer) is formulated as a hierarchical nonconvex mixed-integer nonlinear programming (MINLP) model and is successfully optimized, providing higher profitability and more efficient scheduling operations considering 12 feedstocks, 10 products and multiple scenarios for time horizon and step. Results highlight the importance of tuning scheduling parameters and employing an enhanced computer-aided framework to enable the solution of industrial refinery scheduling operations.

Designing a Reverse Logistics Network for End-of-Life Vehicles in an Uncertain Environment

The strategic development of reverse logistics networks is crucial for addressing the common challenge of low recovery rates for end-of-life vehicles (ELVs) in China. To minimize the total cost of the reverse logistics network for ELVs, this paper proposes a mixed-integer linear programming (MILP) model. The model considers the recycling volume of different vehicle types, facility processing capacity, and the proportions of parts and materials. Building on this foundation, a fuzzy mixed-integer nonlinear programming (FMINLP) model is developed to account for the inherent uncertainty associated with recycling volumes and facility processing capacities. The model was solved using Lingo, and its effectiveness was validated using Jiangsu Province of China as a case study, followed by a sensitivity analysis. The results indicate that dismantling and machining centers incur the highest processing costs. Variations in recycling volume and facility handling capacity significantly impact total costs and site selection, with the former having a more pronounced effect. Increasing facility processing capacity effectively increases the recovery rate. Moreover, a higher confidence level corresponds to higher total costs and a greater demand for facilities.

Ship coping strategies for hurricane-induced port disruptions

PurposeDisruptions at ports may destroy the planned ship schedules profoundly, which is an imperative operation problem that shipping companies need to overcome. This paper attempts to help shipping companies cope with port disruptions through recovery scheduling.Design/methodology/approachThis paper studies the ship coping strategies for the port disruptions caused by severe weather. A novel mixed-integer nonlinear programming model is proposed to solve the ship schedule recovery problem (SSRP). A distributionally robust mean conditional value-at-risk (CVaR) optimization model was constructed to handle the SSRP with port disruption uncertainties, for which we derive tractable counterparts under the polyhedral ambiguity sets.FindingsThe results show that the size of ambiguity set, confidence level and risk-aversion parameter can significantly affect the optimal values, decision-makers should choose a reasonable parameter combination. Besides, sailing speed adjustment and handling rate adjustment are effective strategies in SSRP but may not be sufficient to recover the schedule; therefore, port skipping and swapping are necessary when multiple or longer disruptions occur at ports.Originality/valueSince the port disruption is difficult to forecast, we attempt to take the uncertainties into account to achieve more meaningful results. To the best of our knowledge, there is barely a research study focusing on the uncertain port disruptions in the SSRP. Moreover, this is the first paper that applies distributionally robust optimization (DRO) to deal with uncertain port disruptions through the equivalent counterpart of DRO with polyhedral ambiguity set, in which a robust mean-CVaR optimization formulation is adopted as the objective function for a trade-off between the expected total costs and the risk.

Optimizing City-Scale Demolition Waste Supply Chain Under Different Carbon Policies.

In order to establish a green, low-carbon circular development economic system, imperative goals include achieving carbon peaking and carbon neutrality. This research delves into the resource utilization of city-scale demolition waste (C&DW), aligning with environmental protection needs and sustainable development principles. The paper introduces a unique closed-loop supply chain (CLSC) model tailored for C&DW and employs a distinctive mixed integer nonlinear programming (MINLP) model for optimization. Guangzhou serves as a case study for thorough analysis, verification, and practical application of the proposed model, especially under diverse scenarios of carbon price (CP) and carbon trading (CT) policies. The key conclusions drawn from this study include the following: (1) The cost of carbon emissions is intricately influenced by both carbon emissions and carbon price, with the latter effectively regulating the carbon emissions during C&DW recycling. (2) The implementation of a CT policy, with a fixed carbon price, contributes to a further reduction in the cost of C&DW recycling treatment. (3) Under equivalent conditions, the CT policy demonstrates the potential to decrease costs and enhance the economic benefits within the building environmental protection product market. The research outcomes not only contribute to the advancement of management theory in the C&DW recycling supply chain (SC) but also provide a robust theoretical foundation for governmental initiatives aimed at introducing effective C&DW recycling management policies.

Period-dependent pricing methods of multi-type vehicles for BOT highway projects with pavement rehabilitation effects

In the planning stage of a BOT highway project, the future traffic demand plays a pivotal role on designing contract factors. Considering that the traffic demand generally has variations with time, the constant toll during the concession period obviously is not a good choice. And the different toll charges classified by vehicles types definitely affect the traffic composition and maintenance cost. To address this practical issue, this study proposes the mixed integer nonlinear programming models to select the period-dependent toll charges for multi-type vehicles and highway capacity, by considering both the objectives of the government and private firms. Then two solution algorithms are provided to solve our proposed models. Finally, numerical experiments are conducted to assess the applicability and efficiency of our models and algorithms.

Stochastic ambulance dispatching and routing in mass casualty incident under road vulnerability

This paper investigates the stochastic ambulance dispatching and routing (SADR) problem with multiple casualty collection points in the mass casualty incidents (MCI) under uncertainty of road vulnerability and traffic congestion caused by an earthquake. A two-stage stochastic mixed-integer nonlinear programming model is formulated to derive a reliable and high-quality ambulance scheduling, dispatching and routing solution for maximizing expected number of survivors of all casualties. An integrated simulation optimization approach combining a data-driven travel time scenario generation algorithm, sample average approximation, and a column-generation-based heuristic method is developed to efficiently solve this complicated two-stage SADR model with a nonlinear objective function and continuous travel time distributions. Collaborating with the National Science and Technology Center for Disaster Reduction (NCDR), an empirical study using a potential real-world earthquake scenario occurring in Taiwan is conducted to demonstrate the usefulness and effectiveness of our proposed model and solution approach compared to the deterministic model and two current-practice heuristics.

Matheuristics vs. metaheuristics for joint lot-sizing and dynamic pricing problem with nonlinear demands

This paper considers a joint dynamic pricing and production planning decisions problem for a profit-maximizing firm that produces and sells multiple products. The objective is to develop a coordinated decision approach for multi-product pricing and lot sizing decisions for a manufacturer considering a limited production capacity. The demand for each product is assumed to be iso-elastic and integrates the complementarity and the substitution effects between the products. First, the problem is formulated as a non-convex mixed integer nonlinear programming model (MINLP) incorporating capacity constraints, setup costs, and nonlinear demand functions. Then, since the model is nonlinear and non-convex, a set of approximate approaches based on the Genetic algorithm, Late Acceptance Hill Climbing and Simulated Annealing methods are designed to solve this problem. Based on this study, the performances of two variants of approximate methods: matheuristics and metaheuristics are discussed and analyzed. The extensive experimental study, performed on real-world inspired instances, shows that matheuristic methods with setup-variables encoding scheme outperform the rest of the methods. The research outcomes show that coordinating a decision-making process by optimizing both prices and production plans simultaneously can result in significant profit for a company. However, one must consider the joint effect of the parameters of the demand function as well as the impact of the production capacity. This comprehensive understanding enables the company to avoid excessive investments in less lucrative products with lower sales potential, thereby ensuring resource allocation aligns with profitability and market demand.

Vehicle routing problem for fresh products distribution considering customer satisfaction through adaptive large neighborhood search

To deal with the fresh products distribution problem in city logistics, this paper focuses on the customer psychological behavior considering both delivery time and the fresh products quality from the perspective of customer satisfaction. Thereby, we propose a mixed-integer nonlinear programming model, among which a time satisfaction function depending on customers’ time windows and a quality satisfaction function considering the freshness of the products are introduced as constraints. Then, an improved Adaptive Large Neighborhood Search algorithm is designed with a new strategy to jump out of the local optimal solutions and with new operators considering satisfaction functions. Numerical experiments on benchmark instances of standard Solomon instances demonstrate the effectiveness and the efficiency of the proposed method. The calculation results showed that the proposed algorithm is superior to previous approaches by obtaining better feasible solutions with less calculation time. Furthermore, managemental insights are provided for enterprises on making wise investments while ensuring customer experience through optimizing distribution strategies.

Modeling and Solving the Traveling Salesman Problem with Speed Optimization for a Plug-In Hybrid Electric Vehicle

This paper investigates a variant of the traveling salesman problem (TSP) with speed optimization for a plug-in hybrid electric vehicle (PHEV), simultaneously optimizing the average speed and operation mode for each road segment in the route. Two mixed-integer nonlinear programming models are proposed for the problem: one with continuous speed decision variables and one with discretized variables. Because the models are nonlinear, we propose reformulation schemes and introduce valid inequalities to strengthen them. We also describe a branch-and-cut algorithm to solve these reformulations. Extensive numerical experiments are performed to demonstrate the algorithm’s performance in terms of computing time and energy consumption costs. Specifically, the proposed solution method can efficiently solve instances with a realistic number of customers and outperforms the benchmark approaches from the literature. Integrating speed optimization into the TSP of a PHEV can lead to significant energy savings compared with the fixed-speed TSP. In addition, the proposed model is extended to investigate the impact of the presence of charging stations, which makes the problem harder to solve but has the potential to further reduce energy consumption costs. Funding: F. Wu gratefully acknowledges the support of the National Natural Science Foundation of China [Grant 72271161]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2023.0247 .

Decisions on ship route, refueling, and sailing speed considering ECA regulation and demand uncertainty

This article aims to maximize the shipping company’s profits by jointly optimizing shipping volume, refueling, ship leg option, and sailing speed with considering of stochastic shipping demand and Emission Control Area (ECA) regulation. Shipping demand is inherently stochastic and intermittent characteristic, causing formidable challenges in shipping service. A distributionally robust optimization method is adopted to handle the demand uncertainties. We build an ambiguity set of shipping demand and introduce a chance constraint with the shipping volume and shipping demand balance to guarantee the satisfactory rate of shipping demand. Moreover, the ECA regulation stimulates the diversification of transportation routes, which further increases transportation uncertainty. Thus, a nonlinear mixed integer programming model with nonconvex chance-constraint is developed. The optimal solution is obtained by transforming the original problem into an attractable second-order conic programming problem. An Asia service route is selected as a numerical experiment. The results show that the proposed model is effective, the optimal refueling strategy, leg option, speed, and shipping volume can be determined simultaneously. Meanwhile, it outperforms the state-of-the-art methods in requiring less demand information and producing fewer conservative results under demand uncertainties. Additionally, some managerial insights are revealed according to the case study.

Integrated location and inventory planning in service parts logistics with last-mile delivery outsourcing

Outsourcing last-mile delivery to express companies provides an innovative strategy for capital goods manufacturers to improve service parts logistics. Given delivery dilemma due to geographically dispersed customers and low-frequency uncertain demands, we develop an integrated location and inventory problem with last-mile delivery outsourcing in a two-echelon service parts logistics. The proposed non-linear mixed integer programming model captures important decisions involving (1) location of local warehouses, (2) customer allocation to opened local warehouses, (3) inventory levels at these local warehouses, (4) outsourcing or self-operating last-mile delivery. We then replace service level constraints that cause nonlinearity with two new sets of constraints, and construct a linear mixed integer formulation that could be solved by GUROBI solver for some small-scale problems. It is still a challenging problem for large scale case, for which we propose an adaptive large neighbourhood search (ALNS) algorithm. Our extensive computational experiments verify the effectiveness of our model and algorithm. Computational experiments’ results indicate that outsourcing last-mile delivery amounting to around 12 % total cost savings on average, compared to self-operating in a real-world large instance from our research partner. Furthermore, we test some key parameters with computational experiments, and provide some managerial implications.

Model and solution approach to coordinate production-inventory strategies considering nonlinear price-sensitive demand: application to Canadian pulp and paper industry

This study addresses a practical problem within a multi-level supply chain where a wide range of customers can be served through different strategies such as make-to-stock, make-to-order, or vendor-managed inventory. The customer demand is stochastic, and sensitive to pricing associated with different production-inventory strategies. We propose a two-stage stochastic mixed-integer non-linear programming model. In the first stage, decisions are made regarding the selection of production-inventory strategies and pricing to maximise the expected profit. The second stage involves decisions related to production, inventory, and distribution, which are used to evaluate the first-stage decisions under various scenarios with different levels of accuracy. To solve the model, a metaheuristic approach based on the Simulated Annealing algorithm is developed. To showcase the practical applicability of our model and solution approach, we use a real case study in a Canadian pulp and paper supply chain. The results revealed that both the production-inventory strategy assigned to customers and the sales price underwent changes across scenarios. Furthermore, we demonstrated that by implementing the SA algorithm, we could improve the initial profit by up to 1.43% through slight adjustments in the sales price and assigned strategies for customers.

A bi-objective mixed-integer non-linear programming model with Grasshopper Optimization Algorithm for military-based humanitarian supply chains

Military systems have many components, such as garrisons, border guards, and industries with a large population that may be affected by disasters. Garrisons, hospitals, clinics, and other facilities in military systems are needed to reduce human casualties. However, these facilities may not have enough resources to handle the disruption. There is no mathematical model to manage disasters with military systems including garrisons and border guards as relief forces interacting with non-military medical centers. This study develops a model for coordinating humanitarian supply chains and logistics within the military system. The affected military sites (AMSs) are military hospitals, clinics, industries, airports, military organizational settlements, garrisons, and border guards. A bi-objective mixed integer non-linear programming model is developed to minimize total costs during pre- and post-disaster planning, a weighted average of injury time in military vehicles, and total delay in relief items (RIs) distribution and rescue operations. For small and medium test problems, GAMS software is used, and for large test problems, the Grasshopper Optimization Algorithm (GOA) is proposed. We conducted some numerical examples to demonstrate the proposed model’s efficiency and feasibility and evaluate the metaheuristic. Sensitivity analyses are conducted to evaluate model behavior under various conditions.

Decentralized exam timetabling: A solution for conducting exams during pandemics

Traditionally, exams were scheduled into designated timeslots within a centralized campus. However, the outbreak of COVID-19 disrupted this practice, forcing universities to prioritize the safety of individuals during exams. In response, a shift to online assessments occurred, bringing forth new challenges such as academic dishonesty and inadequate infrastructure. This study presents a novel approach to decentralized in-person exams for geographically dispersed students. A mixed-integer non-linear programming model is developed, followed by a mixed-integer linear reformulation, to optimally determine the scheduling of exams, test center locations, and student allocation, while addressing the incorporation of both hard and soft constraints. Given the complexity of the problem, we have implemented meta-heuristic algorithms to solve larger instances, including genetic and simulated-annealing algorithms. Additionally, we have devised a hybrid algorithm that combines the strengths of population-based and local-search algorithms. For evaluation purposes, we used real data collected from our department during the fall semester of 2022. The results demonstrate the superior performance of the hybrid algorithm compared to others, in terms of both effectiveness and adaptability. It remains consistent across different scenarios, effectively addressing real-world complexities. Finally, a comparison between centralized and decentralized scenarios underscores the advantages of the latter approach during the pandemic.

Rescheduling trains by crossover tracks to promote service quality of urban rail transit under partial blockages.

This study focuses on the rescheduling problem with disruptions that cause partial blockages in the urban rail transit (URT), contributing to extending the relative train rescheduling studies. The alternative driving measure (ADM), which could be regarded as one train rerouting measure, is used to skip the blocked section, and a mixed-integer nonlinear programming (MINLP) model is built based on it. Time-varying passenger flow as well as the turnaround process of rolling stocks is taken into consideration. To solve the model, a customized genetic algorithm is used to quickly generate high-quality solutions. Real-world data is studied and sensitivity analyses are taken to verify the feasibility and advantage of ADM. The results validate the proposed model and algorithm, as well as confirm that ADM shows significantly better performance than the practical operation measure in promoting passenger service quality of URT under partial blockage.

Study on cruise speed optimization and refueling strategy considering emission control areas

In order to alleviate the cost pressure caused by navigation restrictions and rising fuel prices faced by cruise companies, on the basis of considering the constraints of emission control areas and the arrival time constraints of each port, according to the relationship between the speed and fuel consumption of ships during navigation, a mixed integer nonlinear programming model with the minimum sum of the sum of ship fuel replenishment cost, arrival delay penalty cost and carbon tax cost is established to optimize the speed and refueling strategy of cruise ships. The results show that with the change of fuel price and time constraints, the fuel replenishment strategy will also change.

Optimal supply network with vendor managed inventory in a healthcare system with RFID

We consider a supply network for a medical implant manufacturer within the healthcare system, wherein the manufacturer oversees multi-hospital warehouses under a vendor-managed inventory (VMI) policy and utilizes RFID technology to reduce the network’s total costs and enhance customer service. The manufacturer must determine the warehouse locations, inventory policies, and RFID investments to minimize total costs. In this research, we develop a mixed-integer nonlinear program model to formulate the integrated decision problem, which combines the location-inventory problem with the allocation and assignment of RFID technology, including its investment level. A real-world problem is resolved and analyzed using the proposed approach.