Exact Solution Algorithm Research Articles

Optimizing hazardous materials transportation network: A bi-level programming approach considering road blocking

Road transportation serves as the primary mode for hazardous materials (hazmat) transportation in China. However, the current academic literature lacks sufficient exploration of optimizing transportation networks through road-blocking strategies. This research proposed a bi-level programming model, where the government acts as the decision-maker in the upper-level programming, and the transportation enterprise operates at the lower level. Road-blocking strategies are employed by the government to mitigate transportation risks. We developed exact solution algorithms for the upper programming and employed genetic algorithms for the lower model separately. Finally, using a real road network in Beijing as a case study, we showcased the effectiveness of our approach. We computed government decision schemes and enterprise transportation routes for 13 scenarios, and conducted a further discussion and analysis on scenarios with road service levels of 83% and 80.9%. The method and results presented in our study can adeptly dissect the transportation of hazmat in city areas, offering insightful perspectives and robust support for devising more streamlined management strategies.

Due date-oriented picker routing, an efficient exact solution algorithm, and its application to pick-from-store omnichannel retailing

The advent of e-commerce and omnichannel retailing has sparked renewed interest in picker routing in warehouses. This paper presents two significant methodological advances in this well-established field. First, it is a well-known fact that the parallel-aisle layout of warehouses, as opposed to general graphs, allows for polynomial-time solutions of the Traveling Salesman Problem. We show that the parallel-aisle structure can also be exploited when pickers are tasked with visiting storage positions associated with specific due dates. We establish that picker routing in warehouses, subject to soft due date constraints, is a binary NP-hard problem. We also present an exact branch-and-bound algorithm with pseudo-polynomial time complexity. This algorithm effectively solves instances with up to 60 picking positions and five cross aisles within a few seconds while guaranteeing optimality. Second, for even larger pick lists, we demonstrate the successful integration of our algorithm into a real-time framework. This approach allows us to avoid extended solution times that would otherwise delay the picker’s departure, without compromising the quality of the solution. To illustrate the practical relevance of these two methodological innovations, we apply our routing algorithm to the context of pick-from-store omnichannel retailing. By assigning due dates to critical products, we significantly reduce stockout occurrences for online customers. These stockouts occur when the stock level, initially deemed sufficient to confirm an online order, is depleted by walk-in customers before the picker reaches the relevant shelf.

On-demand mobility-as-a-Service platform assignment games with guaranteed stable outcomes

Mobility-as-a-Service (MaaS) systems are two-sided markets, with two mutually exclusive sets of agents, i.e., travelers/users and operators, forming a mobility ecosystem in which multiple operators compete or cooperate to serve customers under a governing platform provider. This study proposes a MaaS platform equilibrium model based on many-to-many assignment games incorporating both fixed-route transit services and mobility-on-demand (MOD) services. The matching problem is formulated as a convex multicommodity flow network design problem under congestion that captures the cost of accessing MOD services. The local stability conditions reflect a generalization of Wardrop's principles that include operators’ decisions. Due to the presence of congestion, the problem may result in non-stable designs, and a subsidy mechanism from the platform is proposed to guarantee local stability. A new exact solution algorithm to the matching problem is proposed based on a branch and bound framework with a Frank-Wolfe algorithm integrated with Lagrangian relaxation and subgradient optimization, which guarantees the optimality of the matching problem but not stability. A heuristic which integrates stability conditions and subsidy design is proposed, which reaches either an optimal MaaS platform equilibrium solution with global stability, or a feasible locally stable solution that may require subsidy. For the heuristic, a worst-case bound and condition for obtaining an exact solution are both identified. Two sets of reproducible numerical experiments are conducted. The first, on a toy network, verifies the model and algorithm, and illustrates the differences local and global stability. The second, on an expanded Sioux Falls network with 82 nodes and 748 links, derives generalizable insights about the model for coopetitive interdependencies between operators sharing the platform, handling congestion effects in MOD services, effects of local stability on investment impacts, and illustrating inequities that may arise under heterogeneous populations.

Maintenance optimization for multi-component systems with a single sensor

We consider a multi-component system in which a single sensor monitors a condition parameter. Monitoring gives the decision maker partial information about the system state, but it does not reveal the exact state of the components. Each component follows a discrete degradation process, possibly correlated with the degradation of other components. The decision maker infers a belief about each component’s exact state from the current condition signal and the past data, and uses that to decide when to intervene for maintenance. A maintenance intervention consists of a complete and perfect inspection, and may be followed by component replacements. We model this problem as a partially observable Markov decision process. For a suitable stochastic order, we show that the optimal policy partitions in at most three regions on stochastically ordered line segments. Furthermore, we show that in some instances, the optimal policy can be partitioned into two regions on line segments. In two examples, we visualize the optimal policy. To solve the examples, we modify the incremental pruning algorithm, an exact solution algorithm for partially observable Markov decision processes. Our modification has the potential to also speed up the solution of other problems formulated as partially observable Markov decision processes.

A non-anticipative learning-optimization framework for solving multi-stage stochastic programs

AbstractWe present a non-anticipative learning- and scenario-based prediction-optimization (ScenPredOpt) framework that combines deep learning, heuristics, and mathematical solvers for solving combinatorial problems under uncertainty. Specifically, we transform neural machine translation frameworks to predict the optimal solutions of scenario-based multi-stage stochastic programs. The learning models are trained efficiently using the input and solution data of the multi-stage single-scenario deterministic problems. Then our ScenPredOpt framework creates a mapping from the inputs used in training into an output of predictions that are close to optimal solutions. We present a Non-anticipative Encoder-Decoder with Attention (NEDA) approach, which ensures the non-anticipativity property of multi-stage stochastic programs and, thus, time consistency by calibrating the learned information based on the problem’s scenario tree and adjusting the hidden states of the neural network. In our ScenPredOpt framework, the percent predicted variables used for the solution are iteratively reduced through a relaxation of the problem to eliminate infeasibility. Then, a linear relaxation-based heuristic is performed to further reduce the solution time. Finally, a mathematical solver is used to generate the complete solution. We present the results on two NP-Hard sequential optimization problems under uncertainty: stochastic multi-item capacitated lot-sizing and stochastic multistage multidimensional knapsack. The results show that the solution time can be reduced by a factor of 599 with an optimality gap of only 0.08%. We compare the results of the ScenPredOpt framework with cutting-edge exact and heuristic solution algorithms for the problems studied and find that our framework is more effective. Additionally, the computational results demonstrate that ScenPredOpt can solve instances with a larger number of items and scenarios than the trained ones. Our non-anticipative learning-optimization approach can be beneficial for stochastic programming problems involving binary variables that are solved repeatedly with various types of dimensions and similar decisions at each period.

A holistic approach to introducing a light electric freight vehicle (LEFV) system in a historic urban environment: The case of Quito

In this paper we propose a comprehensive approach for introducing a first/last mile system based on Light Electric Freight Vehicles (LEFVs) in the historical centre of Quito (Ecuador). The approach includes four crucial steps for successful launching and establishment of a LEFVs system. The first step considers organizational aspects by proposing a number of alternative cooperative business models. In the second step, the LEFVs fleet mix problem is defined as a Multicriteria Decision Making Problem (MCDM) and solved by the Analytic Network Process (ANP). The third step includes the problem of optimal micro hub location as well as the problem of optimal routing of LEFVs. The micro hub location problem is modelled as a capacitated p-median problem and solved by an exact solution algorithm. The problem of optimal LEFVs routing is defined as the capacitated pickup and delivery with time windows addressed by a web-based optimization tool. The fourth step includes an assessment of the existing policy framework and proposing a set of additional legal instruments for facilitating the LEFVs market uptake. Based on the proposed four-step approach, the first/last mile system is currently being implemented in the historical centre of Quito. The initial results appear promising regarding the success of the new LEFVs first/last mile system.

A competitive heuristic algorithm for vehicle routing problems with drones

We propose a heuristic algorithm capable of handling multiple variants of the vehicle routing problem with drones (VRPD). Assuming that the drone may be launched from a node and recovered at another, these variants are characterized by three axes, (1) minimizing the transportation cost or minimizing the makespan, (2) the drone is either allowed or not allowed to land while awaiting recovery, and (3) single or multiple trucks each equipped with a drone. In our algorithm, we represent a VRPD solution as a set of customer sequences and evaluate it via local search procedures solving for each sequence a problem that we refer to as the fixed route drone dispatch problem (FRDDP). Given a sequence of customers to be served by a single truck and its drone, the FRDDP selects a subset of customers to be served by the drone and determines drone launch and recovery nodes, while ensuring that each such customer is positioned between two nodes in the initial sequence. We introduce a heuristic dynamic program (HDP) to solve the FRDDP with reduced computational complexity compared to an exact solution algorithm for the problem. We reinforce our algorithm by developing filtering strategies based on the HDP. We benchmark the performance of our algorithm on nine benchmark sets pertaining to four VRPD variants resulting in 932 instances. Our algorithm computes 651 of 680 optimal solutions and identifies 189 new best-known solutions.

A tight compact quadratically constrained convex relaxation of the Optimal Power Flow problem

In this paper, we consider the Optimal Power Flow (OPF) problem which consists in determining the power production at each bus of an electric network by minimizing the production cost. Our contribution is an exact solution algorithm for the OPF problem. It consists in a spatial branch-and-bound algorithm based on a compact quadratically-constrained convex relaxation. It is computed by solving the semidefinite rank relaxation of OPF once at the root node of the algorithm. An important result is that the optimal value of our compact relaxation is equal to the rank relaxation value. Then, at every sub-nodes of our branch-and-bound, the lower bound is obtained by solving a quadratic convex problem instead of an SDP. Another contribution is that we add only O(n+m) variables that model the squares of the initial variables, where n is the number of buses in the power system and m the number of transmission lines, to construct our relaxation. Then, since the relations between the initial and auxiliary variables are non-convex, we relax them to get a quadratic convex relaxation. Finally, in our branch-and-bound algorithm, we only have to force a reduced number of equalities to prove global optimality. This quadratic convex relaxation approach is here tailored to the OPF problem, but it can address any application whose formulation is a quadratic optimization problem subject to quadratic equalities and ring constraints. Our first experiments on instances of the OPF problem show that our new algorithm Compact OPF (COPF) is more efficient than the standard solvers and other quadratic convex relaxation based methods we compare it with.

A two-stage method for doubly resource constrained elementary shortest path problems

The resource constrained shortest path problem (RCSPP) has been extensively studied. However, its more general version, the doubly resource constrained (elementary) shortest path problem (DRCESPP), is usually not concerned and no exact solution algorithm has been found in the literature. In this work, an exact two-stage method is proposed for solving the DRCESPP. For leveraging resource lower limits, an estimation to the maximum resource consumption of all elementary source–destination paths is proposed in the first stage. This makes the network be reduced more significantly and the algorithm be terminated in the first stage. By introducing an effective search direction into the conventional Pulse algorithm, a LB-first Pulse algorithm is proposed in the second stage to find the exact solution from the reduced network output in the first stage. Extensive computational experiments on various test instances are conducted. The numerical results show that the maximum resource consumption estimation method and the improved Pulse algorithm are effective, and the proposed two-stage method outperforms the Pulse algorithm in solving complex instances.

An algorithm for exact analytical solutions for tilted anisotropic Dirac materials

An algorithm for exact analytical solutions for tilted anisotropic Dirac materials

Chance-Constrained Optimization for a Green Multimodal Routing Problem with Soft Time Window under Twofold Uncertainty

This study investigates a green multimodal routing problem with soft time window. The objective of routing is to minimize the total costs of accomplishing the multimodal transportation of a batch of goods. To improve the feasibility of optimization, this study formulates the routing problem in an uncertain environment where the capacities and carbon emission factors of the travel process and the transfer process in the multimodal network are considered fuzzy. Taking triangular fuzzy numbers to describe the uncertainty, this study proposes a fuzzy nonlinear programming model to deal with the specific routing problem. To make the problem solvable, this study adopts the fuzzy chance-constrained programming approach based on the possibility measure to remove the fuzziness of the proposed model. Furthermore, we use linear inequality constraints to reformulate the nonlinear equality constraints represented by the continuous piecewise linear functions and realize the linearization of the nonlinear programming model to improve the computational efficiency of problem solving. After model processing, we can utilize mathematical programming software to run exact solution algorithms to solve the specific routing problem. A numerical experiment is given to show the feasibility of the proposed model. The sensitivity analysis of the numerical experiment further clarifies how improving the confidence level of the chance constraints to enhance the possibility that the multimodal route planned in advance satisfies the real-time capacity constraint in the actual transportation, i.e., the reliability of the routing, increases both the total costs and carbon emissions of the route. The numerical experiment also finds that charging carbon emissions is not absolutely effective in emission reduction. In this condition, bi-objective analysis indicates the conflicting relationship between lowering transportation activity costs and reducing carbon emissions in routing optimization. The sensitivity of the Pareto solutions concerning the confidence level reveals that reliability, economy, and environmental sustainability are in conflict with each other. Based on the findings of this study, the customer and the multimodal transport operator can organize efficient multimodal transportation, balancing the above objectives using the proposed model.

Performance Assessment of an ITU-T Compliant Machine Learning Enhancements for 5G RAN Network Slicing

Network slicing is a technique introduced by 3GPP to enable multi-tenant operation in 5 G systems. However, the support of slicing at the air interface requires not only efficient optimization algorithms operating in real time but also its tight integration into the 5 G control plane. In this paper, we first present a priority-based mechanism enabling defined performance isolation among slices competing for resources. Then, to speed up the resource arbitration process, we propose and compare several supervised machine learning (ML) techniques. We show how to embed the proposed approach into the ITU-T standardized ML architecture. The proposed ML enhancement is evaluated under realistic traffic conditions with respect to the performance criteria defined by GSMA while explicitly accounting for 5 G millimeter wave channel conditions. Our results show that ML techniques are able to provide suitable approximations for the resource allocation process ensuring slice performance isolation, efficient resource use, and fairness. Among the considered algorithms, polynomial regressions show the best results outperforming the exact solution algorithm by 5–6 orders of magnitude in terms of execution time and both neural network and random forest algorithms in terms of accuracy (by 20–40 %), sensitiveness to workload variations and training sample size. Finally, ML algorithms are generally prone to service level agreements (SLA) violation under high load and time-varying channel conditions, implying that an SLA enforcement system is needed in ITU-T's 5 G ML framework.

An artificial bee colony algorithm for medical goods distribution and pharmacological waste collection by hybrid vehicles considering environmental criteria

PurposeThis paper aims to focus on a medical goods distribution problem and pharmacological waste collection by plug-in hybrid vehicles with some real-world restrictions. In this research, considering alternative energy sources and simultaneous pickup and delivery led to a decrease in greenhouse gas emissions and distribution costs, respectively.Design/methodology/approachHere, this problem has been modeled as mixed-integer linear programming with the traveling and energy consumption costs objective function. The GAMS was used for model-solving in small-size instances. Because the problem in this research is an NP-hard problem and solving real-size problems in a reasonable time is impossible, in this study, the artificial bee colony algorithm is used.FindingsThen, the algorithm results are compared with a simulated annealing algorithm that recently was proposed in the literature. Finally, the results obtained from the exact solution and metaheuristic algorithms are compared, analyzed and reported. The results showed that the artificial bee colony algorithm has a good performance.Originality/valueIn this paper, medical goods distribution with pharmacological waste collection is studied. The paper was focused on plug-in hybrid vehicles with simultaneous pickup and delivery. The problem was modeled with environmental criteria. The traveling and energy consumption costs are considered as an objective function.

Routing Replenishment Workers: The Prize Collecting Traveling Salesman Problem in Scattered Storage Warehouses

Many online retailers apply scattered (or mixed-shelves) storage in the picking areas of their warehouses. Instead of keeping unit loads together, individual pieces of stock keeping units (SKUs) are stored on various shelves throughout the warehouse. This storage strategy increases the probability that—whatever it is that customers order jointly—somewhere in the warehouse these products will be located in close proximity. Hence, a picker can retrieve them without excessive unproductive walking. The price for this advantage on the picking side, however, is additional effort for the replenishment workers (also denoted as stowers) when restocking the shelves. Instead of moving only a single homogeneous unit load toward an SKU’s designated storage position, each stower has to travel along multiple open shelf spaces until all products on the cart are stored on shelves. The resulting stower routing problem is equivalent to the well-known prize collecting traveling salesman problem (PCTSP). While the PCTSP for general graphs is known to be strongly [Formula: see text]-hard, we show that in a warehousing environment, where all open storage positions are located along parallel aisles, it is only binary [Formula: see text]-hard. The special parallel-aisle structure allows us to derive an exact solution algorithm with pseudo-polynomial runtime, which solves even instances with hundreds of open storage positions to proven optimality in just a few seconds. Our computational tests show that the performance gains of an optimized stowing process over the status quo, where stowers operate without decision support, are significant. Especially, when the fill level of a warehouse is high, directing stowers on optimized routes promises huge improvements. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms – Discrete. Funding: This work was supported by the German Science Foundation/Deutsche Forschungsgemeinschaft (DFG) by the grant “Routing of human and automated order pickers in modern warehouses” (BO 3148/14-1 and BO 1972/2-1). Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoc.2022.0173 .

Network flow problem heuristic reduction using machine learning

Most of the supporting tools developed for logistic optimization and processing infrastructure planning are based on the network flow problem. The real-world application of these instruments can provide great insight and help to ensure long-term sustainability. The main limitation of these tools lies in great computing demand when there is the necessity of solving large-scale tasks in great detail. It means that the ability to find the optimal solution for real-world problems is limited. Thus, the detail of infrastructure is often reduced by data aggregation or heuristic approaches are used to find a suboptimal solution. This paper proposes a machine learning classification model to reduce the number of variables for an exact solution algorithm. First, the design of experiments is used to create a set of smaller problems that are possible to solve exactly. Artificial data are used at this stage, while domain knowledge is used to set appropriate distribution and parameters. Second, the classification model estimates the probability of the presence of each arc in the optimal solution. Features, which are related to costs and capacity, of each arc are utilized in the classification model. Models created on a subset of generated problems are then tested on the other problems. Finally, the proposed framework is applied to the waste management problem in the Czech Republic. The results of the verification show, that it is possible to remove 95% of arcs without impact on strategic decisions and without significant change of an objective function. The computing time of the reduced problem takes only 7% of the original task.

A Decomposition Algorithm for Single and Multiobjective Integrated Market Selection and Production Planning

We study an integrated market selection and production planning problem. There is a set of markets with deterministic demand, and each market has a certain revenue that is obtained if the market’s demand is satisfied throughout a planning horizon. The demand is satisfied with a production scheme that has a lot-sizing structure. The problem is to decide on which markets’ demand to satisfy and plan the production simultaneously. We consider both single and multiobjective settings. The single objective problem maximizes the profit, whereas the multiobjective problem includes the maximization of the revenue and the minimization of the production cost objectives. We develop a decomposition-based exact solution algorithm for the single objective setting and show how it can be used in a proposed three-phase algorithm for the multiobjective setting. The master problem chooses a subset of markets, and the subproblem calculates an optimal production plan to satisfy the selected markets’ demand. We investigate the subproblem from a cooperative game theory perspective to devise cuts and strengthen them based on lifting. We also propose a set of valid inequalities and preprocessing rules to improve the proposed algorithm. We test the efficacy of our solution method over a suite of problem instances and show that our algorithm substantially decreases solution times for all problem instances. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms – Discrete. Funding: This work was supported by TUBITAK [Grant 1059B191801782].

A survey on exact algorithms for the maximum flow and minimum‐cost flow problems

AbstractNetwork flow problems form an important and much‐studied family of combinatorial optimization problems, with a huge array of practical applications. Two network flow problems in particular have received a great deal of attention: the maximum flow and minimum‐cost flow problems. We review the progress that has been made on exact solution algorithms for these two problems, with an emphasis on worst‐case running times.

A massively parallel branch-&-bound algorithm for the balanced minimum evolution problem

We build upon recent theoretical advances in the Balanced Minimum Evolution Problem (BMEP) to design a new massively parallel exact solution algorithm that proves to be up to one order of magnitude faster than the current state-of-the-art under the same computing settings and environment.

Exact and heuristic solution approaches for energy-efficient identical parallel machine scheduling with time-of-use costs

Nowadays, energy-efficient scheduling has assumed a key role in ensuring the sustainability of manufacturing processes. In this context, we focus on the bi-objective problem of scheduling a set of jobs on identical parallel machines to simultaneously minimize the maximum completion time and the total energy consumption over a time horizon partitioned into a set of discrete slots. The energy costs are determined by a time-of-use pricing scheme, which plays a crucial role in regulating energy demand and flattening its peaks. First, we uncover a symmetry-breaking property that characterizes the structure of the solution space of the problem. As a consequence, we provide a novel, compact mixed-integer linear programming formulation at the core of an efficient exact solution algorithm. A thorough experimental campaign shows that the use of the novel mathematical programming formulation enables the solution of larger-scale instances and entails a reduction in the computational times as compared to the formulation already available in the literature. Furthermore, we propose a new heuristic that improves the state-of-the-art in terms of required computational effort and quality of solutions. Such a heuristic outperforms the existing heuristics for the problem and is also capable of speeding up the exact solution algorithm when used for its initialization. Finally, we introduce a novel dynamic programming algorithm that is able to compute the optimal timing of the jobs scheduled on each machine to further improve the performance of the new heuristic.

An Exact Solution Algorithm for Integer Bilevel Programming with Application in Energy Market Optimization

We develop an exact cutting plane solution algorithm for a special class of bilevel programming models utilized for optimal price-bidding of energy producers in day-ahead electricity markets. The proposed methodology utilizes a suitable reformulation in which a key prerequisite for global optimality, termed bilevel feasibility, is relaxed. Solving the problem to global optimality involves finding the price-offers of the strategic producer (upper-level decision variables) which maximize his self-profit upon clearing of the market and identification of the optimal energy quantity distribution (lower-level decision variables). To exclude from consideration the encountered bilevel infeasible solutions, the algorithm employs a special type of valid cuts drawn from the theory of integer parametric programming. The generation of these cuts involves finding the truly optimal lower-level solution using the strategic price-offers at the bilevel infeasible solution subject to exclusion and devising range intervals for these offers such that the optimality of this solution is retained when each of them lies in its corresponding interval. Each cut imposes a suitable part of this solution, under the condition that each price-offer belongs to its associated interval, which renders the bilevel infeasible solution invalid. We establish the theoretical framework for the development of the proposed algorithm, we illustrate its application on a small case study, and we present extensive computational results demonstrating its behavior and performance on random problem instances. These results indicate that the algorithm is capable of solving to global optimality considerably larger problems than those that a previous elementary version of the same algorithm could solve. This constitutes significant research contribution, considering the lack of generic optimization software for bilevel programming, as well as the fact that the applicability of specialized algorithms on problems of realistic size is rather limited.