Problem Instances Research Articles

Thompson sampling-based recursive block elimination for dynamic assignment under limited budget in pure-exploration

In this paper, we investigate Thompson sampling-based sequential block elimination approaches for dynamic assignment problems in a pure-exploration Multi-Armed Bandit (MAB) setting with limited budget constraints. The problem can be considered as a bandit game-play between the environment and a decision-maker in a metric space. Many instances of problems in fields such as e-commerce, logistics, mobility management, data management and operations research can be framed as dynamic assignment problems with budget constraints. Given an l-dimensional action space representing l variants of an entity and a budget for exploring the action space, the optimal dynamic assignment problem refers to the task of identifying the values to be assigned to different variants of the entity that maximizes the total reward by utilizing at most the given budget of rounds of play. We contribute a class of block elimination-based MAB algorithms specifically designed for the dynamic assignment problem with limited budget. Our algorithms begin by discretizing the continuous action space into a finite set of discrete actions, then proceed with a recursive block elimination procedure to remove sub-optimal actions. The elimination is carried out by calculating confidence bounds over blocks of actions. We explore two different confidence bound estimation techniques. We perform comprehensive experiments on two problem instances from distributed data management and logistics. Our results showcase that our approach yields a lower misidentification probability (i.e., the probability of recommending a non-optimal action) compared to state-of-the-art elimination-based pure-exploration MAB algorithms.

Crossover Operator Inspired by the Selection Operator for an Evolutionary Task Sequencing Algorithm

This paper proposes a novel crossover operator for evolutionary algorithms in task sequencing and verifies its efficacy. Unlike the conventional blind and entirely stochastic selection of sequence fragments exchanged with the second individual, the proposed operator employs a method where the probability of fragment selection is influenced by the total cost of internal connections within the exchanged fragments. At the same time, the new operator retains its stochastic nature and is not a deterministic operator, which reduces the risk of the evolutionary algorithm getting stuck in a local minimum. The idea of the proposed crossover operator was based on the main mechanism of the evolutionary algorithm that determines the success of this type of algorithm selection. To assess its effectiveness, the new operator was compared against previously employed crossover operators using a traveling salesman problem (TSP) instance in a multidimensional space in order to map the problem of symmetric sequencing tasks described with multiparameters (e.g., a symmetric variant of production tasks sequencing).

Stochastic Vehicle Routing With Delivery Choice

ABSTRACTWe consider the problem of designing delivery routes for vehicles where the vendor has the choice of how much of the demand from a customer to fulfill. The customer demand is known a priori only as a probability distribution. Exact customer demand is known only after visiting the customer. Different customers are able to negotiate different prices for each unit of product with the vendor. Given a route, the objective is to decide at each customer location, how much demand to satisfy so as to maximize expected profit taking into account a linear penalty cost for unfulfilled demand and the vehicle routing costs. In this article, we develop several new structural results for this problem. We illustrate how these structural results can be embedded in different heuristic frameworks commonly used for deterministic vehicle routing problems. This helps develop efficient routes for a single vehicle as well as a multiple vehicle scenario for this stochastic variant. For small‐sized problems that allow for exhaustive enumeration, we demonstrate the effectiveness of the illustrated heuristic. For larger problem instances, based on structural results, we develop methods that allow the heuristic to run more efficiently than otherwise. Results are reported on instances based on benchmark instances drawn from literature for upward of 100 customers and vehicle capacity up to 600 units. Computational times needed to heuristically solve such problems are within 1 100 s.

An Infinite Family of Counterexamples to a Conjecture on Distance Magic Labeling

This work is about a partition problem which is an instance of the distance magic graph labeling problem. Given positive integers n, k and p 1 ≤ p 2 ≤ ⋯ ≤ p k such that p 1 + ⋯ + p k = n and k divides ∑ i = 1 n i , we study the problem of characterizing the cases where it is possible to find a partition of the set { 1 , 2 , … , n } into k subsets of respective sizes p 1 , … , p k , such that the element sum in each subset is equal. Using a computerized search we found examples showing that the necessary condition, ∑ i = 1 p 1 + ⋯ + p j ( n − i + 1 ) ≥ j ( n + 1 2 ) / k for all j = 1 , … , k , is not generally sufficient, refuting a past conjecture. Moreover, we show that there are infinitely many such counter-examples. The question whether there is a simple characterization is left open and for all we know the corresponding decision problem might be NP-complete.

Monte Carlo tree search with spectral expansion for planning with dynamical systems.

The ability of a robot to plan complex behaviors with real-time computation, rather than adhering to predesigned or offline-learned routines, alleviates the need for specialized algorithms or training for each problem instance. Monte Carlo tree search is a powerful planning algorithm that strategically explores simulated future possibilities, but it requires a discrete problem representation that is irreconcilable with the continuous dynamics of the physical world. We present Spectral Expansion Tree Search (SETS), a real-time, tree-based planner that uses the spectrum of the locally linearized system to construct a low-complexity and approximately equivalent discrete representation of the continuous world. We prove that SETS converges to a bound of the globally optimal solution for continuous, deterministic, and differentiable Markov decision processes, a broad class of problems that includes underactuated nonlinear dynamics, nonconvex reward functions, and unstructured environments. We experimentally validated SETS on drone, spacecraft, and ground vehicle robots and one numerical experiment, each of which is not directly solvable with existing methods. We successfully show that SETS automatically discovers a diverse set of optimal behaviors and motion trajectories in real time.

Nondeterminism and the Clique Problem

The Clique problem is known to be NP-Complete [7,3] and the question whether P=NP is unresolved. This paper examines the relative power of nondeterminism versus determinism in a restricted setting. Specifically, we consider solving the clique problem using non-deterministic and deterministic Turing machines. We impose a (reasonable) format in which a problem instance is encoded. We also impose constraints on the computation of both deterministic and non-deterministic Turing machines: both have two tapes, the input tape is read-only and one-way, and once a certain stop point in the input tape is reached, no additional writing on the work tape is allowed. We consider two cases for the position of the stop point: immediately after the number of graph nodes and the size of the clique are specified, or controlled by a parameter q that indicates what portion of the graph nodes' edge specifications have been scanned. The parameter q may be arbitrarily close to 1, e.g., q=0.99999. We show, for both cases in our setting, that a non-deterministic Turing machine can solve the problem in O(n3) time whereas no deterministic Turing machine can solve the problem in polynomial time. However, we exhibit an exponential time deterministic single work tape, two-heads Turing machine that solves the clique problem in our setting.

A reinforcement learning hyper-heuristic algorithm for the distributed flowshops scheduling problem under consideration of emergency order insertion

Large enterprises are composed of several subproduction centers. The production plan is changed based on the procedure of the manufacturing system. The distributed flowshop scheduling problem under consideration of emergence order insertion is challenging as the assignment of the product, and the scheduling of the process are coupled with each other. A general distributed flow shop scheduling problem regarding emergency order insertion (DFSP_EOI) is addressed under rescheduling circumstances in this paper. The Q-learning hyper-heuristic algorithm with dynamic insertion rule (QLHH_DIR) is proposed to solve the DFSP_EOI. Eight low-level heuristics (LLHs) for static job assignment. A dynamic insertion rule based on the state of each production center is designed for emergency order insertion. The Q-learning mechanism at high-level space selects appropriate LLH through learning the experience from the optimization process. The computational simulation is carried out, and the results confirm that the proposed algorithm is superior to the competitors in solving the distributed flow shop rescheduling problem. The results of the 720 problem instances show that the proposed algorithm is highly efficient in rescheduling problems.

Modifying an instance of the super-stable matching problem

The topic of this paper is the stable matching problem in a bipartite graph. Super-stability is one of the stability concepts in the stable matching problem with ties. It is known that there may not exist a super-stable matching, and the existence of a super-stable matching can be checked in polynomial time. In this paper, we consider the problem of modifying an instance of the stable matching problem with ties by deleting some bounded number of agents in such a way that there exists a super-stable matching in the modified instance. First, we consider the setting where we are allowed to delete agents on only one side. We prove that, in this setting, our problem can be solved in polynomial time. Interestingly, this result is obtained by carefully observing the existing algorithm for checking the existence of a super-stable matching. Next, we consider the setting where we are given an upper bound on the number of deleted agents for each side, and we are allowed to delete agents on both sides. We prove that, in this setting, our problem is NP-complete.

Construct, merge, solve and adapt

AbstractThe CMSA algorithm for combinatorial optimization is a hybrid technique based on repeatedly solving sub-instances to the original problem instance. The incumbent sub-instance is extended at each iteration by the probabilistic generation of valid solutions to the original problem instance and by adding the components found in these solutions to the sub-instance. In addition, the incumbent sub-instance is reduced at each iteration by removing seemingly useless solution components. In recent years the usefulness of the CMSA algorithm has been shown by a range of applications to different combinatorial optimization problems. In this work, we provide a gentle introduction to CMSA by describing the application to the so-called minimum global domination problem as an example.

Using Reinforcement Learning in a Dynamic Team Orienteering Problem with Electric Batteries

This paper addresses the team orienteering problem (TOP) with vehicles equipped with electric batteries under dynamic travel conditions influenced by weather and traffic, which impact travel times between nodes and hence might have a critical effect on the battery capacity to cover the planned route. The study incorporates a novel approach for solving the dynamic TOP, comparing two solution methodologies: a merging heuristic and a reinforcement learning (RL) algorithm. The heuristic combines routes using calculated savings and a biased-randomized strategy, while the RL model leverages a transformer-based encoder–decoder architecture to sequentially construct solutions. We perform computational experiments on 50 problem instances, each subjected to 200 dynamic conditions, for a total of 10,000 problems solved. The results demonstrate that while the deterministic heuristic provides an upper bound for rewards, the RL model consistently yields robust solutions with lower variability under dynamic conditions. However, the dynamic heuristic, with a 20 s time limit for solving each instance, outperformed the RL model by 3.35% on average. The study highlights the trade-offs between solution quality, computational resources, and time when dealing with dynamic environments in the TOP.

Joint Design and Pricing Problem for Symbiotic Bioethanol Supply Chain Network: Model and Resolution Approach

To fight climate change, the Province of Quebec, Canada, has set targets to reduce greenhouse gas emissions by reducing fossil fuel consumption and integrating biofuel content into gasoline and diesel fuel. Motivated by a real-world case study, this paper presents a novel distributed decision model for designing a symbiotic supply chain network and supporting pricing decisions. A distributed decision-making problem is formulated as a game theoretic approach considering a Stackelberg–Nash equilibrium. A novel mathematical model is proposed to support the decisions of four actors: corn farms, processing depots, pig farms, and biorefineries. In addition to the configuration of a biofuel-based industrial symbiosis, the model offers the possibility of setting purchase prices and supply levels for biomass (corn stover supplied by farms), as well as determining sales prices and production levels for the main product (the cellulosic sugar used for the bioethanol production) and a coproduct (pig feed sold to pig farmers). A three-step optimization process involving the user is proposed to address the computational challenges posed by large design problem instances. The case study of the Province of Quebec is used to evaluate the performance of the proposed resolution approach.

The Heterogeneous-Fleet Electric Vehicle Routing Problem with Nonlinear Charging Functions

This paper introduces the Heterogeneous-Fleet Electric Vehicle Routing Problem with Nonlinear Charging Functions (HEVRP-NL). This problem involves routing a heterogeneous fleet of electric vehicles, utilizing multiple charging modes, and accounting for time-dependent waiting time functions at charging stations. The problem is modeled using a path-based mixed-integer linear programming formulation. To solve this problem, we present an algorithmic framework that alternates between two components. The first component is an iterated local search algorithm with a problem-specific route evaluation function, which obtains local optimal solutions and generates a pool of high-quality routes. The second component is a set-partitioning model that combines a subset of routes from the pool, which is constructed based on reduced costs, into a feasible solution. We design HEVRP-NL benchmark instances based on the publicly available electric fleet size and mix vehicle routing problem instances, which are used to evaluate our methods. For small-scale HEVRP-NL instances, the proposed model can be employed in a general-purpose mixed integer programming solver to achieve optimal solutions or find good upper bounds. This exact approach serves as an evaluation of our heuristic algorithm’s ability to attain optimal solutions rapidly. Extensive computational results on large-scale HEVRP-NL instances illustrate the advantages of considering non-linear charging functions and show the impact of waiting time at the charging stations. Finally, we conduct experiments on 120 benchmark instances for the E-VRP-NL and 168 benchmark instances for the E-FSMFTW-PR, which are the special cases of our problem. The results indicate that our algorithm outperforms existing approaches from the literature and identifies 32 new best solutions for the E-VRP-NL and 33 new best solutions for the E-FSMFTW-PR, respectively.

From single-objective to multi-objective reinforcement learning-based model transformation

AbstractModel-driven optimization allows to directly apply domain-specific modeling languages to define models which are subsequently optimized by applying a predefined set of model transformation rules. Objectives guide the optimization processes which can range from one single objective formulation resulting in one single solution to a set of objectives that necessitates the identification of a Pareto-optimal set of solutions. In recent years, a multitude of reinforcement learning approaches has been proposed that support both optimization cases and competitive results for various problem instances have been reported. However, their application to the field of model-driven optimization has not gained much attention yet, especially when compared to the extensive application of meta-heuristic search approaches such as genetic algorithms. Thus, there is a lack of knowledge about the applicability and performance of reinforcement learning for model-driven optimization. We therefore present in this paper a general framework for applying reinforcement learning to model-driven optimization problems. In particular, we show how a catalog of different reinforcement learning algorithms can be integrated with existing model-driven optimization approaches that use a transformation rule application encoding. We exemplify this integration by presenting a dedicated reinforcement learning extension for MOMoT. We build on this tool support and investigate several case studies for validating the applicability of reinforcement learning for model-driven optimization and compare the performance against a genetic algorithm. The results show clear advantages of using RL for single-objective problems, especially for cases where the transformation steps are highly dependent on each other. For multi-objective problems, the results are more diverse and case-specific, which further motivates the usage of model-driven optimization to utilize different approaches to find the best solutions.

Comparative analysis of AI-based search algorithms in solving 8 puzzle problems

BackgroundThe 8-puzzle problem is a well-known combinatorial search problem, often used to test the effectiveness of various artificial intelligence (AI) algorithms. This study aims to evaluate the performance of three AI-based search algorithms—Breadth-First Search (BFS), Depth-First Search (DFS), and A* Search—in solving the 8-puzzle problem. The algorithms were implemented in Java, and their performance was measured in terms of solution length, the number of expanded nodes, and execution time.ResultsExperiments were conducted on eight randomly generated instances of the 8-puzzle problem. The findings reveal that both BFS and A* Search consistently identify optimal solutions in all cases. However, BFS was found to use more memory and operate slower than A* Search. Conversely, DFS demonstrated faster execution times and lower memory usage compared to BFS but was less reliable in finding optimal or feasible solutions. The study also explored the impact of different heuristic functions on the performance of A* Search. Among the heuristics tested, the Manhattan distance heuristic was determined to be the most effective, offering the best balance between accuracy and efficiency.ConclusionsIn conclusion, both BFS and A* Search are effective in finding optimal solutions, with A* Search being more efficient in terms of speed and memory usage. DFS, while faster and more memory-efficient, is less consistent in producing optimal solutions. The Manhattan distance heuristic significantly enhances the performance of A* Search, making it the preferred heuristic for the 8-puzzle problem. These results suggest that A* Search, particularly with the Manhattan distance heuristic, is a highly efficient choice for solving the 8-puzzle problem, especially in contexts where computational resources are limited.

The use of continuous action representations to scale deep reinforcement learning for inventory control

Abstract Accepted by: M. Zied Babai Deep reinforcement learning (DRL) can solve complex inventory problems with a multi-dimensional state space. However, most approaches use a discrete action representation and do not scale well to problems with multi-dimensional action spaces. We use DRL with a continuous action representation for inventory problems with a large (multi-dimensional) discrete action space. To obtain feasible discrete actions from a continuous action representation, we add a tailored mapping function to the policy network that maps the continuous outputs of the policy network to a feasible integer solution. We demonstrate our approach to multi-product inventory control. We show how a continuous action representation solves larger problem instances and requires much less training time than a discrete action representation. Moreover, we show its performance matches state-of-the-art heuristic replenishment policies. This promising research avenue might pave the way for applying DRL in inventory control at scale and in practice.

On semi-orthogonal matrices with row vectors of equal lengths

When does a rectangular matrix with an orthonormal set of column vectors have row vectors of equal lengths? The column spaces of such matrices are multidimensional generalizations of the projection plane used in isometric perspective. We show that in the absence of unexpected linear relations, any rectangular matrix can be row-scaled so that if we were to orthonormalize the column vectors, the row vectors would attain equal lengths in the process. We use Grassmann coordinates to reduce the question into an instance of the famous matrix scaling problem, and with the help of existing theory introduce simple numerical solutions.

Modeling and Solving the Knapsack Problem with a Multi-Objective Equilibrium Optimizer Algorithm Based on Weighted Congestion Distance

The knapsack problem is a typical bi-objective combinatorial optimization issue, wherein maximizing the value of the packed items is achieved concurrently with minimizing the weight of the load. Due to the conflicting objectives of the knapsack problem and the typical discrete property of the items involved, swarm intelligence algorithms are commonly employed. The diversity of optimal combinations in the knapsack problem has become a focal point, which involves finding multiple packing solutions at the same value and weight. For this purpose, this paper proposes a Multi-Objective Equilibrium Optimizer Algorithm based on Weighted Congestion Distance (MOEO_WCD). The algorithm employs a non-dominated sorting method to find a set of Pareto front solutions rather than a single optimal solution, offering multiple decision-making options based on the varying needs of the decision-makers. Additionally, MOEO_WCD improves the balance pool generation mechanism and incorporates a weighted congestion incentive, emphasizing the diversity of packing combination solutions under the objectives of value and weight to explore more Pareto front solutions. Considering the discrete characteristics of the knapsack combination optimization problem, our algorithm also incorporates appropriate discrete constraint handling. This paper designs multiple sets of multi-objective knapsack combinatorial optimization problems based on the number of knapsacks, the number of items, and the weights and values of the items. This article compares five algorithms suitable for solving multi-objective problems: MODE, MO-PSO-MM, MO-Ring-PSO-SCD, NSGA-II, and DN-NSGAII. In order to evaluate the performance of the algorithm, this paper proposes a new solution set coverage index for evaluation. We also used the hypervolume indicator to evaluate the diversity of algorithms. The results show that our MOEO-WCD algorithm achieves optimal coverage of the reference composite Pareto front in the decision space of four knapsack problems. The experimental results indicate that our MOEO_WCD algorithm achieves the optimal coverage of the reference composite Pareto front in the decision space for four sets of knapsack problems. Although our MOEO_WCD algorithm covers less of the composite front in the objective space compared with the MODE algorithm for knapsack problem 1, its coverage of the integrated reference solutions in the decision space is greater than that of the MODE algorithm. The experiments demonstrate the superior performance of the MOEO_WCD algorithm on bi-objective knapsack combinatorial optimization problem instances, which provides an important solution to the search for diversity in multi-objective combinatorial optimization solutions.

Nested benders decomposition for a deterministic biomass feedstock logistics problem

AbstractIn this paper, we address a biomass feedstock logistics problem to supply biomass from production fields to satellite storage locations (SSLs) and from there to bioenergy plants (BePs) and then to a biorefinery. It entails a new problem feature of routing load-out equipment sets among the SSLs to perform loading/unloading of biomass and/or its pre-processing operations. The ownership of the loading equipment is a very capital-intensive link of the ethanol production supply chain, which when loaded onto trucks and routed along the logistics chain significantly brings down the ethanol production costs. This will make ethanol a cost-competitive alternative to fossil fuels, lead to sustainable use of fossil fuels and add to the overall relevance of the bioenergy sector. In this regard, the objective of our problem is to minimize the total cost incurred due to the ownership of equipment sets, fixed setups, and land rental cost, as well as the cost of transporting biomass from the fields to the BePs and biocrude oil from the BePs to the refinery. A mixed-integer mathematical model of the problem is presented, and a nested Benders decomposition-based solution approach is developed which involves decomposing this large problem into three stages. Stage 1 deals with the selection of fields, BePs, and SSLs, and assignment of fields to the SSLs. The remaining model consists of multiple Capacitated Vehicle Routing Problems (CVRPs) that are separable over individual BePs. For each BeP, the CVRP is further decomposed into Stage 2 and Stage 3 sub-problems where the Stage 2 problem is an allocation problem that assigns SSLs to tours associated to each BeP, and the Stage 3 problem is a variant of Traveling Salesman Problem (TSP) that determines the sequence in which equipment is routed over the predesignated set of SSLs for each tour. These sub-problems are integer programs rather than linear programs. First novelty of our proposed approach is to effectively handle the integrality of variables arising due to the consideration of the routing of load-out equipment. Second is solution methodology and in the use of proposed multi-cut version of optimality cuts that capture the solution value at an integer solution for the sub-problems. These cuts aid in faster convergence and are shown to be stronger than those proposed in the literature. The applicability of the proposed methodology is demonstrated by applying it to a real-life problem that utilizes available GIS data for the catchment area of regions around Gretna and Bedford in Virginia. We then solved a set of varying problem size instances using the state-of-the-art CPLEX® Branch-and-Bound and Benders Strategy methods. The CPLEX® algorithms struggled to solve instances even 10 times smaller than the real-life problem size instances; with MIP optimality gaps ranging from 5.85% to 82.79% in the allowed time limit of 10,000 s. On the other hand, our proposed nested Benders decomposition algorithm was able to achieve faster convergence and provided optimal solutions for all the considered problem instances with an average CPU run-time of around 3,700 s. This validates the efficacy and superiority of our solution approach. Lastly, we summarize our work and point out some interesting potential future research opportunities.

Energy-aware scheduling in flow shops: a novel artificial neural network-driven multi-objective optimization

Group technology is a managerial strategy used to optimize production by reducing setup times, lead times, and work-in-process inventories. Research on flow-shop sequence-dependent group scheduling problems (FSDGSPs) has primarily focused on minimizing makespan and total flow time to improve efficiency. However, the need for energy-efficient scheduling in FSDGSPs remains underexplored despite increasing sustainability concerns. To address this, the energy-efficient flow-shop sequence-dependent group scheduling problem (EEFSDGSP) is introduced. A novel multi-objective optimization (MOO) technique, the artificial neural network-based multi-objective genetic algorithm (ANN-MOGA), is proposed to minimize makespan and energy consumption in EEFSDGSP. ANN-MOGA advances MOO by using a neural network to evaluate fitness and guide selection, reducing computational complexity versus traditional methods like NSGA-II and SPEA2. A post-processing step (PPANNS) further enhances solution diversity and distribution. Results show ANN-MOGA, especially with PPANNS, outperforms NSGA-II and competes effectively with SPEA2 in larger problem instances.

A Novel Solution Approach Based on Dominance Evaluation Measure for Project Scheduling in Multi-Project Environments

The widely recognized measure for resources called resource strength (RS) does not fully capture the resources complexity of a project. Therefore, it cannot be used as a standalone measure to distinguish the complexity of various instances of project scheduling problems. Consequently, additional resource measures such as total amount of overflow (TAO) have been introduced, which should be used in conjunction with the RS. Extensive experimental studies have shown that as the value of TAO increases in a project, scheduling schemes with higher dimensional scheduling schemes such as bi-directional and tri-directional result in schedules with shorter makespans. In this study, an effective approach is proposed for integrating projects in multi-project environments, called the integrated project approach (IPA), taking into account the influence of TAO and building upon the relation between the TAO and the scheduling generation schemes. To assess the performance of IPA, we develop a new random multi-project generator based on the well-known benchmark sets, which utilizes TAO as a control tool to generate instances. The findings indicate that prioritizing the projects and frequency of the projects integration, facilitated by the proposed IPA, have a positive impact on the quality of multi-project schedules.