Constraint Programming Formulation Research Articles

The Dial-a-Tour Problem

This article delves into the Dial-a-Tour Problem (DATP) frequently encountered in local travel agencies, known for their personalized and hands-on customer service. In the DATP, multiple tourist groups are efficiently matched with different vehicles using vanpooling, maximizing vehicle occupancy while ensuring seamless transfers to the local attractions included in their booked tours. Each tourist group is picked up from and returned to their respective hotels after their journey. The predetermined locations and the sequence of visits for each tour are known in advance. Therefore, the primary objective is to minimize the number of vehicles required to fulfill all requests and maintain the desired level of service.To tackle this challenge, the article presents a comprehensive approach. It introduces a mixed-integer linear model, two constraint programming formulations, and a novel operator called the k-buffer insertion operator. This operator serves as the centerpiece of a descent heuristic proposed in the study.Extensive evaluation across a diverse range of randomly generated instances underscores the superiority of the proposed constraint programming formulations over the mixed integer linear programming model. However, it is noteworthy that the proposed heuristic surpasses even the most effective constraint programming formulation. This highlights its remarkable efficiency in solving real-size instances within notably brief computation times. Such prowess underscores the practical utility of this heuristic in effectively tackling the complexities inherent in the Dial-a-Tour Problem.

Advanced constraint programming formulations for additive manufacturing machine scheduling problems

In comparison with traditional subtractive manufacturing techniques, additive manufacturing (AM) enables fabricating complex parts through a layer-by-layer process. AM makes it possible to produce one-piece and lightweight functional products, which are traditionally made from several parts. This paper introduces constraint programming (CP) models to minimise makespan in single, parallel identical and parallel unrelated AM machine scheduling environments for selective laser melting. Alternative CP formulations are explored to improve efficiency. The proposed CP model significantly benefits from the introduction of interval variables to replace binary assignment variables, and pre-definitions to narrow the search space, resulting in increased search performance. A computational study has been conducted to compare the performance of our proposed CP model with both a mixed-integer programming and a genetic algorithm from existing literature, evaluating improvements made to its search capability. Computational results indicate that the proposed CP model can obtain high-quality solutions in a timely manner even for several large-size instances.

An adapted constraint‐programming formulation of the resource‐constrained project scheduling problem applied to the identical parallel machines group shop and mixed shop scheduling problems

AbstractIn this paper, we study the group shop and the mixed shop scheduling problems with single and identical parallel machines at each workstation with the makespan criterion. We adapted a constraint programming formulation previously presented for the classical resource‐constrained project scheduling problem. The effectiveness of our approach is evident in the fact that it achieved optimality in 107 out of 130 classical group shop scheduling problem instances and in 320 classical mixed shop scheduling problem instances. In the last set, we obtained 13 new optimal solutions.

Resource constraint scheduling on two dedicated machines: Application to avionics

In civil aircraft, two partially redundant hydraulic circuits typically power various systems. During assembly, a critical phase involves simultaneously rinsing and purging these hydraulic circuits using loops. Precedence constraints are necessary to prevent the recontamination of already rinsed loops, leading to increased rinsing time. This paper presents this problem as a unique instance of the Resource Constrained Parallel Machine Scheduling Problem, where each circuit represents a machine, pipe loops to be rinsed represent jobs, and machines share a hydraulic power source. For two dedicated processors and a single resource, an optimal schedule minimizing the makespan can be generated in polynomial time. However, due to the requirement of rinsing certain pipe loops on a circuit before others, there are precedence constraints between some jobs within the same circuit. By employing a reduction of the 3-partition problem, we demonstrate that this situation results in a problem that is NP-hard in the strong sense. We evaluate several Mixed-Integer Linear Programming and Constraint Programming formulations of the problem, using Cplex, CPO, Gurobi, and Z3, against several proposed heuristics. Given that the size of the instances we need to solve exceeds what can be solved in acceptable time by solvers, we propose a heuristic and compare its performance with the optimum.

Optimising the flight turnaround schedules: An improved sliding time windows approach based on MILP and CP models

This paper addresses the scheduling of turnaround operations for flights in civil airports. A series of complex issues, including the heterogeneity of flights and operators, resource constraints on turnaround operations, temporal restrictions and alternative service patterns of turnaround operations, multi-functional operators, multi-trip services and multi-to-one services, are jointedly considered. We propose two formulations for the problem: A mixed-integer linear programming (MILP) model, and a constraint programming (CP) model. For comparison purpose, we also present simplified versions of the models in which the multi-to-one services are prohibited. The uncertainty are incorporated by modifying the models based on robust optimisation principles. The numerical experiments are conducted using instances generated based on real flight data at Beijing Capital International Airport. It is shown that under current computer configurations, small-scale instances can be solved to the optimum by CPLEX within an acceptable period of time using the MILP model, while moderate-scale instances can be tackled using the CP formulation. To cope with large-scale instances, an improved sliding time window method is developed, and evaluated with different parameter combinations, showing that practical turnaround schedule would best be updated two to three times per hour. Besides, after a case study in both static and stochastic scenarios, we illustrate that multi-to-one services should be specifically considered and highlighted, for their existence could significantly improve the operator utilisation rate and reduce flight delay, especially when the uncertainties are taken into account.

A constraint programming model for the scheduling and workspace layout design of a dual-arm multi-tool assembly robot

The generation of a robot program can be seen as a collection of sub-problems, where many combinations of some of these sub-problems are well studied. The performance of a robot program is strongly conditioned by the location of the tasks. However, the scope of previous methods does not include workspace layout design, likely missing high-quality solutions. In industrial applications, designing robot workspace layout is part of the commissioning. We broaden the scope and show how to model a dual-arm multi-tool robot assembly problem. Our model includes more robot programming sub-problems than previous methods, as well as workspace layout design. We propose a constraint programming formulation in MiniZinc that includes elements from scheduling and routing, extended with variable task locations. We evaluate the model on realistic assembly problems and workspaces, utilizing the dual-arm YuMi robot from ABB Ltd. We also evaluate redundant constraints and various formulations for avoiding arm-to-arm collisions. The best model variant quickly finds high-quality solutions for all problem instances. This demonstrates the potential of our approach as a valuable tool for a robot programmer.

Learning cost action planning models with perfect precision via constraint propagation

Data-driven AI is rapidly gaining importance. In the context of AI planning, a constraint programming formulation for learning action models in a data-driven fashion is proposed. Data comprises plan observations, which are automatically transformed into a set of planning constraints which need to be satisfied. The formulation captures the essence of the action model and unifies functionalities that are individually supported by other learning approaches, such as costs, noise/uncertainty on actions, information on intermediate state observations and mutex reasoning.Reliability is a key concern in data-driven learning, but existing approaches usually learn action models that can be imprecise, where imprecision here is an error indicator of learning something incorrect. On the contrary, the proposed approach guarantees reliability in terms of perfect precision by using constraint propagation. This means that what is learned is 100% correct (i.e., error-free), not only for the initial observations, but also for future observations. To our knowledge, this is a novelty in action model learning literature. Although perfect precision might potentially limit the amount of learned information, the exhaustive experiments over 20 planning domains show that such amount is comparable, and even better, to ARMS and FAMA, two state-of-the-art benchmarks in action model learning.

Permutation flowshop problems minimizing core waiting time and core idle time

Waiting time and idle time are among the main cost sources in production systems. They can also affect the feasibility of operations from a technological perspective; hence, both such times have to be kept as small as possible. This paper studies four single-objective variants of the permutation flowshop scheduling problem, where two objectives are considered: the weighted sum of the makespan and the core waiting time, and the weighted sum of the makespan and the core idle time. For each objective, both the problem with the assumption of semi-active solution and the one without it are considered.A general solution framework for tackling the above-mentioned problems is provided. First, two Mixed Integer Linear Programming (MILP) formulations (based on positional and precedence variables, respectively) and one Constraint Programming (CP) formulation are presented. Second, a MILP-based local search approach based on the positional MILP formulation and the concept of sliding windows are defined. An extensive set of computational experiments on benchmark instances show that the positional MILP formulation strongly outperforms the other two formulations in all the considered cases. The experiments also show that the sliding window local search heuristic achieves much better performances than other state-of-the-art local search heuristics. Indeed, it is able to improve the state-of-the-art in 2384 instances out of 2400.

A Constraint Programming Formulation of the Multi-Mode Resource-Constrained Project Scheduling Problem for the Flexible Job Shop Scheduling Problem

A Constraint Programming Formulation of the Multi-Mode Resource-Constrained Project Scheduling Problem for the Flexible Job Shop Scheduling Problem

Packing-based branch-and-bound for discrete malleable task scheduling

This paper addresses the problem of scheduling chain-like structures of tasks on a single multiprocessor resource. In fact, sub-tasks of unit-time length and predefined size are aggregated to composite tasks that have to be scheduled without preemption, but subject to flexibility concerning resource allocation. This setting most closely resembles the problem of malleable task scheduling, with sub-tasks being the smallest atomic unit of allocation. The specific type of malleability is realized using precedence constraints with minimum and maximum time lags. A bin packing model is established for this scheduling problem and a corresponding, dedicated branch-and-bound algorithm is devised, alongside problem-specific bound tightening, symmetry breaking and dominance concepts. The efficacy of the solution approach is demonstrated based on extensive computational experiments, including randomized instances, adapted benchmark instances from the literature, and a small real-world data set. In comparison to mixed-integer and constraint programming formulations, the new method is able to achieve a considerably higher percentage of optimal solutions at computation times that are up to orders of magnitude smaller.

A constraint programming approach for the premarshalling problem

The enormous amount of containers handled at ports hampers the efficiency of terminal operations. The optimization of crane movements is crucial for speeding up the loading and unloading of vessels. To this end, the premarshalling problem aims to reorder a set of containers placed in adjacent stacks with a minimum number of crane movements, so that a container with an earlier retrieval time is not below one with a later retrieval time. In this study, we present a series of constraint programming models to optimally solve the premarshalling problem. Extensive computational comparisons show that the best proposed constraint programming formulation yields better results than the state-of-the-art integer programming approach. A salient finding in this paper is that the logic behind the model construction in constraint programming is radically different from that of more traditional mixed integer linear programming models.

Configuration and Evaluation of Multi-CQF Shapers in IEEE 802.1 Time-Sensitive Networking (TSN)

Time-Sensitive Networking (TSN) is a task group of the IEEE 802.1 standardization working group (WG) developing the IEEE 802.1 TSN communication standards. TSN is developing a “toolbox” of many standards to provide support for enabling the separation of critical and non-critical traffic, the timeliness and dependability, i.e., reliability, fault-tolerance, and security, of critical traffic. In this paper we focus on the Cyclic Queuing and Forwarding (CQF) traffic shapers, such as the original CQF, the Cycle Specified Queuing and Forwarding (CSQF) and an extension of those, the so-called Multi-CQF shaper. We define formally the problem of configuring CQF-based networks. We have developed a Constraint Programming (CP) formulation for Multi-CQF, as well as a Simulated Annealing (SA)-based metaheuristic solution. These solutions can also obtain results for CQF and CSQF, which can be seen as a special case of Multi-CQF. The CQF configuration problem is NP-hard. We evaluate our solutions on several test cases and scenarios. The CP formulation can find optimal solutions for small problem sizes but does not scale for realistic test cases. However, SA is able to handle large test cases and to find good quality solutions, making it suitable for both design-time and runtime network configuration. We also present comprehensive evaluation results comparing the CQF-based variants (CQF, CSQF, Multi-CQF) on industrial use cases, contrast them to the Time Aware Shaper (TAS, 802.1Qbv), and discuss their advantages and disadvantages.

Novel Constraint Programming Formulations for Additive Manufacturing Machine Scheduling Problems

Novel Constraint Programming Formulations for Additive Manufacturing Machine Scheduling Problems

Dial-a-ride problem: mixed integer programming revisited and constraint programming proposed

The dial-a-ride problem with time windows is studied in this article. First, a new efficient mixed integer programming formulation is proposed by merging some of drop-off nodes into paired pick-up nodes when a size of the order forbids ridesharing, so a vehicle must visit the paired drop-off nodes directly after the pick-up, thus reducing the number of decision variables by 40% on benchmark instances. Secondly, three different constraint programming formulations are proposed for the first time, generating efficient schedules within a few seconds. Thirdly, a real-time rescheduling perspective is further studied, since the ridesharing becomes dynamic when a schedule must be provided instantly. In particular, the warm-start search method, in which a precalculated schedule is continuously revised as new orders come in, is proposed.

DORY: Automatic End-to-End Deployment of Real-World DNNs on Low-Cost IoT MCUs

The deployment of Deep Neural Networks (DNNs) on end-nodes at the extreme edge of the Internet-of-Things is a critical enabler to support pervasive Deep Learning-enhanced applications. Low-Cost MCU-based end-nodes have limited on-chip memory and often replace caches with scratchpads, to reduce area overheads and increase energy efficiency -- requiring explicit DMA-based memory transfers between different levels of the memory hierarchy. Mapping modern DNNs on these systems requires aggressive topology-dependent tiling and double-buffering. In this work, we propose DORY (Deployment Oriented to memoRY) - an automatic tool to deploy DNNs on low cost MCUs with typically less than 1MB of on-chip SRAM memory. DORY abstracts tiling as a Constraint Programming (CP) problem: it maximizes L1 memory utilization under the topological constraints imposed by each DNN layer. Then, it generates ANSI C code to orchestrate off- and on-chip transfers and computation phases. Furthermore, to maximize speed, DORY augments the CP formulation with heuristics promoting performance-effective tile sizes. As a case study for DORY, we target GreenWaves Technologies GAP8, one of the most advanced parallel ultra-low power MCU-class devices on the market. On this device, DORY achieves up to 2.5x better MAC/cycle than the GreenWaves proprietary software solution and 18.1x better than the state-of-the-art result on an STM32-F746 MCU on single layers. Using our tool, GAP-8 can perform end-to-end inference of a 1.0-MobileNet-128 network consuming just 63 pJ/MAC on average @ 4.3 fps - 15.4x better than an STM32-F746. We release all our developments - the DORY framework, the optimized backend kernels, and the related heuristics - as open-source software.

An exact approach for the green vehicle routing problem with two-dimensional loading constraints and split delivery

This paper presents a study about the Capacitated Vehicle Routing Problem with Two-Dimensional Loading Constraints (2L-CVRP) and its three variants: allowing split delivery (2L-SDVRP), with green requirements (G2L-CVRP), and integrating split delivery with green requirements (G2L-SDVRP). When considering split delivery, a customer can be served by more than one vehicle. The green variant takes into consideration the CO2 emission. Our objective is to analyze the cost benefits obtained with the aggregation of split delivery and the reduction of CO2 emission. Mathematical models are presented for each variant, and instances are solved with a branch-and-cut approach. We develop a tailored procedure to address the packing subproblem, including the computation of lower bounds, a constructive-based heuristic, and a constraint programming formulation. Computational experiments performed on literature instances and newly created ones show that the proposed approach can outperform previous results. Besides that, the green variant has solutions with low emissions of CO2. The variant with split delivery has solutions with lower cost but at the expense of higher computing time.

The integrated orderline batching, batch scheduling, and picker routing problem with multiple pickers: the benefits of splitting customer orders

Fast delivery is one of the most popular services in e-commerce retail. It consists in shipping the items ordered on-line in short times. Customer orders in this segment come with deadlines, and respecting this latter is pivotal to ensure a high service quality. The most time-consuming process in the warehouse is order picking. It consists in regrouping orders into batches, assigning those batches to order pickers, sequencing the batches assigned to each order picker such that the orders deadlines are satisfied, and the picking time is minimized. To speed up the order picking operations, e-commerce warehouses implement new logistical practices. In this paper, we study the impact of splitting the orders (assigning the orderlines of an order to multiple pickers). We thus generalize the integrated orders batching, batch scheduling, and picker routing problem by allowing the orders splitting and propose a route first-schedule second heuristic to solve the problem. In the routing phase, the heuristic divides the orders into clusters and constructs the picking tours that retrieve the orderlines of each cluster using a split-based procedure. In the scheduling phase, the constructed tours are assigned to pickers such that the orders deadlines are satisfied using a constraint programming formulation. On a publicly available benchmark, we compare our results against a state-of-the-art iterated local search algorithm designed for the non-splitting version of the problem. Results show that splitting the customer orders using our algorithm reduces the picking time by 30% on average with a maximum reduction of 60%.

Constraint programming approaches for the discretizable molecular distance geometry problem

AbstractThe Distance Geometry Problem (DGP) seeks to find positions for a set of points in geometric space when some distances between pairs of these points are known. The so‐called discretization assumptions allow us to discretize the search space of DGP instances. In this paper, we focus on a key subclass of DGP, namely the Discretizable Molecular DGP, and study its associated graph vertex ordering problem, the Contiguous Trilateration Ordering Problem (CTOP), which helps solve DGP. We propose the first constraint programming formulations for CTOP, as well as a set of checks for proving infeasibility, domain reduction techniques, symmetry breaking constraints, and valid inequalities. Our computational results on random and pseudo‐protein instances indicate that our formulations outperform the state‐of‐the‐art integer programming formulations.

A constraint programming approach and a hybrid of genetic and K-means algorithms to solve the p-hub location-allocation problems

ABSTRACT p-Hub location-allocation problem is one of the most interesting subjects in the location theory. Hubs act as switching points to reduce the transportation cost. In this study, two new solution methods, a constraint programming (CP) based model and a hybrid of k-means and genetic algorithm (KGA), are developed to generate exact and approximate solutions, respectively. The proposed CP formulation is more understandable and straightforward in comparison with the MIP model. The experimental results indicate that the CP model uses the memory of the computer (RAM) more efficiently, which enables us to solve the medium size problems. But, in terms of run time, this method cannot be superior to the MIP model. The CP formulation is also extended for the multi allocation p-hub location problem. K-means algorithm, a well-known algorithm for clustering data, is used to generate initial solutions of GA. Furthermore, a new adaptive crossover operator, which is based on the k-means algorithm, is proposed. The experimental results indicate that the KGA algorithm is superior to the GA, regarding time, objective value, and quality of solution measures.

Integer Programming, Constraint Programming, and Hybrid Decomposition Approaches to Discretizable Distance Geometry Problems

Given an integer dimension K and a simple, undirected graph G with positive edge weights, the Distance Geometry Problem (DGP) aims to find a realization function mapping each vertex to a coordinate in [Formula: see text] such that the distance between pairs of vertex coordinates is equal to the corresponding edge weights in G. The so-called discretization assumptions reduce the search space of the realization to a finite discrete one, which can be explored via the branch-and-prune (BP) algorithm. Given a discretization vertex order in G, the BP algorithm constructs a binary tree where the nodes at a layer provide all possible coordinates of the vertex corresponding to that layer. The focus of this paper is on finding optimal BP trees for a class of discretizable DGPs. More specifically, we aim to find a discretization vertex order in G that yields a BP tree with the least number of branches. We propose an integer programming formulation and three constraint programming formulations that all significantly outperform the state-of-the-art cutting-plane algorithm for this problem. Moreover, motivated by the difficulty in solving instances with a large and low-density input graph, we develop two hybrid decomposition algorithms, strengthened by a set of valid inequalities, which further improve the solvability of the problem. Summary of Contribution: We present a new model to solve a combinatorial optimization problem on graphs, MIN DOUBLE, which comes from the highly active area of distance geometry and has applications in a wide variety of fields. We use integer programming (IP) and present the first constraint programming (CP) models and hybrid decomposition methods, implemented as a branch-and-cut procedure, for MIN DOUBLE. Through an extensive computational study, we show that our approaches advance the state of the art for MIN DOUBLE. We accomplish this by not only combining generic techniques from IP and CP but also exploring the structure of the problem in developing valid inequalities and variable fixing rules. Our methods significantly improve the solvability of MIN DOUBLE, which we believe can also provide insights for tackling other problem classes and applications.