Conflict-Free Routing of Mobile Robots

The recent advances in perception have enabled the development of more autonomous mobile robots in the sense that they can operate in a more dynamic environment where obstacles surrounding the robot emerge, disappear, and move. The increased perception of Autonomous Mobile Robots (AMRs) allows them to plan detailed on-line trajectories in order to avoid previously unforeseen obstacles, making AMRs useful in dynamic environments where humans, traditional fork-lifts, and also other mobile robots operate. These abilities contributed to increase automation in logistic applications. This thesis discusses how to efficiently operate a fleet of AMRs and make sure that all tasks are successfully completed.Assigning robots to specific delivery tasks and deciding the routes they have to travel can be modelled as a variant of the classical Vehicle Routing Problem (VRP), the combinatorial optimization problem of designing routes for vehicles. In related research it has been extended to scheduling routes for vehicles to serve customers according to predetermined specifications, such as arrival time at a customer, amount of goods to deliver, etc.In this thesis we consider to schedule a fleet of robots such that areas avoid being congested, delivery time-windows are met, the need for robots to recharge is considered, while at the same time the robots have freedom to use alternative paths to handle changes in the environment. This particular version of the VRP, called CF-EVRP (Conflict-free Electrical Vehicle Routing Problem) is motivated by an industrial need. In this work we consider using optimizing general purpose solvers, in particular, MILP and SMT solvers are investigated. We run extensive computational analysis over well-known combinatorial optimization problems, such as job shop scheduling and bin-packing problems, to evaluate modeling techniques and the relative performance of state-of-the-art MILP and SMT solvers.We propose a monolithic model for the CF-EVRP as well as a compositional approach that decomposes the problem into sub-problems and formulate them as either MILP or SMT problems depending on what fits each particular problem best. The performance of the two approaches is evaluated on a set of CF-EVRP benchmark problems, showing the feasibility of using a compositional approach for solving practical fleet scheduling problems

Compact Representation of Time-Index Job Shop Problems Using a Bit-Vector Formulation

The Job Shop Scheduling Problem (JSP) is a combinatorial optimization problem where jobs visit single-capacity machines while minimizing a cost function, typically the makespan. The problem can be extended to fit typical industrial scenarios such as flexible assembly shop floors or for coordinating fleets of automated vehicles. General purpose optimizers can handle extended versions of the problem that typically arise in industrial problems. Mixed Integer Linear Programming (MILP) solvers and recently optimizing Satisfiability Modulo Theory (SMT) solvers can be used as general solvers for JSP problems. There exist different formulations of JSP problems, among them the time-index (TI) model. The TI offers the advantage of providing strong lower bounds, though its drawback is the model size.In this paper we present a new formulation of the TI model suitable for optimizing SMT-solvers that support bit-vector theories. The new formulation is significantly more compact than the standard TI formulation and is thus reducing one of the major issues with the TI model.We benchmark two different optimizing SMT solvers supporting bit-vector theories, comparing the standard formulation of the TI to the new formulation on a set of benchmark instances. The computational analysis shows that the new formulation outperforms the standard one, being up to twice faster and regardless of the solver employed; moreover the model generated with the new formulation is considerably smaller than with the standard formulation

An SMT Based Compositional Algorithm to Solve a Conflict-Free Electric Vehicle Routing Problem

The Vehicle Routing Problem (VRP) is the combinatorial optimization problem of designing routes for vehicles to visit customers in such a fashion that a cost function, typically the number of vehicles, or the total travelled distance is minimized. The problem finds applications in industrial scenarios, for example where Automated Guided Vehicles run through the plant to deliver components from the warehouse. This specific problem, henceforth called the Electric Conflict-Free Vehicle Routing Problem (CF-EVRP), involves constraints such as limited operating range of the vehicles, time windows on the delivery to the customers, and limited capacity on the number of vehicles the road segments can accommodate at the same time. Such a complex system results in a large model that cannot easily be solved to optimality in reasonable time. We therefore developed a compositional model that breaks down the problem into smaller and simpler sub-problems and provides sub-optimal, feasible solutions to the original problem. The algorithm exploits the strengths of SMT solvers, which proved in our previous work to be an efficient approach to deal with scheduling problems. Compared to a monolithic model for the CF-EVRP, written in the SMT standard language and solved using a state-of-the-art SMT solver the compositional model was found to be significantly faster

Leveraging Conflicting Constraints in Solving Vehicle Routing Problems

The Conflict-Free Electric Vehicle Routing Problem (CF-EVRP) is a combinatorial optimization problem of designing routes for vehicles to visit customers such that a cost function, typically the number of vehicles or the total travelled distance, is minimized. The CF-EVRP involves constraints such as time windows on the delivery to the customers, limited operating range of the vehicles, and limited capacity on the number of vehicles that a road segment can simultaneously accommodate.In previous work, the compositional algorithm ComSat was introduced and that solves the CF-EVRP by breaking it down into sub-problems and iteratively solve them to build an overall solution.Though ComSat showed good performance in general, some problems took significant time to solve due to the high number of iterations required to find solutions that satisfy the road segments\u27 capacity constraints. The bottleneck is the Paths Changing Problem, i.e., the sub-problem of finding a new set of shortest paths to connect a subset of the customers, disregarding previously found shortest paths. This paper presents an improved version of the PathsChanger function to solve the Paths Changing Problem that exploits the unsatisfiable core, i.e., information on which constraints conflict, to guide the search for feasible solutions. Experiments show faster convergence to feasible solutions compared to the previous version of PathsChanger

On the Use of Equivalence Classes for Optimal and Suboptimal Bin Packing and Bin Covering

Bin packing and bin covering are important optimization problems in many industrial fields, such as packaging, recycling, and food processing. The problem concerns a set of items, each with its own value, that are to be sorted into bins in such a way that the total value of each bin, as measured by the sum of its item values, is not above (for packing) or below (for covering) a given target value. The optimization problem concerns minimizing, for bin packing, or maximizing, for bin covering, the number of bins. This is a combinatorial NP-hard problem, for which true optimal solutions can only be calculated in specific cases, such as when restricted to a small number of items. To get around this problem, many suboptimal approaches exist. This article describes the formulations of the bin packing and covering problems that allow finding the true optimum, for instance, counting hundreds of items using general-purpose MILP-solvers. Also presented are suboptimal solutions that come within less than 10% of the optimum while taking significantly less time to calculate, even ten to 100 times faster, depending on the required accuracy. Note to Practitioners - A typical case for bin covering is in food processing where food items are automatically sorted into trays of similar weight so that the overweight is minimized. Another application is in recycling, where items such as batteries should be put in crates of similar weight, so that the crates do not exceed a target weight due to later manual handling, but, at the same time, we want as few crates as possible. This is a bin packing problem. On an industrial scale, these tasks are fully automated. Though modern software tool\u27s efficiency to solve bin sorting problems has increased significantly in later years, the problems are inherently tough in the sense that the solution time grows exponentially with the number of items. This limits the problem sizes that can be solved to optimality within a reasonable time. Therefore, much research has focused on heuristic rules that give reasonable solving times while not giving the true optimal number of bins. However, in many cases, the true optimal solution is preferable, and sometimes even necessary, so this is an industrially interesting problem. This article describes an approach to solve the bin packing and covering problems to the true optimum that increases the limit of the number of items that can typically be handled. This is done by observing that items of the same value need not be distinguished. Instead, we can formulate packing/covering problems over item values rather than individual items and sort integer numbers of these values into bins, which allows us to solve to optimum for more than 500 items in a reasonable time. In addition, by redefining what we mean by the same value, we can consider more items to have the same value and achieve even better calculation efficiency

Comparison of Exact and Approximate methods for the Vehicle Routing Problem with Time Windows

This paper presents a comparison of two approaches for solving the vehicle routing problem with time windows (VRPTW). Scheduling of vehicles for pickup and delivery is a common problem in logistics and may be expressed as VRPTW, for which both exact and approximate techniques are available. It is therefore interesting to compare such techniques to evaluate their performance and figure what is the best option based on the instance features and size. In this work, we compared Mixed Integer Linear Programming (MILP) with Set-Based Particle Swarm optimization (S-PSO). Both algorithms are tested on the full 56 instances of the Solomon dataset. The results show that the two algorithms perform similarly for lower number of customers while there are significant differences for the cases with higher number of customers. For higher number of customers MILP consistently performs as good as or better than S-PSO for the clustered data, both with short and long scheduling horizons, while the S-PSO outperforms MILP in most cases with random and mixed random clustered data with long scheduling horizons. Furthermore when the algorithms perform the same with regards to the main objective (number of vehicles), MILP generally achieves a better result in the second objective (distance traveled)

On scheduling using optimizing SMT-solvers.

Modern production systems are becoming more complex by the year and flexibility of production is one of the key factors to success. Companies want to be able to provide a customized product that fits exactly the customer requirements and, therefore production systems have to be able to produce a wide range of product variants. This introduces additional complexity in the production system,Among the problems that companies have to deal with is the vehicle routing problem (VRP), which is the problem of scheduling routes for vehicles to serve customers according to predetermined specifications, such as arrival time at a customer, amount of goods to deliver, etc. The problem is industrially relevant since material needs to be delivered from warehouses to the production lines and as the plants grow in size, managing an ever growing fleet of vehicles is becoming a challenge. Due to the complexity of the requirements and the increasing size of the transportation systems, it is no longer feasible to solve these problems manually, since the variables and constraints to keep into account are simply to many. Modern computers can provide feasible schedules much faster than human beings and for this reason companies are willing to pay a high fee to use the cutting edge scheduling solvers on the market. Among the class of general purpose solvers, we find mixed integer linear programming (MILP) solvers and satisfiability modulo theory (SMT) solvers. They are not designed to solve one specific problem, but entire classes of problems. In particular, both MILP and SMT solvers can handle mixed integer linear models and, since the VRP can be described by a mixed integer linear model, both MILP and SMT qualify as suitable tools to deal with it

SMT Solvers for Job-Shop Scheduling Problems: Models Comparison and Performance Evaluation

The optimal assignment of jobs to machines is a common problem when implementing automated production systems. A specific variant of this category is the job-shop scheduling problem (JSP) that is known to belong to the class of NP-hard problems. JSPs are typically either formulated as Mixed Integer Linear Programming (MILP) problems and solved by general-purpose-MILP solvers or approached using heuristic algorithms specifically designed for the purpose. During the last decade a new approach, satisfiability (SAT), led to develop solvers with incredible abilities in finding feasible solutions for hard combinatorial problems on Boolean variables. Moreover, an extension of SAT, Satisfability Modulo Theory (SMT), allows to formulate constraints involving linear operations over integers and reals and some SMT-solvers have been also extended with an optimizing tool. Since the JSP is a well-known hard combinatorial problem, it is interesting to evaluate how SMT-solvers perform in solving it and how they compare to traditional MILP-solvers. We therefore evaluate state-of-the-art MILP and SMT solvers on benchmark JSP instances and find that general-purpose open-source SMT-solvers are competitive against commercial MILP-solvers

SMT solvers for flexible job-shop scheduling problems: A computational analysis

In this paper we evaluate different formulations for solving the Flexible Job-shop Scheduling Problem (FJSP) to optimality. Heuristic methods resulting in sub-optimal solutions are traditionally used to solve this problem since it is a known NP-hard problem. Since industrial problems often have additional constraints that need to be considered during optimization, the heuristic methods need to be adapted to deal efficiently with the additional constraints. For this reason general-purpose solvers that are often used in industrial applications. There are different approaches to formulate FJSPs as Mixed-Integer Linear Programming problems (MILP) that can be solved using generic MILP-solvers. In recent years, satisfiability solvers, i.e. SAT- and SMT-solvers, have evolved within the formal verification community and shown to be able to efficiently solve large instances of well-known NP-hard problems. In our previous work we have shown that SMT-solvers extended with optimization techniques can be a competitive alternative to commercial MILP-solvers on traditional job-shop scheduling problems. In this work we have adapted three formulations used for formulating FJSPs for MILP solvers into SMT-formulations. The three formulations are used to solve benchmark FJSPs using the open-source Z3 SMT-solver. We show that the a formulation based on the Manne formulation adapted for SMT-formulations for FJSPs is a competitive alternative for solving large-scale FJSPs, and might be considered as a viable alternative for solving industrial problems

Solving the conflict-free electric vehicle routing problem using SMT solvers

The vehicle routing problem is a combinatorial optimization problem of computing routes to serve customers while minimizing a cost function, typically the traveled distance or the number of vehicles required. Industrial applications of the problem in manufacturing plants are the scheduling and routing of Automated Guided Vehicles (AGVs) to deliver material between storage areas and assembly stations. For in-plant transportation it is necessary to take the limited space of the plant floor into account during scheduling and routing in order to limit the number of AGVs that are at certain areas at a given time. In addition, AGVs are most often powered by batteries and therefore have limited operating range and non-negligible charging time that will also affect the scheduling and routing decisions. In this paper we provide a monolithic model formulation for the scheduling and routing of AGVs with given time-windows for delivering material, restricted by capacity constraints on the path network, and with the need for battery recharge. The problem is modelled and solved using optimizing Satisfiability Modulo Theory (SMT) solvers. The approach is evaluated on a set of generated problem instances, showing that the solver can handle medium size instances in a reasonable amount of time