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

Abstract

The Job Shop Scheduling Problem (JSP) is a combinatorial optimization problem where jobs visit singlecapacity 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.We gratefully acknowledge financial support from Chalmers AI Research Centre (CHAIR) and AB Volvo (Project ViMCoR).

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

“…For the analysis we used Z3 4.8.9. The time limit for MonoMod is set to 10800 seconds (three hours); the model generation time is measured separately, since it is implementation-dependent and can be dealt with using more efficient formulations, as discussed in our previous work [23]. As for ComSat, we only computed ten paths for each pair of customers.…”

Solving the Conflict-Free Electric Vehicle Routing Problem Using SMT Solvers

“…The following constraints are related to the conflict-free routing, specifying how one or more vehicle's location affects the other vehicles' movements: (19) states that if n vehicles are on the same node, they cannot transit on the same edge having capacity n − 1 at the same time-step; (20) states that if n vehicles are on two adjacent nodes and decide to traverse the edge connecting them and whose capacity is n + m, at most m vehicles can traverse the edge from the other node before the n vehicles are done transiting.…”

“…For the analysis we used Z3 4.8.9. The time limit is set to 10800 seconds (three hours); the model generation time is measured separately, since it is implementation-dependent and can be dealt with using more efficient formulations, as discussed in our previous work [20]. All the experiments were performed on an Intel Core i7 6700K, 4.0 GHZ, 32GB RAM running Ubuntu-18.04 LTS.…”

“…Tasks are reported in the figure next to their location; for instance, task 2 of job 1 is at node I. Time windows, if they exist, are reported below the task name, in between square brackets; for instance, the time window for task 1 of job 2 is[20,25]. In this problem jobs 1 to 5 can only be executed by vehicles a to e respectively, while Job x1 can be executed by any vehicle.…”