Heinz Gröflin

A new neighborhood and tabu search for the Blocking Job Shop

The Blocking Job Shop is a version of the job shop scheduling problem with no intermediate buffers, where a job has to wait on a machine until being processed on the next machine. We study a generalization of this problem which takes into account transfer operations between machines and sequence-dependent setup times. After formulating the problem in a generalized disjunctive graph, we develop a neighborhood for local search. In contrast to the classical job shop, there is no easy mechanism for generating feasible neighbor solutions. We establish two structural properties of the underlying disjunctive graph, the concept of closures and a key result on short cycles, which enable us to construct feasible neighbors by exchanging critical arcs together with some other arcs. Based on this neighborhood, we devise a tabu search algorithm and report on extensive computational experience, showing that our solutions improve most of the benchmark results found in the literature.

The flexible blocking job shop with transfer and set-up times

The Flexible Blocking Job Shop (FBJS) considered here is a job shop scheduling problem characterized by the availability of alternative machines for each operation and the absence of buffers. The latter implies that a job, after completing an operation, has to remain on the machine until its next operation starts. Additional features are sequence-dependent transfer and set-up times, the first for passing a job from a machine to the next, the second for change-over on a machine from an operation to the next. The objective is to assign machines and schedule the operations in order to minimize the makespan. We give a problem formulation in a disjunctive graph and develop a heuristic local search approach. A feasible neighborhood is constructed, where typically a critical operation is moved (keeping or changing its machine) together with some other operations whose moves are “implied”. For this purpose, we develop the theoretical framework of job insertion with local flexibility, based on earlier work of Gröflin and Klinkert on insertion. A tabu search that consistently generates feasible neighbor solutions is then proposed and tested on a larger test set. Numerical results support the validity of our approach and establish first benchmarks for the FBJS.

Feasible insertions in job shop scheduling, short cycles and stable sets

Insertion problems arise in scheduling when additional activities have to be inserted into a given schedule. This paper investigates insertion problems in a general disjunctive scheduling framework capturing a variety of job shop scheduling problems and insertion types. First, a class of scheduling problems is introduced, characterized by disjunctive graphs with the so-called short cycle property, and it is shown that in such problems, the feasible selections correspond to the stable sets of maximum cardinality in an associated conflict graph. Two types of insertion problems are then identified where the underlying disjunctive graph is through- or bi-connected. For these cases, it is shown that the short cycle property holds and the conflict graph is bipartite, allowing to derive a polyhedral characterization of all feasible insertions. An efficient method for deciding whether there exists a feasible insertion, and a lower and upper bound procedure for the minimum makespan insertion problem are developed. For bi-connected graphs, this procedure solves the insertion problem to optimality. The obtained results are applied to three extensions of the classical Job Shop, the Multi-Processor Task, Blocking and No-Wait Job Shop, and two types of insertions, job and block insertion.

Optimal job insertion in the no-wait job shop

The no-wait job shop (NWJS) considered here is a version of the job shop scheduling problem where, for any two operations of a job, a fixed time lag between their starting times is given. Also, sequence-dependent set-up times between consecutive operations on a machine can be present. The NWJS problem consists in finding a schedule that minimizes the makespan.We address here the so-called optimal job insertion problem (OJI) in the NWJS. While the OJI is NP-hard in the classical job shop, it was shown by Gröflin & Klinkert to be solvable in polynomial time in the NWJS. We present a highly efficient algorithm with running time

The blocking job shop with rail-bound transportation

\mathcal {O}(n^{2}\cdot\max\{n,m\})

Lattice matrices, intersection of ring families and dicuts

for this problem. The algorithm is based on a compact formulation of the NWJS problem and a characterization of all feasible insertions as the stable sets (of prescribed cardinality) in a derived comparability graph.As an application of our algorithm, we propose a heuristic for the NWJS problem based on optimal job insertion and present numerical results that compare favorably with current benchmarks.

The no-wait job shop with regular objective: a method based on optimal job insertion

The blocking job shop with rail-bound transportation (BJS-RT) considered here is a version of the job shop scheduling problem characterized by the absence of buffers and the use of a rail-bound transportation system. The jobs are processed on machines and are transported from one machine to the next by mobile devices (called robots) that move on a single rail. The robots cannot pass each other, must maintain a minimum distance from each other, but can also “move out of the way”. The objective of the BJS-RT is to determine for each machining operation its starting time and for each transport operation its assigned robot and starting time, as well as the trajectory of each robot, in order to minimize the makespan. Building on previous work of the authors on the flexible blocking job shop and an analysis of the feasible trajectory problem, a formulation of the BJS-RT in a disjunctive graph is derived. Based on the framework of job insertion in this graph, a local search heuristic generating consistently feasible neighbor solutions is proposed. Computational results are presented, supporting the value of the approach.

Optimum partitioning into intersections of ring families

Abstract Lattice matrices are 0/1-matrices used in the description of certain lattice polyhedra and related to dicuts in a graph. The incidence matrix AI of a so-called intersection of two ring families and the incidence matrix AD of all dicuts of a graph are examples of such matrices. After showing that any lattice matrix A can be obtained from some matrix AI by deletion or some AD by contraction, we first describe the convex hull of the rows of AI, CONV(AI), as the solution set of a system x⩾0, Bx⩽1, Rx⩽0, which is tdi. We then derive the main result, the description of CONV(A) by another tdi system. As applications, the polyhedral description of all dicuts in a graph, CONV(AD), and that of all convex sets of bounded length in a poset are established.

Feasible job insertions in the multi-processor-task job shop

The no-wait job shop problem (NWJS-R) considered here is a version of the job shop scheduling problem where, for any two operations of a job, a fixed time lag between their starting times is prescribed. Also, sequence-dependent set-up times between consecutive operations on a machine can be present. The problem consists in finding a schedule that minimizes a general regular objective function. We study the so-called optimal job insertion problem in the NWJS-R and prove that this problem is solvable in polynomial time by a very efficient algorithm, generalizing a result we obtained in the case of a makespan objective. We then propose a large neighborhood local search method for the NWJS-R based on the optimal job insertion algorithm and present extensive numerical results that compare favorably with current benchmarks when available.

Feasible Sequential Decisions and a Flexible Lagrangean-Based Heuristic for Dynamic Multi-Level Lot Sizing

Abstract Given two ring families C and D on a finite ground set V , with both o and V ϵ C and D , consider the family of so-called intersections L = [L ⊂- V¦L = C∩D, C ϵ C, D ϵ D and C∪D = V] and let A be the incidence matrix of L . The minimum partitioning problem: “Given a vector d ϵ Z v + , minimize y 1 s . t . yA = d , y ⩾ 0, y integer”, is solved by a longest path computation. The approach is polyhedral and capitalizes on previous results related to lattice matrices.