A Computational Study of Lower Bounding Schemes for Total Weighted Tardiness Job Shops Roland Braune ∗ , G¨ unther Z¨ apfel ∗ , Michael Affenzeller † ∗ Institute for Production and Logistics Management Johannes Kepler University, Linz, Austria Email: roland.braune@jku.at † Department of Software Engineering Upper Austria University of Applied Sciences, Hagenberg, Austria Abstract—In this paper, we perform a computational study of lower bounding schemes for job shop scheduling problems under special consideration of total weighted tardiness costs. Due to the characteristics of this objective function, lower bounds are much more difficult to derive than for the classical makespan. On the other hand, the practical relevance of tardiness related costs makes it even more important to have corresponding bounds available, especially for rating the results of approximate optimization approaches. Apart from the quality of the bounds, i.e. the tightness with respective to an existing optimal solution or upper bound, a further important focus of our investigations is the required computation time, since this is a determining criterion for the applicability. Computational results are reported based on selected benchmark problems. I. I NTRODUCTION The academic field of job shop scheduling [1] provides theoretical foundations for many machine scheduling related problems in the industrial world. Finding a feasible sequence or schedule of jobs on each machine under a given objective represents a combinatorial optimization problem, which is NP- hard in most configurations and therefore usually very difficult to solve. The most popular optimization objective, the minimization of the maximum completion time (makespan), is primarily of theoretical interest. In the real world, other objective functions, in particular tardiness related ones, play a much more impor- tant role. Despite this fact, the range of solution approaches for such criteria is considerably smaller. Besides the Branch&Bound algorithm proposed by Singer and Pinedo [2], mainly heuristic solution methods appear in scientific literature. Among them, especially the Genetic Algorithm, neighborhood search and Shifting Bottleneck based ones are worth mentioning. Solving problems with tardiness related objectives to opti- mality, however, seems to be even more challenging than in the makespan case, and problem size limits are reached much earlier. In the absence of optimal solutions, it is difficult to assess the performance of heuristic solution approaches, as they are often the only reasonable way of tackling problems of greater or even industrial dimensions. In such situations, lower bounds on the optimal objective function value may serve as a replacement - as long as they are sufficiently tight (cf. e.g. [3]). The scope of this paper is an experimental comparison of lower bounding approaches for the total weighted tardiness objective. Our main interest is directed towards the tightness of the bounds and the computational effort required to compute them. The latter is important for a second potential purpose of lower bounds: The integration into a Branch&Bound algo- rithm, which of course requires a high degree of efficiency. The paper is organized as follows: In Section II, we formally introduce the job shop scheduling problem with total weighted tardiness costs. A short survey on the examined lower bound- ing approaches is given in Section III. The corresponding computational results are summarized in Section IV followed by concluding remarks and an outlook on future research (cf. Section V). II. PROBLEM STATEMENT A job shop scheduling problem is specified by a finite set J of n jobs, J = {J 1 ,...,J n }, which have to be scheduled on a finite set M of m machines, M = {M 1 ,...,M m }. In the standard case, each job J i ∈ J has to be processed on each machine M u ∈ M , suggesting the fragmentation of jobs into separate operations. The order in which operations pass the machines may be different for each job and is predefined by the precedence constraints. Hence for each job J i ∈ J we are given a fixed sequence of operations (o i1 ,...,o im ), where o ik denotes the k-th operation of job J i , indicating the routing of J i . Each operation o ik is assigned a nonnegative integer processing time p ik . Each machine can only process one operation at a time (capacity constraints) and the processing of an operation cannot be interrupted until it is finished, thus no preemption is allowed. The goal is to determine a schedule represented by a set of operation starting times S = {s ik | 1 ≤ i ≤ n, 1 ≤ k ≤ m} which is feasible with respect to precedence and capacity constraints and minimizes a predefined cost function. 978-1-4244-3958-4/09/$25.00 ©2009 IEEE