Wiebke Höhn

On the Performance of Smith’s Rule in Single-Machine Scheduling with Nonlinear Cost

We consider a single-machine scheduling problem. Given some continuous, nondecreasing cost function, we aim to compute a schedule minimizing the weighted total cost, where the cost of each job is determined by the cost function value at its completion time. This problem is closely related to scheduling a single machine with nonuniform processing speed. We show that for piecewise linear cost functions it is strongly NP-hard. The main contribution of this article is a tight analysis of the approximation guarantee of Smith’s rule under any convex or concave cost function. More specifically, for these wide classes of cost functions we reduce the task of determining a worst-case problem instance to a continuous optimization problem, which can be solved by standard algebraic or numerical methods. For polynomial cost functions with positive coefficients, it turns out that the tight approximation ratio can be calculated as the root of a univariate polynomial. We show that this approximation ratio is asymptotically equal to k(k − 1)/(k + 1), denoting by k the degree of the cost function. To overcome unrealistic worst-case instances, we also give tight bounds for the case of integral processing times that are parameterized by the maximum and total processing time.

On the performance of smith's rule in single-machine scheduling with nonlinear cost

We consider the problem of scheduling jobs on a single machine. Given some continuous cost function, we aim to compute a schedule minimizing the weighted total cost, where the cost of each individual job is determined by the cost function value at the jobs completion time. This problem is closely related to scheduling a single machine with nonuniform processing speed. We show that for piecewise linear cost functions it is strongly NP-hard. The main contribution of this article is a tight analysis of the approximation factor of Smiths rule under any particular convex or concave cost function. More specifically, for these wide classes of cost functions we reduce the task of determining a worst case problem instance to a continuous optimization problem, which can be solved by standard algebraic or numerical methods. For polynomial cost functions with positive coefficients it turns out that the tight approximation ratio can be calculated as the root of a univariate polynomial. To overcome unrealistic worst case instances, we also give tight bounds that are parameterized by the minimum, maximum, and total processing time.

Integrated Sequencing and Scheduling in Coil Coating

We consider a complex planning problem in integrated steel production. A sequence of coils of sheet metal needs to be color coated in consecutive stages. Different coil geometries and changes of colors necessitate time-consuming setup work. In most coating stages one can choose between two parallel color tanks. This can either reduce the number of setups needed or enable setups concurrent with production. A production plan comprises the sequencing of coils and the scheduling of color tanks and setup work. The aim is to minimize the makespan for a given set of coils. We present an optimization model for this integrated sequencing and scheduling problem. A core component is a graph theoretical model for concurrent setup scheduling. It is instrumental for building a fast heuristic that is embedded into a genetic algorithm to solve the sequencing problem. The quality of our solutions is evaluated via an integer program based on a combinatorial relaxation, showing that our solutions are within 10% of the optimum. Our algorithm is implemented at Salzgitter Flachstahl GmbH, a major German steel producer. This has led to an average reduction in makespan by over 13% and has greatly exceeded expectations. This paper was accepted by Dimitris Bertsimas, optimization.

On Eulerian extensions and their application to no-wait flowshop scheduling

We consider a variant of no-wait flowshop scheduling that is motivated by continuous casting in the multistage production process in steel manufacturing. The task is to find a feasible schedule with a minimum number of interruptions, i.e., continuous idle time intervals on the last production stage. Based on an interpretation as Eulerian Extension Problems, we fully settle the complexity status of any particular problem case: We give a very intuitive optimal algorithm for scheduling on two processing stages with one machine in the first stage, and we show that all other problem variants are strongly NP-hard. We also discuss alternative idle time related scheduling models and their justification in the considered steel manufacturing environment. Here, we derive constant factor approximations.

Sequencing and scheduling for filling lines in dairy production

We consider an integrated sequencing and scheduling problem arising at filling lines in dairy industry. Even when a processing sequence is decided, still a scheduling problem has to be solved for the sequence. This incorporates typical side constraints as they occur also in other sequencing problems in practice. Previously, we proposed a framework for general sequencing and scheduling problems: A genetic algorithm is utilized for the sequencing, incorporating a problem specific algorithm for the fixed-sequence scheduling. In this paper, we investigate how this approach performs for filling lines. Based on insights into structural properties of the problem, we propose different scheduling algorithms. In cooperation with Sachsenmilch GmbH, the algorithm was implemented for their bottleneck filling line, and evaluated in an extensive computational study. For the real data from production, our algorithm computes almost optimal solutions. However, as a surprising result, our simple greedy algorithms outperform the more elaborate ones in many aspects, showing interesting directions for future research.

How Unsplittable-flow-covering Helps Scheduling with Job-dependent Cost Functions

Generalizing many well-known and natural scheduling problems, scheduling with job-specific cost functions has gained a lot of attention recently. In this setting, each job incurs a cost depending on its completion time, given by a private cost function, and one seeks to schedule the jobs to minimize the total sum of these costs. The framework captures many important scheduling objectives such as weighted flow time or weighted tardiness. Still, the general case as well as the mentioned special cases are far from being very well understood yet, even for only one machine. Aiming for better general understanding of this problem, in this paper we focus on the case of uniform job release dates on one machine for which the state of the art is a 4-approximation algorithm. This is true even for a special case that is equivalent to the covering version of the well-studied and prominent unsplittable flow on a path problem, which is interesting in its own right. For that covering problem, we present a quasi-polynomial time (1 + e)-approximation algorithm that yields an (e + e)-approximation for the above scheduling problem. Moreover, for the latter we devise the best possible resource augmentation result regarding speed: a polynomial time algorithm which computes a solution with optimal cost at 1 + e speedup. Finally, we present an elegant QPTAS for the special case where the cost functions of the jobs fall into at most logn many classes. This algorithm allows the jobs even to have up to logn many distinct release dates. All proposed quasi-polynomial time algorithms require the input data to be quasi-polynomially bounded.

How Unsplittable-Flow-Covering Helps Scheduling with Job-Dependent Cost Functions

Generalizing many well-known and natural scheduling problems, scheduling with job-specific cost functions has gained a lot of attention recently. In this setting, each job incurs a cost depending on its completion time, given by a private cost function, and one seeks to schedule the jobs to minimize the total sum of these costs. The framework captures many important scheduling objectives such as weighted flow time or weighted tardiness. Still, the general case as well as the mentioned special cases are far from being very well understood yet, even for only one machine. Aiming for better general understanding of this problem, in this paper we focus on the case of uniform job release dates on one machine for which the state of the art is a 4-approximation algorithm. This is true even for a special case that is equivalent to the covering version of the well-studied and prominent unsplittable flow on a path problem, which is interesting in its own right. For that covering problem, we present a quasi-polynomial time

An experimental and analytical study of order constraints for single machine scheduling with quadratic cost

(1+\varepsilon )