A survey of scheduling problems with no-wait in process

In this paper we address the production scheduling and distribution planning problem in a yoghurt production line of the multi-product dairy plants. A mixed integer linear programming model is developed for the considered problem. The objective function aims to maximize the benefit by considering the shelf life dependent pricing component and costs such as processing, setup, storage, overtime, backlogging, and transportation costs. Key features of the model include sequence dependent setup times, minimum and maximum lot sizes, overtime, shelf life requirements, machine speeds, dedicated production lines, typically arising in the dairy industry. The model obtains the optimal production plan for each product type, on each production line, in each period together with the delivery plan. The hybrid modelling approach is adopted to explore the dynamic behavior of the real world system. In the hybrid approach, operation time is considered as a dynamic factor and it is adjusted by the results of the simulation and optimization model iteratively. Thus, more realistic solutions are obtained for the scheduling problem in yoghurt industry by using the iterative hybrid optimization-simulation procedure. The efficiency and applicability of the proposed model and approach are demonstrated in a case study for a leading dairy manufacturing company in Turkey.

Integrated production scheduling and distribution planning in dairy supply chain by hybrid modelling

We study an integrated charge batching and casting width selection problem arising in the continuous casting operation of the steelmaking process at Shanghai, China based Baosteel. This decision-making problem is not unique to Baosteel; it exists in every large iron and steel company in the world. We collaborated with Baosteel on this problem from 2006 to 2008 by developing and implementing a decision support system DSS that replaced their manual planning method. The DSS is still in active use at Baosteel. This paper describes the solution algorithms we developed and imbedded in the DSS. For the general problem that is strongly NP-hard, a column generation-based branch-and-price B&P solution approach is developed to obtain optimal solutions. By exploiting the problem structure, efficient dynamic programming algorithms are designed to solve the subproblems involved in the column generation procedure. Branching strategies are designed in a way that ensures that after every stage of branching the structure of the subproblems is preserved such that they can still be solved efficiently. We also consider a frequently occurring case of the problem where each steel grade is incompatible with any other grade. For this special case, a two-level polynomial-time algorithm is developed to obtain optimal solutions. Computational tests on a set of real production data as well as on a more diverse set of randomly generated problem instances show that our algorithms outperform the manual planning method that Baosteel used to use by a significant margin both in terms of tundish utilization for almost every case, and in terms of total cost for most cases. Consequently, by replacing their manual method with our DSS, the estimated benefits to Baosteel include an annual cost saving of about US $1.6 million and an annual revenue increase of about US $3.25 million.

Integrated Charge Batching and Casting Width Selection at Baosteel

We study scheduling problems on a machine of varying speed. Assuming a known speed function (given through an oracle) we ask for a cost-efficient scheduling solution. Our main result is a PTAS for minimizing the total weighted completion time on a machine of varying speed. This implies also a PTAS for the closely related problem of scheduling to minimize generalized global cost functions. The key to our results is a re-interpretation of the problem within the well-known two-dimensional Gantt chart: instead of the standard approach of scheduling in the time-dimension, we construct scheduling solutions in the weight-dimension.

We also consider a dynamic problem variant in which deciding upon the speed is part of the scheduling problem and we are interested in the tradeoff between scheduling cost and speed-scaling cost, which is typically the energy consumption. We obtain two insightful results: (1) the optimal scheduling order is independent of the energy consumption and (2) the problem can be reduced to the setting where the speed of the machine is fixed, and thus admits a PTAS.

Dual techniques for scheduling on a machine with varying speed

Numerous applications in scheduling, such as resource allocation or steel manufacturing, can be modeled using the NP-hard Independent Set problem (given an undirected graph and an integer $$k$$k, find a set of at least $$k$$k pairwise non-adjacent vertices). Here, one encounters special graph classes like 2-union graphs (edge-wise unions of two interval graphs) and strip graphs (edge-wise unions of an interval graph and a cluster graph), on which Independent Set remains $$\mathrm{NP}$$NP-hard but admits constant ratio approximations in polynomial time. We study the parameterized complexity of Independent Set on 2-union graphs and on subclasses like strip graphs. Our investigations significantly benefit from a new structural "compactness" parameter of interval graphs and novel problem formulations using vertex-colored interval graphs. Our main contributions are as follows:1.We show a complexity dichotomy: restricted to graph classes closed under induced subgraphs and disjoint unions, Independent Set is polynomial-time solvable if both input interval graphs are cluster graphs, and is $$\mathrm{NP}$$NP-hard otherwise.2.We chart the possibilities and limits of effective polynomial-time preprocessing (also known as kernelization).3.We extend Halldorsson and Karlsson (2006)'s fixed-parameter algorithm for Independent Set on strip graphs parameterized by the structural parameter "maximum number of live jobs" to show that the problem (also known as Job Interval Selection) is fixed-parameter tractable with respect to the parameter $$k$$k and generalize their algorithm from strip graphs to 2-union graphs. Preliminary experiments with random data indicate that Job Interval Selection with up to 15 jobs and $$5\cdot 10^5$$5·105 intervals can be solved optimally in less than 5 min.

https://hal-lirmm.ccsd.cnrs.fr/lirmm-01349213/document

Interval scheduling and colorful independent sets

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 job's 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 Smith's 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.

/pdf/on-the-performance-of-smith-s-rule-in-single-machine-14un97na1z.pdf

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)s(k p 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.

/pdf/on-the-performance-of-smith-s-rule-in-single-machine-2nimta7mi0.pdf

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

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.

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

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.

Integrated Sequencing and Scheduling in Coil Coating

We consider the problem of scheduling jobs on a single machine. Given a quadratic cost function, we aim to compute a schedule minimizing the weighted total cost, where the cost of each job is defined as the cost function value at the job's completion time. Throughout the past decades, great effort has been made to develop fast exact algorithms for the case of quadratic costs. The efficiency of these methods heavily depends on the utilization of structural properties of optimal schedules such as order constraints, i.e., sufficient conditions for pairs of jobs to appear in a certain order. A considerable number of different kinds of such constraints have been proposed. In this work we enhance the map of known order constraints by proving an extended version of a constraint that has been conjectured by Mondal and Sen more than a decade ago.

Besides proving this conjecture, our main contribution is an extensive experimental study where the inuence of different kinds order constraints on the performance of exact algorithms is systematically evaluated. In addition to a best-first graph search algorithm, we test a Quadratic Integer Programming formulation that admits to add order constraints as additional linear constraints. We also evaluate the optimality gap of well known Smith's rule for different monomial cost functions. Our experiments are based on sets of problem instances that have been generated using a new method which allows us to adjust a certain degree of difficulty of the instances.

Wiebke Höhn

Papers

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

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

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

Integrated Sequencing and Scheduling in Coil Coating

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