Maximization Problems in Single Machine Scheduling

Problems of scheduling n jobs on a single machine to maximize regular objective functions are studied. Precedence constraints may be given on the set of jobs and the jobs may have different release times. Schedules of interest are only those for which the jobs cannot be shifted to start earlier without changing job sequence or violating release times or precedence constraints. Solutions to the maximization problems provide an information about how poorly such schedules can perform. The most general problem of maximizing maximum cost is shown to be reducible to n similar problems of scheduling n?1 jobs available at the same time. It is solved in O(mn+n ²) time, where m is the number of arcs in the precedence graph. When all release times are equal to zero, the problem of maximizing the total weighted completion time or the weighted number of late jobs is equivalent to its minimization counterpart with precedence constraints reversed with respect to the original ones. If there are no precedence constraints, the problem of maximizing arbitrary regular function reduces to n similar problems of scheduling n?1 jobs available at the same time.     

Optimization-based heuristics for underground mine scheduling

Underground mine production scheduling possesses mathematical structure similar to and yields many of the same challenges as general scheduling problems. That is, binary variables represent the time at which various activities are scheduled. Typical objectives seek to minimize costs or some measure of production time, or to maximize net present value; two principal types of constraints exist: (i) resource constraints and (ii) precedence constraints. In our setting, we maximize “discounted metal production” for the remaining life of an underground lead and zinc mine that uses three different underground methods to extract the ore. Resource constraints limit the grade, tonnage, and backfill paste (used for structural stability) in each time period, while precedence constraints enforce the sequence in which extraction (and backfill) is performed in accordance with the underground mining methods used. We tailor exact and heuristic approaches to reduce model size, and develop an optimization-based decomposition heuristic; both of these methods transform a computationally intractable problem to one for which we obtain solutions in seconds, or, at most, hours for problem instances based on data sets from the Lisheen mine near Thurles, Ireland.     

Time complexity of single- and identical parallel-machine scheduling with GERT network precedence constraints

In this paper we deal with the time complexity of single- and identical parallel-machine scheduling problems in which the durations and precedence constraints of the activities are stochastic. The stochastic precedence constraints are given by GERT networks. First, we sketch the basic concepts of GERT networks and machine scheduling with GERT network precedence constraints. Second, we discuss the time complexity of some open single-machine scheduling problems with GERT network precedence constraints. Third, we investigate the time complexity of identical parallel-machine scheduling problems with GERT network precedence constraints. Finally, we present an efficient reduction algorithm for the problem of computing the expected makespan for the latter type of scheduling problem.     

An exact algorithm for the precedence-constrained single-machine scheduling problem

This study proposes an efficient exact algorithm for the precedence-constrained single-machine scheduling problem to minimize total job completion cost where machine idle time is forbidden. The proposed algorithm is based on the SSDP (Successive Sublimation Dynamic Programming) method and is an extension of the authors’ previous algorithms for the problem without precedence constraints. In this method, a lower bound is computed by solving a Lagrangian relaxation of the original problem via dynamic programming and then it is improved successively by adding constraints to the relaxation until the gap between the lower and upper bounds vanishes. Numerical experiments will show that the algorithm can solve all instances with up to 50 jobs of the precedence-constrained total weighted tardiness and total weighted earliness–tardiness problems, and most instances with 100 jobs of the former problem.     

Preemptive Hybrid Flowshop Scheduling problem of interval orders

The Preemptive Hybrid (multi-processor) Flowshop Scheduling (PHFS) problem consists in preemptively scheduling n jobs in a flowshop subject to precedence constraints with the objective of minimizing the makespan. A special case of the general precedence constraints problems is NP-hard in the strong sense, Hoogeveen et al. [J.A. Hoogeveen, J.K. Lenstra, B. Veltman, European Journal of Operational Research 89 (1996) 172]. In this paper a class of precedence constraints is proposed for which the problem is polynomially solvable. The reported results demonstrate the feasibility and reliability of the proposed approach. This should open future prospects for developing approximation algorithms for any class of precedence constraints.     

Lower Bounds for Scheduling a Single Robot in a Job-Shop Environment

We consider a single-machine scheduling problem which arises as a subproblem in a job-shop environment where the jobs have to be transported between the machines by a single transport robot. The robot scheduling problem may be regarded as a generalization of the traveling salesman problem with time windows, where additionally generalized precedence constraints (minimal time-lags) have to be respected. The objective is to determine a sequence of all nodes and corresponding starting times in the given time windows in such a way that all generalized precedence relations are respected and the sum of all traveling and waiting times is minimized.We calculate lower bounds for this problem using constraint propagation techniques and a linear programming formulation which is solved by a column generation procedure. Computational results are presented for test data arising from job-shop instances with a single transport robot and some modified traveling salesman instances.     

Multi-machine scheduling lower bounds using decision diagrams

We consider parallel multi-machine scheduling with due times, where a partition of jobs is given where jobs in the same partition have a common release time, possibly precedence constraints, and cannot overlap. A formulation of decision diagrams for this problem greatly improves upon a more natural extension of the state-of-the-art for single-machine scheduling, and can provide decent lower bounds, outperforming existing solvers given the same short runtime limit, for problem instances with large time scales and tight due times.     

Two machine preemptive scheduling problem with release dates,equal processing times and precedence constraints

We consider a scheduling problem with two identical parallel machines and n jobs. For each job we are given its release date when job becomes available for processing. All jobs have equal processing times. Preemptions are allowed. There are precedence constraints between jobs which are given by a (di)graph consisting of a set of outtrees and a number of isolated vertices. The objective is to find a schedule minimizing mean flow time. We suggest an O(n²) algorithm to solve this problem.The suggested algorithm also can be used to solve the related two-machine open shop problem with integer release dates, unit processing times and analogous precedence constraints.     

A sliding time window heuristic for open pit mine block sequencing

The open pit mine block sequencing problem (OPBS) seeks a discretetime production schedule that maximizes the net present value of the orebody extracted from an open-pit mine. This integer program (IP) discretizes the mine’s volume into blocks, imposes precedence constraints between blocks, and limits resource consumption in each time period. We develop a “sliding time window heuristic” to solve this IP approximately. The heuristic recursively defines, solves and partially fixes an approximating model having: (i) fixed variables in early time periods, (ii) an exact submodel defined over a “window” of middle time periods, and (iii) a relaxed submodel in later time periods. The heuristic produces near-optimal solutions (typically within 2% of optimality) for model instances that standard optimization software fails to solve. Furthermore, it produces these solutions quickly, even though our OPBS model enforces standard upper-bounding constraints on resource consumption along with less standard, but important, lower-bounding constraints.     

A mathematical programming viewpoint for solving the ultimate pit problem

In 1965 Helmut Lerchs and Ingo Grossmann presented to the mining community an algorithm to find the optimum design for an open pit mine. In their words, “the objective is to design the contour of a pit so as to maximize the difference between total mine value of the ore extracted and the total extraction cost of ore and waste”. They modeled the problem in graph theoretic terms and showed that an optimal solution of the ultimate pit problem is equivalent to finding the maximum closure of their graph based model. In this paper, we develop a network flow algorithm based on the dual to solve the same problem. We show how this algorithm is closely related to Lerchs and Grossmann's and how the steps in their algorithm can be viewed in mathematical programming terms. This analysis adds insight to the algorithm of Lerchs and Grossmann and shows where it can be made more efficient. As in the case Lerchs and Grossmann, our algorithm allows us to use very efficient data structures based on graphs that represent the data and constraints.. 1782 1528 V 3     

Scheduling precedence graphs of bounded height

The existence of a schedule for a partially ordered set of unit length tasks on m identical processors is known to be NP-complete (J. D. Ullman, NP-complete scheduling problems, J. Comput. System Sci., 10 (1975), 384–393). The problem remains NP-complete even if we restrict the precedence graph to be of height bounded by a constant. (J. K. Lenkstra and A. H. G. Rinnooy Kan, Complexity of scheduling under precedence constraints, Operations Res., 26 (1978), 22–35; D. Dolev and M. K. Warmuth, “Scheduling Flat Graphs,” IBM Research Report RJ 3398, 1982). In these NP-completeness proofs the upper bound on the number of available processors varies with the problem instance. We present a polynomial algorithm for the case where the upper bound on the number of available processors and the height of the precedence graph are both constants.     

On the single machine serial batching scheduling problem to minimize total completion time with precedence constraints, release dates and identical processing times

We consider the single machine, serial batching, total completion time scheduling problem with precedence constraints, release dates and identical processing times in this paper. The complexity of this problem is reported as open in the literature. We provide an O(n⁵) time algorithm to solve this problem.     

Complexity of scheduling of coupled tasks with chains precedence constraints and any constant length of gap

Coupled tasks scheduling was originally introduced for modelling complex radar devices. It is still used for controlling such devices and applied in similar applications. This paper considers a problem of coupled tasks scheduling on one processor, under the assumptions that all processing times are equal to 1, the gap has a constant exact length and the precedence constraints are strict. Although it is proven that the problem stated above is NP-hard in the strong sense if the precedence constraints have a form of a general graph, it is possible to solve some of its relaxed versions in polynomial time. This paper contains a solution for the problem of coupled tasks scheduling with an assumption that the precedence constraints graph has a form of chains and it presents an algorithm that can solve the problem with such assumption in time O(n?log?n).     

Implementing a parametric maximum flow algorithm for optimal open pit mine design under uncertain supply and demand

Conventional open pit mine optimization models for designing mining phases and ultimate pit limit do not consider expected variations and uncertainty in metal content available in a mineral deposit (supply) and commodity prices (market demand). Unlike the conventional approach, a stochastic framework relies on multiple realizations of the input data so as to account for uncertainty in metal content and financial parameters, reflecting potential supply and demand. This paper presents a new method that jointly considers uncertainty in metal content and commodity prices, and incorporates time-dependent discounted values of mining blocks when designing optimal production phases and ultimate pit limit, while honouring production capacity constraints. The structure of a graph representing the stochastic framework is proposed, and it is solved with a parametric maximum flow algorithm. Lagragnian relaxation and the subgradient method are integrated in the proposed approach to facilitate producing practical designs. An application at a copper deposit in Canada demonstrates the practical aspects of the approach and quality of solutions over conventional methods, as well as the effectiveness of the proposed stochastic approach in solving mine planning and design problems.     

Reduction of job-shop problems to flow-shop problems with precedence constraints

We study the problem of scheduling N independent jobs in a job-shop environment. Each job must be processed on M machines according to individual routes. The objective is to minimize the maximum completion time of the jobs. First, the job-shop problem is reduced to a flow-shop problem with job precedence constraints. Then, a set of flow-shop algorithms are modified to solve it. To evaluate the quality of these heuristics, several lower bounds on the optimal solution have been computed and compared with the heuristic solutions for 3040 problems. The heuristics appear especially promising for job-shop problems with ‘flow-like’ properties.     

Minimizing the completion time of a project under resource constraints and feeding precedence relations: a Lagrangian relaxation based lower bound

In this paper we study the Resource Constrained Project Scheduling Problem (RCPSP) with “Feeding Precedence” (FP) constraints and minimum makespan objective. This problem typically arises in production planning environment, like make-to-order manufacturing, where the effort associated with the execution of an activity is not univocally related to its duration percentage and the traditional finish-to-start precedence constraints or the generalized precedence relations cannot completely represent the overlapping among activities. In this context, we need to introduce in the RCPSP the FP constraints. For this problem we propose a new mathematical formulation and define a lower bound based on the Lagrangian relaxation of the resource constraints. A computational experimentation on randomly generated instances of sizes of up to 100 activities shows a better performance of this lower bound when compared to other lower bounds. Moreover, for the optimally solved instances, its value is very close to the optimal one. Furthermore, in order to show the effectiveness of the proposed lower bound on large instances for which the optimal solution is known, we adapted our approach to solve the benchmarks of the basic RCPSP from the PSLIB with 120 activities.     

Three stage generalized flowshop: Scheduling civil engineering projects

We model the working of a civil engineering firm concerned with land development as a three stage flexible flowshop with weak chain precedence constraints and where preemption is allowed. The scheduling objective is to minimize the total tardiness for all the projects. Since solving this problem optimally is very hard, we propose a number of heuristic scheduling procedures which are evaluated extensively on real-life data and artificial problem instances.     

Surrogate duality relaxation for job shop scheduling

Surrogate duality bounds for the job shop scheduling problem are obtained by replacing certain constraints by their weighted sum and strengthening the aggregate constraint by iterating over all possible weights. The constraints successively considered for this purpose are the capacity constraints on the machines and the precedence constraints determining the machine order for each job. The resulting relaxations are investigated from a theoretical and a computational point of view.     

A genetic approach to solving the problem of cyclic job shop scheduling with linear constraints

This study concerns the domain of cyclic scheduling. More precisely we consider the cyclic job shop scheduling problem with linear constraints. The main characteristic of this problem is that the tasks of each job are cyclic and are subjected to linear precedence constraints. First we review some approaches in the field of cyclic scheduling and present the cyclic job shop scheduling problem definition, which has an open complexity. Then we present a general approach for solving it, based on the coupling of a genetic algorithm and a scheduler. This scheduler utilises a Petri-net modelling the linear precedence constraints between cyclic tasks. The goal of this genetic algorithm is to propose an order of priority for jobs on the machines, to be used by the scheduler for solving resource conflicts. Finally a benchmark and some preliminary results of this approach are presented.     

S3 sets. An extension of the Beale-Tomlin special ordered sets

In this work an extension of the Beale-Tomlin special ordered sets is introduced that has proved to be efficient for solving certain types of open shop scheduling problems. Besides their usual characteristics, exclusivity constraints in the jobs are allowed, more general than tree-like precedence structures are considered, and semi-active schedules that cannot be labeled as non-optimal solutions may occur. The problem is formulated as a large-scale 0–1 model. Computational experience on some real-life problems is reported.