Generating the best K sequences in relocation problems

The blocks relocation problem (BRP) may be defined as follows: given a set of homogeneous blocks stored in a two-dimensional stock, which relocations are necessary to retrieve the blocks from the stock in a pre-defined order while minimizing the number of those relocations? In this paper, we first prove NP-hardness of the BRP as well as a special case, closing open research questions. Moreover, we propose different solution approaches. First, a mathematical model is presented that provides optimal solutions to the general BRP in cases where instances are small. To overcome such limitation, some realistic assumption taken from the literature is introduced, leading to the definition of a binary linear programming model. In terms of computational time, this approach is reasonably fast to be used to solve medium-sized instances. In addition, we propose a simple heuristic based upon a set of relocation rules. This heuristic is used to generate ''good'' quality solutions for larger instances in very short computational time, and, consequently, is proposed for tackling problem instances where solutions are required (almost) immediately. Solution quality of the heuristic is measured against optimal solutions obtained using a state-of-the-art commercial solver and both of them are compared with reference results from literature.

Lecture Notes in Computer Science, 2014

In modern data centers and cloud computing systems, jobs often require resources distributed across nodes providing a wide variety of services. Motivated by this, we study the Coupled Placement problem, in which we place jobs into computation and storage nodes with capacity constraints, so as to optimize some costs or profits associated with the placement. The coupled placement problem is a natural generalization of the widely-studied generalized assignment problem (GAP), which concerns the placement of jobs into single nodes providing one kind of service. We also study a further generalization, the k-Sided Placement problem, in which we place jobs into k-tuples of nodes, each node in a tuple offering one of k services. For both the coupled and k-sided placement problems, we consider minimization and maximization versions. In the minimization versions (MinCP and MinkSP), the goal is to achieve minimum placement cost, while incurring a minimum blowup in the capacity of the individual nodes. Our first main result is an algorithm for MinkSP that achieves optimal cost while increasing capacities by at most a factor of k + 1, also yielding the first constantfactor approximation for MinCP. In the maximization versions (MaxCP and MaxkSP), the goal is to maximize the total weight of the jobs that are placed under hard capacity constraints. MaxkSP can be expressed as a k-column sparse integer program, and can be approximated to within a factor of O(k)

Computers & Operations Research, 1996

Scope and Purpose-One of the key problems in manufacturing operations is to determine the assignment of jobs to machines and the sequence of the jobs on each machine. In this paper we consider an environment in which the machines are identical, changeover times in between the processing of two consecutive jobs are jobdependent and the objective is to maximize the utilization of the machines which is equivalent to minimizing the makespan. Unfortunately, such problems fall into the strongly NP-hard category. However, it is the purpose of this paper to show that the computational complexity drastically improves if the jobs are divided into a small number of classes where each class consists of "similar" jobs. Such situations can be found in many applications where the jobs in a group are identical, such as for example in the Aircraft Sequencing Problem (where an optimal sequence has to be determined for the landing of airplanes which are divided into several classes according to their size) or if the jobs in a group require a similar state of the machine, like e.g. color for a painting machine or tool-loading in flexible manufacturing systems. In such cases the changeover times are actually determined by the class of the jobs. For a fixed number of job classes, we give fast algorithms for the onemachine problem and the problem with multiple machines and identical processing times.

Acta Informatica, 2012

We study the problem of scheduling unit execution time (UET) jobs with release dates and precedence constraints on two identical processors. We say that a schedule is ideal if it minimizes both maximum and total completion time simultaneously. We give an instance of the problem where the min-max completion time is exceeded in every preemptive schedule that minimizes total completion time for that instance, even if the precedence constraints form an intree. This proves that ideal schedules do not exist in general when preemptions are allowed. On the other hand, we prove that, when preemptions are not allowed, then ideal schedules do exist for general precedence constraints, and we provide an algorithm for finding ideal schedules in O(n 3 ) time, where n is the number of jobs. In finding such ideal schedules we resolve a conjecture of Baptiste and Timkovsky [1]. Further, our algorithm for finding min-max completion-time schedules requires only O(n 3 ) time, while the most efficient solution to date has required O(n 9 ) time.

This is a printed edition of a special issue on Algorithms for Scheduling Problems published in the journal Algorithms. The Ebook is freely downloadable from the publisher's website given in the link.

Kirkuk University Journal-Scientific Studies, 2010

The problem of scheduling n jobs on a single machine is considered, where the jobs are divided into two classes and a machine set up is necessary between jobs of different classes. Jobs i (i= 1,…, n) becomes available for processing at time zero, requires a positive processing time i P. Disjoint subsets N 1 and N 2 define the partition of jobs into two classes. If two jobs in the same class are sequenced in adjacent positions, then no set up time between these jobs in necessary. We address the bicriterion (multi objective) scheduling problem, the two criteria are the minimization of flow time ( N i i c) and the minimization maximum Tardiness (max T). We characterized the set of all efficient points and the optimal solution. A modified algorithm presented to find efficient solutions for the problem with set up times. A relation found between number of efficient solutions and range of 'tardiness of shortest processing time (SPT T), tardiness of early due date (EDD T)'. This algorithm treats with a case that the set up time in  SPT rule is in increasing order. A counter example presented to show that the algorithm will fail if the set up time in  SPT rule is in decreasing order. Our task is to present the decision makers with all possible solutions and let them make the final selection. The decision maker has two objectives in mind ( N i i c) , (max T) and some solutions (efficient), we will choose the best one from the efficient solutions depending on his experiences.

Management Science

The search for a solution to the problem of finding an optimal or near optimal se- quence of jobs being scheduled in a flow shop type situation has given consideration to both exact and approximate techniques of solution. Exact techniques, which usually require an electronic computer, ...

Operations Research Letters, 2006

We study the problem of minimizing the vector norm ||·|| p of the workloads. We examine move-optimal assignments and prove a performance guarantee of

This special issue of Algorithms is a follow-up issue of an earlier one, entitled 'Algorithms for Scheduling Problems'. In particular, the new issue is devoted to the development of exact and heuristic scheduling algorithms. Submissions were welcome both for traditional scheduling problems as well as for new practical applications. In the Call for Papers, we mentioned topics such as single-criterion and multi-criteria scheduling problems with additional constraints including setup times (costs), precedence constraints, batching (lot sizing), resource constraints as well as scheduling problems arising in emerging applications.

ArXiv, 2012

In this paper, we consider the problem of choosing a minimum cost set of resources for executing a specified set of jobs. Each input job is an interval, determined by its start-time and end-time. Each resource is also an interval determined by its start-time and end-time; moreover, every resource has a capacity and a cost associated with it. We consider two versions of this problem. In the partial covering version, we are also given as input a number k, specifying the number of jobs that must be performed. The goal is to choose k jobs and find a minimum cost set of resources to perform the chosen k jobs (at any point of time the capacity of the chosen set of resources should be sufficient to execute the jobs active at that time). We present an O(log n)-factor approximation algorithm for this problem. We also consider the prize collecting version, wherein every job also has a penalty associated with it. The feasible solution consists of a subset of the jobs, and a set of resources, t...