Approximation Algorithms for a Symmetric Quadratic Knapsack Problem and its Scheduling Applications

We consider a knapsack problem to minimize a symmetric quadratic function. We demonstrate that this symmetric quadratic knapsack problem is relevant to two problems of single machine scheduling: the problem of minimizing the weighted sum of the completion times with a single machine non-availability interval under the non-resumable scenario; and the problem of minimizing the total weighted earliness and tardiness with respect to a common small due date. We develop a polynomial-time approximation algorithm that delivers a constant worst-case performance ratio for a special form of the symmetric quadratic knapsack problem. We adapt that algorithm to our scheduling problems and achieve a better performance. For the problems under consideration no fixed-ratio approximation algorithms have been previously known.

Nested heuristic methods for the location-routeing problem

The concept of 'nested methods' is adopted to solve the location-routeing problem. Unlike the sequential and iterative approaches, in this method we treat the routeing element as a sub-problem within the larger problem of location. Efficient techniques that take into account the above concept and which use a neighbourhood structure inspired from computational geometry are presented. A simple version of tabu search is also embedded into our methods to improve the solutions further. Computational testing is carried out on five sets of problems of 400 customers with five levels of depot fixed costs, and the results obtained are encouraging.

The open shop scheduling problem with a given sequence of jobs on one machine

The paper considers the open shop scheduling problem to minimize the make-span, provided that one of the machines has to process the jobs according to a given sequence. We show that in the preemptive case the problem is polynomially solvable for an arbitrary number of machines. If preemption is not allowed, the problem is NP-hard in the strong sense if the number of machines is variable, and is NP-hard in the ordinary sense in the case of two machines. For the latter case we give a heuristic algorithm that runs in linear time and produces a schedule with the makespan that is at most 5/4 times the optimal value. We also show that the two-machine problem in the nonpreemptive case is solvable in pseudopolynomial time by a dynamic programming algorithm, and that the algorithm can be converted into a fully polynomial approximation scheme. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 705–731, 1998

Earliness penalties on a single machine subject to precedence constraints: SLK due date assignment

The paper considers the single machine due date assignment and scheduling problems with n jobs in which the due dates are to be obtained from the processing times by adding a positive slack q. A schedule is feasible if there are no tardy jobs and the job sequence respects given precedence constraints. The value of q is chosen so as to minimize a function ϕ(F,q) which is non-decreasing in each of its arguments, where F is a certain non-decreasing earliness penalty function. Once q is chosen or fixed, the corresponding scheduling problem is to find a feasible schedule with the minimum value of function F. In the case of arbitrary precedence constraints the problems under consideration are shown to be NP-hard in the strong sense even for F being total earliness. If the precedence constraints are defined by a series-parallel graph, both scheduling and due date assignment problems are proved solvable in time, provided that F is either the sum of linear functions or the sum of exponential functions. The running time of the algorithms can be reduced to if the jobs are independent. Scope and purpose We consider the single machine due date assignment and scheduling problems and design fast algorithms for their solution under a wide range of assumptions. The problems under consideration arise in production planning when the management is faced with a problem of setting the realistic due dates for a number of orders. The due dates of the orders are determined by increasing the time needed for their fulfillment by a common positive slack. If the slack is set to be large enough, the due dates can be easily maintained, thereby producing a good image of the firm. This, however, may result in the substantial holding cost of the finished products before they are brought to the customer. The objective is to explore the trade-off between the size of the slack and the arising holding costs for the early orders.

The many-to-many location-routing problem

In this paper the many to many location routing problem is introduced, and its relationship to various problems in distribution management is emphasised. Useful mathematical formulations which can be easily extended to cater for other related problems are produced. Techniques for tackling this complex distribution problem are also outlined.

Analysis of species hyper-diversity in the tropical rain forests of Indonesia: the problem of non-observance

The main interest in the assessment of forest species diversity for conservation purposes is in the rare species. The main problem in the tropical rain forests is that most of the species are rare. Assessment of species diversity in the tropical rain forests is therefore often concerned with estimating that which is not observed in recorded samples. Statistical methodology is therefore required to try to estimate the truncated tail of the species frequency distribution, or to estimate the asymptote of species/diversity-area curves. A Horvitz-Thompson estimator of the number of unobserved (“virtual”) species in each species intensity class is proposed. The approach allows a definition of an extended definition of diversity, ( or generalised Renyi entropy). The paper presents a case study from data collected in Jambi, Sumatra, and the “extended diversity measure” is used on the species data.

The centrality of groups and classes

This paper extends the standard network centrality measures of degree, closeness and betweenness to apply to groups and classes as well as individuals. The group centrality measures will enable researchers to answer such questions as ‘how central is the engineering department in the informal influence network of this company?’ or ‘among middle managers in a given organization, which are more central, the men or the women?’ With these measures we can also solve the inverse problem: given the network of ties among organization members, how can we form a team that is maximally central? The measures are illustrated using two classic network data sets. We also formalize a measure of group centrality efficiency, which indicates the extent to which a group's centrality is principally due to a small subset of its members.

Heuristics for the two-stage job shop scheduling problem with a bottleneck machine

The paper considers the job shop scheduling problem to minimize the makespan. It is assumed that each job consists of at most two operations, one of which is to be processed on one of m⩾2 machines, while the other operation must be performed on a single bottleneck machine, the same for all jobs. For this strongly NP-hard problem we present two heuristics with improved worst-case performance. One of them guarantees a worst-case performance ratio of 3/2. The other algorithm creates a schedule with the makespan that exceeds the largest machine workload by at most the length of the largest operation.

Group technology approach to the open shop scheduling problem with batch setup times

This paper studies the problem of scheduling jobs in a two-machine open shop to minimize the makespan. Jobs are grouped into batches and are processed without preemption. A batch setup time on each machine is required before the first job is processed, and when a machine switches from processing a job in some batch to a job of another batch. For this NP-hard problem, we propose a linear-time heuristic algorithm that creates a group technology schedule, in which no batch is split into sub-batches. We demonstrate that our heuristic is a -approximation algorithm. Moreover, we show that no group technology algorithm can guarantee a worst-case performance ratio less than 5/4.

An open shop scheduling problem with a non-bottleneck machine

This paper considers the problem of processing n jobs in a two-machine non-preemptive open shop to minimize the makespan, i.e., the maximum completion time. One of the machines is assumed to be non-bottleneck. It is shown that, unlike its flow shop counterpart, the problem is NP-hard in the ordinary sense. On the other hand, the problem is shown to be solvable by a dynamic programming algorithm that requires pseudopolynomial time. The latter algorithm can be converted into a fully polynomial approximation scheme that runs in time. An O(n log n) approximation algorithm is also designed whi finds a schedule with makespan at most 5/4 times the optimal value, and this bound is tight.

A heuristic for the two-machine open-shop scheduling problem with transportation times

The paper considers a problem of scheduling n jobs in a two-machine open shop to minimize the makespan, provided that preemption is not allowed and the interstage transportation times are involved. This problem is known to be unary NP-hard. We present an algorithm that requires O (n log n) time and provides a worst-case performance ratio of 3/2.

Proportionate flow shop with controllable processing times

This paper considers a special class of flow-shop problems, known as the proportionate flow shop. In such a shop, each job flows through the machines in the same order and has equal processing times on the machines. The processing times of different jobs may be different. It is assumed that all operations of a job may be compressed by the same amount which will incur an additional cost. The objective is to minimize the makespan of the schedule together with a compression cost function which is non-decreasing with respect to the amount of compression. For a bicriterion problem of minimizing the makespan and a linear cost function, an O(n log n) algorithm is developed to construct the Pareto optimal set. For a single criterion problem, an O(n2) algorithm is developed to minimize the sum of the makespan and compression cost. Copyright © 1999 John Wiley & Sons, Ltd.

Scheduling batches with sequential job processing for two-machine flow and open shops

In this paper, we study a problem of scheduling and batching on two machines in a flow-shop and open-shop environment. Each machine processes operations in batches, and the processing time of a batch is the sum of the processing times of the operations in that batch. A setup time, which depends only on the machine, is required before a batch is processed on a machine, and all jobs in a batch remain at the machine until the entire batch is processed. The aim is to make batching and sequencing decisions, which specify a partition of the jobs into batches on each machine, and a processing order of the batches on each machine, respectively, so that the makespan is minimized. The flow-shop problem is shown to be strongly NP-hard. We demonstrate that there is an optimal solution with the same batches on the two machines; we refer to these as consistent batches. A heuristic is developed that selects the best schedule among several with one, two, or three consistent batches, and is shown to have a worst-case performance ratio of 4/3. For the open-shop, we show that the problem is NP-hard in the ordinary sense. By proving the existence of an optimal solution with one, two or three consistent batches, a close relationship is established with the problem of scheduling two or three identical parallel machines to minimize the makespan. This allows a pseudo-polynomial algorithm to be derived, and various heuristic methods to be suggested.