Yakov M. Shafransky

Scheduling for Parallel Dedicated Machines with a Single Server

This paper examines scheduling problems in which the setup phase of each operation needs to be attended by a single server, common for all jobs and different from the processing machines. The objective in each situation is to minimize the makespan. For the processing system consisting of two parallel dedicated machines we prove that the problem of finding an optimal schedule isNP-hard in the strong sense even if all setup times are equal or if all processing times are equal. For the case of m parallel dedicated machines, a simple greedy algorithm is shown to create a schedule with the makespan that is at most twice the optimum value. For the two machine case, an improved heuristic guarantees a tight worst-case ratio of 3/2. We also describe several polynomially solvable cases of the later problem. The two-machine flow shop and the open shop problems with a single server are also shown to be NP-hard in the strong sense. However, we reduce the two-machine flow shop no-wait problem with a single server to the Gilmore{Gomory traveling salesman problem and solve it in polynomial time. c 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 304{328, 2000

Batch scheduling with deadlines on parallel machines

The problem of scheduling G groups of jobs on m parallel machines is considered. Each group consists of several identical jobs. We have to find splittings of groups into batches (i.e. sets of jobs to be processed contiguously) and to schedule the batches on the machines. It is possible for different batches of the same group to be processed concurrently on different machines. However, at any time, a batch can be processed on at most one machine. A sequence-independent machine set-up time is required immediately before a batch of a group is processed. A deadline is associated with each group. The objective is to find a schedule which is feasible with respect to deadlines. The problem is shown to be NP-hard even for the case of two identical machines, unit processing times, unit set-up times and a common deadline. It is strongly NP-hard if machines are uniform, the number of jobs in each group is equal and processing times, set-up times and deadlines are unit. Special cases which are polynomially solvable are discussed. For the general problem, a family {DPɛ} of approximation algorithms is constructed. A new dynamic rounding technique is used to develop DPɛ. For any ɛ > 0, DPɛ delivers a schedule in which the completion time of each group is at most (1 + ɛ) times the value of its deadline if there exists a schedule which is feasible with respect to the deadlines. The time complexity of DPɛ is O(G2m+1/ɛ2m).

An optimization algorithm for the clearing of interbank payments

We consider the clearing of interbank payments under limited amount of money used by banks for the maintenance of the clearing process. We show that the corresponding discrete optimization problem is non-approximable and propose a fast heuristic for solving the problem. The results of computational experiments demonstrate the high quality of solutions.

Sequencing with Ordered Criteria, Precedence and Group Technology Constraints

Multicriteria sequencing problems with criteria ordered according to their importance are considered. Additional precedence and group technology constraints are imposed. We intro- duce a notion of a priority-generating vector function and suggest general techniques that form a base for the construction of polynomial time algorithms for numerous sequencing problems includ- ing all known polynomially solvable problems. A comprehensive survey of results for sequencing problems with ordered criteria is given as well.

Uniform machine scheduling of unit-time jobs subject to resource constraints

The problem of scheduling a set of unit-time jobs on m uniform machines is studied. Some jobs may require a unit of an additional single resource during their execution. The resource is renewable but the total resource consumption is limited by the same value at each time instant. The objective is to find a feasible schedule minimizing the maximum job completion time. We show that an approach suggested in the literature to solve this problem is incorrect. Then we present an O(m log m) algorithm for the problem with no machine idle times and a linear-time algorithm for the problem with identical machines.

Minimizing maximum weight of subsets of a maximum matching in a bipartite graph

In a complete bipartite graph G = ( U , V , E ) with weighted edges, set U of vertices is partitioned into disjoint subsets called components. The aim is to find an inclusion-wise maximal matching that minimizes the maximum weight of a component (sum of the weights of those edges incident to a vertex of the component). The well-known assignment problem is a special case of this problem when the number of components equals one. We prove that the problem is strongly NP-hard and that finding an approximate solution with the approximation ratio less than 2 is strongly NP-hard as well. We develop heuristics to find solutions that are less than one percent away from the optimum on average. The problem has been encountered while planning container transportation in rail-road transshipment yards.

Flow Shop Scheduling Problems Under Machine–Dependent Precedence Constraints

The paper considers the flow shop scheduling problems to minimize the makespan, provided that an individual precedence relation is specified on each machine. A fairly complete complexity classification of problems with two and three machines is obtained.

Batch scheduling with deadlines on parallel machines: an NP-hard case

The problem of scheduling groups of jobs on unrelated parallel machines in batches subject to group deadlines was studied by Brucker et al. (1997) and Kovalyov and Shafransky (1994). A classification of computational complexities of special cases was provided only for the situation when all groups have equal deadlines. We prove that the problem is strongly NP-hard for the case of different deadlines, identical machines, unit processing times and unit set-up times, and equal numbers of jobs in each group. The problem where machine deadlines are given instead of job deadlines is proved to be strongly NP-complete.

Lawler's minmax cost algorithm: optimality conditions and uncertainty

The well-known