3 research outputs found
Planar 3-dimensional assignment problems with Monge-like cost arrays
Given an cost array we consider the problem -P3AP
which consists in finding pairwise disjoint permutations
of such that
is minimized. For the case
the planar 3-dimensional assignment problem P3AP results.
Our main result concerns the -P3AP on cost arrays that are layered
Monge arrays. In a layered Monge array all matrices that result
from fixing the third index are Monge matrices. We prove that the -P3AP
and the P3AP remain NP-hard for layered Monge arrays. Furthermore, we show that
in the layered Monge case there always exists an optimal solution of the
-3PAP which can be represented as matrix with bandwidth . This
structural result allows us to provide a dynamic programming algorithm that
solves the -P3AP in polynomial time on layered Monge arrays when is
fixed.Comment: 16 pages, appendix will follow in v
Open Shop Scheduling with Synchronization
In this paper, we study open shop scheduling problems with synchronization. This model has the same features as the classical open shop model, where each of the n jobs has to be processed by each of the m machines in an arbitrary order. Unlike the classical model, jobs are processed in synchronous cycles, which means that the m operations of the same cycle start at the same time. Within one cycle, machines which process operations with smaller processing times have to wait until the longest operation of the cycle is finished before the next cycle can start. Thus, the length of a cycle is equal to the maximum processing time of its operations. In this paper, we continue the line of research started by Weiß et al. (Discrete Appl Math 211:183–203, 2016). We establish new structural results for the two-machine problem with the makespan objective and use them to formulate an easier solution algorithm. Other versions of the problem, with the total completion time objective and those which involve due dates or deadlines, turn out to be NP-hard in the strong sense, even for m=2 machines. We also show that relaxed models, in which cycles are allowed to contain less than m jobs, have the same complexity status
Heuristic Solution Approaches to the Solid Assignment Problem
The 3-dimensional assignment problem, also known as the Solid Assignment Problem (SAP), is a challenging problem in combinatorial optimisation. While the ordinary or 2-dimensional assignment problem is in the P-class, SAP which is an extension of it, is NP-hard. SAP is the problem of allocating n jobs to n machines in n factories such that exactly one job is allocated to one machine in one factory. The objective is to minimise the total cost of getting these n jobs done. The problem is commonly solved using exact methods of integer programming such as Branch-and-Bound B&B. As it is intractable, only approximate solutions are found in reasonable time for large instances. Here, we suggest a number of approximate solution approaches, one of them the Diagonals Method (DM), relies on the Kuhn-Tucker Munkres algorithm, also known as the Hungarian Assignment Method. The approach was discussed, hybridised, presented and compared with other heuristic approaches such as the Average Method, the Addition Method, the Multiplication Method and the Genetic Algorithm. Moreover, a special case of SAP which involves Monge-type matrices is also considered. We have shown that in this case DM finds the exact solution efficiently.
We sought to provide illustrations of the models and approaches presented whenever appropriate. Extensive experimental results are included and discussed. The thesis ends with a conclusions and some suggestions for further work on the same and related topics