Constrained partitioning problems

AbstractWe consider partitioning problems subject to the constraint that the subsets in the partition are independent sets or bases of given matroids. We derive conditions for the functions F and [fnof] such that an optimal partition (S∗1, S∗2,…, S∗k) which minimizes F([fnof](S1),…, [fnof](S k)) has certain order properties. These order properties allow to determine optimal partitions by Greedy-like algorithms. In particular balancing partitioning problems can be solved in this way

Concavity and Efficient Points of Discrete Distributions in Probabilistic Programming

We consider stochastic programming problems with probabilistic constraints involving integer-valued random variables. The concept of a p-efficient point of a probability distribution is used to derive various equivalent problem formulations. Next we introduce the concept of r-concave discrete probability distributions and analyse its relevance for problems under consideration. These notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables. The results are illustrated with numerical examples

Revisiting the Evolution and Application of Assignment Problem: A Brief Overview

The assignment problem (AP) is incredibly challenging that can model many real-life problems. This paper provides a limited review of the recent developments that have appeared in the literature, meaning of assignment problem as well as solving techniques and will provide a review on   a lot of research studies on different types of assignment problem taking place in present day real life situation in order to capture the variations in different types of assignment techniques. Keywords: Assignment problem, Quadratic Assignment, Vehicle Routing, Exact Algorithm, Bound, Heuristic etc

Improving group role assignment problem by incremental assignment algorithm

The Assignment Problem is a basic combinatorial optimization problem. In a weighted bipartite graph, the Assignment Problem is to find a largest sum of weights matching. The Hungarian method is a well-known algorithm which is combinatorial optimization. Adding a new row and a new column to a weighted bipartite graph is called the Incremental Assignment Problem (IAP). The maximum weighted matching (the optimal solution) of the weighted bipartite graph has been given. The algorithm of the Incremental Assignment Problem utilizes the given optimal solution (the maximum weighted matching) and the dual variables to solve the matrix after extended bipartite graph. This thesis proposes an improvement of the Incremental Assignment Algorithm (IAA), named the Improved Incremental Assignment Algorithm. The improved algorithm will save the operation time and operation space to find the optimal solution (the maximum weighted matching) of the bipartite graph. We also present the definition of the Incremental Group Role Assignment Problem that based on the Group Role Assignment Problem (GRAP) and Incremental Assignment Problem (IAP). A solution has been designed to solve it by using the Improved Incremental Assignment Algorithm (IIAA). In this thesis, simulation results are presented. We utilize the tests to compare the algorithm of the Incremental Assignment Problem and the Improved Incremental Assignment Algorithm (IIAA) to show the advantages of IIAA.Master of Science (MSc) in Computational Science

Scheduling Models with Additional Features: Synchronization, Pliability and Resiliency

In this thesis we study three new extensions of scheduling models with both practical and theoretical relevance, namely synchronization, pliability and resiliency. Synchronization has previously been studied for flow shop scheduling and we now apply the concept to open shop models for the first time. Here, as opposed to the traditional models, operations that are processed together all have to be started at the same time. Operations that are completed are not removed from the machines until the longest operation in their group is finished. Pliability is a new approach to model flexibility in flow shops and open shops. In scheduling with pliability, parts of the processing load of the jobs can be re-distributed between the machines in order to achieve better schedules. This is applicable, for example, if the machines represent cross-trained workers. Resiliency is a new measure for the quality of a given solution if the input data are uncertain. A resilient solution remains better than some given bound, even if the original input data are changed. The more we can perturb the input data without the solution losing too much quality, the more resilient the solution is. We also consider the assignment problem, as it is the traditional combinatorial optimization problem underlying many scheduling problems. Particularly, we study a version of the assignment problem with a special cost structure derived from the synchronous open shop model and obtain new structural and complexity results. Furthermore we study resiliency for the assignment problem. The main focus of this thesis is the study of structural properties, algorithm development and complexity. For synchronous open shop we show that for a fixed number of machines the makespan can be minimized in polynomial time. All other traditional scheduling objectives are at least as hard to optimize as in the traditional open shop model. Starting out research in pliability we focus on the most general case of the model as well as two relevant special cases. We deliver a fairly complete complexity study for all three versions of the model. Finally, for resiliency, we investigate two different questions: `how to compute the resiliency of a given solution?' and `how to find a most resilient solution?'. We focus on the assignment problem and single machine scheduling to minimize the total sum of completion times and present a number of positive results for both questions. The main goal is to make a case that the concept deserves further study