An algorithm for the optimal solution of variable knockout problems

In this paper we consider a class of problems related to variable knockout. Given an optimisation problem formulated as an integer program the question we face in problems of this type is what might be an appropriate set of variables to delete, i.e. knockout of the problem, in order that the optimal solution to the problem that remains after variable knockout has a desired property. We present an algorithm for the optimal solution of the problem. We indicate how our algorithm can be adapted when the number of variables knocked out is specified (i.e. when we have a cardinality constraint). Computational results are given for the problem of finding the minimal number of arcs to knockout from a directed network such that, after knockout, the shortest path from an origin node to a destination node is of length at least a specified value. We also present results for shortest path cardinality constrained knockout

Set-theoretic duality: A fundamental feature of combinatorial optimisation

The duality between conflicts and diagnoses in the field of diagnosis, or between plans and landmarks in the field of planning, or between unsatisfiable cores and minimal co-satisfiable sets in SAT or CSP solving, has been known for many years. Recent wo

Chemical reaction optimization for the set covering problem

The set covering problem (SCP) is one of the representative combinatorial optimization problems, having many practical applications. This paper investigates the development of an algorithm to solve SCP by employing chemical reaction optimization (CRO), a general-purpose metaheuristic. It is tested on a wide range of benchmark instances of SCP. The simulation results indicate that this algorithm gives outstanding performance compared with other heuristics and metaheuristics in solving SCP. © 2014 IEEE.postprin

Multi-objective Database Queries in Combined Knapsack and Set Covering Problem Domains

Database queries are one of the most important functions of a relational database. Users are interested in viewing a variety of data representations, and this may vary based on database purpose and the nature of the stored data. The Air Force Institute of Technology has approximately 100 data logs which will be converted to the standardized Scorpion Data Model format. A relational database is designed to house this data and its associated sensor and non-sensor metadata. Deterministic polynomial-time queries were used to test the performance of this schema against two other schemas, with databases of 100 and 1000 logs of repeated data and randomized metadata. Of these approaches, the one that had the best performance was chosen as AFIT’s database solution, and now more complex and useful queries need to be developed to enable filter research. To this end, consider the combined Multi-Objective Knapsack/Set Covering Database Query. Algorithms which address The Set Covering Problem or Knapsack Problem could be used individually to achieve useful results, but together they could offer additional power to a potential user. This paper explores the NP-Hard problem domain of the Multi-Objective KP/SCP, proposes Genetic and Hill Climber algorithms, implements these algorithms using Java, populates their data structures using SQL queries from two test databases, and finally compares how these algorithms perform

Skills management heuristics.

A common problem faced by most organizations in today\u27s world is one of worker-task assignments. Assigning a large number of complex tasks to workers at various training levels can be a complicated process which has the potential to cost or to save a company large sums of money. The aim of this project is to develop a heuristic tool designed to match tasks to workers given the workers skills proficiency profiles. This heuristic should also provide a training plan which will rectify current worker skills gaps while minimizing training costs. Prior research maintained a focus on utilizing mathematical models of this skills management problem. The main difficulty with these mathematical models is that they were unable to reach feasible solutions in a reasonable amount of time when the problem size became large. It is therefore wise to investigate possible heuristic solution techniques. This research will compare and contrast three specific heuristic techniques: a Greedy Assignment Algorithm, Meta-RaPS Greedy Heuristic, and Meta-RaPS Shortest Augmenting Path (SAP) Heuristic. Meta-RaPS is a meta-heuristic that is used to improve the performance of algorithms by strategically infusing randomness which allows the exploration of more of the solution space. The skills management heuristics developed in this research were tested using 47 randomly generated data sets generating results within 0.03% of optimal for the recommended Meta-RaPS SAP solution methodology

The set covering problem revisited: an empirical study of the value of dual information

This paper investigates the role of dual information on the performances of heuristics designed for solving the set covering problem. After solving the linear programming relaxation of the problem, the dual information is used to obtain the two main approaches proposed here: (i) The size of the original problem is reduced and then the resulting model is solved with exact methods. We demonstrate the effectiveness of this approach on a rich set of benchmark instances compiled from the literature. We conclude that set covering problems of various characteristics and sizes may reliably be solved to near optimality without resorting to custom solution methods. (ii) The dual information is embedded into an existing heuristic. This approach is demonstrated on a well-known local search based heuristic that was reported to obtain successful results on the set covering problem. Our results demonstrate that the use of dual information significantly improves the efficacy of the heuristic in terms of both solution time and accuracy

The assignment problem with dependent costs.

Assigning workers, each with their own skill set, to tasks which demand different skills in an efficient manner is a challenging problem that often requires workers to receive additional training. The training of workers is very costly with Training Magazine’s Annual Industry Report stating 58.5 billion dollars were spent in 2007 on employee training in the United States. Therefore assigning workers to tasks in such a way as to minimize the overall training costs is an important problem in many organizations. In this research, the assignment problem with dependent cost is considered, i.e. the training cost associated with assigning a worker to a particular task depends on the training the worker receives for their other assigned tasks. Once a worker is trained in a skill that training will available for any additional tasks that may be assigned. The problem is formulated mathematically as an integer linear program. Based on past research, high quality solutions to large-size problems are difficult to obtain. This research develops and upper bound approach and three heuristic solution methodologies. The basic idea of the heuristics is to form groups of tasks which require similar skills, then assign a worker to the task group. The Shortest Augmenting Path (SAP) algorithm of Jonker and Volgenant is known to quickly find the optimal assignment of N workers to N tasks. This SAP algorithm will be used in this research after grouping the tasks into N groups which can then be assigned to the N workers. The task grouping heuristic methods developed in this research were tested for several randomly generated large-sized data sets. Results showed an average 7.34% improvement compared to previous solution methods. Additionally to consider workers’ preferences, a multiple-objective model is presented for the skills management problem to maximize workers’ preferences and aggregate training while minimizing training cost. The model is demonstrated for randomly generated data sets