Searching for optimal integer solutions to set partitioning problems using column generation

We describe a new approach to produce integer feasible columns to a set partitioning problem directly in solving the linear programming (LP) relaxation using column generation. Traditionally, column generation is aimed to solve the LP relaxation as quick as possible without any concern of the integer properties of the columns formed. In our approach we aim to generate the columns forming the optimal integer solution while simultaneously solving the LP relaxation. By this we can remove column generation in the branch and bound search. The basis is a subgradient technique applied to a Lagrangian dual formulation of the set partitioning problem extended with an additional surrogate constraint. This extra constraint is not relaxed and is used to better control the subgradient evaluations. The column generation is then directed, via the multipliers, to construct columns that form feasible integer solutions. Computational experiments show that we can generate the optimal integer columns in a large set of well known test problems as compared to both standard and stabilized column generation and simultaneously keep the number of columns smaller than standard column generation

Modeling and solving the periodic maintenance problem.

We study the problem of scheduling maintenance services. Given is a set of m machines and integral cost-coefficients a(i) and b(i) for each machine i (1Branch-and-price; Column generation; Costs; Linear programming; Model; Models; Optimal; Scheduling; Structure; Studies; Time;

Genetic algorithm approach to find the best input variable partitioning

Conference PaperThis paper presents a variable partition algorithm which combines the quasi-reduced ordered multiple-terminal multiple-valued decision diagrams and genetic algorithms (GAs). The algorithm is better than the previous techniques which find a good functional decomposition by non-exhaustive search and expands the range of searching for the best decomposition providing the optimal subtable multiplicity. The possible solutions are evaluated using the gain of decomposition for a multiple-output multiple-valued logic function. The distinct feature of GA is the possible solutions being coded by real numbers. Here the simplex-based crossover is proposed to use for the recombination stage of GA. It permits to increase the GA coverag

A Literature Review On Combining Heuristics and Exact Algorithms in Combinatorial Optimization

There are several approaches for solving hard optimization problems. Mathematical programming techniques such as (integer) linear programming-based methods and metaheuristic approaches are two extremely effective streams for combinatorial problems. Different research streams, more or less in isolation from one another, created these two. Only several years ago, many scholars noticed the advantages and enormous potential of building hybrids of combining mathematical programming methodologies and metaheuristics. In reality, many problems can be solved much better by exploiting synergies between these approaches than by “pure” classical algorithms. The key question is how to integrate mathematical programming methods and metaheuristics to achieve such benefits. This paper reviews existing techniques for such combinations and provides examples of using them for vehicle routing problems

Scheduling trainees at a hospital department using a branch-and-price approach.

Scheduling trainees (graduate students) is a complicated problem that has to be solved frequently in many hospital departments. We will describe a trainee scheduling problem encountered in practice (at the ophthalmology department of the university hospital Gasthuisberg, Leuven). In this problem a department has a certain number of trainees at its disposal, which assist specialists in their activities (surgery, consultation, etc.). For each trainee one has to schedule the activities in which (s)he will assist during a certain time horizon, usually one year. Typically, these kind of scheduling problems are characterized by both hard and soft constraints. The hard constraints consist of both work covering constraints and formation requirements, whereas the soft constraints include trainees' preferences and setup restrictions. In this paper we will describe an exact branch-and-price method to solve the problem to optimality.Branch-and-price; Constraint; Health care; Problems; Requirements; Scheduling; Staff scheduling; Time; University;