Melih Ozlen

Multi-Objective Integer Programming: An Improved Recursive Algorithm

This paper introduces an improved recursive algorithm to generate the set of all nondominated objective vectors for the Multi-Objective Integer Programming (MOIP) problem. We significantly improve the earlier recursive algorithm of Özlen and Azizoğlu by using the set of already solved subproblems and their solutions to avoid solving a large number of IPs. A numerical example is presented to explain the workings of the algorithm, and we conduct a series of computational experiments to show the savings that can be obtained. As our experiments show, the improvement becomes more significant as the problems grow larger in terms of the number of objectives.

Computing the Crosscap Number of a Knot Using Integer Programming and Normal Surfaces

The crosscap number of a knot is an invariant describing the nonorientable surface of smallest genus that the knot bounds. Unlike knot genus (its orientable counterpart), crosscap numbers are difficult to compute and no general algorithm is known. We present three methods for computing crosscap number that offer varying trade-offs between precision and speed: (i) an algorithm based on Hilbert basis enumeration and (ii) an algorithm based on exact integer programming, both of which either compute the solution precisely or reduce it to two possible values, and (iii) a fast but limited precision integer programming algorithm that bounds the solution from above. The first two algorithms advance the theoretical state-of-the-art, but remain intractable for practical use. The third algorithm is fast and effective, which we show in a practical setting by making significant improvements to the current knowledge of crosscap numbers in knot tables. Our integer programming framework is general, with the potential for further applications in computational geometry and topology.

Rescheduling unrelated parallel machines with total flow time and total disruption cost criteria

In this paper, we consider a rescheduling problem where a set of jobs has already been assigned to unrelated parallel machines. When a disruption occurs on one of the machines, the affected jobs are rescheduled, considering the efficiency and the schedule deviation measures. The efficiency measure is the total flow time, and the schedule deviation measure is the total disruption cost caused by the differences between the initial and current schedules. We provide polynomial-time solution methods to the following hierarchical optimization problems: minimizing total disruption cost among the minimum total flow time schedules and minimizing total flow time among the minimum total disruption cost schedules. We propose exponential-time algorithms to generate all efficient solutions and to minimize a specified function of the measures. Our extensive computational tests on large size problem instances have revealed that our optimization algorithm finds the best solution by generating only a small portion of all efficient solutions.

A tree traversal algorithm for decision problems in knot theory and 3-manifold topology

In low-dimensional topology, many important decision algorithms are based on normal surface enumeration, which is a form of vertex enumeration over a high-dimensional and highly degenerate polytope. Because this enumeration is subject to extra combinatorial constraints, the only practical algorithms to date have been variants of the classical double description method. In this paper we present the first practical normal surface enumeration algorithm that breaks out of the double description paradigm. This new algorithm is based on a tree traversal with feasibility and domination tests, and it enjoys a number of advantages over the double description method: incremental output, significantly lower time and space complexity, and a natural suitability for parallelisation.

A model for solving the prescribed burn planning problem

The increasing frequency of destructive wildfires, with a consequent loss of life and property, has led to fire and land management agencies initiating extensive fuel management programs. This involves long-term planning of fuel reduction activities such as prescribed burning or mechanical clearing. In this paper, we propose a mixed integer programming (MIP) model that determines when and where fuel reduction activities should take place. The model takes into account multiple vegetation types in the landscape, their tolerance to frequency of fire events, and keeps track of the age of each vegetation class in each treatment unit. The objective is to minimise fuel load over the planning horizon. The complexity of scheduling fuel reduction activities has led to the introduction of sophisticated mathematical optimisation methods. While these approaches can provide optimum solutions, they can be computationally expensive, particularly for fuel management planning which extends across the landscape and spans long term planning horizons. This raises the question of how much better do exact modelling approaches compare to simpler heuristic approaches in their solutions. To answer this question, the proposed model is run using an exact MIP (using commercial MIP solver) and two heuristic approaches that decompose the problem into multiple single-period sub problems. The Knapsack Problem (KP), which is the first heuristic approach, solves the single period problems, using an exact MIP approach. The second heuristic approach solves the single period sub problem using a greedy heuristic approach. The three methods are compared in term of model tractability, computational time and the objective values. The model was tested using randomised data from 711 treatment units in the Barwon-Otway district of Victoria, Australia. Solutions for the exact MIP could be obtained for up to a 15-year planning only using a standard implementation of CPLEX. Both heuristic approaches can solve significantly larger problems, involving 100-year or even longer planning horizons. Furthermore there are no substantial differences in the solutions produced by the three approaches. It is concluded that for practical purposes a heuristic method is to be preferred to the exact MIP approach.

Transmission expansion planning considering energy storage

In electricity transmission networks, energy storage systems (ESS) provide a means of upgrade deferral by smoothing supply and matching demand. We develop a mixed integer programming (MIP) extension to the transmission network expansion planning (TEP) problem that considers the installation and operation of ESS as well as additional circuits. The model is demonstrated on the well known Carvers 6-bus and IEEE 25-bus test circuits for two 24 hour operating scenarios; a short peak, and a long peak. We show optimal location and capacity of storage is sensitive not only to cost, but also variability of demand in the network.

Optimising a nonlinear utility function in multi-objective integer programming

In this paper we develop an algorithm to optimise a nonlinear utility function of multiple objectives over the integer efficient set. Our approach is based on identifying and updating bounds on the individual objectives as well as the optimal utility value. This is done using already known solutions, linear programming relaxations, utility function inversion, and integer programming. We develop a general optimisation algorithm for use with k objectives, and we illustrate our approach using a tri-objective integer programming problem.

A Tree Traversal Algorithm for Decision Problems in Knot Theory and 3-Manifold Topology

In low-dimensional topology, many important decision algorithms are based on normal surface enumeration, which is a form of vertex enumeration over a high-dimensional and highly degenerate polytope. Because this enumeration is subject to extra combinatorial constraints, the only practical algorithms to date have been variants of the classical double description method. In this paper we present the first practical normal surface enumeration algorithm that breaks out of the double description paradigm. This new algorithm is based on a tree traversal with feasibility and domination tests, and it enjoys a number of advantages over the double description method: incremental output, significantly lower time and space complexity, and a natural suitability for parallelisation. Experimental comparisons of running times are included.

A parallel approach to bi-objective integer programming

The real world applications of optimisation algorithms often are only interested in the running time of an algorithm, which can frequently be significantly reduced through parallelisation. We present two methods of parallelising the recursive algorithm presented by Ozlen, Burton and MacRae [J. Optimization Theory and Applications; 160:470--482, 2014]. Both new methods utilise two threads and improve running times. One of the new methods, the Meeting algorithm, halves running time to achieve near-perfect parallelisation, allowing users to solve bi-objective integer problems with more variables.

A Probabilistic Tree-Based Representation for Non-convex Minimum Cost Flow Problems.

Network flow optimisation has many real-world applications. The minimum cost flow problem (MCFP) is one of the most common network flow problems. Mathematical programming methods often assume the linearity and convexity of the underlying cost function, which is not realistic in many real-world situations. Solving large-sized MCFPs with nonlinear non-convex cost functions poses a much harder problem. In this paper, we propose a new representation scheme for solving non-convex MCFPs using genetic algorithms (GAs). The most common representation scheme for solving the MCFP in the literature using a GA is priority-based encoding, but it has some serious limitations including restricting the search space to a small part of the feasible set. We introduce a probabilistic tree-based representation scheme (PTbR) that is far superior compared to the priority-based encoding. Our extensive experimental investigations show the advantage of our encoding compared to previous methods for a variety of cost functions.