Heuristic approaches for multi-criteria optimisation in kidney exchange programs

The Kidney Exchange Problem (KEP) is an optimisation problem that was first discussed in Rapaport (1986) but has only more recently been the subject of much work by combinatorial optimisation re-searchers. This has been in parallel with its increased prevalence in the medical community. In the basic formulation of a KEP, each instance of the problem features a directed graph D = (V,A) . Each node i ∈ V represents an incompatible pair wherein the patient needs to trade kidneys with the patient of another incompatible pair. The goal is to find an optimal set of cycles such that as many patients as possible receive a transplant. The problem is further complicated by the imposition of a cycle-size constraint, usually considered to be 3 or 4. Kidney exchange programs around the world implement different algorithms to solve the allocation problem by matching up kidneys from potential donors to patients. In some systems all transplants are considered equally desirable, whereas in others, ranking criteria such as the age of the patient or distance they will need to travel are applied, hence the multi-criteria nature of the KEP. To address the multi-criteria aspect of the KEP, in this paper we propose a two-stage approach for the kidney exchange optimisation problem. In the first stage the goal is to find the optimal number of exchanges, and in the second stage the goal is to maximise the weighted sum of the kidney matches, subject to the added constraint that the number of exchanges must remain optimal. The idea can potentially be extended to multiple-objectives, by repeating the process in multiple runs. In our preliminary numerical experiments, we first find the maximum number of kidney matches by using an existing open source exact algorithm of Anderson et al. (2015). The solution will then be used as an initial solution for the stage two optimisation problem, wherein two heuristic methods, steepest ascent and random ascent, are implemented in obtaining good quality solutions to the objective of maximizing total weight of exchanges. The neighbourhood is obtained by two-swaps. It is our intention in the future to implement a varying neighbourhood scheme within the same two heuristic framework, or within other meta-heuristic framework.

Facets of the polytope of the asymmetric travelling salesman problem with replenishment arcs

The Asymmetric Travelling Salesman Problem with Replenishment Arcs (RATSP) is a new class of problems arising from work related to aircraft routing. Given a digraph with cost on the arcs, a solution of the RATSP, like that of the Asymmetric Travelling Salesman Problem, induces a directed tour in the graph which minimises total cost. However the tour must satisfy additional constraints: the arc set is partitioned into replenishment arcs and ordinary arcs, each node has a non-negative weight associated with it, and the tour cannot accumulate more than some weight limit before a replenishment arc must be used. To enforce this requirement, constraints are needed. We refer to these as replenishment constraints.

In this paper, we review previous polyhedral results for the RATSP and related problems, then prove that two classes of constraints developed in V. Mak and N. Boland [Polyhedral results and exact algorithms for the asymmetric travelling salesman problem with replenishment arcs, Technical Report TR M05/03, School of Information Technology, Deakin University, 2005] are, under appropriate conditions, facet-defining for the RATS polytope.

Polyhedral results and exact algorithms for the asymmetric travelling salesman problem with replenishment arcs

The asymmetric travelling salesman problem with replenishment arcs (RATSP), arising from work related to aircraft routing, is a generalisation of the well-known ATSP. In this paper, we introduce a polynomial size mixed-integer linear programming (MILP) formulation for the RATSP, and improve an existing exponential size ILP formulation of Zhu [The aircraft rotation problem, Ph.D. Thesis, Georgia Institute of Technology, Atlanta, 1994] by proposing two classes of stronger cuts. We present results that under certain conditions, these two classes of stronger cuts are facet-defining for the RATS polytope, and that ATSP facets can be lifted, to give RATSP facets. We implement our polyhedral findings and develop a Lagrangean relaxation (LR)-based branch-and-bound (BNB) algorithm for the RATSP, and compare this method with solving the polynomial size formulation using ILOG Cplex 9.0, using both randomly generated problems and aircraft routing problems. Finally we compare our methods with the existing method of Boland et al. [The asymmetric traveling salesman problem with replenishment arcs, European J. Oper. Res. 123 (2000) 408–427]. It turns out that both of our methods are much faster than that of Boland et al. [The asymmetric traveling salesman problem with replenishment arcs, European J. Oper. Res. 123 (2000) 408–427], and that the LR-based BNB method is more efficient for problems that resemble the aircraft rotation problems.

On the kidney exchange problem: cardinality constrained cycle and chain problems on directed graphs: a survey of integer programming approaches

The Kidney Exchange Problem (KEP) is a combinatorial optimization problem and has attracted the attention from the community of integer programming/combinatorial optimisation in the past few years. Defined on a directed graph, the KEP has two variations: one concerns cycles only, and the other, cycles as well as chains on the same graph. We call the former a Cardinality Constrained Multi-cycle Problem (CCMcP) and the latter a Cardinality Constrained Cycles and Chains Problem (CCCCP). The cardinality for cycles is restricted in both CCMcP and CCCCP. As for chains, some studies in the literature considered cardinality restrictions, whereas others did not. The CCMcP can be viewed as an Asymmetric Travelling Salesman Problem that does allow subtours, however these subtours are constrained by cardinality, and that it is not necessary to visit all vertices. In existing literature of the KEP, the cardinality constraint for cycles is usually considered to be small (to the best of our knowledge, no more than six). In a CCCCP, each vertex on the directed graph can be included in at most one cycle or chain, but not both. The CCMcP and the CCCCP are interesting and challenging combinatorial optimization problems in their own rights, particularly due to their similarities to some travelling salesman- and vehicle routing-family of problems. In this paper, our main focus is to review the existing mathematical programming models and solution methods in the literature, analyse the performance of these models, and identify future research directions. Further, we propose a polynomial-sized and an exponential-sized mixed-integer linear programming model, discuss a number of stronger constraints for cardinality-infeasible-cycle elimination for the latter, and present some preliminary numerical results.

Efficient serial and parallel implementations of the cutting angle global optimisation technique

We examine efficient computer implementation of one method of deterministic global optimisation, the cutting angle method. In this method the objective function is approximated from values below the function with a piecewise linear auxiliary function. The global minimum of the objective function is approximated from the sequence of minima of this auxiliary function. Computing the minima of the auxiliary function is a combinatorial problem, and we show that it can be effectively parallelised. We discuss the improvements made to the serial implementation of the cutting angle method, and ways of distributing computations across multiple processors on parallel and cluster computers.

Combinatorial aspects of jump codes

We develop an approach to jump codes concentrating on their combinatorial and symmetry properties. The main result is a generalization of a theorem previously proved in the context of isodual codes. We show that several previously constructed jump codes are instances of this theorem.

On work in progress optimisation

In this paper, we investigate how to best optimise the level of work in progress (WIP) in a real world factory. Using a simulation model of the factory, we show that an optimum level of WIP can be attained. By systematically varying the maximum allowable level of WIP within different model runs, results show that the throughput reaches a high level very quickly and then tapers off. The production lead times, in contrast, begin at relatively low levels and increase after the optimum WIP level has been reached.

Global optimisation of neural networks using a deterministic hybrid approach

Selection of the topology of a neural network and correct parameters for the learning algorithm is a tedious task for designing an optimal artificial neural network, which is smaller, faster and with a better generalization performance. In this paper we introduce a recently developed cutting angle method (a deterministic technique) for global optimization of connection weights. Neural networks are initially trained using the cutting angle method and later the learning is fine-tuned (meta-learning) using conventional gradient descent or other optimization techniques. Experiments were carried out on three time series benchmarks and a comparison was done using evolutionary neural networks. Our preliminary experimentation results show that the proposed deterministic approach could provide near optimal results much faster than the evolutionary approach.

A medium-temperature solar thermal power system and its efficiency optimisation

This paper firstly expounds that the reheat-regenerative Rankine power cycle is a suitable cycle for the parabolic trough collector, a popular kind of collector in the power industry. In a thermal power cycle, the higher the temperature at which heat is supplied, the higher the efficiency of the cycle. On the other hand, for a given kind of collector at the same exiting temperature, the higher the temperature of the fluid entering the collector, the lower the efficiency of the collector. With the same exiting temperature of the solar field and the same temperature differences at the hottest end of the superheater/reheater and at the pinch points in the heat exchangers (e.g., the boiler) in the cycle, the efficiencies of the system are subject to the temperature of the fluid entering the collector or the saturation temperature at the boiler. This paper also investigates the optimal thermal and exergetic efficiencies for the combined system of the power cycle and collector. To make most advantage of the collector, the exiting fluid is supposed to be at the maximum temperature the collector can harvest. Hence, the thermal and exergetic efficiencies of the system are related to the saturation temperature at the boiler here.

Knowledge based optimisation techniques for sheet metal forming die design

A knowledge based optimism technique has been developed to predict solutions for quality issues found in an initial draw die design. Post processing of the initial design yields all the features applying forces, and major quality issues. Using the geometric relationship between the two, a knowledge-base is interrogated to determine the possible corrective actions. These actions are then passed through a fast semi-analytical model to determine the level of change required. Results from a 2D forming are presented to highlight the advantage of the new algorithms over current optimisation techniques.

A winner determination algorithm for multi-unit combinatorial auctions with reserve prices

Combinatorial auction mechanisms have been used in many applications such as resource and task allocation, planning and time scheduling in multi-agent systems, in which the items to be allocated are complementary or substitutable. The winner determination in combinatorial auction itself is a NP-complete problem, and has attracted many attentions of researchers world wide. Some outstanding achievements have been made including CPLEX and CABOB algorithms on this topic. To our knowledge, the research into multi-unit combinatorial auctions with reserve prices considered is more or less ignored. To this end, we present a new algorithm for multi-unit combinatorial auctions with reserve prices, which is based on Sandholm's work. An efficient heuristic function is developed for the new algorithm. Experiments have been conducted. The experimental results show that auctioneer agent can find the optimal solution efficiently for a reasonable problem scale with our algorithm.