Jan H. van Vuuren

New and improved level heuristics for the rectangular strip packing and variable-sized bin packing problems

We consider two types of orthogonal, oriented, rectangular, two-dimensional packing problems. The first is the strip packing problem, for which four new and improved level-packing algorithms are presented. Two of these algorithms guarantee a packing that may be disentangled by guillotine cuts. These are combined with a two-stage heuristic designed to find a solution to the variable-sized bin packing problem, where the aim is to pack all items into bins so as to minimise the packing area. This heuristic packs the levels of a solution to the strip packing problem into large bins and then attempts to repack the items in those bins into smaller bins in order to reduce wasted space. The results of the algorithms are compared to those of seven level-packing heuristics from the literature by means of a large number of strip-packing benchmark instances. It is found that the new algorithms are an improvement over known level-packing heuristics for the strip packing problem. The advancements made by the new and improved algorithms are limited in terms of utilised space when applied to the variable-sized bin packing problem. However, they do provide results faster than many existing algorithms.

Finding good nurse duty schedules: a case study

Abstract Constructing duty schedules for nurses at large hospitals is a difficult problem. The objective is usually to ensure that there is always sufficient staff on duty, while taking into account individual preferences with respect to work patterns, requests for leave and financial restrictions, in such a way that all employees are treated fairly. The problem is typically solved via mixed integer programming or heuristic (local) search methods in the operations research literature. In this paper the problem is solved using a tabu search approach as a case study at Stikland Hospital, a large psychiatric hospital in the South African Western Cape, for which a computerized decision support system with respect to nurse scheduling was developed. This decision support system, called NuRoDSS (short for Nurse Rostering Decision Support System) is described in some detail.

Robust multi-objective optimization for water distribution system design using a meta-metaheuristic

The design of an urban water distribution system (WDS) is a challenging problem involving multiple objectives. The goal of robust multi-objective optimization for WDS design is to find the set of solutions which embodies an acceptable trade-off between system cost and reliability, so that the ideal solution may be selected for a given budget. In addition to satisfying consumer needs, a system must be built to accommodate multiple demand loading conditions, withstand component failures and allow surplus capacity for growth. In a developmental setting, WDS robustness becomes even more crucial, owing to the limited availability of resources, especially for maintenance. Recent optimization studies have achieved success using multi-objective evolutionary algorithms, such as the Non-dominated Sorting Genetic Algorithm II (NSGA-II). However, the multi-objective design of a large WDS within a reasonable timeframe remains a formidable problem, owing to the extremely high computational complexity of the problem. In this paper, a meta-algorithm called AMALGAM is applied for the first time to WDS design. AMALGAM uses multiple metaheuristics simultaneously in an attempt to improve optimization performance. Additionally, a Jumping-gene Genetic Algorithm (NSGA-II-JG) is also applied for the first time to WDS design. These two algorithms were tested against some other metaheuristics (including NSGA-II and a new greedy algorithm) with respect to a number of benchmark systems documented in the literature, and AMALGAM demonstrated the best performance overall, while NSGA-II-JG fared worse than the ordinary NSGA-II. Large cost savings and reliability improvements are demonstrated for a real WDS developmental case study in South Africa.

Multiobjective Optimization for Water Distribution System Design Using a Hyperheuristic

Several modern multiobjective algorithms are compared with respect to the biobjective optimization of water distribution systems’ design, considering the objectives of cost minimization and reliability maximization. A new unbiased algorithmic comparison strategy is suggested using hypervolume convergence. A recent hyperheuristic named AMALGAM, which dynamically combines the strengths of multiple metaheuristics, is applied to this problem for the first time, demonstrating superior performance. The shortcomings of a novel multiobjective greedy algorithm are also demonstrated.

Vertex Covers and Secure Domination in Graphs

Let G = (V,E) be a graph and let S ⊆ V. The set S is a dominating set of G if every vertex in V \ S is adjacent to some vertex in S. The set S is a secure dominating set of G if for each u ∈ V \ S, there exists a vertex v ∈ S such that uv ∈ E and (S \ {v}) ∪ {u} is a dominating set of G. The minimum cardinality of a secure dominating set in G is the secure domination number γ s(G) of G. We show that if G is a connected graph of order n with minimum degree at least two that is not a 5-cycle, then γ s(G) ≤ n/2 and this bound is sharp. Our proof uses a covering of a subset of V(G) by vertex-disjoint copies of subgraphs each of which is isomorphic to K 2 or to an odd cycle.

A comparison between continuous and discrete modelling of cables with bending stiffness

Abstract The static configuration in space of an inflexible two-dimensional cable anchored at its endpoints is obtained by the numerical solution of both continuous and discrete models. The results are compared and validated by experiment. It is shown that although the continuous model yields more accurate results than the discrete model, as expected, the effort and cost of numerically integrating the continuous model do not compare favourably with the relative ease and efficiency of solving the discrete model, which yields perhaps surprisingly accurate results.

Lions in the Kgalagadi Transfrontier Park: modelling the effect of human‐caused mortality

An age-structured demographic matrix model is developed to determine the long-term viability of the Southern African Kgalagadi Transfrontier Park lion ( Panthera leo) population that is subjected to persecution along the park boundaries. In doing so, maximum sustainable persecution rates are established, so that efficient conservation measures may be taken, if required, to ensure the continued survival of the species in the park. Sensitivity analyses indicate that adult female survival ability alone is the most important component of the model in terms of long-term population survival. Furthermore, these analyses show that the boundary prides of the park are able to increase their chances of survival and are able to sustain significantly higher human-caused mortality rates by inducing birth-sex ratio biases in favour of females. The model suggests that the persecution rate experienced by the study population over the period 1998–2001 cannot be sustained in the long term, provided that there is no immigration of adult females into the population from the park interior. To ensure the survival of existing park boundary prides, adult lionesses should, as far as possible, be afforded protection from persecution, so that park boundary populations do not merely serve as population sinks for lions from elsewhere, or become extinct.

Ramsey numbers in complete balanced multipartite graphs. Part II: Size numbers

The notion of a graph theoretic Ramsey number is generalised by assuming that both the original graph whose edges are arbitrarily bi-coloured and the sought after monochromatic subgraphs are complete, balanced, multipartite graphs, instead of complete graphs as in the classical definition. We previously confined our attention to diagonal multipartite Ramsey numbers. In this paper the definition of a multipartite Ramsey number is broadened still further, by incorporating off-diagonal numbers, fixing the number of vertices per partite set in the larger graph and then seeking the minimum number of such partite sets that would ensure the occurrence of certain specified monochromatic multipartite subgraphs.

Simulation modelling as a decision support in developing a sterile insect-inherited sterility release strategy for Eldana saccharina (Lepidoptera: Pyralidae)

Abstract A user-friendly simulation tool for determining the impact of the sterile insect technique/inherited sterility technique (SIT/IS) on populations of the African sugarcane stalk borer, Eldana saccharina Walker (Lepidoptera: Pyralidae) is described in this paper. The simulation tool is based on a spatio-temporal model. The design of the simulation tool is such that it is applicable for use in a number of pest/crop and pest control scenarios. It uses 4 interacting subsystems (pest species population dynamics, crop dynamics, environmental dynamics and economics) within a specified spatial domain. Furthermore, the spatial domain describes the layout of the agricultural crop (position, size, shape, crop age and variety of the different fields contained within the crop area). The pest species population subsystem describes E. saccharina population dynamics (but is designed to also include population dynamics of other pest species) under the influence of the IS technique. The E. saccharina module developed utilizes mean-field and spatio-temporal models, and includes dynamics of all E. saccharina life stages under the influence of the control measure. Only temperature and damage caused by E. saccharina are currently included as variables in the sugarcane dynamics subsystem. This subsystem estimates stalk length as a function of time and temperature, and sucrose percentage as a function of damage caused by E. saccharina boring. Interaction between E. saccharina population growth and sugarcane growth is described by a decreasing s-shaped density-dependent mortality function—the older the cane, the higher the carrying capacity (more food resources) and corresponding infestation and damage levels. The only environmental factor considered as an independent variable in the environmental dynamics subsystem is temperature. Possible extensions to this subsystem are discussed. The economics subsystem developed includes the estimation of the recoverable value, percentage, expected revenue and the cost of control. No other farm expenditures are taken into account. As such only profit or loss expected from applying the IS technique is estimated. The profit or loss is defined as the increase in revenue expected less the cost of applying a pest control measure. An example of using the simulation tool is presented in the context of a real field scenario of a simulated SIT/IS program against E. saccharina at a pilot site near the Eston area of KwaZulu-Natal, South Africa.

Scheduling multi-colour print jobs with sequence-dependent setup times

In this paper, a scheduling problem is considered which arises in the packaging industry. Plastic and foil wrappers used for packaging candy bars, crisps and other snacks typically require overlay printing with multiple colours. Printing machines used for this purpose usually accommodate a small number of colours which are loaded into a magazine simultaneously. If two consecutively scheduled print jobs require significantly different colour overlays, then substantial down times are incurred during the transition from the former magazine colour configuration to the latter, because ink cartridges corresponding to colours not required for the latter job have to be cleaned after completion of the former job. The durations of these down times are therefore sequence dependent (the washing and refilling time is a function of the number of colours in which two consecutive printing jobs differ). It is consequently desirable to schedule print jobs so that the accumulated down times associated with all magazine colour transitions are as short as possible for each printing machine. We show that an instance of this scheduling problem can be modelled as the well-known tool switching problem, which is tractable for small instances only. The problem can, however, be solved rather effectively in heuristic fashion by decomposing it into two subproblems: a job grouping problem (which can be modelled as a unicost set covering problem) and a group sequencing problem (which is a generalisation of the celebrated travelling salesman problem). We solve the colour print scheduling problem both exactly and heuristically for small, randomly generated test problem instances, studying the trade-off between the time efficiency and solution quality of the two approaches. Finally, we apply both solution approaches to real problem data obtained from a printing company in the South African Western Cape as a special case study.