Optimization applications of Goldbach's conjecture.

Goldbach's conjecture is an unsolved legendary problem. In this paper, we investigate an optimization problem as an attempt to discuss potential applications of the conjecture. In an optimization perspective, the research question, modelled from design of base components, is to select a minimum number of primes to span a given set of target even integers. We formally define the problem, propose integer programming formulations, and develop solution procedures, which are then tested through a computational study.

Unifying colors by primes.

RGB and CYMK are two major coloring schemes currently available for light colors and pigment colors, respectively. Both systems use letter-based color codes that require a large range of values to represent different colors. The problem is that these two systems are hard to use for manipulating any operations involving combinations of colors, and they lack the capacity for inter-changeability or unification. Based on prime number theory and Goldbach's conjecture, this study presents a universal color system (C235) using a number-based structure to encode, compute and unify all colors on a color wheel. The proposed C235 system offers a unified representation for the efficient encoding and effective manipulation of color. It can be applied to designing a high-rate LCD system and colorizing objects with multiple attributes and DNA codons, opening the door to manipulating colors and lights for even broader applications.

Scheduling of Anaesthesia Operations in Operating Rooms.

This paper considers scheduling of surgical operations across multiple operating rooms subject to the limited availability of anaesthetists. The objective is to construct a feasible operations schedule that has the minimum makespan, i.e., the completion time of all operations. We abstract the problem into a theoretical server scheduling problem and formulate it in a mathematical form by proposing an integer programming model. Due to the intractability of its computing time, we circumvent the exact approaches and develop two approximation methods. Then, the steepest descent search is adopted for improving the solutions. Computational study suggests that the proposed methods can produce quality solutions in a few seconds.

Discrete particle swarm optimization with scout particles for library materials acquisition.

Materials acquisition is one of the critical challenges faced by academic libraries. This paper presents an integer programming model of the studied problem by considering how to select materials in order to maximize the average preference and the budget execution rate under some practical restrictions including departmental budget, limitation of the number of materials in each category and each language. To tackle the constrained problem, we propose a discrete particle swarm optimization (DPSO) with scout particles, where each particle, represented as a binary matrix, corresponds to a candidate solution to the problem. An initialization algorithm and a penalty function are designed to cope with the constraints, and the scout particles are employed to enhance the exploration within the solution space. To demonstrate the effectiveness and efficiency of the proposed DPSO, a series of computational experiments are designed and conducted. The results are statistically analyzed, and it is evinced that the proposed DPSO is an effective approach for the studied problem.