Evolutionary combinatorial optimization on the grain mixing problem

Abstract

Combinatorial optimization is an important area in computer science that uses combinatorics to solve discrete optimization problems. In this thesis, we considered a combinatorial optimization problem in the wheat supply chain known as grain mixing. The grain mixing problem involves mixing two or more collections of grain with different protein content to produce collections of grain with a weighted average protein content that improves the overall profit to the farmer. The presence of non-linearity in the objective function and some of the constraints in the grain mixing problem makes the problem difficult to solve exactly using linear programming (LP) or mixed-integer LP models. First, we presented an NP-Hardness proof for the grain mixing problem to justify the use of approximation algorithms. Then we explored several approaches to solve the grain mixing problem. For the approximation algorithms, we adapted two evolutionary approaches (EA) for the grain mixing problem: Genetic Algorithm (GA) and Differential Evolution (DE). We developed a pseudo-permutation-based representation for the EAs for which the conventional crossover operator for GA and mutation operator for DE had to be adapted to fit our problem representation. Specifically, we adapted two crossover operators for GA: Ordered Crossover (OX) and Partially Mapped Crossover (PMX), and two discrete mutation operators for DE: Relative Position Indexing (RPI) and Global Best Perturbation (GBP). Moreover, we introduced and compared three baseline approaches: no mixing, greedy mixing, and random mixing to evaluate the solution quality of the proposed EA approaches. The experimental results demonstrate that solutions obtained from the evolutionary approaches consistently provided a higher overall profit compared to the non-evolutionary baseline methods for both real and simulated datasets. It also suggests that grain mixing is beneficial to improve farmers' overall wheat selling profitability.