Advances in Agriculture

Volume 2016, Article ID 2313695, 6 pages

http://dx.doi.org/10.1155/2016/2313695

## A Mixed Integer Programming Poultry Feed Ration Optimisation Problem Using the Bat Algorithm

^{1}Department of Applied Mathematics, National University of Science & Technology, P.O. Box AC939, Ascot, Bulawayo, Zimbabwe^{2}Department of Agricultural Economics and Extension, University of Zimbabwe, P.O. Box MP167, Mt Pleasant, Harare, Zimbabwe^{3}Department of Agricultural Economics, Education and Extension, Bindura University of Science Education, Private Bag Box 1020, Bindura, Zimbabwe

Received 12 June 2016; Accepted 29 November 2016

Academic Editor: Christos Tsadilas

Copyright © 2016 Godfrey Chagwiza et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, a feed ration problem is presented as a mixed integer programming problem. An attempt to find the optimal quantities of* Moringa oleifera* inclusion into the poultry feed ration was done and the problem was solved using the Bat algorithm and the Cplex solver. The study used findings of previous research to investigate the effects of* Moringa oleifera* inclusion in poultry feed ration. The results show that the farmer is likely to gain US0.89 more if* Moringa oleifera* is included in the feed ration. Results also show superiority of the Bat algorithm in terms of execution time and number of iterations required to find the optimum solution as compared with the results obtained by the Cplex solver. Results revealed that there is a significant economic benefit of* Moringa oleifera* inclusion into the poultry feed ration.

#### 1. Introduction

The cost of feed significantly contributes to the profitability of animal farming and has been estimated to constitute of the total costs of poultry production [1, 2]. If a farmer manages to reduce the cost of feed then a significant amount of profit can be realised. It is important to develop methods that can effectively cut down the costs of feeds. In this research, an attempt to cut down the costs of poultry feeds is carried out. The research has been motivated by the success of the experiment carried out by Gadzirayi et al. [3] to substitute soyabean as source of proteins by* Moringa oleifera*. Most economies of developing countries are agro-based and this study is likely to improve the livelihood of farmers in poultry production.

Feed ration formulation is a particular challenge especially if new nutritive mixtures are involved. Feed mixing can be classified as an hard problem. Vast literature exists that deals with the formulation of feed ration and modelling of these formulations. Several mathematical tools have been used to solve feed ration formulation problems such as Pearson’s Square method [4], goal programming (GP) [5–7], multiobjective goal programming (MOP) [8], multiobjective fractional programming (MOFP) [9], nonlinear programming (NLP) [10], chance constrained programming (CCP) [11, 12], quadratic programming (QP) [13], risk formulation (RF) [14], and linear programming (LP) [15–18].

Most of these methods have been found to have difficulties in finding optimal solutions because of the following variables. Ingredient variability due to varying, fluctuating, and unstable nutrient components has made it difficult to find optimal solution of the feed mixing problems [19, 20]. Variability of price of the ingredient due to changes in price of the ingredients makes the feed formulation problem complex. Optimisation algorithms have been also used, to a limited extent, for animal diet formulation such as genetic algorithm (GA) [21] and evolutionary algorithm [22].

In this paper, researchers seek to formulate a mixed integer programming (MIP) problem for the feed ration formulation with the objective of finding the optimal quantities of* Moringa oleifera* inclusion in the poultry feed ration to substitute, to a certain level, soyabean which is expensive as compared to* Moringa oleifera*. Bat algorithm is used to solve the MIP. The remainder of the paper is arranged as follows. In Section 2, a review of the Bat algorithm is presented and in Section 3 formulation of the MIP is presented. Implementation of the Bat algorithm, results, and discussions are presented in Section 4 and conclusions are drawn in Section 5.

#### 2. Review of the Bat Algorithm

The Bat algorithm was first introduced by Yang [23] and the algorithm is inspired by the echolocation behaviour of bats. Microbats are believed to use echolocation in most of their time to detect prey and avoiding obstacles. The bats emit a loud sound pulse and listen to the echo bouncing from the surrounding objects. The Bat algorithm assumes that all bats use echolocation and bats fly randomly and that the loudness varies from positive to a minimum constant value. The following are the steps of the Bat algorithm.(i)Bat population initialisation(ii)Definition of pulse frequency(iii)Initialisation of pulse rate and loudness(iv)Generation of the new solution through adjusting frequency(v)Updating velocities and locations or solutions and if the current pulse rate is greater than the initial pulse, then a solution is found among the best solutions(vi)Generation of a new solution by flying randomly(vii)Accepting the new solution and increasing the pulse and reducing loudness if the current pulse is less than the previous loudness(viii)Ranking the bats and finding the best current solution(ix)Postprocessing results and visualizationThe Bat algorithm has been modified and applied to various optimisation problems such as the studies by Khooban and Niknam [24], Taha et al. [25], He et al. [26], Alihodzic and Tuba [27], Sathya and Mohamed Thameem Ansari [28], Huang et al. [29], and Goyal and Patterh [30]. This research is the first of its kind to apply the Bat algorithm to solve a feed ration formulation optimisation problem.

#### 3. Formulation of the Feed Ration Problem

In this section, a feed ration formulation problem that provides an alternative source of proteins is formulated. The research is motivated by the success of the experiment carried out by Gadzirayi et al. [3] on the inclusion of* Moringa oleifera* as a source of proteins for the broiler feed ration. Table 1 presents definitions of the parameters and variables that are used in this research.