Applied Engineering in Agriculture Vol. 23(1): 105-110 E 2007 American Society of Agricultural and Biological Engineers ISSN 0883−8542 105 MARKOVIAN MODEL OF FRUIT GROWTH A. Clausse, F. J. Mayorano, A. J. Rubiales, V. A. Herrero ABSTRACT. A computer model for the assessment of fruit growth, based on Markov chains, is presented in this article. The model is aimed to support commercial and logistic decisions in fruit productions, providing forecasting of the size distribution at harvest, given the regional current climatic parameters. The model applies historical correlations between the fruit growth and temperatures, and requires the construction of a specific database. A corresponding methodology and software support for the fruit-size database was also developed with the model. The forecasting performance was validated against experimental data of Granny Smith apples in the Rio Negro Valley (Argentina). Keywords. Simulation, Markov process, Fruit size, Granny Smith. he knowledge of the fruit size distribution is crucial in the fruit industry, good forecasts of this data im- proves the firm position in several aspects. From a commercial standpoint, the exporting companies often sign contracts with clients in advance, and therefore ev- ery piece of information about the fruit characteristics at har- vest time gives an important strategic advantage. From a logistic and packaging perspective, fruit sizes forecasts help reduce raw material inventories and related additional costs incurred due to stock breaks. There are several classes of fruit growth models in the literature. Most of these models addressing biological processes are detailed and involve the use of differential and algebraic equations to represent energy and mass balances (Gandar et al., 1996). On the other hand, when sufficient data is available, fruit size forecasts can be tackled based on mere statistical considerations (Hall and Gandar, 1996). In the first approach, fruit growth is represented by functional dependencies on environmental factors, which generally end in average equations. Sigmoidal curves are often used to adjust the fruit size temporal evolution, containing parameters depending on climatic variables (e.g., mean temperature). Nevertheless, if detailed knowledge of the size distribution is required the forecast of the average size is not sufficient. Some approaches introduced stochastic differential equations to represent the temporal evolution of the size distribution (Hall and Gandar, 1996). However, these approaches involve some mathematical difficulty (e.g., solution of partial differential equations) and need a huge amount of data in order to validate the model. In this article a new approach to simulate fruit growth is presented, based on stochastic Markov processes. The basic concept of this approach is to represent the fruit size information by discrete probability distributions and to Submitted for review in May 2006 as manuscript number IET 6406; approved for publication by the Information & Electrical Technologies Division of ASABE in November 2006. The authors are Alejandro Clausse, Professor, Fernando J. Mayorano, Graduate Student, Aldo J. Rubiales, Graduate Student, and Victor A. Herrero, Professor, Universidad Nacional del Centro and CNEA-CONICET, Argentina Corresponding author: Clausse Alejandro, Pinto 399, 7000 Tandil, Argentina; phone: +54-2293-439690; fax: +54-2293-439690; e-mail: clausse@exa.unicen.edu.ar. assess the growth process through transition probabilities governed by environmental parameters. The resulting model predicts the final fruit size distribution based on statistical samplings and existing climatic data. MATERIALS AND METHODS MARKOVIAN MODEL OF FRUIT GROWTH Markovian chains are frequently used to represent time evolving processes involving certain inherent randomness (Papoulis , 1991). A classical example of a Markov chain is the random walk, representing the position of an individual that in every step  called transition  moves to a neighbor cell based in certain a-priori direction probability. Other typical examples are renewal processes (Chiang , 1980). Fruit growth presents several stochastic elements that can be modeled with Markov chains. There are a number of random factors affecting the growth of individual fruits, such as local weather variations, plant location in the farm, soil humidity, and insect’s action. Considering that the annual set of fruits handled by companies is very large, it is reasonable to represent the population of a given variety by means of a fruit size distribution that changes smoothly from bloom to harvest. Obviously this type of representation will be more precise if fruit populations corresponding to a homogeneous region are considered. Since the fruit size is a continuous variable, a discretiza- tion method should be introduced to be able to model the problem with a discrete Markov chain. In order to do that, the size domain (0, 8) is partitioned in a finite number of ranges. This is a classification procedure regularly applied in the fruit industry for commercialization and packing purposes. A probability p i that the size of a fruit picked at random at a certain time is within the i-range can be then defined. The distribution p i will be different at different growth stages (e.g., 40 and 80 d after bloom). This change can be written as a mathematical relation between past (t 1 ) and future (t 2 ) distributions, that is: ) ( ) ( 1 2 t p M t p = ³ ³ (1) where T