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Abstract

Different types of crops are planted in winter grain regions. These crops can be planted in even more combinations. Crops that are planted on the same land in successive years do influence one another. In this situation the problem of which sequence of crops to plant arises. If appropriate assumptions are made, this problem can be solved with linear programming (LP). The complexity of this problem, however, increases dramatically with an increase in the number of crops as well as the number of years considered. Another problem with this formulation is how to choose the boundary conditions. Furthermore the solution to this problem is degenerate. In practice it is found that farmers prefer a cycle of crops in a rotary crop system, because it simplifies the management of the system. If the assumption is made that a farmer uses a cycle of at most three years, the problem of the boundary conditions is solved. If the dual formulation of this problem is considered it is evident that the problem has a specific structure. The structure of the problem as well as the fact that the solution to the problem is highly degenerate can be exploited to simplify the solution of the problem substantially. If the correct selection of solutions from this degenerate dual problem is chosen this selection can be used to formulate a simplified model in terms of strategies for the farmer. If the problem is formulated in terms of these strategies, the problem is further reduced to an LP with one constraint (the knapsack problem), which makes it possible to solve this problem by hand for a relative large number of crops in the rotary crop system. These strategies can also be used to solve a more generalised problem. Consider nature to be a player in Game Theory. The strategies of nature can be defined in many ways. The amount of rain per annum, for example, could be strategies used by nature to play a game against the farmer. The farmer in return can then use his strategies and the knowledge of Game Theory to maximise his profit. The problem can be generalised further if it is formulated in terms of Game Theory. In this generalisation the Game Theory model is expanded to incorporate a number of different situations with the use of extra constraints. The first of these situations is to incorporate other enterprises with the rotary crop system. The farmer could, for instance want the solution to contain at least a certain amount of feed for his cattle. The second situation is when the farmer has knowledge of the weather patterns, which implies for the Game Theory model that the farmer has knowledge of his opponent's strategies. Both these situations are incorporated in the model. Finally, a case study is presented in which the use of the model is demonstrated. Relevant data from the Swartland region are used. A dry year, an average year and a wet year were used as strategies for nature. The solutions drawn from these data were confirmed with a farmer from the region, who agreed that the solutions appear to be reasonable if it is compared to what is found in practice. The most frequently chosen strategies were the ones that contain wheat and clover. The solutions indicate that a farmer should plant less clover and more grain if it is a dry year, whilst it is better to increase the amount of clover planted if the rainfall is higher.

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