Thesis (M.A.)--Boston University / Probability problems in which a time parameter is involved are known as stochastic processes. The simplest time dependent stochastic processes are those in which the probabilities of a system changing to various states are solely dependent upon the present state of the system. These processes are known as Markov processes, or for the case where only discrete time intervals are considered, as Markov chains. A Markov chain may be completely defined by the matrix of its transition probabilities. This matrix is called a stochastic matrix and is characterized by the facts that it is a square matrix, that the elements of each column sum to one and that all the elements are non-negative. An important consideration in most Markov chain problems is the effect of a number of transitions as defined by the stochastic matrix. Performing this operation requires determining the higher powers of the stochastic matrix. Two modal matrices are defined, where k is the matrix of the column characteristic vectors of the stochastic matrix and K is the matrix of the row characteristic vectors. It is shown that with proper normalization of these vectors, the stochastic matrix P is equal to kAK, where A is the matrix of the characteristic roots along the diagonal and zeroes elsewhere. .The higher powers of the stochastic matrix, Pm, are then found to be equal to kAmk. The stochastic matrix is found always to have a characteristic root one, and all the other roots are shown to be less than one in absolute value. The limiting transition matrix P ∞ is found to have identical columns, each consisting of the characteristic column vector associated with the characteristic root one. The limiting distribution is the same vector and is independent of the initial conditions.[TRUNCATED]

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Abstract This article reports on the mechanisms of business-to-business regulation as governance approach in an agricultural supply chain. The article explores the activities of NGOs, government and business in promoting sustainable production. It focuses on the requirements set for producers in agro-food chains and the effects of NGO perceptions on these issues. The subsequent business standards set by retailers and other market role-players and the responses of producers are reported and the responsibility for achieving desired outputs is discussed. The research methodology included open interviews with various stakeholders in the chain, as well as structured questionnaires to determine practices on farms. From the fi ndings it is clear that perceptions of European stakeholders differ substantially from the reality found in South Africa. This has consequences for producers, from both a social and a fi nancial perspective. Understanding the role and function of NGOs, market roleplayers and national government in the governance of sustainable development in the South African–European table grape supply chain contributes to identifying barriers and opportunities to achieving sustainable practices in global supply chains. Copyright © 2009 John Wiley & Sons, Ltd and ERP Environment.

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