Question:

An oil company produces oil from two wells. Well 1 can produce up to 150,000 barrels per day, and well 2 can produce up to 200,000 barrels per day. It is possible to ship oil directly from the wells to the company's customers in Los Angeles and New York. Alternatively, the company could transport oil to the ports of Mobile and Galveston and then ship it by tanker to New York or Los Angeles, respectively. Los Angeles requires 160,000 barrels per day, and New York requires 140,000 barrels per day. The costs of shipping 1000 barrels between various locations are shown in the file P14_88.xlsx, where a blank indicates shipments that are not allowed. Determine how to minimize the transport costs in meeting the oil demands of Los Angeles and New York.

Consider a Markov chain {Xn, n = 0, 1, . .. } on the state space S = {0, 1, 2}. Suppose that the Markov chain has the transition matrix 0 1 - P = 1 0 0 N/H 1. Show that the state space is irreducible. 2. Show that the Markov chain is periodic. Find the period of the Markov chain. 3. Let h denote a stationary mass of the Markov chain. Find h(x) for all x E S.For each of the following transition matrices, determine whether the Markov chain with that transition matrix is regular: (1) Is the Markov chain whose transition matrix whose transition matrix is 1 0.2 0.8 0 0 regular? (Yes or No) (2) Is the Markov chain whose transition matrix whose transition matrix is 0 1 0.1 0.4 0.5 0 1 regular? (Yes or No) (3) Is the Markov chain whose transition matrix whose transition matrix is 0 0 0.8 0 0.2 0.8 0.2 0 regular? (Yes or No) (4) Is the Markov chain whose transition matrix whose transition matrix is 0 1 0.8 0 0.2 0 1 0 regular? (Yes or No)Consider a standard chessboard with an 8 x 8 grid of possible locations. We define a Markov chain by randomly moving a single chess piece on this board. The initial location Xo is sampled uniformly among the 82 = 64 squares. At time t, the piece then chooses Xt+1 by sampling uniformly from the set of legal moves given its current location Xt. For a description of legal chess moves, see: http://en. wikipedia. org/wiki/Rules_of_chess#Basic_moves. a) Suppose the chess piece is a king, which can move to any of the 8 adjacent squares. Is the Markov chain irreducible? Is the Markov chain aperiodic? b) Suppose the chess piece is a bishop. Is the Markov chain irreducible? Is the Markov chain aperiodic? c) Suppose the chess piece is a knight. Is the Markov chain irreducible? Is the Markov chain aperiodic