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Extra resources for Introduction to Operations Research
90. 68. Next we shall calculate F1,2,3(A) in order to discover the optimal distribution when A millions are invested in I, 11, and I11 regions. 3 gives the results. 35 million. 4. 5. 26 2. HOW SHOULD INVESTMENTS BE DISTRIBUTED ? 1(A), F3,1,2(A)? 81 decides to invest. Here we must again stress that these results depend entirely on the validity of the first table. A useful graph is given in Fig. 2, which shows how the marginal optimal profit decreases as a function of A , and it can be stated ) 1 2 3 4 5 6 7 FIG.
38. As N increases, the difference P,(N) - P,(N) approaches zero, as we would readily discover if our calculations were pursued over a greater period than 20 weeks. 6 6 T h e method to be used will be found in Howard and in Kaufmann and Cruon (see the Bibliography for Chapter 2). This small increase of profit will enable him, for example, to take his family for lunch to the floating garden of Xochimilco, a pleasant reward for a more scientific management. MARKOV CHAIN-PROBLEM OF SEQUENTIAL DECISIONS 45 Thus, by associating the theory of Markov decision chains with dynamic programming (it should be observed that in passing from the management for one week to that for two weeks, then for three weeks, and so on, we have strictly applied Bellman's principle of optimality, as defined in Chapter 2), it is possible to treat numerous problems of management with random influences and discover the optimal policy or policies.
Whether the first week was a success or a failure, a state of equilibrium rapidly develops and the probabilities of success or failure remain constant. Manuel himself had discovered this state of equilibrium and had found satisfaction in the discovery. All he needed to support himself and his extensive family was that over a fairly large number of weeks, the probability of success should be appreciably greater than the probability of failure. It is extremely important to realize the possibility of a state of equilibrium existing in an economic system, whether we are considering a small-scale manufacturer of puppets or a nation.