By William Feller

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M, t = 1, . . 8) t yij ≤ ai (git + wit−∆i ), i = 1, . . , n, t = 1, . . 9) i=1 m j=1 w, x, y ≥ 0, where the expectation is taken with respect to the random vector ξ = (ξξ 2 , . . , ξ H ). Here, the elements forming ξ t are the demands, (dt1 , . . , dtk ), and the cost vectors, (rt , qt ). In some cases, ξ t can also contain the lifetimes Li , the delay factors ∆i , and the availability factors ai , depending on the elements deemed uncertain in the future. 9) is a convenient representation of the stochastic program.

T. 0 ≤ w ≤ w ,0≤x≤x . 10) In stochastic programming terms, this formulation gives the deterministic equivalent problem to the stochastic program for minimizing the current value for the design decision plus future reactions to deviations in the axle diameter. Standard optimization procedures can be used to solve 40 1. Introduction and Examples this problem. 94. The graphs of z as a function of w for x = x∗ and as a function of x for w = w∗ appear in Figures 8 and 9. 27)1/3 . FIGURE 8. 038 inches.

We can still compute the analytical expression of Q3 (x3 ) for the other situations. For example, if the surface x3 is such that the production exceeds the quota for any possible yield (l3 x3 > 6000), then the optimal second-stage decisions are simply w3 (ξξ ) = 6000, w4 (ξξ ) = t3 (ξξ )x3 − 6000, for all ξ . The second-stage value for a given ξ is now Q3 (x3 , ξ) = −216000 − 10(t3 (ξ)x3 − 6000) = −156000 − 10t3 (ξ)x3 , and the expected value is simply Q3 (x3 ) = −156000 − 10t¯3 x3 . 11) Similarly, if the surface devoted to sugar beets is so small that for any yield the production is lower than the quota, the second-stage value function is Q3 (x3 ) = −36t¯3 x3 .

### An introduction to probability theory and its applications by William Feller

by William

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