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STOCHASTIC PROGRAMMING
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Monte Carlo Sampling
In stochastic programming where one or more stochastic parameters have continuous or discrete but
infinite event space, there will be too many scenarios, thus making the model computationally
intractable. For such cases Monte Carlo sampling (also called pre-sampling) can be used to
approximate the problem to work with a finite scenario tree. As illustrated in the figure below, if the
model has a single stochastic parameter with a continuous distribution such as the Normal
Distribution; one can discretize the event space simply by generating N sample points and construct a
finite and tractable scenario tree. This is also true for discrete distributions with infinite event space
like the Poisson distribution.
Note:
Sampling a scenario tree prior to the optimization process is also called pre-sampling. This is
to distinguish this type of sampling from the one that is used during optimization process. In
LINDO API, sampling refers to pre-sampling unless otherwise is stated.
Note:
Since the point probability of each scenario in the original model is zero, it is customary to
set the probabilities of sampled scenarios to 1/N. However, the user can always define
customized sampling approaches to work with different scenario probabilities.
Given the parametric distribution of each stochastic parameter, LINDO API’s sampling routines can be
used to generate univariate samples from these distributions efficiently. The user has the option to use
antithetic-variates or Latin-hyper-square sampling to reduce the sample variance. See Appendix 8c at
the end of this chapter for a brief definition of these techniques. Appendix 8b gives a general account
of pseudo-random number generation in LINDO API.