Deciding Whether Deterministic Reoptimization Suffices

Use the Value of the Stochastic Solution (VSS) to decide whether a stochastic optimization model is worth building, or whether a deterministic mean-value model — solved repeatedly as data arrives — suffices. Apply both relative (%VSS) and cost-benefit criteria.

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The Relative VSS Criterion

The Value of the Stochastic Solution is the expected penalty for using the deterministic mean-value first-stage solution instead of the true stochastic optimum. For a minimization problem,

\text{VSS} = \text{EEV} - \text{RP} \ge 0,

where $\text{RP}$ is the optimal value of the recourse problem and $\text{EEV}$ is the expected cost of implementing the expected-value solution.

When $\text{VSS}$ is small, building and solving the full stochastic model buys us very little, and we can get away with deterministic reoptimization — solving the cheap mean-value problem and re-solving it as new data arrives.

To decide "how small is small enough," we compare $\text{VSS}$ to $\text{RP}$ :

\%\text{VSS} = \dfrac{\text{VSS}}{\text{RP}} \times 100\%.

Given a pre-specified tolerance $\varepsilon$ (often $1\%$ – $5\%$ ), deterministic reoptimization suffices when

\%\text{VSS} < \varepsilon.

Illustrative: if $\text{RP} = 1000$ and $\text{EEV} = 1020$ , then $\text{VSS} = 20$ and $\%\text{VSS} = 2\%$ . With $\varepsilon = 5\%$ , deterministic reoptimization suffices.