On the number of stages in multistage stochastic programs

Abstract

Multistage stochastic programs serve to make sequential decisions conditional on a gradual realization of some stochastic process. The progressive arrival of new information divides a problem into stages. As the number of stages increases, however, the applicability of stochastic programming is halted by the curse of dimensionality. A commonly used approximation is obtained by replacing the random parameters by their expected values, which results in a problem with fewer stages. The present paper suggests metrics to support the choice of the number of stages: The value of an extended expected value (EV) problem which accounts for stochastic approximations, the Expected result of using the expected value solution (EEV), the value of the stochastic solution, and the values of two Wait-and-See (WS) approximations. We show that for linear minimization problems with random right-hand-side, the value of the EV problem provides a lower bound that improves with the number of stages. The same holds for the WS approximations for general linear minimization problems and with one of the approximations improving the EV bound. In contrast, we show that the upper bound provided by the EV solution may not improve with the number of stages. Finally, we define the marginal benefit of including an additional stage in an EV approximation and demonstrate how to use it in a heuristic. We apply our approach to a hedging problem and confirm that increasing the number of stages does not necessarily foster better decisions.

Additional results

Table 9 Results for the tracking problem with fixed asset returns and ${\tilde{G}}_t\approx T[45,55,65]$, $t=2,\ldots ,T$, varying the number of stages T in the problem and $T'$ in the approximation. $VSS^{1,T'}[\%]$ represents the percentage increase in objective value generated by the approximation and is computed as $100\times VSS^{1,T'}/RP^1$. $MSV^{1,T'}[\%]$ represents the percentage increase in the VSS by including stage $T'$ and is computed as $100\times MSV^{1,T'}/VSS^{1,T'-1}$. The remaining values are expressed in thousands of dollars

Table 10 Results for the tracking problem with fixed asset returns and ${\tilde{G}}_t\approx T[45(1+\xi )^{t-1},55(1+\xi )^{t-1},65(1+\xi )^{t-1}]$, $t=2,\ldots ,T$, and $\xi =6.485\%$, varying the number of stages T in the problem and $T'$ in the approximation. $VSS^{1,T'}[\%]$ represents the percentage increase in objective value generated by the approximation and is computed as $100\times VSS^{1,T'}/RP^1$. $MSV^{1,T'}[\%]$ represents the percentage increase in the VSS by including stage $T'$ and is computed as $100\times MSV^{1,T'}/VSS^{1,T'-1}$. The remaining values are expressed in thousands of dollars

We provide additional results for problems with longer planning horizons, namely $T=10,12$ and 15 stages, for the case with fixed asset returns and for approximations with up to $T'=10$ stages. Table 9 shows the results for $\kappa =10$ and Table 10 with $\kappa =10$ and a target growth of $6.485\%$ per stage. To be able to solve the original problem, the scenario trees are obtained by sampling two realizations of the random target per stage. The resulting scenario trees count $2^{T-1}$ scenarios, with $T=10,12,15$.