A node formulation for multistage stochastic programs with endogenous uncertainty

Abstract

This paper introduces a node formulation for multistage stochastic programs with endogenous (i.e., decision-dependent) uncertainty. Problems with such structure arise when the choices of the decision maker determine a change in the likelihood of future random events. The node formulation avoids an explicit statement of non-anticipativity constraints and, as such, keeps the dimension of the model sizeable. An exact solution algorithm for a special case is introduced and tested on a case study. Results show that the algorithm outperforms a commercial solver as the size of the instances increases.

Appendices

A Notation table

Table 4 Notation of problem (1)

B A big-M reformulation

In this appendix a big-M reformulation that linearizes model (1) is introduced. In addition to the notation introduced in Sect. 2, let $M_{nd}\in {\mathbb {R}}^1$ be a suitably high constant. The linearized MSPEU is thus

$$\begin{aligned}&\max r_0^Tx_{0}+\mathop {\sum }\limits _{{d\in {\mathcal {D}}_0}}q_{0d}\delta _{0d}+\theta _0 \end{aligned}$$

(7a)

$$\begin{aligned}&\text {s.t.} \mathop {\sum }\limits _{{d\in {\mathcal {D}}_n}}\delta _{nd} = 1&n\in {\mathcal {N}}, \end{aligned}$$

(7b)

$$\begin{aligned}&A_nx_n + \mathop {\sum }\limits _{{d\in {\mathcal {D}}_n}}B_{nd}\delta _{nd}+ C_{a(n)}x_{a(n)} + \mathop {\sum }\limits _{{d\in {\mathcal {D}}_{a(n)}}}D_{a(n),d}\delta _{a(n),d}= h_n&\quad n\in {\mathcal {N}}, \end{aligned}$$

(7c)

$$\begin{aligned}&\theta _n\le \mathop {\sum }\limits _{{m\in {\mathcal {N}}_{nd}}}\pi _m\left( r_m^Tx_{m}+\mathop {\sum }\limits _{{d\in {\mathcal {D}}_m}}q_{md}\delta _{md}+\theta _m) + M_{nd} (1 - \delta _{nd}\right)&\quad n\in {\mathcal {N}}{\setminus }{{\mathcal {N}}_T}, d\in {\mathcal {D}}_n \end{aligned}$$

(7d)

$$\begin{aligned}&\theta _{n} = \Theta _n&\quad n\in {\mathcal {N}}_{T}, \end{aligned}$$

(7e)

$$\begin{aligned}&x_n \in X_{t(n)}&\quad n\in {\mathcal {N}}, \end{aligned}$$

(7f)

$$\begin{aligned}&\delta _{nd} \in \{0,1\}&\quad n\in {\mathcal {N}},d\in {\mathcal {D}}_n, \end{aligned}$$

(7g)

$$\begin{aligned}&\theta _{n} \in {\mathcal {R}}&\quad n\in {\mathcal {N}}. \end{aligned}$$

(7h)

Note, particularly, that constraints (7d) are equivalent to (1d). Consider a given node n, other than a leaf node. Observe that only for one distribution d there will be a $\delta _{nd}$ which takes value one at n (see (7b)). For the same n and for the same d, the second term on the right-hand-side of (7d) will be zero (i.e., the big-M will not be enforced), and the resulting right-hand-side will be the most binding among the $|{\mathcal {D}}_n|$ constraints for node n. Since we are maximizing, at optimality $\theta _n$ will take value of the expectation according to the distribution d for which $\delta _{nd}=1$, as it happens in model (1).

C Finding big-M values

An efficient implementation of model (7) requires tight big-M values. Observe that, for $n\in \mathcal {N}{\setminus }\mathcal {N}_T$ and $d\in \mathcal {D}_n$, constant $M_{nd}$ must be a valid upper bound for constraints (7d), that is:

$$\begin{aligned} \theta _{n} - \mathop {\sum }\limits _{{m\in {\mathcal {N}}_{nd}}}\pi _m\left( r_m^Tx_{m}+\mathop {\sum }\limits _{{d\in {\mathcal {D}}_m}}q_{md}\delta _{md}+\theta _m\right) \le M_{nd} \end{aligned}$$

Let us introduce $\phi _{nd}$ to represent the expectation at the children of node n for distribution d, that is:

$$\begin{aligned} \phi _{nd} = \mathop {\sum }\limits _{{m\in {\mathcal {N}}_{nd}}}\pi _m\left( r_m^Tx_{m}+\mathop {\sum }\limits _{{d\in {\mathcal {D}}_m}}q_{md}\delta _{md}+\theta _m\right) \end{aligned}$$

Consider the numerical example shown in Table 5 for a given node n and three possible distributions d. The table reports the highest and lowest values the expectation at the following stage can take for each possible distribution, and the corresponding value of $\theta _n$ should a specific distribution be chosen. When choosing $M_{n,d_1}$, notice that the maximum value $\theta _n$ can reach for other distributions is 9, and that the least value $\phi _{n,d_1}$ can reach is 5. Therefore, we can set $M_{n,d_1}=9-5$. In fact, if distribution $d_2$ is chosen, $\theta _n$ will be at most 9 and $\phi _{n,d_1}$ at least 5, thus adding 4 to $\phi _{n,d_1}$ will ensure that $\theta _n$ is correctly set to 9. Similarly, we choose $M_{n,d_2}=6$ as the highest value $\theta _n$ can take if $d_2$ is not selected is 10, while the least value of $\phi _{n,d_2}$ is 4. Finally, with a similar reasoning we can set $M_{n,d_3}=7$.

Table 5 Numerical example for the calculation of constants $M_{nd}$

From the example in Table 5 we understand that finding values for $M_{nd}$ amounts to finding highest values for $\phi _{nd}$ and differences $\theta _n - \phi _{nd}$. In what follows we illustrate how these values can be found for $t=T-1$ in Section C.1 and $t=T-2,\ldots ,1$ in Section C.2.

1.1 C.1 Big-M values for stages $t=T-1$

We start at the second-last stage, $t=T-1$. Our task is that of finding, for each node ${\bar{n}}\in {\mathcal {N}}_{T-1}$ and for each distribution ${\bar{d}}\in {\mathcal {D}}_{{\bar{n}}}$, a constant $M_{{\bar{n}}{\bar{d}}}$ which is slightly higher than the highest difference $\theta _{{\bar{n}}}-\phi _{{\bar{n}}{\bar{d}}}$, where again

$$\begin{aligned} \phi _{{\bar{n}}{\bar{d}}} = \mathop {\sum }\limits _{{m\in {\mathcal {N}}_{{\bar{n}}{\bar{d}}}}}\pi _m\left( r_m^Tx_{m}+\mathop {\sum }\limits _{{d\in {\mathcal {D}}_m}}q_{md}\delta _{md}+\theta _m\right) \end{aligned}$$

Now, the highest difference can be found solving the following optimization problem:

$$\begin{aligned}&M_{{\bar{n}}{\bar{d}}}^*=\max \theta _{{\bar{n}}} - \sum _{m\in {\mathcal {N}}_{{\bar{n}}{\bar{d}}}}\pi _m\left( r_m^Tx_{m}+\mathop {\sum }\limits _{{k\in {\mathcal {D}}_m}}q_{mk}\delta _{mk}+\Theta _m\right) \end{aligned}$$

(8a)

$$\begin{aligned}&\text {s.t.} \mathop {\sum }\limits _{{d\in {\mathcal {D}}_n}}\delta _{nd} = 1&\quad n\in {\mathcal {N}}, \end{aligned}$$

(8b)

$$\begin{aligned}&A_nx_n + \mathop {\sum }\limits _{{d\in {\mathcal {D}}_n}}B_{nd}\delta _{nd}+ C_{a(n)}x_{a(n)} +\mathop {\sum }\limits _{{d\in {\mathcal {D}}_{a(n)}}}D_{a(n),d}\delta _{a(n),d}= h_n&\quad n\in {\mathcal {N}}, \end{aligned}$$

(8c)

$$\begin{aligned}&\theta _{{\bar{n}}} \le \Theta _{{\bar{n}}{\bar{d}}}^*,&\quad \end{aligned}$$

(8d)

$$\begin{aligned}&x_n \in X_{t(n)}&\quad n\in {\mathcal {N}}, \end{aligned}$$

(8e)

$$\begin{aligned}&\delta _{nd} \in \{0,1\}&\quad n\in {\mathcal {N}},d\in {\mathcal {D}}_n, \end{aligned}$$

(8f)

$$\begin{aligned}&\delta _{{\bar{n}}{\bar{d}}} = 0&\end{aligned}$$

(8g)

Problem (8) consists of finding the feasible solution to problem (7) which yields the highest value for the left-hand-side of constraint (7d) for ${\bar{n}}$ and ${\bar{d}}$. The following two elements must be noted in (8). Constraints (7d) of the original problem, which determine the correct expectations at the stages before $T-1$, are not included as they are irrelevant for stage $T-1$. The second element to note is constraint (8d) which sets an upper bound $\theta _{{\bar{n}}}$. This upper bound represents the highest value $\theta _{{\bar{n}}}$ can take for the distributions other than ${\bar{d}}$. This value can, in turn, be obtained solving optimization problems. The highest expectation for stage T, given distribution $d'\in {D}_{{\bar{n}}}$ is the optimal value to problem (9):

$$\begin{aligned} \Phi _{{\bar{n}}d'}^*&=\max ~ \sum _{m\in {\mathcal {N}}_{{\bar{n}}d'}}\pi _m\left( r_m^Tx_{m}+\mathop {\sum }\limits _{{k\in {\mathcal {D}}_m}}q_{mk}\delta _{mk}+\Theta _m\right) \end{aligned}$$

(9a)

$$\begin{aligned} \text {s.t.}&\mathop {\sum }\limits _{{d\in {\mathcal {D}}_n}}\delta _{nd} = 1&\quad n\in {\mathcal {N}}, \end{aligned}$$

(9b)

$$\begin{aligned}&A_nx_n + \mathop {\sum }\limits _{{d\in {\mathcal {D}}_n}}B_{nd}\delta _{nd}+ C_{a(n)}x_{a(n)} + \mathop {\sum }\limits _{{d\in {\mathcal {D}}_{a(n)}}}D_{a(n),d}\delta _{a(n),d}= h_n&\quad n\in {\mathcal {N}}, \end{aligned}$$

(9c)

$$\begin{aligned}&x_n \in X_{t(n)}&\quad n\in {\mathcal {N}}, \end{aligned}$$

(9d)

$$\begin{aligned}&\delta _{nd} \in \{0,1\}&\quad n\in {\mathcal {N}},d\in {\mathcal {D}}_n, \end{aligned}$$

(9e)

$$\begin{aligned}&\delta _{{\bar{n}}d'} = 1&\end{aligned}$$

(9f)

Therefore, when calculating $M_{{\bar{n}}{\bar{d}}}$, the upper bound $\Theta _{{\bar{n}}}^*$ in (8d) is given by:

$$\begin{aligned} \Theta _{{\bar{n}}{\bar{d}}}^* = \max _{ d\in \mathcal {D}_{{\bar{n}}} : d\ne {\bar{d}}}\Phi _{{\bar{n}}d}^* \end{aligned}$$

Clearly, solving problems (8) and (9) amounts to solving integer programs of size comparable with the original problem (7). However, the tightest $\Theta _{{\bar{n}}{\bar{d}}}^*$ and $M_{{\bar{n}}{\bar{d}}}$ are not necessary, and higher values would still provide correct results. A suitable value for $M_{{\bar{n}}{\bar{d}}}$ can be obtained by solving any relaxation of problem (8), yielding $M_{{\bar{n}}{\bar{d}}}^{R}$, and (9), yielding $\Phi _{{\bar{n}}{\bar{d}}}^{R}$ and in turn $\Theta _{{\bar{n}}{\bar{d}}}^{R}$. As an example, one might solve the linear programming relaxation of problems (8) and (9) or, if the size of the problems is excessively high, one might choose to relax constraints (8c) and (9c) for some stages. Finally, since the procedure outlined might return negative values for some $M_{nd}$, we set $M_{nd}=\max \{0,M_{nd}^{R}\}$ to reduce, when possible, high big-M absolute values. The procedure is summarized in Algorithm 1.

1.2 C.2 Big-M values for stages $t=T-2,\ldots ,1$

Once constants $M_{nd}$ are available for every $n\in \mathcal {N}_{T-1}$ and $d\in \mathcal {D}_n$ we can proceed in a similar way to calculate big-Ms for stages $T-2,\ldots ,1$. Given a stage ${\bar{t}}\in \{T-2,\ldots ,1\}$, a node at that stage, ${\bar{n}}\in \mathcal {N}_{{\bar{t}}}$, and distribution available at that node ${\bar{d}}\in \mathcal {D}_{{\bar{n}}}$, the tightest value of constant $M_{{\bar{n}}{\bar{d}}}$, namely $M_{{\bar{n}}{\bar{d}}}^*$, is

$$\begin{aligned}&M_{{\bar{n}}{\bar{d}}}^*=\max \theta _{{\bar{n}}} - \sum _{m\in {\mathcal {N}}_{{\bar{n}}{\bar{d}}}}\pi _m\left( r_m^Tx_{m}+\mathop {\sum }\limits _{{k\in {\mathcal {D}}_m}}q_{mk}\delta _{mk}+\theta _m\right) \end{aligned}$$

(10a)

$$\begin{aligned}&\text {s.t.}\mathop {\sum }\limits _{{d\in {\mathcal {D}}_n}}\delta _{nd} = 1&\quad n\in {\mathcal {N}}, \end{aligned}$$

(10b)

$$\begin{aligned}&A_nx_n + \mathop {\sum }\limits _{{d\in {\mathcal {D}}_n}}B_{nd}\delta _{nd}+ C_{a(n)}x_{a(n)} +\mathop {\sum }\limits _{{d\in {\mathcal {D}}_{a(n)}}}D_{a(n),d}\delta _{a(n),d}= h_n&\quad n\in {\mathcal {N}}, \end{aligned}$$

(10c)

$$\begin{aligned}&\theta _{n} \ge \sum _{m\in {\mathcal {N}}_{nd}}\pi _m\left( r_m^Tx_{m}+\mathop {\sum }\limits _{{k\in {\mathcal {D}}_m}}q_{mk}\delta _{mk}+\theta _m\right) \nonumber \\&\quad - M_{nd} (1 - \delta _{nd}) t={\bar{t}}+1,\ldots ,T-1,&\quad n\in \mathcal {N}_t,d\in {\mathcal {D}}_{n}, \end{aligned}$$

(10d)

$$\begin{aligned}&\theta _{n} = \Theta _{n}&\quad n\in \mathcal {N}_T \end{aligned}$$

(10e)

$$\begin{aligned}&\theta _{{\bar{n}}} \le \Theta _{{\bar{n}}{\bar{d}}}^*,&\quad \end{aligned}$$

(10f)

$$\begin{aligned}&x_n \in X_{t(n)}&\quad n\in {\mathcal {N}}, \end{aligned}$$

(10g)

$$\begin{aligned}&\delta _{nd} \in \{0,1\}&\quad n\in {\mathcal {N}},d\in {\mathcal {D}}_n, \end{aligned}$$

(10h)

$$\begin{aligned}&\delta _{{\bar{n}}{\bar{d}}} = 0&\end{aligned}$$

(10i)

Problem (10) consists of finding the feasible solution to problem (7) which yields the highest value of the left-hand-side of constraint (7d) for node ${\bar{n}}$ and distribution ${\bar{d}}$. Notice that, unlike in problem (8), problem (10) includes constraints (10d) which are necessary to ensure that $\theta _n$ values are set to the lowest expectation for all stages between ${\bar{t}}$ and $T-1$. Note that constant $M_{nd}$ in constraints (10d) is also an upper bound to the quantity $\theta _{n} - \sum _{m\in {\mathcal {N}}_{nd}}\pi _m\left( r_m^Tx_{m}+\sum _{k\in {\mathcal {D}}_m}q_{mk}\delta _{mk}+\theta _m\right)$ and can be thus set to the quantities determined at previous iterations. Also in this case, $\Theta _{{\bar{n}}}^*$ in (10f) represents the highest possible value $\theta _{{\bar{n}}}$ can take for distributions other than ${\bar{d}}$. The highest expectation for stage ${\bar{t}}+1$, given distribution $d'\in {D}_{{\bar{n}}}$ is the optimal value to problem (11)

$$\begin{aligned}&\Phi _{{\bar{n}}d'}^*=\max \sum _{m\in {\mathcal {N}}_{{\bar{n}}d'}}\pi _m\left( r_m^Tx_{m}+\mathop {\sum }\limits _{{k\in {\mathcal {D}}_m}}q_{mk}\delta _{mk}+\theta _m\right) \end{aligned}$$

(11a)

$$\begin{aligned}&\text {s.t.} \mathop {\sum }\limits _{{d\in {\mathcal {D}}_n}}\delta _{nd} = 1&\quad n\in {\mathcal {N}}, \end{aligned}$$

(11b)

$$\begin{aligned}&A_nx_n + \mathop {\sum }\limits _{{d\in {\mathcal {D}}_n}}B_{nd}\delta _{nd}+ C_{a(n)}x_{a(n)} + \mathop {\sum }\limits _{{d\in {\mathcal {D}}_{a(n)}}}D_{a(n),d}\delta _{a(n),d}= h_n&\quad n\in {\mathcal {N}}, \end{aligned}$$

(11c)

$$\begin{aligned}&\theta _{n} \le \sum _{m\in {\mathcal {N}}_{nd}}\pi _m\left( r_m^Tx_{m}+\mathop {\sum }\limits _{{k\in {\mathcal {D}}_m}}q_{mk}\delta _{mk}+\theta _m\right) \nonumber \\&\quad +M_{nd} (1 - \delta _{nd}) t={\bar{t}}+1,\ldots ,T-1,&\quad n\in \mathcal {N}_t,d\in {\mathcal {D}}_{n}, \end{aligned}$$

(11d)

$$\begin{aligned}&\theta _{n} = \Theta _{n}&\quad n\in \mathcal {N}_T \end{aligned}$$

(11e)

$$\begin{aligned}&x_n \in X_{t(n)}&\quad n\in {\mathcal {N}}, \end{aligned}$$

(11f)

$$\begin{aligned}&\delta _{nd} \in \{0,1\}&\quad n\in {\mathcal {N}},d\in {\mathcal {D}}_n, \end{aligned}$$

(11g)

$$\begin{aligned}&\delta _{{\bar{n}}d'} = 1&\quad \end{aligned}$$

(11h)

Therefore, the upper bound $\Theta _{{\bar{n}}{\bar{d}}}^*$ in (10f) is given by:

$$\begin{aligned} \Theta _{{\bar{n}}}^* = \max _{d\in \mathcal {D}_{{\bar{n}}}:d\ne {\bar{d}}}\Phi _{{\bar{n}}d}^* \end{aligned}$$

Similarly to Section C.1, calculating the optimal $M_{{\bar{n}}{\bar{d}}}^*$ is cumbersome as well as not strictly necessary. Therefore, any computationally suitable relaxation to problems (10) and (11) can be adopted. The procedure for obtaining constants $M_{nd}$ for stages $T-2,\ldots ,1$ is sketched in Algorithm 2.

D Computation time for finding big-M values for the FTCP

Table 6 Average elapsed time in seconds for the computations of big-M values using the procedure in Sect. 4.2