<figure><img src="image_9.png" /><figcaption>**Figure 8** Optimality gaps for the location-transportation budgeted-uncertainty instances, between $\bar{\nu}^{2S-LDR}$ and $\bar{\nu}_R^{NA}$.</figcaption></figure>

cutting-plane method for obtaining the dual bound $\bar{\nu}_R^{NA}$, using a sample of size $50(T - 2)$. For both algorithms, we stop when the optimality gap of the method falls below 5% or we reach the 10-hour time limit. Figure 8 shows the value of $\text{OPT}^N$, the optimality gap between $\nu^{2S-LDR}$ and $\bar{\nu}_R^{NA}$, for our 85 instances (detailed results are provided in Appendix B.2). In this figure, every bar represents an instance. All instances of a block (characterized by $T$, $(I, J)$ and $\alpha^d$) share the same parameters except $\alpha^u$, and the instances within a block appear in an increasing order of the value of their $\alpha^u$. Our methods return an average optimality gap of 17.4% among all the instances, with a minimum gap of 5.2% and a maximum gap of 29.2%. Given the strength of the primal bound in the scenario-tree instances, and the fact that the bound remains strong despite increasing the size of the tree, we believe that the larger gaps in most instances are caused by weak $\bar{\nu}_R^{NA}$ bounds. Since the value of $\bar{\nu}_R^{NA}$ depends on the sample on which it is solved, not having a large enough sample can lead to a poor bound. However, because of the bilinear form of the cutting-plane master problem, we were not able to solve the model with large samples using off-the-shelf commercial solvers. By employing specialized algorithms developed for bilinear problems, it could be possible to increase the size of the sample and improve the $\bar{\nu}_R^{NA}$ bound, which we leave for future research.

Next we analyze the performance of the algorithms in solving our instances. Solution times and number of iterations for both algorithms are given in Appendix B.2. Here, we study the final optimality gap of the algorithms (C&CG and the cutting-plane), that could have a negative impact on the quality of the bounds. Figure 9 shows that in all instances, the C&CG stops at a solution of the 2ARO model with less than 5% gap. In fact, the method proves to be quite powerful in detecting the significant scenarios for the 2ARO approximation, such that in 69 out of 85 instances it achieves this gap after only two iterations. On the other hand, in 39 instances the cutting-plane method is not able to reach the optimality gap of 5% within the time limit of 10 hours, meaning
costs are drawn from $f_i \in [100, 1000]$, $a_i \in [10, 100]$, $c_{ij} \in [1, 1000]$, respectively. Maximal capacity is set to $K_i = 2 \times 10^4$ based on preliminary experiments to make sure the instances are feasible. Customer demands are random parameters with support $[\mu_{jt}, (1 + \alpha^d)\mu_{jt}]$, where $\mu_{jt} \in [10, 500]$ and $\alpha^d \in \{0.1, 0.3, 0.5\}$ are given, with $\alpha^d$ a parameter controlling the variation among demand realizations of customer $j$. For this problem, we have generated 32 instances with $\alpha^d = 0.5$ over small scenario trees to compare the bounds with the optimal objective value. In addition, we have generated 83 instances with demands $d_{jt} = \mu_{jt} + \xi_{jt}\sigma_{jt}$, $j \in [J]$, $t \in [2, T]$, where $\xi$ belongs to the following budgeted uncertainty set:

$$ \Xi^T = \left\{ \xi \in \mathbb{R}_+^{J \times T-1} \mid \xi_{jt} \in [0, 1], j \in [J], t \in [2, T], \sum_{t \in [2, T]} \sum_{j \in [J]} \xi_{jt} \le \Gamma \right\}. $$

Parameter $\Gamma$ correlates the demands of all customers and stages together, which results in stagewise (temporal) dependence between the decision stages. We use $\Gamma = \alpha^u I$, with $\alpha^u \in \{0.1, 0.4, 0.7, 1\}$.

### 5.3.2. Scenario-Tree Instances
In solving the primal (2S-LDR) and exact models of the scenario-tree instances, we have used their respective extensive forms, while for computing our dual bound we implemented the cutting-plane method described in Section 4.3.

Table 2 presents the gap between the bound and the optimal value of the exact solution, $100(\frac{\nu^{(\cdot)} - \nu^*}{\nu^*})$ and $100(\frac{\nu^* - \nu^{(\cdot)}}{\nu^*})$ for the primal and dual problems, respectively. Results show that, among the group of instances with similar characteristics, as the branching factor or the number of stages increases, the $\nu^{LDR}$ bound gets noticeably worse. In contrast, $\nu^{2S-LDR}$ stays very close to the

<table><caption>**Table 2** Quality of the bounds for the scenario-tree instances of the location-transportation problem</caption><thead><tr><th>T</th><th>I</th><th>J</th><th>BR</th><th>|Ξ<sup>T</sup>|</th><th>ν<sup>LDR</sup></th><th>ν<sup>2S-LDR</sup></th><th>ν̄<sup>NA-DO</sup></th><th>ν<sup>PI</sup></th></tr></thead><tbody><tr><td rowspan="10">3</td><td rowspan="10">5</td><td rowspan="10">10</td><td>3</td><td>9</td><td>2.9%</td><td>0.0%</td><td>1.4%</td><td>4.7%</td></tr><tr><td>4</td><td>16</td><td>0.8%</td><td>0.0%</td><td>3.3%</td><td>3.4%</td></tr><tr><td>5</td><td>25</td><td>4.6%</td><td>0.0%</td><td>2.5%</td><td>13.2%</td></tr><tr><td>6</td><td>36</td><td>5.6%</td><td>0.0%</td><td>17.1%</td><td>27.5%</td></tr><tr><td>7</td><td>49</td><td>8.2%</td><td>0.0%</td><td>9.4%</td><td>9.4%</td></tr><tr><td>8</td><td>64</td><td>8.8%</td><td>0.0%</td><td>4.8%</td><td>14.6%</td></tr><tr><td>9</td><td>81</td><td>6.5%</td><td>0.0%</td><td>6.1%</td><td>6.1%</td></tr><tr><td>10</td><td>100</td><td>6.6%</td><td>0.0%</td><td>8.0%</td><td>10.2%</td></tr><tr><td rowspan="10">3</td><td rowspan="10">10</td><td rowspan="10">10</td><td>3</td><td>9</td><td>2.3%</td><td>0.0%</td><td>2.5%</td><td>13.2%</td></tr><tr><td>4</td><td>16</td><td>2.7%</td><td>0.1%</td><td>1.2%</td><td>8.4%</td></tr><tr><td>5</td><td>25</td><td>6.1%</td><td>0.0%</td><td>12.0%</td><td>19.3%</td></tr><tr><td>6</td><td>36</td><td>9.8%</td><td>0.0%</td><td>10.4%</td><td>22.6%</td></tr><tr><td>7</td><td>49</td><td>13.7%</td><td>0.1%</td><td>6.7%</td><td>14.1%</td></tr><tr><td>8</td><td>64</td><td>13.9%</td><td>0.3%</td><td>5.5%</td><td>5.5%</td></tr><tr><td>9</td><td>81</td><td>12.2%</td><td>0.2%</td><td>15.5%</td><td>15.5%</td></tr><tr><td>10</td><td>100</td><td>12.3%</td><td>0.2%</td><td>10.4%</td><td>19.0%</td></tr><tr><td rowspan="10">3</td><td rowspan="10">20</td><td rowspan="10">20</td><td>3</td><td>9</td><td>2.5%</td><td>0.7%</td><td>9.5%</td><td>11.3%</td></tr><tr><td>4</td><td>16</td><td>3.9%</td><td>0.5%</td><td>7.0%</td><td>7.0%</td></tr><tr><td>5</td><td>25</td><td>5.8%</td><td>0.5%</td><td>28.4%</td><td>28.4%</td></tr><tr><td>6</td><td>36</td><td>6.6%</td><td>0.5%</td><td>13.0%</td><td>19.2%</td></tr><tr><td>7</td><td>49</td><td>9.1%</td><td>0.5%</td><td>13.3%</td><td>31.3%</td></tr><tr><td rowspan="10">3</td><td rowspan="10">20</td><td rowspan="10">20</td><td>3</td><td>9</td><td>2.5%</td><td>0.7%</td><td>9.5%</td><td>11.3%</td></tr><tr><td>4</td><td>16</td><td>3.9%</td><td>0.5%</td><td>7.0%</td><td>7.0%</td></tr><tr><td>5</td><td>25</td><td>5.8%</td><td>0.5%</td><td>28.4%</td><td>28.4%</td></tr><tr><td>6</td><td>36</td><td>6.6%</td><td>0.5%</td><td>13.0%</td><td>19.2%</td></tr><tr><td>7</td><td>49</td><td>9.1%</td><td>0.5%</td><td>13.3%</td><td>31.3%</td></tr></tbody></table>

<table><thead><tr><th>T</th><th>I</th><th>J</th><th>BR</th><th>|Ξ<sup>T</sup>|</th><th>ν<sup>LDR</sup></th><th>ν<sup>2S-LDR</sup></th><th>ν̄<sup>NA-DO</sup></th><th>ν<sup>PI</sup></th></tr></thead><tbody><tr><td rowspan="10">4</td><td rowspan="10">5</td><td rowspan="10">10</td><td>3</td><td>27</td><td>8.0%</td><td>0.0%</td><td>2.6%</td><td>8.0%</td></tr><tr><td>4</td><td>64</td><td>6.8%</td><td>0.0%</td><td>2.0%</td><td>2.0%</td></tr><tr><td>5</td><td>125</td><td>13.5%</td><td>0.1%</td><td>5.0%</td><td>5.0%</td></tr><tr><td>3</td><td>27</td><td>11.9%</td><td>0.1%</td><td>17.3%</td><td>17.3%</td></tr><tr><td>4</td><td>64</td><td>13.3%</td><td>0.0%</td><td>10.2%</td><td>21.0%</td></tr><tr><td>5</td><td>125</td><td>13.1%</td><td>0.4%</td><td>10.0%</td><td>10.0%</td></tr><tr><td>4</td><td>20</td><td>20</td><td>3</td><td>27</td><td>9.0%</td><td>0.9%</td><td>11.7%</td><td>18.1%</td></tr><tr><td>4</td><td>20</td><td>20</td><td>3</td><td>64</td><td>11.9%</td><td>0.5%</td><td>9.8%</td><td>21.7%</td></tr><tr><td>5</td><td>5</td><td>10</td><td>3</td><td>81</td><td>14.7%</td><td>1.0%</td><td>4.4%</td><td>6.3%</td></tr><tr><td>5</td><td>10</td><td>10</td><td>3</td><td>81</td><td>16.8%</td><td>0.7%</td><td>8.2%</td><td>8.2%</td></tr></tbody></table>
<figure><img src="image_10.png" /><figcaption>Figure 9 Optimality gaps of the solution algorithms for $\nu^{2S-LDR}$ and $\bar{\nu}_R^{NA}$ at convergence/interruption.</figcaption></figure>

that we use the best lower bound on $\bar{\nu}_R^{NA}$ obtained at the end of 10 hours in calculating OPT$^N$. This can further contribute to a weak dual bound. This suggests that, in addition to having a difficult nonlinear master problem, the cutting-plane method itself requires algorithmic enhancements, such as the design of stronger cuts.

## 4.4. Robust Capital Budgeting Problem

In the capital budgeting problem, a company wants to invest in a subset of $I$ projects with uncertain cost and profit, subject to an initial budget of $B$ that can be increased by getting a loan. A variant of the two-stage problem is studied by Subramanyam et al. (2020). In the following, we formulate the multistage capital budgeting problem as an MSARO. Over a planning horizon of $T$ stages, let $x_{it}(\xi^t)$ be a binary decision variable taking the value of 1 if the company decides to invest in the project $i \in [I]$ at stage $t \in [T]$, with a cost of $c_{it}(\xi^t)$ and profit of $r_{it}(\xi^t)$. Further, let $L_t(\xi^t)$ be a continuous decision variable determining the amount of loan the company decides to get at stage $t \in [T]$ with a unit cost of $c_L\mu^{t-1}$, $\mu > 1$. The MSARO model is as follows:

$$ \max z \qquad (32a) $$

$$ \text{s.t.} \quad z \le \sum_{t \in [T]} \sum_{i \in [I]} r_{it}(\xi^t) x_{it}(\xi^t) - \sum_{t \in [T]} c_L \mu^{t-1} L_t(\xi^t) \qquad \xi^T \in \Xi \qquad (32b) $$

$$ B_t(\xi^t) - B_{t-1}(\xi^{t-1}) + C_{t-1}(\xi^{t-1}) - L_t(\xi^t) = 0 \qquad t \in [T], \xi^t \in \Xi^t \qquad (32c) $$

$$ \sum_{i \in [I]} c_{it}(\xi^t) x_{it}(\xi^t) = C_t(\xi^t) \qquad t \in [T], \xi^t \in \Xi^t \qquad (32d) $$

$$ B_t(\xi^t) - C_t(\xi^t) \ge 0 \qquad t \in [T], \xi^t \in \Xi^t \qquad (32e) $$

$$ x_t(\xi^t) \in \{0, 1\}^I, L_t(\xi^t) \in \mathbb{R}_+ \qquad t \in [T], \xi^t \in \Xi^t, \qquad (32f) $$
optimal value, with an average relative distance of 0.2% among all the instances (compared to 8.7% for $v^{LDR}$). We do not observe the same trend for the dual bounds, and their relative distance to the optimal value fluctuates even between two instances that only differ in the number of realizations. Nevertheless, $\bar{v}_R^{NA-DO}$ considerably outperforms PI, with an average improvement of 36.2%.

### 5.3.3. Budgeted-Uncertainty Instances
For the instances with the budgeted uncertainty set, we use, as benchmarks, $v^{LDR}$ as an upper bound and $v^{\Omega(LDR)}$ as a lower bound, obtained by solving problem (33) using only the binding realizations identified from the benchmark primal solution. We calculate $v^{2S-LDR}$ using the C&CG method described in Section 3.2 (detailed models are given in Appendix C.3.1). Similar to the scenario-tree instances, we use the cutting-plane method to obtain the dual bound $\bar{v}_R^{NA-DO}$. To do so, we use a sample of size at least $50(T-2)$, which includes both the binding realizations from the two-stage LDR solution and additional randomly generated realizations from the uncertainty set. For both algorithms, we stop when the optimality gap of the method falls below 5% or we reach the 10-hour time limit. Table 3 presents the optimality gaps between the benchmark bounds and the proposed bounds, respectively, for the 83 instances considered. Detailed results for each instance are provided in Appendix D.2. Our methods return an average optimality gap of 17.3% across all instances, compared to an average gap of 27% from the benchmarks. Given the strength of the two-stage LDR bound observed in the scenario-tree instances, and its resilience to increases in the size of the tree, it is likely that the dual bounds are further from the optimal value.

Figure 7 illustrates the improvement achieved from $\bar{v}_R^{NA-DO}$ over the benchmark dual bound, $v^{\Omega(LDR)}$. Each bar in the figure represents an individual instance. Instances are grouped into blocks based on their shared parameters $T$, $(I, J)$, and $\alpha^d$, and within each block, instances are arranged in ascending order of their $\alpha^u$ values. Across all instances, $\bar{v}_R^{NA-DO}$ demonstrates an average improvement of 7%. This figure also highlights that improved identification of critical realizations, guided by the two-stage LDR solution, and the subsequent solution of the discretized relaxation of the problem using these realizations, independently contribute to an average 3.6% enhancement of the dual bound. A key consideration here is that the value of $\bar{v}_R^{NA-DO}$ depends on the sample used for its computation. An insufficiently large sample can lead to a poor bound. However, because of the bilinear form of the cutting-plane master problem, we were not able to solve the model with large samples using off-the-shelf commercial solvers. By employing specialized algorithms developed for bilinear problems, it could be possible to increase the size of the sample and improve the $\bar{v}_R^{NA-DO}$ bound, which we leave for future research.

Analyzing the performance of the C&CG and cutting-plane algorithms in solving our instances provides further insight into the quality of the bounds. Solution times, number of iterations, and
where $B_t(\xi^t)$ is the amount of available funds at stage $t \in [T]$, and is determined by constraints (32c), while $C_t(\xi^t)$ is the expenditure calculated through constraints (32d), with initial values of $B_0(\xi^0) = B$ and $C_0(\xi^0) = 0$. Constraints (32e) make sure that the expenditure amount is bounded by the amount of available funds. The objective is to maximize the worst-case profit over the planning horizon, measured by constraints (32b).

**4.4.1. Problem Instances** Our instance generation loosely follows the procedure of Subramanyam et al. (2020) for the two-stage problem. For $T \in [3, 5]$ and $I \in \{5, 8, 10\}$, uncertain costs and profits $(c_{it}(\xi^t), r_{it}(\xi^t))$ belong to the following uncertainty set

$$ \Xi = \{ (c, r) \in \mathbb{R}_+^{I \times T} \times \mathbb{R}_+^{I \times T} \mid c_{it} = c_{it}^0 + \delta_{it}^c \sigma_{it}, r_{it} = \frac{c_{it}^0}{5} + \delta_{it}^r \sigma_{it}, \delta_{it}^c, \delta_{it}^r \in [0, 1], i \in [I], t \in [T] \}, $$

where the nominal costs $c^0$ and perturbations $\sigma$ are drawn uniformly from $[0, 10]^5$ and $[0, 5]^5$, respectively. Loan purchase cost is $c_L \mu^{t-1} = 0.8(1.2)^{t-1}$ per unit of loan. For every combinations of $T$ and $I$ we have considered two levels of initial budget to analyze the impact of this parameter that impacts the dependence between stages.

**4.4.2. Optimality Gap** Due to the presence of binary variables $x_t(\xi^t)$, with the existing methods in the literature of 2ARO we cannot calculate $\nu^{2S-LDR}$ exactly. Therefore, we solve an approximation of it using the K-adaptability method of Subramanyam et al. (2020) with $K=2$ and use the obtained bound $\nu^K$ in measuring the optimality gap. On the dual side, we study three options: (i) $\nu^\Omega$, the upper bound from solving the model (32) with a sample $|\Omega| = 100$, (ii) the NA bound $\bar{\nu}_R^{NA}$ with the same sample $\Omega$, (iii) the decomposable NA bound $\bar{\nu}_R^{DNA}$ with a sample of size 300. For each option, when the model is not solved to optimality within the time limit, the best valid bound is used in the calculations. Note that, if we can solve both options (i) and (ii) to optimality, we should expect a better bound from option (i).

Table 3 presents the optimality gaps of the capital budgeting instances between the bound $\nu^K$ and the three choices of upper bound. Results show that, even though the bound $\nu^\Omega$ should theoretically be at least as good as the bound $\bar{\nu}_R^{NA}$, on average the NA bound returns a better upper bound within the same time limit. This is testament to the difficulty of the multistage problem even when it is solved for a discrete set of scenarios. The decomposable NA bound $\bar{\nu}_R^{DNA}$ further improves the results by using a larger sample which is viable because of its superior computational performance. Note that, the best gaps in Table 3, ranging between 15% to 34%, are obtained from approximations over approximations on both primal and dual side. Accordingly, these rather large gaps can be attributed to both bounding methods. $\nu^K$ can be further improved by using larger values of $K$, whereas the NA bounds might benefit from implicit enumeration techniques in solving the relaxation models without resorting to sampling.
<table><caption>**Table 3** Optimality gaps for the budgeted-uncertainty instances, calculated using both benchmark and proposed bounding methods. For this sampling instance, OPT<sup>T</sup> is defined as the gap between $\nu^{LDR}$ and $\nu^{\Omega(LDR)}$. As before, OPT<sup>N</sup> represents the gap between $\nu^{2S-LDR}$ and $\bar{\nu}_{R}^{NA-DO}$.</caption><thead><tr><th>(T, I, J)</th><th>α<sup>d</sup></th><th>α<sup>u</sup></th><th>OPT<sup>T</sup></th><th>OPT<sup>N</sup></th><th>(T, I, J)</th><th>α<sup>d</sup></th><th>α<sup>u</sup></th><th>OPT<sup>T</sup></th><th>OPT<sup>N</sup></th></tr></thead><tbody><tr><td rowspan="12">(3,10,10)</td><td rowspan="4">0.1</td><td>0.1</td><td>9.0%</td><td>7.4%</td><td rowspan="4">0.1</td><td>0.1</td><td>22.4%</td><td>16.8%</td></tr><tr><td>0.4</td><td>20.8%</td><td>12.4%</td><td>0.4</td><td>32.1%</td><td>28.1%</td></tr><tr><td>0.7</td><td>32.0%</td><td>22.5%</td><td>0.7</td><td>15.2%</td><td>10.2%</td></tr><tr><td>1</td><td>30.6%</td><td>22.2%</td><td>1</td><td>31.2%</td><td>19.0%</td></tr><tr><td rowspan="4">0.3</td><td>0.1</td><td>15.3%</td><td>12.0%</td><td rowspan="4">0.3</td><td>0.1</td><td>28.5%</td><td>20.8%</td></tr><tr><td>0.4</td><td>27.0%</td><td>16.7%</td><td>0.4</td><td>37.7%</td><td>30.6%</td></tr><tr><td>0.7</td><td>30.3%</td><td>21.5%</td><td>0.7</td><td>42.9%</td><td>27.6%</td></tr><tr><td>1</td><td>29.1%</td><td>21.3%</td><td>1</td><td>24.3%</td><td>14.2%</td></tr><tr><td rowspan="4">0.5</td><td>0.1</td><td>17.8%</td><td>12.9%</td><td rowspan="4">0.5</td><td>0.1</td><td>33.0%</td><td>27.7%</td></tr><tr><td>0.4</td><td>28.3%</td><td>15.1%</td><td>0.4</td><td>22.2%</td><td>11.4%</td></tr><tr><td>0.7</td><td>32.7%</td><td>28.4%</td><td>0.7</td><td>27.7%</td><td>17.9%</td></tr><tr><td>1</td><td>26.8%</td><td>7.4%</td><td>1</td><td>33.1%</td><td>23.5%</td></tr><tr><td rowspan="12">(3,10,15)</td><td rowspan="4">0.1</td><td>0.1</td><td>32.8%</td><td>24.4%</td><td rowspan="4">0.1</td><td>0.1</td><td>19.4%</td><td>17.5%</td></tr><tr><td>0.4</td><td>31.5%</td><td>27.1%</td><td>0.4</td><td>32.1%</td><td>26.6%</td></tr><tr><td>0.7</td><td>12.1%</td><td>7.2%</td><td>0.7</td><td>38.7%</td><td>32.7%</td></tr><tr><td>1</td><td>31.5%</td><td>17.3%</td><td>1</td><td>21.7%</td><td>10.8%</td></tr><tr><td rowspan="4">0.3</td><td>0.1</td><td>12.8%</td><td>8.3%</td><td rowspan="4">0.3</td><td>0.1</td><td>11.1%</td><td>6.6%</td></tr><tr><td>0.4</td><td>35.3%</td><td>25.6%</td><td>0.4</td><td>19.8%</td><td>16.5%</td></tr><tr><td>0.7</td><td>17.2%</td><td>5.7%</td><td>0.7</td><td>22.6%</td><td>17.4%</td></tr><tr><td>1</td><td>26.2%</td><td>13.6%</td><td>1</td><td>38.1%</td><td>28.0%</td></tr><tr><td rowspan="4">0.5</td><td>0.1</td><td>32.4%</td><td>28.2%</td><td rowspan="4">0.5</td><td>0.1</td><td>24.9%</td><td>12.4%</td></tr><tr><td>0.4</td><td>25.8%</td><td>15.3%</td><td>0.4</td><td>23.3%</td><td>11.0%</td></tr><tr><td>0.7</td><td>29.4%</td><td>18.2%</td><td>0.7</td><td>31.4%</td><td>18.2%</td></tr><tr><td>1</td><td>29.7%</td><td>20.7%</td><td>1</td><td>31.2%</td><td>25.7%</td></tr><tr><td rowspan="12">(4,5,5)</td><td rowspan="4">0.1</td><td>0.1</td><td>14.3%</td><td>6.0%</td><td rowspan="4">0.1</td><td>0.1</td><td>17.2%</td><td>13.7%</td></tr><tr><td>0.4</td><td>33.2%</td><td>25.7%</td><td>0.4</td><td>24.5%</td><td>20.6%</td></tr><tr><td>0.7</td><td>27.7%</td><td>10.8%</td><td>0.7</td><td>34.2%</td><td>21.8%</td></tr><tr><td>1</td><td>12.7%</td><td>5.6%</td><td>1</td><td>20.3%</td><td>8.6%</td></tr><tr><td rowspan="4">0.3</td><td>0.1</td><td>13.5%</td><td>7.8%</td><td rowspan="4">0.3</td><td>0.1</td><td>28.9%</td><td>25.6%</td></tr><tr><td>0.4</td><td>25.3%</td><td>18.1%</td><td>0.4</td><td>32.5%</td><td>19.3%</td></tr><tr><td>0.7</td><td>23.6%</td><td>13.8%</td><td>0.7</td><td>39.1%</td><td>31.9%</td></tr><tr><td>1</td><td>26.2%</td><td>9.6%</td><td>1</td><td>37.5%</td><td>29.4%</td></tr><tr><td rowspan="4">0.5</td><td>0.1</td><td>31.8%</td><td>27.1%</td><td rowspan="4">0.5</td><td>0.1</td><td>36.6%</td><td>31.8%</td></tr><tr><td>0.4</td><td>22.6%</td><td>14.1%</td><td>0.4</td><td>39.8%</td><td>26.6%</td></tr><tr><td>0.7</td><td>29.8%</td><td>19.6%</td><td>0.7</td><td>39.2%</td><td>17.6%</td></tr><tr><td>1</td><td>29.8%</td><td>5.4%</td><td>1</td><td>41.6%</td><td>27.1%</td></tr><tr><td rowspan="12">(4,5,7)</td><td rowspan="4">0.1</td><td>0.1</td><td>15.7%</td><td>13.6%</td><td rowspan="4">0.1</td><td>0.1</td><td>28.9%</td><td>25.6%</td></tr><tr><td>0.4</td><td>24.0%</td><td>19.4%</td><td>0.4</td><td>32.5%</td><td>19.3%</td></tr><tr><td>1</td><td>38.3%</td><td>24.5%</td><td>0.4</td><td>32.5%</td><td>19.3%</td></tr><tr><td>0.1</td><td>23.2%</td><td>19.3%</td><td>0.4</td><td>32.5%</td><td>19.3%</td></tr><tr><td rowspan="4">0.3</td><td>0.1</td><td>23.2%</td><td>19.3%</td><td rowspan="4">0.3</td><td>0.1</td><td>28.9%</td><td>25.6%</td></tr><tr><td>0.4</td><td>30.8%</td><td>21.6%</td><td>0.4</td><td>32.5%</td><td>19.3%</td></tr><tr><td>0.7</td><td>10.2%</td><td>6.4%</td><td>0.4</td><td>32.5%</td><td>19.3%</td></tr><tr><td>1</td><td>30.7%</td><td>18.7%</td><td>0.4</td><td>32.5%</td><td>19.3%</td></tr><tr><td rowspan="4">0.5</td><td>0.1</td><td>24.2%</td><td>21.5%</td><td rowspan="4">0.5</td><td>0.1</td><td>36.6%</td><td>31.8%</td></tr><tr><td>0.4</td><td>39.5%</td><td>27.1%</td><td>0.4</td><td>39.8%</td><td>26.6%</td></tr><tr><td>0.7</td><td>32.7%</td><td>17.9%</td><td>0.7</td><td>39.2%</td><td>17.6%</td></tr><tr><td>1</td><td>41.0%</td><td>23.3%</td><td>1</td><td>41.6%</td><td>27.1%</td></tr></tbody></table>
<table><caption>**Table 3** Optimality gaps for the capital budgeting instances</caption><thead><tr><th>T</th><th>I</th><th>B</th><th>(ν<sup>Ω</sup> - ν<sup>K</sup>)%</th><th>(ν<sup>NA</sup><sub>B</sub> - ν<sup>K</sup>)%</th><th>(ν<sup>DNA</sup><sub>B</sub> - ν<sup>K</sup>)%</th></tr></thead><tbody><tr><td rowspan="6">3</td><td rowspan="2">5</td><td>50</td><td>17%</td><td>20%</td><td>16%</td></tr><tr><td>100</td><td>27%</td><td>25%</td><td>18%</td></tr><tr><td rowspan="2">8</td><td>100</td><td>24%</td><td>25%</td><td>24%</td></tr><tr><td>150</td><td>34%</td><td>21%</td><td>21%</td></tr><tr><td rowspan="2">10</td><td>100</td><td>33%</td><td>28%</td><td>16%</td></tr><tr><td>150</td><td>38%</td><td>29%</td><td>19%</td></tr><tr><td rowspan="6">4</td><td rowspan="2">5</td><td>100</td><td>35%</td><td>26%</td><td>20%</td></tr><tr><td>150</td><td>46%</td><td>33%</td><td>15%</td></tr><tr><td rowspan="2">8</td><td>150</td><td>43%</td><td>30%</td><td>34%</td></tr><tr><td>200</td><td>57%</td><td>37%</td><td>30%</td></tr><tr><td rowspan="2">10</td><td>150</td><td>48%</td><td>36%</td><td>31%</td></tr><tr><td>200</td><td>44%</td><td>33%</td><td>20%</td></tr><tr><td rowspan="6">5</td><td rowspan="2">5</td><td>200</td><td>65%</td><td>42%</td><td>20%</td></tr><tr><td>250</td><td>42%</td><td>39%</td><td>37%</td></tr><tr><td rowspan="2">8</td><td>250</td><td>40%</td><td>39%</td><td>31%</td></tr><tr><td>300</td><td>43%</td><td>40%</td><td>21%</td></tr><tr><td rowspan="2">10</td><td>250</td><td>55%</td><td>31%</td><td>31%</td></tr><tr><td>300</td><td>65%</td><td>31%</td><td>25%</td></tr></tbody></table>

## 5. Conclusion

Robust optimization models are built on a different premise than stochastic programming in the sense that they do not assume any knowledge about the probability distribution, focusing instead on optimizing the worst-case outcomes. We adapt two decision rules from the stochastic programming literature (namely two-stage LDRs and LDDRs) to MSARO, which result in two-stage approximation models for which we design several appropriate solution methods. More specifically, for two-stage linear decision rules we present sufficient conditions that make the approximation models amenable to the constraint-and-column generation method. In order to solve LDDR-induced models, we build a cutting-plane method for mixed-integer MSAROs, as well as a monolithic bilinear program for continuous problems. Our numerical experiments on the newsvendor, location-transportation and capital budgeting problems reveal that our methods considerably reduce the optimality gap from commonly used approaches in the literature.

## Acknowledgments

This work was supported by Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2018-04984] and Agence Nationale de la Recherche of France [Grant ANR-22-CE048-0018-01]. Computations were performed on the Niagara supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation; the Government of Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto.
<figure><img src="image_9.png" /><figcaption>**Figure 7** Improvement over the benchmark bound $\nu^{\Omega(\text{LDR})}$ in the budgeted-uncertainty instances. Here, $\nu^{\Omega(\text{LDR})}$ and $\nu^{\Omega(2\text{S-LDR})}$ are defined as the bounds obtained by solving problem (33) using only the binding realizations of $\nu^{\text{LDR}}$ and $\nu^{2\text{S-LDR}}$, respectively.</figcaption></figure>

final optimality gaps at the time limit (namely the termination gap) are provided in Appendix D.2. Our findings show that while our methods do require extra computational effort, they yield stronger bounds as a result. Another key aspect is that a large termination gap can negatively impact the quality of the bounds. Our results show that in all instances, the C&CG stops at a solution of the 2ARO model with less than 5% termination gap. In fact, the method proves to be quite powerful in detecting the significant realizations for the 2ARO approximation, such that in 69 out of 83 instances it achieves this gap after only two iterations. On the other hand, in 39 instances the cutting-plane method is not able to reach the optimality gap of 5% within the time limit of 10 hours albeit having less than 10% optimality gap in all instances with one exception of 13.4%. Consequently, for calculating $\text{OPT}^N$, we use the best lower bound on $\bar{\nu}_R^{\text{NA-DO}}$ obtained at the end of 10 hours. This can further contribute to an increased optimality gap. This suggests that, in addition to having a difficult nonlinear master problem, the cutting-plane method itself requires algorithmic enhancements, such as the design of stronger cuts.

## 5.4. Robust Capital Budgeting Problem

In the capital budgeting problem, a company wants to invest in a subset of *I* projects with uncertain cost and profit, subject to an initial budget of *B* that can be increased by getting a loan. A variant of the two-stage problem is studied by Subramanyam et al. (2020). In the following, we formulate the multistage capital budgeting problem as an MSARO. Over a planning horizon of *T* stages, let $x_{it}(\xi^t)$ be a binary decision variable taking the value of 1 if the company decides to invest in the
For the purpose of Open Access, a CC-BY public copyright licence has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.

<figure><img src="image_11.png" /></figure>

## References

ApS M (2022) MOSEK modeling cookbook.

Ardestani-Jaafari A, Delage E (2021) Linearized robust counterparts of two-stage robust optimization problems with applications in operations management. *INFORMS Journal on Computing* 33(3):1138–1161.

Arslan AN, Detienne B (2022) Decomposition-based approaches for a class of two-stage robust binary optimization problems. *INFORMS Journal on Computing* 34(2):857–871.

Bampou D, Kuhn D (2011) Scenario-free stochastic programming with polynomial decision rules. *50th IEEE Conference on Decision and Control and European Control Conference*, 7806–7812.

Ben-Tal A, El Ghaoui L, Nemirovski A (2009) Robust optimization. *Robust optimization* (Princeton university press).

Ben-Tal A, El Housni O, Goyal V (2020) A tractable approach for designing piecewise affine policies in two-stage adjustable robust optimization. *Mathematical Programming* 182(1):57–102.

Ben-Tal A, Goryashko A, Guslitzer E, Nemirovski A (2004) Adjustable robust solutions of uncertain linear programs. *Mathematical Programming* 99(2):351–376.

Ben-Tal A, Nemirovski A (1999) Robust solutions of uncertain linear programs. *Operations Research Letters* 25(1):1–13.

Bertsimas D, Brown DB, Caramanis C (2011a) Theory and applications of robust optimization. *SIAM review* 53(3):464–501.

Bertsimas D, Caramanis C (2010) Finite adaptability in multistage linear optimization. *IEEE Transactions on Automatic Control* 55(12):2751–2766.

Bertsimas D, de Ruiter FJ (2016) Duality in two-stage adaptive linear optimization: Faster computation and stronger bounds. *INFORMS Journal on Computing* 28(3):500–511.

Bertsimas D, Dunning I (2016) Multistage robust mixed-integer optimization with adaptive partitions. *Operations Research* 64(4):980–998.

Bertsimas D, Georghiou A (2015) Design of near optimal decision rules in multistage adaptive mixed-integer optimization. *Operations Research* 63(3):610–627.

Bertsimas D, Georghiou A (2018) Binary decision rules for multistage adaptive mixed-integer optimization. *Mathematical Programming* 167(2):395–433.
project $i \in [I]$ at stage $t \in [T]$, with a cost of $c_{it}(\xi^t)$ and profit of $r_{it}(\xi^t)$. Further, let $L_t(\xi^t)$ be a continuous decision variable determining the amount of loan the company decides to get at stage $t \in [T]$ with a unit cost of $c_L \mu^{t-1}, \mu > 1$. The MSARO model is as follows:

$$ \max z \qquad (34a) $$

$$ \text{s.t.} \quad z \le \sum_{t \in [T]} \sum_{i \in [I]} r_{it}(\xi^t) x_{it}(\xi^t) - \sum_{t \in [T]} c_L \mu^{t-1} L_t(\xi^t) \qquad \xi^T \in \Xi^T \qquad (34b) $$

$$ B_t(\xi^t) - B_{t-1}(\xi^{t-1}) + C_{t-1}(\xi^{t-1}) - L_t(\xi^t) = 0 \qquad t \in [T], \xi^t \in \Xi^t \qquad (34c) $$

$$ \sum_{i \in [I]} c_{it}(\xi^t) x_{it}(\xi^t) = C_t(\xi^t) \qquad t \in [T], \xi^t \in \Xi^t \qquad (34d) $$

$$ B_t(\xi^t) - C_t(\xi^t) \ge 0 \qquad t \in [T], \xi^t \in \Xi^t \qquad (34e) $$

$$ x_t(\xi^t) \in \{0, 1\}^I, L_t(\xi^t) \in \mathbb{R}_+ \qquad t \in [T], \xi^t \in \Xi^t, \qquad (34f) $$

where $B_t(\xi^t)$ is the amount of available funds at stage $t \in [T]$, determined by constraints (34c), while $C_t(\xi^t)$ is the expenditure calculated through constraints (34d), with initial values of $B_0(\xi^0) = B$ and $C_0(\xi^0) = 0$. Constraints (34e) bound the expenditure amount by available funds. The objective is to maximize the worst-case profit over the planning horizon, measured by constraints (34b).

**5.4.1. Problem Instances** Our instance generation follows the procedure of Subramanyam et al. (2020) for the two-stage problem. To incorporate the dynamic nature of multistage capital budgeting, project costs and profits are modeled as affine functions of evolving risk factors $\xi_t, t \in [2, T]$:

$$ c_{it}(\xi_t) = c_{it}^0 \left(1 + \Phi_{it}^\top \xi_t / 2\right), \quad r_{it}(\xi_t) = r_{it}^0 \left(1 + \Psi_{it}^\top \xi_t / 2\right), $$

where $c_{it}^0$ and $r_{it}^0$ represent the baseline cost and profit, respectively, assuming all risk factors are held at their neutral value of zero; $\Phi_{it}$ and $\Psi_{it}$ are factor loading vectors governing the sensitivity of cost and profit to deviations from the neutral state; and $c_{i1}$ and $r_{i1}$ are set to the nominal values. We consider four risk factors at each stage $t \in [2, T]$ such that $\xi_t \in [-1, 1]^4$. Thus, $\Phi_{it} \in \mathbb{R}^4$ and $\Psi_{it} \in \mathbb{R}^4$, quantify the influence of each of the four risk factors on the project's financial outcome. When sampling from $\mathbb{R}^4$, we ensure that $\Phi_{it}^\top e = \Psi_{it}^\top e = 1$ for all $i \in \mathcal{I}$ and $t \in [2, T]$, where $e$ is a vector of ones.

For $T \in \{3, 4, 5\}$ and $I \in \{5, 8, 10\}$, the nominal cost vector $c_{it}^0, t \in [2, T], i \in [I]$ is drawn uniformly from $[0, 10]^I$, and nominal profits are set as $r_{it}^0 = \frac{c_{it}^0}{5}$. Loan purchase cost is $c_L \mu^{t-1} = 0.12(1.2)^{t-1}$ per unit of loan. For each combination of $T$ and $I$ we consider different levels of initial budget which impacts the dependence between stages.
Bertsimas D, Goyal V (2012) On the power and limitations of affine policies in two-stage adaptive optimization. *Mathematical Programming* 134(2):491–531.

Bertsimas D, Goyal V, Lu BY (2015) A tight characterization of the performance of static solutions in two-stage adjustable robust linear optimization. *Mathematical Programming* 150(2):281–319.

Bertsimas D, Iancu DA, Parrilo PA (2010) Optimality of affine policies in multistage robust optimization. *Mathematics of Operations Research* 35(2):363–394.

Bertsimas D, Iancu DA, Parrilo PA (2011b) A hierarchy of near-optimal policies for multistage adaptive optimization. *IEEE Transactions on Automatic Control* 56(12):2809–2824.

Bertsimas D, Litvinov E, Sun XA, Zhao J, Zheng T (2012) Adaptive robust optimization for the security constrained unit commitment problem. *IEEE Transactions on Power Systems* 28(1):52–63.

Bertsimas D, Sim M, Zhang M (2019) Adaptive distributionally robust optimization. *Management Science* 65(2):604–618.

Bertsimas D, Thiele A (2006) A robust optimization approach to inventory theory. *Operations Research* 54(1):150–168.

Bodur M, Luedtke JR (2018) Two-stage linear decision rules for multi-stage stochastic programming. *Mathematical Programming* 1–34.

Chen X, Sim M, Sun P, Zhang J (2008) A linear decision-based approximation approach to stochastic programming. *Operations Research* 56(2):344–357.

Chen X, Zhang Y (2009) Uncertain linear programs: Extended affinely adjustable robust counterparts. *Operations Research* 57(6):1469–1482.

Daryalal M, Bodur M, Luedtke JR (2021) Lagrangian dual decision rules for multistage stochastic mixed integer programming. *Operations Research* To appear.

Feige U, Jain K, Mahdian M, Mirrokni V (2007) Robust combinatorial optimization with exponential scenarios. *International Conference on Integer Programming and Combinatorial Optimization*, 439–453 (Springer).

Geoffrion AM (1974) *Lagrangean relaxation for integer programming*, volume 2, 82–114 (Berlin, Heidelberg: Springer Berlin Heidelberg).

Georghiou A, Tsoukalas A, Wiesemann W (2019) Robust dual dynamic programming. *Operations Research* 67(3):813–830.

Georghiou A, Tsoukalas A, Wiesemann W (2020) A primal-dual lifting scheme for two-stage robust optimization. *Operations Research* 68(2):572–590.

Goh J, Sim M (2010) Distributionally robust optimization and its tractable approximations. *Operations Research* 58(4-part-1):902–917.
**5.4.2. Optimality Gap** Due to presence of binary variables $x_t(\xi^t)$, with the existing methods in the literature of 2ARO we cannot calculate $\nu^{2S-LDR}$ exactly. Therefore, we solve an approximation of it using the K-adaptability method of Subramanyam et al. (2020) with $K = 2$ and use the obtained bound $\nu^K$ in measuring the optimality gap. On the dual side, we study three options: (i) $\nu^\Omega$, the upper bound from solving the model (34) with a sample $|\Omega| = 250$, (ii) the NA bound $\bar{\nu}_R^{\text{NA-DO}}$ with the same sample $\Omega$, (iii) the decomposable NA bound $\bar{\nu}_R^{\text{DNA-DO}}$ with a sample of size 500, which includes the sample $\Omega$. For each option, when the model is not solved to optimality within the time limit, the best valid bound is used in the calculations. Note that, if we can solve both options (i) and (ii) to optimality, we should expect a better bound from option (i). Figure 8 presents the optimality gaps of the capital budgeting instances between the bound $\nu^K$ and the three choices of upper bound (detailed results are given in Appendix D.3). Results show that, even though the bound $\nu^\Omega$ should theoretically be at least as good as the bound $\bar{\nu}_R^{\text{NA-DO}}$, on average the NA bound returns a better upper bound within the same time limit. This is a testament to the difficulty of the multistage problem even when it is solved for a discrete set of realizations. The decomposable NA bound $\bar{\nu}_R^{\text{DNA-DO}}$ further improves the results by using a larger sample which is viable because of its superior computational performance. Note that, the best gaps in Figure 8, ranging between 7% to 33%, are obtained from approximations over approximations on both primal and dual side. Accordingly, these rather large gaps can be attributed to both bounding methods.

*General Integer Recourse Variables.* To assess our algorithms’ performance with general integer recourse, we conducted a set of experiments where loan amounts, $L_t(\xi^t)$, were restricted to integer values. Figure 9 in Appendix D.4 presents the same analysis as in Figure 8 for the capital budgeting problem, but with the added constraint of $L_t(\xi^t) \in \mathbb{Z}_+$. Interestingly, requiring integer loan amounts did not significantly alter the optimal investment decisions compared to the continuous case. This explains the visual similarity between the two figures, with both exhibiting similar patterns in optimal decisions. While there are slight differences in objective function values, the overall investment strategies remain largely unaffected by the integrality constraints.

It is important to note that although the advantage of our dual methods appears to diminish with an increasing number of stages, even the initial gap relies on the strength of our primal solution, $\nu^K$. Therefore, these figures demonstrate our ability to achieve further improvements beyond the initial strong primal bound.

Appendix D.3 provides a detailed analysis of the computational requirements for bounding the capital budgeting problem, considering both fractional and integral loans. For the primal side, we present the solution times of our approach, as no alternative primal method is available for comparison. On the dual side, we compare the solution times of the three previously discussed bounding
Guslitser E (2002) *Uncertainty-immunized solutions in linear programming*. Ph.D. thesis, Citeseer.

Hadjiyiannis MJ, Goulart PJ, Kuhn D (2011) A scenario approach for estimating the suboptimality of linear decision rules in two-stage robust optimization. *50th IEEE Conference on Decision and Control and European Control Conference*, 7386-7391 (IEEE).

Hanasusanto GA, Kuhn D, Wiesemann W (2015) K-adaptability in two-stage robust binary programming. *Operations Research* 63(4):877-891.

Hashemi Doulabi H, Jaillet P, Pesant G, Rousseau LM (2021) Exploiting the structure of two-stage robust optimization models with exponential scenarios. *INFORMS Journal on Computing* 33(1):143-162.

Hendrix EM, Boglárka G, et al. (2010) *Introduction to nonlinear and global optimization*, volume 37 (Springer).

Iancu DA, Sharma M, Sviridenko M (2013) Supermodularity and affine policies in dynamic robust optimization. *Operations Research* 61(4):941-956.

Kuhn D, Wiesemann W, Georghiou A (2011) Primal and dual linear decision rules in stochastic and robust optimization. *Mathematical Programming* 130(1):177-209.

Loken C, Gruner D, Groer L, Peltier R, Bunn N, Craig M, Henriques T, Dempsey J, Yu CH, Chen J, et al. (2010) SciNet: lessons learned from building a power-efficient top-20 system and data centre. *Journal of Physics-Conference Series*, volume 256, 012026.

Marandi A, Den Hertog D (2018) When are static and adjustable robust optimization problems with constraint-wise uncertainty equivalent? *Mathematical Programming* 170(2):555-568.

Pereira MV, Pinto LM (1991) Multi-stage stochastic optimization applied to energy planning. *Mathematical Programming* 52(1-3):359-375.

Philpott AB, de Matos VL, Kapelevich L (2018) Distributionally robust SDDP. *Computational Management Science* 15(3):431-454.

Ponce M, van Zon R, Northrup S, Gruner D, Chen J, Ertinaz F, Fedoseev A, Groer L, Mao F, Mundim BC, et al. (2019) Deploying a top-100 supercomputer for large parallel workloads: The Niagara supercomputer. *Proceedings of the Practice and Experience in Advanced Research Computing on Rise of the Machines (learning)*, 1-8 (Association for Computing Machinery, New York, NY, United States).

Postek K, Hertog Dd (2016) Multistage adjustable robust mixed-integer optimization via iterative splitting of the uncertainty set. *INFORMS Journal on Computing* 28(3):553-574.

Romeijnders W, Postek K (2020) Piecewise constant decision rules via branch-and-bound based scenario detection for integer adjustable robust optimization. *INFORMS Journal on Computing*.

See CT, Sim M (2010) Robust approximation to multiperiod inventory management. *Operations Research* 58(3):583-594.

Shapiro A, Dentcheva D, Ruszczyński A (2009) *Lectures on Stochastic Programming: Modeling and Theory* (SIAM).
<figure><img src="image_10.png" /><figcaption>**Figure 8** Optimality gap improvements for capital budgeting problems when using LDDR-based methods. In the legend, $\nu^\Omega$ denotes the bound obtained by solving model (34) over a sample set.</figcaption></figure>

approaches. The solution times demonstrate that the extra computational effort associated with the decomposable NA model yields a demonstrably improved dual bound.

# 6. Conclusion

Robust optimization models are built on a different premise than stochastic programming in the sense that they do not assume any knowledge about the probability distribution, focusing instead on optimizing the worst-case outcomes. In this paper, we study general MSAROs for which we
Subramanyam A, Gounaris CE, Wiesemann W (2020) K-adaptability in two-stage mixed-integer robust optimization. *Mathematical Programming Computation* 12(2):193–224.

Wolsey LA, Nemhauser GL (1999) *Integer and combinatorial optimization*, volume 55 (John Wiley & Sons).

Xu G, Hanasusanto GA (2021) Improved decision rule approximations for multi-stage robust optimization via copositive programming. *arXiv preprint arXiv:1808.06231*.

Yanıkoğlu İ, Gorissen BL, den Hertog D (2019) A survey of adjustable robust optimization. *European Journal of Operational Research* 277(3):799–813.

Yu X, Shen S (2020) Multistage distributionally robust mixed-integer programming with decision-dependent moment-based ambiguity sets. *Mathematical Programming* 1–40.

Zeng B, Zhao L (2013) Solving two-stage robust optimization problems using a column-and-constraint generation method. *Operations Research Letters* 41(5):457–461.

Zhen J, Den Hertog D, Sim M (2018) Adjustable robust optimization via fourier-motzkin elimination. *Operations Research* 66(4):1086–1100.
develop primal and dual bounding methods by adapting two decision rule approximations from the stochastic programming literature (namely two-stage LDRs and LDDRs). These approximations allow us to reduce MSARO to a 2ARO from the primal side and a two-stage stochastic programming problem from the dual side. As such, the resulting approximations drastically reduce the theoretical complexity of the studied problems. Since our dual bounds are dependent on the choice of a probability distribution while deriving the dual model, we propose to solve a distribution optimization problem to obtain a stronger bound. We develop various solution methods for our proposed bounding problems where we also leverage existing methods from both the robust optimization and the stochastic programming literature. Our extensive numerical study demonstrates that our methods considerably improve both primal and dual bounds compared to the commonly used approaches in the literature. Our work opens the door to the direct application of existing two-stage robust optimization and stochastic programming algorithms and other future algorithmic developments in these areas to MSAROs. For instance, as the algorithms such as C&CG and K-adaptability for 2ARO improve, our models can be solved more efficiently.

We believe that our work can initiate additional methodological and numerical developments. Both from the primal and the dual side the question of solving the problems we pose in a more numerically efficient manner definitely merits more attention. Further methodological work may also explore the synergies between primal and dual decision rules. Finally, the following directions can be the subject of future research: exploiting problem structure in order to approximate MSAROs with numerically more favorable, e.g., decomposable models, identifying special cases of MSAROs in which the proposed approximations can be proven to be exact, and application of similar approaches in related fields such as distributionally robust optimization.

## Acknowledgments

This work was supported by Natural Sciences and Engineering Research Council of Canada [Grants RGPIN-2018-04984 and RGPIN-2024-05908] and Agence Nationale de la Recherche of France [Grant ANR-22-CE048-0018]. Computations were performed on the Niagara supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation; the Government of Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto.

For the purpose of Open Access, a CC-BY public copyright licence has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.
References

ApS M (2022) *MOSEK modeling cookbook*.

Ardestani-Jaafari A, Delage E (2021) Linearized robust counterparts of two-stage robust optimization problems with applications in operations management. *INFORMS Journal on Computing* 33(3):1138–1161.

Arslan AN, Detienne B (2022) Decomposition-based approaches for a class of two-stage robust binary optimization problems. *INFORMS Journal on Computing* 34(2):857–871.

Ayoub J, Poss M (2016) Decomposition for adjustable robust linear optimization subject to uncertainty polytope. *Computational Management Science* 13(2):219–239.

Ben-Tal A, El Ghaoui L, Nemirovski A (2009) Robust optimization. *Robust optimization* (Princeton university press).

Ben-Tal A, El Housni O, Goyal V (2020) A tractable approach for designing piecewise affine policies in two-stage adjustable robust optimization. *Mathematical Programming* 182(1):57–102.

Ben-Tal A, Goryashko A, Guslitzer E, Nemirovski A (2004) Adjustable robust solutions of uncertain linear programs. *Mathematical Programming* 99(2):351–376.

Ben-Tal A, Nemirovski A (1999) Robust solutions of uncertain linear programs. *Operations Research Letters* 25(1):1–13.

Bertsimas D, Brown DB, Caramanis C (2011a) Theory and applications of robust optimization. *SIAM review* 53(3):464–501.

Bertsimas D, Caramanis C (2010) Finite adaptability in multistage linear optimization. *IEEE Transactions on Automatic Control* 55(12):2751–2766.

Bertsimas D, Dunning I (2016) Multistage robust mixed-integer optimization with adaptive partitions. *Operations Research* 64(4):980–998.

Bertsimas D, Georghiou A (2015) Design of near optimal decision rules in multistage adaptive mixed-integer optimization. *Operations Research* 63(3):610–627.

Bertsimas D, Georghiou A (2018) Binary decision rules for multistage adaptive mixed-integer optimization. *Mathematical Programming* 167(2):395–433.

Bertsimas D, Goyal V (2012) On the power and limitations of affine policies in two-stage adaptive optimization. *Mathematical Programming* 134(2):491–531.

Bertsimas D, Goyal V, Lu BY (2015) A tight characterization of the performance of static solutions in two-stage adjustable robust linear optimization. *Mathematical Programming* 150(2):281–319.

Bertsimas D, Iancu DA, Parrilo PA (2010) Optimality of affine policies in multistage robust optimization. *Mathematics of Operations Research* 35(2):363–394.

Bertsimas D, Iancu DA, Parrilo PA (2011b) A hierarchy of near-optimal policies for multistage adaptive optimization. *IEEE Transactions on Automatic Control* 56(12):2809–2824.
Bertsimas D, Litvinov E, Sun XA, Zhao J, Zheng T (2012) Adaptive robust optimization for the security constrained unit commitment problem. *IEEE Transactions on Power Systems* 28(1):52–63.

Bertsimas D, Sim M, Zhang M (2019) Adaptive distributionally robust optimization. *Management Science* 65(2):604–618.

Bertsimas D, Thiele A (2006) A robust optimization approach to inventory theory. *Operations Research* 54(1):150–168.

Bodur M, Luedtke JR (2018) Two-stage linear decision rules for multi-stage stochastic programming. *Mathematical Programming* 1–34.

Chen X, Sim M, Sun P, Zhang J (2008) A linear decision-based approximation approach to stochastic programming. *Operations Research* 56(2):344–357.

Chen X, Zhang Y (2009) Uncertain linear programs: Extended affinely adjustable robust counterparts. *Operations Research* 57(6):1469–1482.

Cheramin M, Cheng J, Jiang R, Pan K (2022) Computationally efficient approximations for distributionally robust optimization under moment and wasserstein ambiguity. *INFORMS Journal on Computing* 34(3):1768–1794.

Daryalal M, Bodur M, Luedtke JR (2024) Lagrangian dual decision rules for multistage stochastic mixed-integer programming. *Operations Research* 72(2):717–737.

Dumouchelle J, Julien E, Kurtz J, Khalil EB (2023) Neur2RO: Neural two-stage robust optimization. *arXiv preprint arXiv:2310.04345* .

Geoffrion AM (1974) *Lagrangean relaxation for integer programming*, volume 2, 82–114 (Berlin, Heidelberg: Springer Berlin Heidelberg).

Georghiou A, Tsoukalas A, Wiesemann W (2019) Robust dual dynamic programming. *Operations Research* 67(3):813–830.

Georghiou A, Tsoukalas A, Wiesemann W (2020) A primal-dual lifting scheme for two-stage robust optimization. *Operations Research* 68(2):572–590.

Goerigk M, Lendl S, Wulf L (2024) On the complexity of robust multi-stage problems with discrete recourse. *Discrete Applied Mathematics* 343:355–370.

Goh J, Sim M (2010) Distributionally robust optimization and its tractable approximations. *Operations Research* 58(4-part-1):902–917.

Gurobi Optimization, LLC (2022) Gurobi Optimizer Reference Manual. URL https://www.gurobi.com.

Hadjiyiannis MJ, Goulart PJ, Kuhn D (2011) A scenario approach for estimating the suboptimality of linear decision rules in two-stage robust optimization. *50th IEEE Conference on Decision and Control and European Control Conference*, 7386–7391 (IEEE).
$$ \sum_{j \in [J]} x_{ij2}^k = s_{i1} - \beta_{i2}^0 - \sum_{j \in [J]} d_{j2}^k \beta_{i2}^{j2} \qquad i \in [I], k \in [\kappa] $$

$$ \sum_{j \in [J]} x_{ijt}^k = \beta_{i,t-1}^0 - \beta_{it}^0 - \sum_{j \in [J]} \sum_{t' \in [2,t]} d_{jt'}^k \beta_{it}^{jt'} + \sum_{j \in [J]} \sum_{t' \in [2,t-1]} d_{jt'}^k \beta_{i,t-1}^{jt'} \qquad i \in [I], t \in [3,T], k \in [\kappa] $$

$$ \sum_{i \in [I]} x_{ijt}^k \geq d_{jt}^k \qquad j \in [J], t \in [2, T], k \in [\kappa] $$

$$ \beta_{it}^0 + \sum_{j \in [J]} \sum_{t' \in [2,t]} d_{jt'}^k \beta_{it}^{jt'} \geq 0 \qquad i \in [I], t \in [2, T], k \in [\kappa] $$

$$ x^k \geq 0 \qquad k \in [\kappa] $$

$$ y \in \{0, 1\}^I, z \geq 0, $$

where $x_{ijt}^k$ and $d_{jt}^k$ are the column and significant scenario generated at iteration $k$.

## A.2. Restricted NA Dual

The $Q(\beta^s, \beta^x)$ function for the location-transportation problem:

$$ \begin{array}{ll} \min & z + \sum_{t \in [2,T]} \sum_{i \in [I]} \beta_{it}^s (\boldsymbol{\xi}^T) s_{it} (\boldsymbol{\xi}^T) + \sum_{t \in [2,T]} \sum_{i \in [I]} \sum_{j \in [J]} \beta_{ijt}^x (\boldsymbol{\xi}^T) x_{ijt} (\boldsymbol{\xi}^T) \\ \text{s.t.} & z \geq \sum_{i \in [I]} y_i + \sum_{t \in [T]} \sum_{i \in [I]} a_i s_{it} (\boldsymbol{\xi}^T) + \sum_{t \in [2,T]} \sum_{i \in [I]} \sum_{j \in [J]} c_{ij} x_{ijt} (\boldsymbol{\xi}^T) \qquad \boldsymbol{\xi}^T \in \Xi \\ & s_{i1} \leq K_i y_i, \qquad i \in [I] \\ & s_{it} (\boldsymbol{\xi}^T) = s_{i,t-1} (\boldsymbol{\xi}^{T-1}) - \sum_{j \in [J]} x_{ijt} (\boldsymbol{\xi}^T) \qquad i \in [I], t \in [2, T], \boldsymbol{\xi}^T \in \Xi \\ & \sum_{i \in [I]} x_{ijt} (\boldsymbol{\xi}^T) \geq d_{jt} (\boldsymbol{\xi}^t) \qquad i \in [I], t \in [2, T], \boldsymbol{\xi}^T \in \Xi \\ & y \in \{0, 1\}^I, z \in \mathbb{R}_+^{I \times T}, x \in \mathbb{R}_+^{I \times J \times (T-1)}. \end{array} $$

For this mixed-integer subproblem, the cutting-plane method presented in Section 3.3 is applicable.
Hanasusanto GA, Kuhn D, Wiesemann W (2015) K-adaptability in two-stage robust binary programming. *Operations Research* 63(4):877–891.

Hashemi Doulabi H, Jaillet P, Pesant G, Rousseau LM (2021) Exploiting the structure of two-stage robust optimization models with exponential scenarios. *INFORMS Journal on Computing* 33(1):143–162.

Hendrix EM, Boglárka G, et al. (2010) *Introduction to nonlinear and global optimization*, volume 37 (Springer).

Iancu DA, Sharma M, Sviridenko M (2013) Supermodularity and affine policies in dynamic robust optimization. *Operations Research* 61(4):941–956.

Julien E, Postek K, Birbil SÎ (2022) Machine learning for K-adaptability in two-stage robust optimization. *arXiv preprint arXiv:2210.11152*.

Kuhn D, Wiesemann W, Georghiou A (2011) Primal and dual linear decision rules in stochastic and robust optimization. *Mathematical Programming* 130(1):177–209.

Loken C, Gruner D, Groer L, Peltier R, Bunn N, Craig M, Henriques T, Dempsey J, Yu CH, Chen J, et al. (2010) SciNet: lessons learned from building a power-efficient top-20 system and data centre. *Journal of Physics-Conference Series*, volume 256, 012026.

Mohajerin Esfahani P, Kuhn D (2018) Data-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations. *Mathematical Programming* 171(1-2):115–166, URL http://dx.doi.org/10.1007/s10107-017-1172-1.

Pereira MV, Pinto LM (1991) Multi-stage stochastic optimization applied to energy planning. *Mathematical Programming* 52(1-3):359–375.

Philpott AB, de Matos VL, Kapelevich L (2018) Distributionally robust SDDP. *Computational Management Science* 15(3):431–454.

Ponce M, van Zon R, Northrup S, Gruner D, Chen J, Ertinaz F, Fedoseev A, Groer L, Mao F, Mundim BC, et al. (2019) Deploying a top-100 supercomputer for large parallel workloads: The Niagara supercomputer. *Proceedings of the Practice and Experience in Advanced Research Computing on Rise of the Machines (learning)*, 1–8 (Association for Computing Machinery, New York, NY, United States).

Postek K, Hertog Dd (2016) Multistage adjustable robust mixed-integer optimization via iterative splitting of the uncertainty set. *INFORMS Journal on Computing* 28(3):553–574.

Romeijnders W, Postek K (2020) Piecewise constant decision rules via branch-and-bound based scenario detection for integer adjustable robust optimization. *INFORMS Journal on Computing*.

See CT, Sim M (2010) Robust approximation to multiperiod inventory management. *Operations Research* 58(3):583–594.

Shapiro A, Dentcheva D, Ruszczyński A (2009) *Lectures on Stochastic Programming: Modeling and Theory* (SIAM).
# B. Detailed Results

## B.1. Newsvendor Problem

<table><caption>Table 4: Detailed results for the newsvendor problem</caption><thead><tr><th>T</th><th>BR</th><th>|Ξ|</th><th>I</th><th>B</th><th>ν<sup>LDR</sup></th><th>ν<sup>2S</sup></th><th>ν<sup>*</sup></th><th>ν̄<sup>NA</sup></th><th>ν<sup>PI</sup></th></tr></thead><tbody><tr><td rowspan="10">3</td><td rowspan="10">5</td><td rowspan="10">25</td><td>3</td><td>100</td><td>656.6</td><td>970.2</td><td>975.3</td><td>975.3</td><td>1457.0</td></tr><tr><td>3</td><td>150</td><td>6668.5</td><td>8410.4</td><td>8647.2</td><td>9794.5</td><td>10036.0</td></tr><tr><td>3</td><td>200</td><td>8662.0</td><td>11446.2</td><td>11569.7</td><td>13709.0</td><td>13709.0</td></tr><tr><td>4</td><td>150</td><td>5154.8</td><td>5384.4</td><td>5466.8</td><td>5466.8</td><td>5781.0</td></tr><tr><td>4</td><td>200</td><td>10049.4</td><td>12452.0</td><td>12759.9</td><td>14501.9</td><td>14525.0</td></tr><tr><td>5</td><td>200</td><td>5944.7</td><td>7120.6</td><td>7402.4</td><td>7402.4</td><td>7713.0</td></tr><tr><td>3</td><td>150</td><td>5752.1</td><td>7620.9</td><td>7984.6</td><td>8464.9</td><td>8895.0</td></tr><tr><td>3</td><td>200</td><td>6279.6</td><td>9199.0</td><td>9700.9</td><td>11833.0</td><td>11833.0</td></tr><tr><td>4</td><td>150</td><td>3502.7</td><td>4886.5</td><td>4910.2</td><td>5002.5</td><td>5323.0</td></tr><tr><td>4</td><td>200</td><td>7922.9</td><td>10961.1</td><td>11273.4</td><td>14036.5</td><td>14060.0</td></tr><tr><td>5</td><td>200</td><td>2865.5</td><td>5877.7</td><td>6063.1</td><td>6063.1</td><td>6737.0</td></tr><tr><td rowspan="10">4</td><td rowspan="10">4</td><td rowspan="10">64</td><td>2</td><td>100</td><td>669</td><td>1047</td><td>1214</td><td>1214</td><td>1337</td></tr><tr><td>3</td><td>200</td><td>7807</td><td>9198</td><td>9491</td><td>9492</td><td>10157</td></tr><tr><td>3</td><td>300</td><td>13421</td><td>16414</td><td>17317</td><td>19188</td><td>19188</td></tr><tr><td>4</td><td>200</td><td>2034</td><td>2297</td><td>2447</td><td>2447</td><td>2640</td></tr><tr><td>4</td><td>300</td><td>13645</td><td>17860</td><td>18721</td><td>20095</td><td>20139</td></tr><tr><td>3</td><td>200</td><td>6643.8</td><td>8070.1</td><td>8295.6</td><td>8760.0</td><td>9249.0</td></tr><tr><td>3</td><td>300</td><td>11854.3</td><td>15035.4</td><td>16606.8</td><td>18240.0</td><td>18240.0</td></tr><tr><td>4</td><td>200</td><td>1443.7</td><td>1700.8</td><td>1739.8</td><td>1917.7</td><td>2122.0</td></tr><tr><td>4</td><td>300</td><td>13186.4</td><td>17579.4</td><td>18450.8</td><td>19618.0</td><td>19635.0</td></tr><tr><td>5</td><td>300</td><td>5142.0</td><td>10494.4</td><td>11299.5</td><td>11607.6</td><td>12343.0</td></tr><tr><td rowspan="10">4</td><td rowspan="10">10</td><td rowspan="10">1000</td><td>3</td><td>200</td><td>5648.8</td><td>8142.9</td><td></td><td>8876.4</td><td>9353.0</td></tr><tr><td>3</td><td>300</td><td>9143.5</td><td>13687.6</td><td></td><td>17853.0</td><td>17853.0</td></tr><tr><td>4</td><td>200</td><td>-104.4</td><td>440.0</td><td></td><td>642.4</td><td>919.0</td></tr><tr><td>4</td><td>300</td><td>11029.5</td><td>15854.7</td><td></td><td>18400.7</td><td>18432.0</td></tr><tr><td>5</td><td>300</td><td>724.9</td><td>6125.5</td><td></td><td>6748.6</td><td>7368.0</td></tr><tr><td>3</td><td>200</td><td>5111.3</td><td>7072.0</td><td></td><td>7646.6</td><td>8222.0</td></tr><tr><td>3</td><td>300</td><td>9030.9</td><td>13040.7</td><td></td><td>17615.0</td><td>17615.0</td></tr><tr><td>2</td><td>150</td><td>2490.9</td><td>2673.5</td><td>2763.4</td><td>2763.3</td><td>2975.4</td></tr><tr><td>2</td><td>200</td><td>8888.4</td><td>11192.5</td><td>11313.7</td><td>11448.6</td><td>11525.6</td></tr><tr><td>3</td><td>250</td><td>7809.0</td><td>8800.3</td><td>9065.4</td><td>9065.5</td><td>9382.3</td></tr><tr><td>3</td><td>300</td><td>14702.3</td><td>16898.8</td><td>17214.3</td><td>17308.7</td><td>18156.1</td></tr><tr><td>4</td><td>256</td><td>300</td><td>12367.7</td><td>15591.0</td><td>15989.0</td><td>16252.5</td><td>17192.6</td></tr><tr><td>5</td><td>5</td><td>625</td><td>300</td><td>11134.5</td><td>15053.4</td><td></td><td>16191.6</td><td>16680.0</td></tr><tr><td>5</td><td>6</td><td>1296</td><td>300</td><td>3651.5</td><td>9494.8</td><td></td><td>10553.8</td><td>10628.0</td></tr><tr><td>5</td><td>6</td><td>1296</td><td>300</td><td>15013.4</td><td>20271.7</td><td></td><td>25157.0</td><td>25157.0</td></tr><tr><td rowspan="3">6</td><td rowspan="3">4</td><td rowspan="3">1024</td><td>3</td><td>400</td><td>15492.3</td><td>21457.9</td><td></td><td>28163.0</td><td>28163.0</td></tr><tr><td>4</td><td>400</td><td>7124.3</td><td>14805.2</td><td></td><td>15426.2</td><td>15473.0</td></tr><tr><td>4</td><td>500</td><td>14445.4</td><td>24887.7</td><td></td><td>32946.9</td><td>32973.0</td></tr><tr><td rowspan="3">7</td><td rowspan="3">3</td><td rowspan="3">729</td><td>3</td><td>300</td><td>24.3</td><td>1495.9</td><td></td><td>2137.3</td><td>2383.0</td></tr><tr><td>3</td><td>400</td><td>12994.7</td><td>17774.1</td><td></td><td>19554.5</td><td>19983.0</td></tr><tr><td>4</td><td>400</td><td>3267.5</td><td>4303.1</td><td></td><td>5354.3</td><td>5427.0</td></tr></tbody></table>
Subramanyam A, Gounaris CE, Wiesemann W (2020) K-adaptability in two-stage mixed-integer robust optimization. *Mathematical Programming Computation* 12(2):193–224.

Tsang MY, Shehadeh KS, Curtis FE (2023) An inexact column-and-constraint generation method to solve two-stage robust optimization problems 51(1):92–98.

Wolsey LA, Nemhauser GL (1999) *Integer and combinatorial optimization*, volume 55 (John Wiley & Sons).

Xu G, Hanasusanto GA (2021) Improved decision rule approximations for multi-stage robust optimization via copositive programming. *arXiv preprint arXiv:1808.06231*.

Yamkoğlu İ, Gorissen BL, den Hertog D (2019) A survey of adjustable robust optimization. *European Journal of Operational Research* 277(3):799–813.

Zeng B, Zhao L (2013) Solving two-stage robust optimization problems using a column-and-constraint generation method. *Operations Research Letters* 41(5):457–461.

Zhao L, Zeng B (2012) An exact algorithm for two-stage robust optimization with mixed integer recourse problems.

Zhen J, Den Hertog D, Sim M (2018) Adjustable robust optimization via Fourier–Motzkin elimination. *Operations Research* 66(4):1086–1100.
<table><caption>Table 4: Detailed results for the newsvendor problem (continued)</caption><thead><tr><th>T</th><th>BR</th><th>|Ξ|</th><th>I</th><th>B</th><th>ν<sup>LDR</sup></th><th>ν<sup>2S</sup></th><th>ν<sup>*</sup></th><th>ν̄<sub>R</sub><sup>NA</sup></th><th>PI</th></tr></thead><tbody><tr><td rowspan="2">8</td><td rowspan="2">3</td><td rowspan="2">2187</td><td>3</td><td>500</td><td>16149.1</td><td>25321.4</td><td></td><td>30259.0</td><td>30259.0</td></tr><tr><td>4</td><td>600</td><td>16892.3</td><td>27608.1</td><td></td><td>28717.7</td><td>28747.0</td></tr></tbody></table>

<table><caption>Table 5 Running time for the larger size newsvendor problem instances</caption><thead><tr><th rowspan="2">T</th><th rowspan="2">BR</th><th rowspan="2">|Ξ|</th><th rowspan="2">I</th><th rowspan="2">B</th><th colspan="4">Time (s)</th></tr><tr><th>ν<sup>LDR</sup></th><th>ν<sup>2S</sup></th><th>ν̄<sub>R</sub><sup>NA</sup></th><th>PI</th></tr></thead><tbody><tr><td rowspan="5">4</td><td rowspan="5">10</td><td rowspan="5">1000</td><td>3</td><td>200</td><td>38</td><td>31</td><td>7561</td><td>1</td></tr><tr><td>3</td><td>300</td><td>50</td><td>31</td><td>24</td><td>1</td></tr><tr><td>4</td><td>200</td><td>47</td><td>28</td><td>625</td><td>1</td></tr><tr><td>4</td><td>300</td><td>58</td><td>30</td><td>2385</td><td>1</td></tr><tr><td>5</td><td>300</td><td>59</td><td>33</td><td>3311</td><td>2</td></tr><tr><td>4</td><td>15</td><td>3375</td><td>3</td><td>200</td><td>158</td><td>140</td><td>642</td><td>3</td></tr><tr><td>4</td><td>20</td><td>8000</td><td>3</td><td>300</td><td>726</td><td>675</td><td>1077</td><td>7</td></tr><tr><td rowspan="2">5</td><td rowspan="2">5</td><td rowspan="2">625</td><td>3</td><td>300</td><td>16</td><td>15</td><td>1426</td><td>1</td></tr><tr><td>4</td><td>300</td><td>22</td><td>19</td><td>5053</td><td>1</td></tr><tr><td>5</td><td>6</td><td>1296</td><td>3</td><td>400</td><td>54</td><td>54</td><td>53</td><td>1</td></tr><tr><td rowspan="3">6</td><td rowspan="3">4</td><td rowspan="3">1024</td><td>3</td><td>400</td><td>57</td><td>53</td><td>48</td><td>1</td></tr><tr><td>4</td><td>400</td><td>70</td><td>67</td><td>5911</td><td>2</td></tr><tr><td>4</td><td>500</td><td>74</td><td>69</td><td>3846</td><td>2</td></tr><tr><td rowspan="3">7</td><td rowspan="3">3</td><td rowspan="3">729</td><td>3</td><td>300</td><td>37</td><td>38</td><td>525</td><td>1</td></tr><tr><td>3</td><td>400</td><td>37</td><td>36</td><td>13010</td><td>1</td></tr><tr><td>4</td><td>400</td><td>50</td><td>50</td><td>12874</td><td>1</td></tr><tr><td rowspan="2">8</td><td rowspan="2">3</td><td rowspan="2">2187</td><td>3</td><td>500</td><td>364</td><td>335</td><td>358</td><td>4</td></tr><tr><td>4</td><td>600</td><td>462</td><td>464</td><td>37409</td><td>5</td></tr></tbody></table>
# Appendices

## A. Obtaining 2ARO Model via Two-stage Decision Rules

In this section, we detail the transition from the MSARO model (3) to the 2ARO model (4) as a result of applying two-stage decision rules proposed in Section 3.1. Substituting the state variables in the MSARO problem with the decision rules $x_t^s(\xi^t) = \Theta_t(\xi^t, \beta_t)$, $t \in [2, T]$, we obtain the following model:

$$ \min c_1^\top x_1 + z^{\text{rest}} \qquad (35a) $$

$$ \text{s.t. } x_1 \in X_1 \qquad (35b) $$

$$ \beta_t \in \mathbb{R}^{K_t} \qquad t \in [2, T] \qquad (35c) $$

$$ z^{\text{rest}} \ge \sum_{t \in [2,T]} c_t^s(\xi^t)^\top \Theta_t(\xi^t, \beta_t) + \sum_{t \in [2,T]} c_t^r(\xi^t)^\top x_t^r(\xi^t) \qquad \xi^T \in \Xi^T \qquad (35d) $$

$$ A_t^r(\xi^t)x_t^r(\xi^t) \le b_t(\xi^t) - (A_t^s(\xi^t)\Theta_t(\xi^t, \beta_t) + B_t^s(\xi^t)x_1^s) \qquad t=2, \xi^t \in \Xi^t \qquad (35e) $$

$$ A_t^r(\xi^t)x_t^r(\xi^t) \le b_t(\xi^t) - (A_t^s(\xi^t)\Theta_t(\xi^t, \beta_t) + B_t^s(\xi^t)\Theta_{t-1}(\xi^{t-1}, \beta_{t-1})) \qquad t \in [3, T], \xi^t \in \Xi^t \qquad (35f) $$

$$ (\Theta_t(\xi^t, \beta_t), x_t^r(\xi^t)) \in X_t(\xi^t) \qquad t \in [2, T], \xi^t \in \Xi^t \qquad (35g) $$

Given the first-stage decisions $x_1$ and $\beta$, observe that the feasible set of the recourse variables, defined by (35e)-(35g) decomposes by stage $t$ and history $\xi^t$ (as both the decision variables, $x_t^r(\xi^t)$, and the constraints, (35e)-(35g), are separately defined for each stage and history, i.e., there is no link between them). For convenience, let us represent this feasible space in a decomposed form via $x_t^r(\xi^t) \in \mathcal{X}_t^r(x_1^s, \beta, \xi^t)$, $t \in [2, T]$, $\xi^t \in \Xi^t$, and write (35) more compactly as follows:

$$ \min_{x_1 \in X_1, \beta} c_1^\top x_1 + z^{\text{rest}} \qquad (36a) $$

$$ \text{s.t. } z^{\text{rest}} \ge \sum_{t \in [2,T]} c_t^s(\xi^t)^\top \Theta_t(\xi^t, \beta_t) + \sum_{t \in [2,T]} c_t^r(\xi^t)^\top x_t^r(\xi^t) \qquad \xi^T \in \Xi^T \qquad (36b) $$

$$ x_t^r(\xi^t) \in \mathcal{X}_t^r(x_1^s, \beta, \xi^t) \qquad t \in [2, T], \xi^t \in \Xi^t \qquad (36c) $$

In order to minimize $z^{\text{rest}}$, the second summation term in (36b) should be minimized over the recourse decisions $x_t^r(\xi^t)$. Since this term is additively separable over stages $t \in [2, T]$ and their associated history $\xi^t \in \Xi^t$, the recourse decisions of each stage can be optimized separately, yielding the following equivalent model:

$$ \min_{x_1 \in X_1, \beta} c_1^\top x_1 + z^{\text{rest}} \qquad (37a) $$

$$ \text{s.t. } z^{\text{rest}} \ge \sum_{t \in [2,T]} c_t^s(\xi^t)^\top \Theta_t(\xi^t, \beta_t) + \sum_{t \in [2,T]} \min_{x_t^r(\xi^t) \in \mathcal{X}_t^r(x_1^s, \beta, \xi^t)} c_t^r(\xi^t)^\top x_t^r(\xi^t) \qquad \xi^T \in \Xi^T \qquad (37b) $$
## B.2. The Location-Transportation Problem

<table><caption>Table 6: Detailed results for the location-allocation problem</caption><thead><tr><th>(T, I, J, α<sup>d</sup>)</th><th>α<sup>u</sup></th><th>ν<sup>2S</sup></th><th>ν̂<sup>NA</sup><sub>R</sub></th></tr></thead><tbody><tr><td rowspan="4">(3, 10, 10, 0.1)</td><td>0.1</td><td>763055.2</td><td>682934.4</td></tr><tr><td>0.4</td><td>763053.2</td><td>651647.4</td></tr><tr><td>0.7</td><td>763052.3</td><td>577630.6</td></tr><tr><td>1</td><td>763054.7</td><td>569238.8</td></tr><tr><td rowspan="4">(3, 10, 10, 0.3)</td><td>0.1</td><td>1577694.3</td><td>1464100.3</td></tr><tr><td>0.4</td><td>1577694.3</td><td>1364705.6</td></tr><tr><td>0.7</td><td>1519050.9</td><td>1206126.4</td></tr><tr><td>1</td><td>1575438.6</td><td>1224115.8</td></tr><tr><td rowspan="4">(3, 10, 10, 0.5)</td><td>0.1</td><td>1575813.5</td><td>1426111.2</td></tr><tr><td>0.4</td><td>1577694.3</td><td>1363127.9</td></tr><tr><td>0.7</td><td>1577694.3</td><td>1181693.0</td></tr><tr><td>1</td><td>1555386.7</td><td>1474506.6</td></tr><tr><td rowspan="4">(3, 10, 15, 0.1)</td><td>0.1</td><td>1171035.9</td><td>857198.3</td></tr><tr><td>0.4</td><td>1171035.9</td><td>831435.5</td></tr><tr><td>0.7</td><td>1158076.3</td><td>1073536.8</td></tr><tr><td>1</td><td>1158076.1</td><td>961203.2</td></tr><tr><td rowspan="4">(3, 10, 10, 0.3)</td><td>0.1</td><td>1171035.9</td><td>1067984.7</td></tr><tr><td>0.4</td><td>1171035.9</td><td>863053.5</td></tr><tr><td>0.7</td><td>1171035.9</td><td>1092576.5</td></tr><tr><td>1</td><td>1171035.9</td><td>994209.5</td></tr><tr><td rowspan="4">(3, 10, 10, 0.5)</td><td>0.1</td><td>1171035.9</td><td>829093.4</td></tr><tr><td>0.4</td><td>1171035.9</td><td>978986.0</td></tr><tr><td>0.7</td><td>1158076.4</td><td>942674.2</td></tr><tr><td>1</td><td>1171035.9</td><td>930973.5</td></tr><tr><td rowspan="4">(3, 10, 20, 0.5)</td><td>0.4</td><td>1672246.0</td><td>1282612.7</td></tr><tr><td>1</td><td>1675102.0</td><td>1457338.7</td></tr><tr><td rowspan="4">(4, 5, 5, 0.1)</td><td>0.1</td><td>781934.0</td><td>724852.8</td></tr><tr><td>0.4</td><td>781934.0</td><td>573157.6</td></tr><tr><td>0.7</td><td>781934.0</td><td>687320.0</td></tr><tr><td>1</td><td>781934.0</td><td>727980.6</td></tr><tr><td rowspan="4">(4, 5, 5, 0.3)</td><td>0.1</td><td>781934.0</td><td>710778.0</td></tr><tr><td>0.4</td><td>781934.0</td><td>627111.1</td></tr><tr><td>0.7</td><td>781934.0</td><td>666989.7</td></tr><tr><td>1</td><td>781934.0</td><td>701394.8</td></tr><tr><td rowspan="4">(4, 5, 5, 0.5)</td><td>0.1</td><td>781934.0</td><td>583322.8</td></tr><tr><td>0.4</td><td>781934.0</td><td>711559.9</td></tr><tr><td>0.7</td><td>781934.0</td><td>620073.7</td></tr><tr><td>1</td><td>781934.0</td><td>728762.5</td></tr><tr><td rowspan="3">(4, 5, 7, 0.1)</td><td>0.1</td><td>1044144.0</td><td>909449.4</td></tr><tr><td>0.4</td><td>1044144.0</td><td>848889.1</td></tr><tr><td>1</td><td>1044144.0</td><td>829050.3</td></tr></tbody></table>
Inspecting the minimization problems in (37b), we observe that the objective function coefficients and the feasible set are parametrized by the history, as such changing the parametrization of the recourse decisions from the history $\xi^t$, to a full uncertainty realization $\xi^T$ should not change the optimal objective value. Thus, instead, we can consider the following equivalent model:

$$ \min_{x_1 \in X_1, \beta} c_1^\top x_1 + z^{\text{rest}} \qquad (38a) $$

$$ \text{s.t. } z^{\text{rest}} \ge \sum_{t \in [2, T]} c_t^s(\xi^t)^\top \Theta_t(\xi^t, \beta_t) + \sum_{t \in [2, T]} \min_{x_t^r(\xi^T) \in \mathcal{X}_t^r(x_1^s, \beta, \xi^t)} c_t^r(\xi^t)^\top x_t^r(\xi^T) \quad \xi^T \in \Xi^T \quad (38b) $$

More formally, we can show that given an optimal solution to the inner minimization problem in (38b), $\hat{x}_t^r : \xi^T \to \mathbb{R}^{p_t}$, we can construct a feasible solution $\tilde{x}_t^r : \xi^t \to \mathbb{R}^{p_t}$ to the inner minimization problem in (37b) which attains the same objective value. Since the minimization problem in (38b) is a relaxation of that in (37b), having more flexible decision variables, the aforementioned construction is sufficient to conclude the proof of our equivalence claim. Given stage $t \in [2, T]$ and history $\xi^t \in \Xi^t$, we let $\tilde{x}_t^r(\xi^t) := \hat{x}_t^r(\xi^T)$ where $\xi^T$ is an arbitrarily selected element from the set of realizations with the same history $\{\xi^T \in \Xi^T : \xi^t = \xi^t\}$. The feasibility of the constructed policy is straightforward since the two inner minimization problems have the same feasibility set, defined by $\mathcal{X}_t^r(x_1^s, \beta, \xi^t), t \in [2, T]$. Next, we observe that $c_t^r(\xi^t)^\top \hat{x}_t^r(\xi^T)$ is the same for all realizations in $\{\xi^T \in \Xi^T : \xi^t = \xi^t\}$ since the minimization problem in (38b) has the same feasible set and objective coefficient vector for any of those realizations and $\hat{x}_t^r(\cdot)$ is chosen to be an optimal policy. Lastly, by construction, the policy $\tilde{x}_t^r(\cdot)$ achieves the same objective value.

As there is no link between the different stage optimization problems in (38b), we can swap the summation and minimization operators and optimize over the recourse variables associated with all the stages together given an uncertainty realization:

$$ \min_{x_1 \in X_1, \beta} c_1^\top x_1 + z^{\text{rest}} \qquad (39a) $$

$$ \text{s.t. } z^{\text{rest}} \ge \sum_{t \in [2, T]} c_t^s(\xi^t)^\top \Theta_t(\xi^t, \beta_t) + \min_{(x_t^r(\xi^T))_{t \in [2, T]} \in \mathcal{X}(x_1^s, \beta, \xi^T)} \sum_{t \in [2, T]} c_t^r(\xi^t)^\top x_t^r(\xi^T) \quad \xi^T \in \Xi^T \quad (39b) $$

where $\mathcal{X}(x_1^s, \beta, \xi^T) = \times_{t \in [2, T]} \mathcal{X}_t^r(x_1^s, \beta, \xi^t)$. Since $z^{\text{rest}}$ is equal to the maximum of the right-hand side of (38b) over the realizations $\xi^T \in \Xi^T$ at an optimal solution, we can turn this problem into the following nested formulation:

$$ \min c_1^\top x_1 + \max_{\xi^T \in \Xi^T} \min_{x^r \in \mathcal{X}(x_1^s, \beta, \xi^T)} \sum_{t \in [2, T]} c_t^s(\xi^t)^\top \Theta_t(\xi^t, \beta_t) + c_t^r(\xi^t)^\top x_t^r \quad (40a) $$

where the parametrization of the recourse variables is omitted since $\xi^T$ is given as an input to the inner minimization problem.
<table><caption>Table 6: Detailed results for the location-allocation problem (continued)</caption><thead><tr><th>(T, I, J, α<sup>d</sup>)</th><th>α<sup>u</sup></th><th>ν<sup>2S</sup></th><th>ν̃<sub>R</sub><sup>NA</sup></th></tr></thead><tbody><tr><td rowspan="4">(4, 5, 7, 0.3)</td><td>0.1</td><td>1044144.0</td><td>847844.9</td></tr><tr><td>0.4</td><td>1044144.0</td><td>825917.9</td></tr><tr><td>0.7</td><td>1044144.0</td><td>984627.8</td></tr><tr><td>1</td><td>1044144.6</td><td>855154.4</td></tr><tr><td rowspan="4">(4, 5, 7, 0.5)</td><td>0.1</td><td>1044144.0</td><td>829050.3</td></tr><tr><td>0.4</td><td>1044144.8</td><td>772667.2</td></tr><tr><td>0.7</td><td>1044144.3</td><td>864551.5</td></tr><tr><td>1</td><td>1044144.7</td><td>816521.2</td></tr><tr><td rowspan="4">(4, 5, 10, 0.1)</td><td>0.1</td><td>2327093.7</td><td>2008281.8</td></tr><tr><td>0.4</td><td>2327093.7</td><td>1738339.0</td></tr><tr><td>0.7</td><td>2327093.7</td><td>2168851.3</td></tr><tr><td>1</td><td>2327093.7</td><td>1954758.7</td></tr><tr><td rowspan="4">(4, 5, 10, 0.3)</td><td>0.1</td><td>2327093.7</td><td>1912871.0</td></tr><tr><td>0.4</td><td>2327093.7</td><td>1663872.0</td></tr><tr><td>0.7</td><td>2327093.7</td><td>1752301.5</td></tr><tr><td>1</td><td>2327094.0</td><td>2064132.4</td></tr><tr><td rowspan="4">(4, 5, 10, 0.5)</td><td>0.1</td><td>2327093.7</td><td>1733684.8</td></tr><tr><td>0.4</td><td>2330349.6</td><td>2078671.9</td></tr><tr><td>0.7</td><td>2327093.7</td><td>1959412.9</td></tr><tr><td>1</td><td>2327093.7</td><td>1824441.4</td></tr><tr><td rowspan="4">(4, 10, 10, 0.1)</td><td>0.1</td><td>2408515.07</td><td>2071323.0</td></tr><tr><td>0.4</td><td>2408515.14</td><td>1854556.7</td></tr><tr><td>0.7</td><td>2408515.1</td><td>1719679.8</td></tr><tr><td>1</td><td>2408515.4</td><td>2237510.8</td></tr><tr><td rowspan="4">(4, 10, 10, 0.3)</td><td>0.1</td><td>2359826.5</td><td>2189919.0</td></tr><tr><td>0.4</td><td>2408515.1</td><td>2083365.6</td></tr><tr><td>0.7</td><td>2408515.5</td><td>2083365.9</td></tr><tr><td>1</td><td>2377021.4</td><td>1739979.6</td></tr><tr><td rowspan="4">(4, 10, 10, 0.5)</td><td>0.1</td><td>2359826.5</td><td>2130923.4</td></tr><tr><td>0.4</td><td>2408515.1</td><td>2203791.3</td></tr><tr><td>0.7</td><td>2359826.5</td><td>1923258.6</td></tr><tr><td>1</td><td>2359826.5</td><td>1781669.0</td></tr><tr><td rowspan="4">(5, 5, 10, 0.1)</td><td>0.1</td><td>3436537.5</td><td>3075701.1</td></tr><tr><td>0.4</td><td>3478066.7</td><td>2855492.7</td></tr><tr><td>0.7</td><td>3442732.0</td><td>2785170.2</td></tr><tr><td>1</td><td>3442731.5</td><td>3256824.0</td></tr><tr><td rowspan="4">(5, 5, 10, 0.3)</td><td>0.1</td><td>3478066.1</td><td>2695501.3</td></tr><tr><td>0.4</td><td>3442731.5</td><td>2840253.5</td></tr><tr><td>0.7</td><td>3478066.1</td><td>2483339.2</td></tr><tr><td>1</td><td>3427190.6</td><td>2450441.3</td></tr><tr><td rowspan="4">(5, 5, 10, 0.5)</td><td>0.1</td><td>3442732.0</td><td>2478767.0</td></tr><tr><td>0.4</td><td>3478066.2</td><td>2625940.0</td></tr><tr><td>0.7</td><td>3469376.2</td><td>2976724.8</td></tr><tr><td>1</td><td>3436537.5</td><td>2543037.8</td></tr></tbody></table>
## B. Proofs

In this section, we present the proofs of the three propositions and one lemma mentioned in the body of the paper, for which we also restate the claims for convenience.

**PROPOSITION 1.** Consider an MSARO with only right-hand-side uncertainty, continuous recourse, and (bounded) polyhedral uncertainty set. If the basis functions $\Phi_t(\xi^t)$ are chosen to be affine in $\xi^t$ for all $t \in [2, T]$, the C&CG algorithm converges to $\nu^{2S-LDR}$ in a finite number of iterations.

*Proof* Denote by $f(x_1^s, \beta, \xi^T)$, the objective function of the inner minimization in the two-stage problem (6). Then, we can rewrite problem (6) as:

$$ \nu^{2S-LDR} = \min \{c_1^\top x_1 + \max_{\xi^T \in \Xi^T} f(x_1^s, \beta, \xi^T) \mid x_1 \in X_1, \beta_t \in \mathbb{R}^{K_t}, t \in [2, T]\}. $$

First, we show that $f(x_1^s, \beta, \xi^T)$ is convex in $\xi^T$ for given $x_1=(x_1^s, x_1^r) \in X_1$ and $\beta$. For $\hat{\xi}^T, \tilde{\xi}^T \in \Xi^T$ and $\lambda \in [0, 1]$, the following (in)equalities hold:

$$ \lambda f(x_1^s, \beta, \hat{\xi}^T) + (1-\lambda) f(x_1^s, \beta, \tilde{\xi}^T) = \sum_{t \in [2,T]} c_t^{s\top} (\lambda \Phi_t(\hat{\xi}^T) + (1-\lambda) \Phi_t(\tilde{\xi}^T)) \beta_t + \lambda \left( \min_{x^r \in \mathcal{X}(x_1^s, \beta, \hat{\xi}^T)} \sum_{t \in [2,T]} c_t^{r\top} x_t^r + (1-\lambda) \left( \min_{x^r \in \mathcal{X}(x_1^s, \beta, \tilde{\xi}^T)} \sum_{t \in [2,T]} c_t^{r\top} x_t^r \right) \right) = \quad (41a) $$

$$ \lambda \left( \min_{x^r \in \mathcal{X}(x_1^s, \beta, \hat{\xi}^T)} \sum_{t \in [2,T]} c_t^{r\top} x_t^r + (1-\lambda) \left( \min_{x^r \in \mathcal{X}(x_1^s, \beta, \tilde{\xi}^T)} \sum_{t \in [2,T]} c_t^{r\top} x_t^r \right) \right) = \sum_{t \in [2,T]} c_t^{s\top} \Phi_t(\lambda \hat{\xi}^t + (1-\lambda) \tilde{\xi}^t) \beta_t + \quad (41b) $$

$$ \lambda \left( \min_{x^r \in \mathcal{X}(x_1^s, \beta, \hat{\xi}^T)} \sum_{t \in [2,T]} c_t^{r\top} x_t^r + (1-\lambda) \left( \min_{x^r \in \mathcal{X}(x_1^s, \beta, \tilde{\xi}^T)} \sum_{t \in [2,T]} c_t^{r\top} x_t^r \right) \right) \geq \sum_{t \in [2,T]} c_t^{s\top} \Phi_t(\lambda \hat{\xi}^t + (1-\lambda) \tilde{\xi}^t) \beta_t + \min_{x^r \in \mathcal{X}(x_1^s, \beta, \lambda \hat{\xi}^T + (1-\lambda) \tilde{\xi}^T)} \sum_{t \in [2,T]} c_t^{r\top} x_t^r = f(x_1^s, \beta, \lambda \hat{\xi}^T + (1-\lambda) \tilde{\xi}^T). $$

Equality (41a) holds because $\Phi_t(\xi^t)$ is affine in $\xi^t$. Inequality (41b) follows from the convexity of the optimal value of the inner minimization problem as a function of $\xi^T$, since uncertainty appears only on the right-hand-sides of constraints and recourse variables are continuous by assumption.

Now, let $\Xi^{\text{EXT}}$ be the set of extreme points of $\Xi^T$. In maximization of a convex function over a compact polyhedral set, there is an optimal solution that is an extreme point (Hendrix et al. 2010). Then, the two-stage problem becomes:

$$ \nu^{2S-LDR} = \min \{c_1^\top x_1 + \eta \mid \eta \ge f(x_1^s, \xi^T, \beta), \xi^T \in \Xi^{\text{EXT}}, x_1 \in X_1, \beta_t \in \mathbb{R}^{K_t}, t \in [2, T]\}. $$

C&CG is then the process of gradually adding constraints for each extreme point. Because $|\Xi^{\text{EXT}}| < +\infty$, iterating over all extreme points takes finitely many steps, which concludes the proof. $\square$
<table><caption>Table 7: Algorithmic details for the location-allocation problem</caption><thead><tr><th rowspan="2">(T, I, J, α<sup>d</sup>)</th><th rowspan="2">α<sup>u</sup></th><th colspan="2">ν<sup>2S-LDR</sup>, C&CG</th><th colspan="2">ν&#773;<sub>R</sub><sup>NA</sup>, Cutting-plane</th></tr><tr><th>#iterations</th><th>Time (s)</th><th>#iterations</th><th>Time (s)</th></tr></thead><tbody><tr><td rowspan="4">(3, 10, 10, 0.1)</td><td>0.1</td><td>14</td><td>147.3</td><td>612</td><td>1077.9</td></tr><tr><td>0.4</td><td>2</td><td>55.6</td><td>795</td><td>1109.5</td></tr><tr><td>0.7</td><td>2</td><td>84.0</td><td>718</td><td>955.4</td></tr><tr><td>1</td><td>2</td><td>104.8</td><td>1313</td><td>&gt; 10h</td></tr><tr><td rowspan="4">(3, 10, 10, 0.3)</td><td>0.1</td><td>12</td><td>834.2</td><td>1000</td><td>&gt; 10h</td></tr><tr><td>0.4</td><td>2</td><td>47.9</td><td>222</td><td>937.4</td></tr><tr><td>0.7</td><td>2</td><td>64.1</td><td>726</td><td>&gt; 10h</td></tr><tr><td>1</td><td>2</td><td>66.6</td><td>976</td><td>541.0</td></tr><tr><td rowspan="4">(3, 10, 10, 0.5)</td><td>0.1</td><td>2</td><td>104.6</td><td>267</td><td>&gt; 10h</td></tr><tr><td>0.4</td><td>2</td><td>102.8</td><td>655</td><td>&gt; 10h</td></tr><tr><td>0.7</td><td>2</td><td>81.2</td><td>1045</td><td>679.4</td></tr><tr><td>1</td><td>2</td><td>94.5</td><td>1387</td><td>&gt; 10h</td></tr><tr><td rowspan="4">(3, 10, 15, 0.1)</td><td>0.1</td><td>2</td><td>146.1</td><td>240</td><td>2584.8</td></tr><tr><td>0.4</td><td>2</td><td>107.6</td><td>333</td><td>&gt; 10h</td></tr><tr><td>0.7</td><td>8</td><td>91.6</td><td>414</td><td>&gt; 10h</td></tr><tr><td>1</td><td>5</td><td>28.3</td><td>808</td><td>&gt; 10h</td></tr><tr><td rowspan="4">(3, 10, 10, 0.3)</td><td>0.1</td><td>2</td><td>125.7</td><td>766</td><td>4155.8</td></tr><tr><td>0.4</td><td>2</td><td>93.6</td><td>959</td><td>4206.4</td></tr><tr><td>0.7</td><td>2</td><td>90.6</td><td>455</td><td>4640.6</td></tr><tr><td>1</td><td>2</td><td>113.9</td><td>1186</td><td>5003.7</td></tr><tr><td rowspan="4">(3, 10, 10, 0.5)</td><td>0.1</td><td>2</td><td>93.1</td><td>1240</td><td>3458.0</td></tr><tr><td>0.4</td><td>4</td><td>151.3</td><td>802</td><td>2402.3</td></tr><tr><td>0.7</td><td>2</td><td>108.0</td><td>870</td><td>&gt; 10h</td></tr><tr><td>1</td><td>2</td><td>85.9</td><td>1085</td><td>3419.8</td></tr><tr><td rowspan="4">(3, 10, 20, 0.5)</td><td>0.4</td><td>2</td><td>3558.8</td><td>1137</td><td>8746.5</td></tr><tr><td>1</td><td>2</td><td>1915.8</td><td>492</td><td>7273.0</td></tr><tr><td>0.1</td><td>2</td><td>0.4</td><td>950</td><td>887.6</td></tr><tr><td>0.4</td><td>2</td><td>0.5</td><td>1477</td><td>&gt; 10h</td></tr><tr><td rowspan="3">(4, 5, 5, 0.1)</td><td>0.7</td><td>2</td><td>0.5</td><td>626</td><td>1182.8</td></tr><tr><td>1</td><td>2</td><td>0.5</td><td>1241</td><td>&gt; 10h</td></tr><tr><td>0.1</td><td>2</td><td>0.4</td><td>1360</td><td>1186.9</td></tr><tr><td rowspan="3">(4, 5, 5, 0.3)</td><td>0.4</td><td>2</td><td>0.4</td><td>552</td><td>&gt; 10h</td></tr><tr><td>0.7</td><td>2</td><td>0.4</td><td>1010</td><td>739.6</td></tr><tr><td>1</td><td>2</td><td>0.4</td><td>543</td><td>&gt; 10h</td></tr><tr><td rowspan="4">(4, 5, 5, 0.5)</td><td>0.1</td><td>2</td><td>0.3</td><td>546</td><td>&gt; 10h</td></tr><tr><td>0.4</td><td>2</td><td>0.5</td><td>282</td><td>&gt; 10h</td></tr><tr><td>0.7</td><td>2</td><td>0.8</td><td>873</td><td>791.2</td></tr><tr><td>1</td><td>2</td><td>0.4</td><td>900</td><td>&gt; 10h</td></tr><tr><td rowspan="3">(4, 5, 7, 0.1)</td><td>0.1</td><td>11</td><td>76.2</td><td>315</td><td>1184.0</td></tr><tr><td>0.4</td><td>2</td><td>131.8</td><td>419</td><td>&gt; 10h</td></tr><tr><td>1</td><td>7</td><td>153.8</td><td>1049</td><td>&gt; 10h</td></tr><tr><td rowspan="3">(4, 5, 7, 0.3)</td><td>0.1</td><td>2</td><td>113.2</td><td>851</td><td>838.7</td></tr><tr><td>0.4</td><td>2</td><td>189.4</td><td>1341</td><td>916.0</td></tr><tr><td>0.7</td><td>2</td><td>174.7</td><td>783</td><td>&gt; 10h</td></tr><tr><td></td><td>1</td><td>2</td><td>150.8</td><td>965</td><td>&gt; 10h</td></tr></tbody></table>
LEMMA 1. For any $\mathbb{P} \in \mathcal{P}^>$, constraints (12e) are equivalent to the following:

$$y_t(\xi^T) = \mathbb{E}_{\xi'^T \sim \mathbb{P}} [y_t(\xi'^T) \mid \xi'^t = \xi^t], \quad t \in [T], \xi^T \in \Xi^T. \qquad (13)$$

*Proof* For a fixed $t$ and $\xi^T$, we start by multiplying both sides of (12e) by the density $p^\mathbb{P}(\xi'^T)$ for every $\xi'^T$ (including $\xi^T$) that shares the same history with $\xi^T$ up to $t$:

$$p^\mathbb{P}(\xi'^T)y_t(\xi^T) = p^\mathbb{P}(\xi'^T)y_t(\xi'^T) \quad t \in [T], \xi^T, \xi'^T \in \Xi^T \text{ with } \xi^t = \xi'^t.$$

Since $p^\mathbb{P}(\xi'^T) > 0$ for every realization $\xi'^T$ under any probability distribution $\mathbb{P} \in \mathcal{P}^>$, the feasible set of model (12) remains the same. Then, we integrate (or sum if $\Xi^T$ is discrete) the scaled constraints over all such $\xi'^T$ realizations to obtain

$$y_t(\xi^T) \int_{\xi'^T: \xi'^t = \xi^t} p^\mathbb{P}(\xi'^T) d\xi'^T = \int_{\xi'^T: \xi'^t = \xi^t} p^\mathbb{P}(\xi'^T) y_t(\xi'^T) d\xi'^T \quad t \in [T], \xi^T \in \Xi^T.$$

Let $\delta := \int_{\xi'^T: \xi'^t = \xi^t} p^\mathbb{P}(\xi'^T) d\xi'^T$ where $\delta > 0$ since $\xi^T \in \{\xi'^T: \xi'^t = \xi^t\}$ with $p^\mathbb{P}(\xi^T) > 0$. Then, we obtain (13) via

$$y_t(\xi^T) = \int_{\xi'^T: \xi'^t = \xi^t} \frac{p^\mathbb{P}(\xi'^T)}{\delta} y_t(\xi'^T) d\xi'^T = \mathbb{E}_{\xi'^T \sim \mathbb{P}} [y_t(\xi'^T) \mid \xi'^t = \xi^t] \quad t \in [T], \xi^T \in \Xi^T.$$

Since constraints (13) are obtained as an aggregation of the original constraints (12e), they are valid for model (12). Further, they imply the nonanticipativity constraints (12e) since for any given $t$ along with $\xi^T$ and $\hat{\xi}^T$ such that $\xi^t = \hat{\xi}^t$, the right-hand side of (13) is the same, i.e., $\mathbb{E}_{\xi'^T \sim \mathbb{P}} [y_t(\xi'^T) \mid \xi'^t = \xi^t] = \mathbb{E}_{\xi'^T \sim \mathbb{P}} [y_t(\xi'^T) \mid \xi'^t = \hat{\xi}^t]$, enforcing $y_t(\xi^T) = y_t(\hat{\xi}^T)$. $\square$

PROPOSITION 2. Let $\mathbb{P}$ be any probability measure in $\mathcal{P}^>$. For MSARO problems with continuous recourse, (15) is a strong dual of (2), i.e., $\mathcal{L}^{\text{NA}}(\mathbb{P}) = \nu^*$.

*Proof* The strength of a Lagrangian dual problem can be studied by its primal characterization, derived by Geoffrion (1974) for a mixed-integer linear optimization problem. Then, using standard Lagrangian duality theory (see, for instance, Wolsey and Nemhauser (1999)), the primal characterization of (15) is:

$$\min z \qquad (42a)$$

$$\text{s.t.} \quad (z, y_1(\xi^T), \dots, y_T(\xi^T)) \in \text{conv}(Y(\xi^T)) \qquad \xi^T \in \Xi^T \qquad (42b)$$

$$y_t(\xi^T) = \mathbb{E}_{\xi'^T \sim \mathbb{P}} [y_t(\xi'^T) \mid \xi'^t = \xi^t] \quad t \in [T], \xi^T \in \Xi^T. \qquad (42c)$$

The result follows from the equivalence of (13) and (12e), as well as the fact that for MSARO problems with continuous recourse, we have that $\text{conv}(Y(\xi^T)) = Y(\xi^T)$ for all $\xi^T \in \Xi^T$. $\square$
<table><caption>Table 7: Algorithmic details for the location-allocation problem (continued)</caption><thead><tr><th rowspan="2">(T, I, J, α<sup>d</sup>)</th><th rowspan="2">α<sup>u</sup></th><th colspan="2">ν<sup>2S-LDR</sup>, C&CG</th><th colspan="2">ν&#773;<sub>R</sub><sup>NA</sup>, Cutting-plane</th></tr><tr><th>#iterations</th><th>Time (s)</th><th>#iterations</th><th>Time (s)</th></tr></thead><tbody><tr><td rowspan="4">(4, 5, 7, 0.5)</td><td>0.1</td><td>2</td><td>111.9</td><td>1226</td><td>742.0</td></tr><tr><td>0.4</td><td>2</td><td>97.3</td><td>381</td><td>831.4</td></tr><tr><td>0.7</td><td>17</td><td>131.8</td><td>559</td><td>704.9</td></tr><tr><td>1</td><td>2</td><td>196.7</td><td>588</td><td>&gt; 10h</td></tr><tr><td rowspan="4">(4, 5, 10, 0.1)</td><td>0.1</td><td>5</td><td>78.7</td><td>1499</td><td>&gt; 10h</td></tr><tr><td>0.4</td><td>11</td><td>87.8</td><td>721</td><td>&gt; 10h</td></tr><tr><td>0.7</td><td>11</td><td>100.2</td><td>478</td><td>4062.0</td></tr><tr><td>1</td><td>2</td><td>115.4</td><td>547</td><td>&gt; 10h</td></tr><tr><td rowspan="4">(4, 5, 10, 0.3)</td><td>0.1</td><td>2</td><td>196.3</td><td>1021</td><td>7742.2</td></tr><tr><td>0.4</td><td>2</td><td>99.5</td><td>652</td><td>11814.1</td></tr><tr><td>0.7</td><td>2</td><td>48.2</td><td>572</td><td>11408.6</td></tr><tr><td>1</td><td>2</td><td>106.1</td><td>403</td><td>8880.7</td></tr><tr><td rowspan="4">(4, 5, 10, 0.5)</td><td>0.1</td><td>2</td><td>179.7</td><td>358</td><td>&gt; 10h</td></tr><tr><td>0.4</td><td>2</td><td>34.2</td><td>1427</td><td>&gt; 10h</td></tr><tr><td>0.7</td><td>2</td><td>71.6</td><td>608</td><td>11996.6</td></tr><tr><td>1</td><td>2</td><td>39.1</td><td>737</td><td>10706.6</td></tr><tr><td rowspan="4">(4, 10, 10, 0.1)</td><td>0.1</td><td>2</td><td>13054.8</td><td>1822</td><td>&gt; 10h</td></tr><tr><td>0.4</td><td>2</td><td>12710.6</td><td>2147</td><td>&gt; 10h</td></tr><tr><td>0.7</td><td>2</td><td>4352.5</td><td>758</td><td>&gt; 10h</td></tr><tr><td>1</td><td>2</td><td>7139.0</td><td>1238</td><td>13663.5</td></tr><tr><td rowspan="4">(4, 10, 10, 0.3)</td><td>0.1</td><td>6</td><td>10709.5</td><td>868</td><td>9490.9</td></tr><tr><td>0.4</td><td>10</td><td>7592.3</td><td>746</td><td>&gt; 10h</td></tr><tr><td>0.7</td><td>2</td><td>12070.4</td><td>1330</td><td>&gt; 10h</td></tr><tr><td>1</td><td>2</td><td>7577.7</td><td>842</td><td>12937.5</td></tr><tr><td rowspan="4">(4, 10, 10, 0.5)</td><td>0.1</td><td>2</td><td>8261.0</td><td>758</td><td>10538.8</td></tr><tr><td>0.4</td><td>2</td><td>8988.9</td><td>791</td><td>&gt; 10h</td></tr><tr><td>0.7</td><td>2</td><td>5248.9</td><td>1461</td><td>14107.7</td></tr><tr><td>1</td><td>2</td><td>3049.6</td><td>684</td><td>&gt; 10h</td></tr><tr><td rowspan="4">(5, 5, 10, 0.1)</td><td>0.1</td><td>2</td><td>349.6</td><td>1401</td><td>&gt; 10h</td></tr><tr><td>0.4</td><td>2</td><td>978.5</td><td>248</td><td>&gt; 10h</td></tr><tr><td>0.7</td><td>2</td><td>647.6</td><td>586</td><td>14988.1</td></tr><tr><td>1</td><td>2</td><td>757.6</td><td>884</td><td>&gt; 10h</td></tr><tr><td rowspan="4">(5, 5, 10, 0.3)</td><td>0.1</td><td>2</td><td>863.5</td><td>779</td><td>&gt; 10h</td></tr><tr><td>0.4</td><td>2</td><td>589.0</td><td>1053</td><td>16150.2</td></tr><tr><td>0.7</td><td>2</td><td>402.9</td><td>757</td><td>12702.8</td></tr><tr><td>1</td><td>2</td><td>941.7</td><td>997</td><td>14375.6</td></tr><tr><td rowspan="4">(5, 5, 10, 0.5)</td><td>0.1</td><td>2</td><td>1008.4</td><td>409</td><td>&gt; 10h</td></tr><tr><td>0.4</td><td>4</td><td>721.7</td><td>1310</td><td>12800.1</td></tr><tr><td>0.7</td><td>12</td><td>379.6</td><td>911</td><td>15108.2</td></tr><tr><td>1</td><td>4</td><td>720.3</td><td>518</td><td>13741.1</td></tr></tbody></table>
PROPOSITION 3. $\tilde{\nu}_R^{\text{NA-DO}}$ is a lower bound for $\nu^*$.

Proof We have shown that $\mathcal{L}_R^{\text{NA}}(\mathbb{P}) \le \nu^*$ for any $\mathbb{P} \in \mathcal{P}^>$. Now we claim that, for any $\hat{\mathbb{P}} \in \mathcal{P}^> \setminus \mathcal{P}^> = \{\mathbb{P} \in \mathcal{P}^> | \exists \xi^T \in \Xi^T : p^\mathbb{P}(\xi^T) = 0\}$ where $p^\mathbb{P}: \Xi^T \to \mathbb{R}_+$ is the density function of $\hat{\mathbb{P}}$, the inequality $\mathcal{L}_R^{\text{NA}}(\hat{\mathbb{P}}) \le \nu^*$ also holds. Given $\hat{\mathbb{P}}$, let $\hat{\Xi}^T := \{\xi^T \in \Xi^T : p^\hat{\mathbb{P}}(\xi^T) > 0\}$. Using $\hat{\Xi}^T$, we create a relaxation of (12), the NA reformulation of the MSARO problem, as

$$ \hat{\nu}(\hat{\mathbb{P}}) := \min z \qquad (43a) $$

$$ \text{s.t.} \quad \sum_{t \in [T]} c_t(\xi^t)^\top y_t(\xi^T) \le z \qquad \xi^T \in \Xi^T \qquad (43b) $$

$$ A_t(\xi^t)y_t(\xi^T) + B_t(\xi^t)y_{t-1}(\xi^T) \le b_t(\xi^t) \quad t \in [2, T], \xi^T \in \Xi^T \qquad (43c) $$

$$ D_t(\xi^t)y_t(\xi^T) \le d_t(\xi^t) \quad t \in [T], \xi^T \in \Xi^T \qquad (43d) $$

$$ y_t(\xi^T) = y_t(\xi'^T) \qquad t \in [T], \xi^T \in \hat{\Xi}^T, \xi'^T \in \hat{\Xi}^T \text{ with } \xi^T = \xi'^T \qquad (43e) $$

$$ y_t(\xi^T) \in \mathbb{R}^{n_t - n_t^i} \times \mathbb{Z}^{n_t^i} \qquad t \in [T], \xi^T \in \Xi^T \qquad (43f) $$

where the NA constraints are only imposed for the pairs of realizations in $\hat{\Xi}^T \subset \Xi^T$. Therefore, we have $\hat{\nu}(\hat{\mathbb{P}}) \le \nu^*$.

Now that $\hat{\mathbb{P}}$ assigns a positive density to all the realizations from $\hat{\Xi}^T$, via Lemma 1, the NA constraints in (43e) can be equivalently reformulated as

$$ y_t(\xi^T) = \mathbb{E}_{\xi'^T \sim \hat{\mathbb{P}}} [y_t(\xi'^T) \mid \xi'^t = \xi^t], \quad t \in [T], \xi^T \in \hat{\Xi}^T. $$

Subsequently, relaxation of these reformulated NA constraints and construction of the Lagrangian dual problem (15) with respect to $\hat{\mathbb{P}}$ yields a relaxation of a relaxation of a minimization problem, where for each fixed $\lambda_t(\cdot)$ we have $\mathcal{L}_{\text{LR}}^{\text{NA}}(\hat{\mathbb{P}}, \lambda_1(\cdot), \dots, \lambda_T(\cdot)) \le \hat{\nu}(\hat{\mathbb{P}})$, and consequently $\mathcal{L}_R^{\text{NA}}(\hat{\mathbb{P}}) \le \hat{\nu}(\hat{\mathbb{P}})$. Furthermore, restricting the Lagrangian duals to follow LDRs, we obtain $\mathcal{L}_R^{\text{NA}}(\hat{\mathbb{P}}) \le \mathcal{L}_R^{\text{NA}}(\hat{\mathbb{P}})$.

Since we showed $\mathcal{L}_R^{\text{NA}}(\hat{\mathbb{P}}) \le \nu^*$ for all $\mathbb{P} \in \mathcal{P}^>$, we have $\tilde{\nu}_R^{\text{NA-DO}} := \max_{\mathbb{P} \in \mathcal{P}^>} \mathcal{L}_R^{\text{NA}}(\mathbb{P}) \le \nu^*$, which completes the proof. $\square$

## C. Detailed Models

### C.1. Monolithic Form of Model (8)

Consider the case mentioned in Remark 1, namely an MSARO where the uncertainty set is a polytope, the basis functions $\Phi_t(\xi^t)$ are chosen to be affine in $\xi^t$ for all $t \in [2, T]$, all the recourse variables are continuous, we have fixed parameters associated with the state variables, and the 2ARO problem (6) has relatively complete recourse. We next detail how a monolithic mixed-integer linear programming formulation of the inner minimization problem (8) can be derived as in (Ayoub and Poss 2016, Zeng and Zhao 2013).
Let $\pi^s$ and $\pi^R$ be the linear programming dual variables associated with the state and recourse constraints in $\mathcal{X}(\hat{x}_1^s, \hat{\beta}, \xi^T)$, in the inner minimization problem of (8), respectively. Then, using KKT conditions, subproblem (8) can be modelled as follows:

$$ \max \sum_{t \in [2, T]} c_t^{s\top} \Phi_t(\xi^t) \hat{\beta}_t + c_t^{r\top} x_t^r \quad (44a) $$

$$ \text{s.t.} \quad A_t^r x_t^r + A_t^s \Phi_t(\xi^t) \hat{\beta}_t + B_t^s \hat{x}_1^s - b_t(\xi^t) \le 0 \qquad t=2 \quad (44b) $$

$$ A_t^r x_t^r + A_t^s \Phi_t(\xi^t) \hat{\beta}_t + B_t^s \Phi_{t-1}(\xi^{t-1}) \hat{\beta}_{t-1} - b_t(\xi^t) \le 0 \qquad t \in [3, T] \quad (44c) $$

$$ D_t^s \Phi_t(\xi^t) \hat{\beta}_t + D_t^r x_t^r - d_t(\xi^t) \le 0 \qquad t \in [2, T] \quad (44d) $$

$$ A_t^{r\top} \pi_t^s + D_t^{r\top} \pi_t^R = c_t^r \qquad t \in [2, T] \quad (44e) $$

$$ (A_t^r x_t^r + A_t^s \Phi_t(\xi^t) \hat{\beta}_t + B_t^s \hat{x}_1^s - b_t(\xi^t))^{\top} \pi_t^s = 0 \qquad t=2 \quad (44f) $$

$$ (A_t^r x_t^r + A_t^s \Phi_t(\xi^t) \hat{\beta}_t + B_t^s \Phi_{t-1}(\xi^{t-1}) \hat{\beta}_{t-1} - b_t(\xi^t))^{\top} \pi_t^s = 0 \qquad t \in [3, T] \quad (44g) $$

$$ (D_t^s \Phi_t(\xi^t) \hat{\beta}_t + D_t^r x_t^r - d_t(\xi^t))^{\top} \pi_t^R = 0 \qquad t \in [2, T] \quad (44h) $$

$$ x_t^r \in \mathbb{R}^{p_t}, \quad \pi_t^s \in \mathbb{R}_-^{m_t^s}, \quad \pi_t^R \in \mathbb{R}_-^{m_t^R} \qquad t \in [2, T] \quad (44i) $$

$$ \xi^T \in \Xi^T. \quad (44j) $$

Inequalities (44b)-(44d) are the primal feasibility constraints at $\hat{\beta}_t$, (44e) the dual feasibility constraints, and (44f)-(44h) are the complementary slackness constraints. The latter include bilinear terms, as basis functions $\Phi_t(\xi^t)$ are functions of $\xi^T$ which are decision variables in (44). They can be linearized with the addition of binary decision variables via the so-called big-M constraints. Since the basis functions are chosen to be affine, the resulting model is a mixed-integer linear program. A detailed example of building a mixed-integer linear model for a multistage location-allocation problem is provided in Appendix C.3.

## C.2. Two-stage Decision Rules for MSAROs with Mixed-integer State Variables

As mentioned in Remark 3, for MSAROs with mixed-integer state variables, linear and piecewise constant decision rules can be combined to obtain a 2ARO approximation. To this end, consider the partition of the index set of the state variables into sets $\mathcal{I}_t^i$ and $\mathcal{I}_t^c$ for integer and continuous variables, respectively, i.e., $\mathcal{I}_t^i \cup \mathcal{I}_t^c = [q_t], \mathcal{I}_t^i \cap \mathcal{I}_t^c = \emptyset, |\mathcal{I}_t^i| = q_t^i$. Similarly, let $x_t^s = (x_t^{s,i}, x_t^{s,c})$ and $\beta = (\beta^i, \beta^c)$ be the vectors of state variables and decision rule design variables with sub-vectors corresponding to the integer and continuous state variables. Then, the application of LDRs (5) to the continuous state variables and PCDRs (9) to the integer state variables yields the following 2ARO model:

$$ \nu^{2S-LDR-PCDR} := \min c_1^{\top} x_1 + SP^{2S-LDR-PCDR}(\beta, x_1^s) \quad (45a) $$
$$ \text{s.t. } x_1 \in X_1 \qquad (45b) $$

$$ \beta_t^c \in \mathbb{R}^{K_t^c} \qquad t \in [2, T] \qquad (45c) $$

$$ \beta_t^i \in \mathbb{R}^{K_t^i} \qquad t \in [2, T] \qquad (45d) $$

$$ -1 \le \hat{\Upsilon}_{ti}(\xi^t)^{\top} \beta_{ti}^{i} \le 1 \qquad t \in [2, T], i \in \mathcal{I}_t^i, \xi^T \in \Xi^T \qquad (45e) $$

where

$$ \mathcal{SP}^{2S-LDR-PCDR}(\beta, x_1^s) := \max_{\xi^T \in \Xi^T} \sum_{t \in [2, T]} c_t^s(\xi^t)^{\top} x_t^s + \min_{x^r \in \mathcal{X}(x^s, \xi^T)} \sum_{t \in [2, T]} c_t^r(\xi^t)^{\top} x^r \qquad (46a) $$

$$ \text{s.t.} \quad \Phi_t(\xi^t)\beta_t^c = x_t^{s,c} \qquad t \in [2, T] \qquad (46b) $$

$$ \sum_{j \in [J_i]} (\underline{\kappa}_{ti} + j - 1) v_{tij} = x_{ti}^{s,i} \qquad t \in [2, T], i \in \mathcal{I}_t^i \qquad (46c) $$

$$ \sum_{j \in [J_i]} \omega_{tij} = \hat{\Upsilon}_{ti}(\xi^t)^{\top} \beta_t^i \qquad t \in [2, T], i \in \mathcal{I}_t^i \qquad (46d) $$

$$ (a_{tj}^i + \epsilon_j) v_{tij} \le \omega_{tij} \le b_{tj}^i v_{tij} \qquad t \in [2, T], i \in \mathcal{I}_t^i, j \in [J_i] \qquad (46e) $$

$$ \sum_{j \in [J_i]} v_{tij} = 1 \qquad t \in [2, T], i \in \mathcal{I}_t^i \qquad (46f) $$

$$ v_{tij} \in \{0, 1\} \qquad t \in [2, T], i \in \mathcal{I}_t^i, j \in [J_i]. \qquad (46g) $$

Given the state variables and an uncertainty realization, the recourse feasible set is defined as

$$ \mathcal{X}(x^s, \xi^T) := \left\{ (x_t^r)_{t \in [2, T]} \in \mathbb{R}^{p_2} \times \mathbb{R}^{p_3} \times \cdots \times \mathbb{R}^{p_T} : \begin{array}{l} A_t^r(\xi^t)x_t^r \le b_t(\xi^t) - (A_t^s(\xi^t)x_t^s + B_t^s(\xi^t)x_1^s) \quad t=2 \\ A_t^r(\xi^t)x_t^r \le b_t(\xi^t) - (A_t^s(\xi^t)x_t^s + B_t^s(\xi^t)x_{t-1}^s) \quad t \in [3, T] \\ (x_t^s, x_t^r) \in X_t(\xi^t) \quad t \in [2, T] \end{array} \right\}. $$

Lastly, we note that in the special case where the assumptions of Remarks 1 and 2 are satisfied, the subproblem (46) can be similarly reformulated as a monolithic mixed-integer linear program.

### C.3. Models for the Location-Transportation

#### C.3.1. Column-and-constraint Generation with Two-stage Linear Decision Rules

Applying LDRs on the state variables of model (33), $s_{it}(\xi^t) = \beta_{it}^0 + \sum_{t' \in [2, t]} \sum_{j \in [J]} d_{jt'}(\xi^t) \beta_{it'}^{j t'}$, $i \in [I], t \in [2, T]$, results in the following 2ARO problem in monolithic form:

min $z$

$$ \text{s.t.} \quad z \ge \sum_{i \in [I]} f_i y_i + \sum_{i \in [I]} a_i s_{i1} + \sum_{t \in [2, T]} \sum_{i \in [I]} a_i (\beta_{it}^0 + \sum_{t' \in [2, t]} \sum_{j \in [J]} d_{jt'}(\xi^{t'}) \beta_{it'}^{j t'}) + $$
$$ \sum_{t \in [2,T]} \sum_{i \in [I]} \sum_{j \in [J]} c_{ij} x_{ijt}(\xi^t) \qquad \xi^T \in \Xi^T \\ s_{i1} \le K_i y_i \qquad i \in [I] $$

$$ \beta_{i2}^0 + \sum_{j \in [J]} d_{j2}(\xi^2) \beta_{i2}^{j2} = s_{i1} - \sum_{j \in [J]} x_{ij2}(\xi^2) \qquad i \in [I], \xi^2 \in \Xi^2 \\ \beta_{it}^0 + \sum_{t' \in [2,t]} \sum_{j \in [J]} d_{jt'}(\xi^t) \beta_{it}^{jt'} = \beta_{i,t-1}^0 + \sum_{t' \in [2,t-1]} \sum_{j \in [J]} d_{jt'}(\xi^{t-1}) \beta_{i,t-1}^{jt'} - \sum_{j \in [J]} x_{ijt}(\xi^t) \qquad i \in [I], t \in [3,T], \xi^t \in \Xi^t $$

$$ \sum_{i \in [I]} x_{ijt}(\xi^t) \ge d_{jt}(\xi^t) \qquad j \in [J], t \in [2,T], \xi^t \in \Xi^t \\ \beta_{it}^0 + \sum_{t' \in [2,t]} \sum_{j \in [J]} d_{jt'}(\xi^t) \beta_{it}^{jt'} \ge 0 \qquad i \in [I], t \in [2,T], \xi^t \in \Xi^t \\ y \in \{0,1\}^I, z, x \ge 0. $$

For better clarity, let us re-write the above formulation in the more common min-max-min form of 2ARO:

$$ \min \sum_{i \in [I]} f_i y_i + \sum_{i \in [I]} a_i (s_{i1} + \sum_{t \in [2,T]} \beta_{it}^0) + \max_{\xi^T \in \Xi^T} \min \sum_{t' \in [2,t]} a_i d_{jt'}(\xi^t) \beta_{it}^{jt'} + \sum_{t \in [2,T]} \sum_{i \in [I]} \sum_{j \in [J]} c_{ij} x_{ijt} \\ \text{s.t.} \quad s_{i1} \le K_i y_i \quad i \in [I] \qquad \text{s.t.} \quad \sum_{j \in [J]} x_{ij2} = s_{i1} - \beta_{i2}^0 - \sum_{j \in [J]} d_{j2}(\xi^2) \beta_{i2}^{j2} \quad i \in [I] \\ \beta_{it}^0 + \sum_{t' \in [2,t]} \sum_{j \in [J]} d_{jt'}(\xi^t) \beta_{it}^{jt'} \ge 0 \quad i \in [I], t \in [2,T], \xi^t \in \Xi^t \\ y \in \{0,1\}^I. \\ \sum_{j \in [J]} x_{ijt} = \beta_{i,t-1}^0 - \beta_{it}^0 - \sum_{j \in [J]} \sum_{t' \in [2,t]} d_{jt'}(\xi^t) \beta_{it}^{jt'} \\ + \sum_{j \in [J]} \sum_{t' \in [2,t-1]} d_{jt'}(\xi^{t-1}) \beta_{i,t-1}^{jt'} \quad i \in [I], t \in [3,T] \\ \sum_{i \in [I]} x_{ijt} \ge d_{jt}(\xi^t) \quad j \in [J], t \in [2,T] \\ x \ge 0 $$

Note that we have strengthened the outer minimization problem by adding the non-negativity constraints from the inner minimization problem as robust constraints. This constraint can be
rewritten as follows:

$$ \beta_{it}^{0} + \sum_{t' \in [2,t]} \sum_{j \in [J]} d_{jt'} (\xi^{t}) \beta_{it}^{jt'} \geq 0 \qquad i \in [I], t \in [2,T], \xi^{t} \in \Xi^{t} \rightarrow $$

$$ \beta_{it}^{0} + \sum_{t' \in [2,t]} \sum_{j \in [J]} \mu_{jt'} \beta_{it}^{jt'} + \min \left\{ \sum_{t' \in [2,t]} \sum_{j \in [J]} \xi_{jt'} \sigma_{jt'} \beta_{it}^{jt'} : \sum_{t' \in [2,T]} \sum_{j \in [J]} \xi_{jt'} \leq \Gamma, \right. \\ \left. 0 \leq \xi_{jt'} \leq 1, j \in [J], t' \in [2,T] \right\} \geq 0 \qquad i \in [I], t \in [2,T] \rightarrow $$

$$ \beta_{it}^{0} + \sum_{t' \in [2,t]} \sum_{j \in [J]} \mu_{jt'} \beta_{it}^{jt'} + \max \left\{ u\Gamma + \sum_{j \in [J]} \sum_{t' \in [2,T]} \omega_{jt'} : \begin{array}{l} u + \omega_{jt'} \leq \sigma_{jt'} \beta_{it}^{jt'}, j \in [J], t' \in [2,t], \\ u \leq 0, \omega_{jt'} \leq 0, j \in [J], t' \in [2,T] \end{array} \right\} \geq 0 \qquad i \in [I], t \in [2,T] \rightarrow $$

$$ \left\{ \begin{array}{l} \beta_{it}^{0} + \sum_{t' \in [2,t]} \sum_{j \in [J]} \mu_{jt'} \beta_{it}^{jt'} + u\Gamma + \sum_{j \in [J]} \sum_{t' \in [2,T]} \omega_{jt'} \geq 0 \qquad i \in [I], t \in [2,T] \\ u + \omega_{jt'} \leq \sigma_{jt'} \beta_{it}^{jt'} \qquad \qquad \qquad \qquad \qquad \qquad \qquad i \in [I], j \in [J], t \in [2,T], t' \in [2,t] \\ \omega_{jt'} \leq 0 \qquad \qquad \qquad \qquad \qquad \qquad \qquad i \in [I], j \in [J], t \in [2,T], t' \in [2,T] \\ u \leq 0 \end{array} \right. $$

In order to establish relatively complete recourse, we add the following constraints to the outer minimization problem:

$$ \sum_{i \in [I]} \left[ s_{i1} - \beta_{i2}^{0} - \sum_{j \in [J]} d_{j2}(\xi^{2}) \beta_{i2}^{j2} \right] \geq \sum_{j \in [J]} d_{j2}(\xi^{2}) \qquad \xi^{2} \in \Xi^{2} $$

$$ \sum_{i \in [I]} \left[ \beta_{i,t-1}^{0} - \beta_{it}^{0} - \sum_{j \in [J]} \sum_{t' \in [2,t]} d_{jt'}(\xi^{t}) \beta_{it}^{jt'} + \sum_{j \in [J]} \sum_{t' \in [2,t-1]} d_{jt'}(\xi^{t-1}) \beta_{i,t-1}^{jt'} \right] \geq \sum_{j \in [J]} d_{jt}(\xi^{t}) \quad t \in [3,T], \xi^{t} \in \Xi^{t}, $$

which can be reformulated as follows:

$$ \sum_{i \in [I]} \left[ s_{i1} - \beta_{i2}^{0} \right] \geq \max_{\xi^{2} \in \Xi^{2}} \left\{ \sum_{j \in [J]} d_{j2}(\xi^{2}) \left[ 1 + \sum_{i \in [I]} \beta_{i2}^{j2} \right] \right\} $$

$$ \sum_{i \in [I]} \left[ \beta_{i,t-1}^{0} - \beta_{it}^{0} \right] \geq \max_{\xi^{t} \in \Xi^{t}} \left\{ \sum_{i \in [I]} \left[ \sum_{j \in [J]} \sum_{t' \in [2,t]} d_{jt'}(\xi^{t}) \beta_{it}^{jt'} - \sum_{j \in [J]} \sum_{t' \in [2,t-1]} d_{jt'}(\xi^{t-1}) \beta_{i,t-1}^{jt'} \right] + \sum_{j \in [J]} d_{jt}(\xi^{t}) \right\} \quad t \in [3,T] $$

The linearization process is similar to the one previously discussed for non-negativity constraints. Denote by $\pi^1, \pi^2, \pi^3$ the dual variables associated with three constraint sets of the inner minimization problem. After taking its linear programming dual and the adding the KKT conditions, we get the following as the subproblem of the column-and-constraint generation:

$$ \max \sum_{t \in [2,T]} \sum_{i \in [I]} \sum_{j \in [J]} \sum_{t' \in [2,t]} a_i d_{jt'} \beta_{it}^{jt'} + \sum_{t \in [2,T]} \sum_{i \in [I]} \sum_{j \in [J]} c_{ij} x_{ijt} $$
$$ \text{s.t.} \quad \sum_{j \in [J]} x_{ij2} = s_{i1} - \beta_{i2}^0 - \sum_{j \in [J]} d_{j2} \beta_{i2}^{j2} \qquad i \in [I] $$

$$ \sum_{j \in [J]} x_{ijt} = \beta_{i,t-1}^0 - \beta_{it}^0 - \sum_{j \in [J]} \sum_{t' \in [2,t]} d_{jt'} \beta_{it}^{j't'} + \sum_{j \in [J]} \sum_{t' \in [2,t-1]} d_{jt'} \beta_{i,t-1}^{j't'} \quad i \in [I], t \in [3,T] $$

$$ \sum_{i \in [I]} x_{ijt} \geq d_{jt} \qquad j \in [J], t \in [2,T] $$

$$ \pi_i^1 + \pi_{j2}^3 \leq c_{ij} \qquad i \in [I], j \in [J] $$

$$ \pi_{it}^2 + \pi_{jt}^3 \leq c_{ij} \qquad i \in [I], j \in [J], t \in [3,T] $$

$$ (d_{jt} - \sum_{i \in [I]} x_{ijt}) \pi_{jt}^3 = 0 \xrightarrow{\text{linearization}} \pi_{jt}^3 \leq M(1 - \ell_{jt}^R), \quad \sum_{i \in [I]} x_{ijt} - d_{jt} \leq M \ell_{jt}^R \qquad j \in [J], t \in [2,T] $$

$$ (\pi_i^1 + \pi_{j2}^3 - c_{ij}) x_{ij2} = 0 \xrightarrow{\text{linearization}} x_{ij2} \leq M(1 - \ell_{ij2}^D), \quad c_{ij} - \pi_i^1 - \pi_{j2}^3 \leq M \ell_{ij2}^D \qquad i \in [I], j \in [J] $$

$$ (\pi_{it}^2 + \pi_{jt}^3 - c_{ij}) x_{ijt} = 0 \xrightarrow{\text{linearization}} x_{ijt} \leq M(1 - \ell_{ijt}^D), \quad c_{ij} - \pi_{it}^2 - \pi_{jt}^3 \leq M \ell_{ijt}^D \qquad i \in [I], j \in [J], t \in [3,T] $$

$$ d_{jt} = \mu_{jt} + \xi_{jt} \sigma_{jt} \qquad j \in [J], t \in [2,T] $$

$$ \sum_{t \in [2,T]} \sum_{j \in [J]} \xi_{jt} \leq \Gamma \\ x_{ijt} \geq 0, \quad \ell_{ijt}^D \in \{0,1\} \qquad i \in [I], j \in [J], t \in [2,T] $$

$$ 0 \leq \xi_{jt} \leq 1, \quad 0 \leq \pi_{jt}^3, \quad \ell_{jt}^R \in \{0,1\} \qquad j \in [J], t \in [2,T]. $$

**C.3.2. Restricted NA Dual** The $Q(\beta^s, \beta^x)$ function for the location-transportation problem:

$$ \begin{array}{ll} \mathcal{Q}(\beta^s, \beta^x) = & \min z + \sum_{t \in [2,T]} \sum_{i \in [I]} \beta_{it}^s(\xi^T) s_{it}(\xi^T) + \sum_{t \in [2,T]} \sum_{i \in [I]} \sum_{j \in [J]} \beta_{ijt}^x(\xi^T) x_{ijt}(\xi^T) \\ \text{s.t.} & z \ge \sum_{i \in [I]} y_i + \sum_{t \in [T]} \sum_{i \in [I]} a_i s_{it}(\xi^T) + \sum_{t \in [2,T]} \sum_{i \in [I]} \sum_{j \in [J]} c_{ij} x_{ijt}(\xi^T) \quad \xi^T \in \Xi^T \\ & s_{i1} \le K_i y_i, \qquad i \in [I] \\ & s_{it}(\xi^T) = s_{i,t-1}(\xi^{T-1}) - \sum_{j \in [J]} x_{ijt}(\xi^T) \qquad i \in [I], t \in [2,T], \xi^T \in \Xi^T \\ & \sum_{i \in [I]} x_{ijt}(\xi^T) \ge d_{jt}(\xi^t) \qquad i \in [I], t \in [2,T], \xi^T \in \Xi^T \\ & y \in \{0,1\}^I, z \in \mathbb{R}_+^{I \times T}, x \in \mathbb{R}_+^{I \times J \times (T-1)}. \end{array} $$

For this mixed-integer subproblem, the cutting-plane method presented in Section 4.3 is applicable.
# D. Detailed Results

## D.1. Newsvendor Problem

<table><caption>Table 4 Detailed results for the newsvendor problem</caption><thead><tr><th>T</th><th>BR</th><th>|Ξ<sup>T</sup>|</th><th>I</th><th>B</th><th>ν<sup>LDR</sup></th><th>ν<sup>2S-LDR</sup></th><th>ν<sup>*</sup></th><th>ν̄<sup>NA-DO</sup><sub>R</sub></th><th>ν<sup>PI</sup></th></tr></thead><tbody><tr><td rowspan="6">3</td><td rowspan="6">5</td><td rowspan="6">25</td><td>3</td><td>100</td><td>656.6</td><td>970.2</td><td>975.3</td><td>975.3</td><td>1457.0</td></tr><tr><td>3</td><td>150</td><td>6668.5</td><td>8410.4</td><td>8647.2</td><td>9802.6</td><td>10036.0</td></tr><tr><td>3</td><td>200</td><td>8662.0</td><td>11446.2</td><td>11569.7</td><td>13709.0</td><td>13709.0</td></tr><tr><td>4</td><td>150</td><td>5154.8</td><td>5384.4</td><td>5466.8</td><td>5466.8</td><td>5781.0</td></tr><tr><td>4</td><td>200</td><td>10049.4</td><td>12452.0</td><td>12759.9</td><td>14501.9</td><td>14525.0</td></tr><tr><td>5</td><td>200</td><td>5944.7</td><td>7120.6</td><td>7402.4</td><td>7402.4</td><td>7713.0</td></tr><tr><td rowspan="6">3</td><td rowspan="6">10</td><td rowspan="6">100</td><td>3</td><td>150</td><td>5752.1</td><td>7620.9</td><td>7984.6</td><td>8590.3</td><td>8895.0</td></tr><tr><td>3</td><td>200</td><td>6279.6</td><td>9199.0</td><td>9700.9</td><td>11833.0</td><td>11833.0</td></tr><tr><td>4</td><td>150</td><td>3502.7</td><td>4886.5</td><td>4910.2</td><td>5110.1</td><td>5323.0</td></tr><tr><td>4</td><td>200</td><td>7922.9</td><td>10961.1</td><td>11273.4</td><td>14036.5</td><td>14060.0</td></tr><tr><td>5</td><td>200</td><td>2865.5</td><td>5877.7</td><td>6063.1</td><td>6063.1</td><td>6737.0</td></tr><tr><td rowspan="6">4</td><td rowspan="6">4</td><td rowspan="6">64</td><td>2</td><td>100</td><td>669</td><td>1047</td><td>1214</td><td>1214</td><td>1337</td></tr><tr><td>3</td><td>200</td><td>7807</td><td>9198</td><td>9491</td><td>9492</td><td>10157</td></tr><tr><td>3</td><td>300</td><td>13421</td><td>16414</td><td>17317</td><td>19105</td><td>19188</td></tr><tr><td>4</td><td>200</td><td>2034</td><td>2297</td><td>2447</td><td>2447</td><td>2640</td></tr><tr><td>4</td><td>300</td><td>13645</td><td>17860</td><td>18721</td><td>20139</td><td>20139</td></tr><tr><td rowspan="6">4</td><td rowspan="6">5</td><td rowspan="6">125</td><td>3</td><td>200</td><td>6643.8</td><td>8070.1</td><td>8295.6</td><td>8785.8</td><td>9249.0</td></tr><tr><td>3</td><td>300</td><td>11854.3</td><td>15035.4</td><td>16606.8</td><td>18240.0</td><td>18240.0</td></tr><tr><td>4</td><td>200</td><td>1443.7</td><td>1700.8</td><td>1739.8</td><td>1917.7</td><td>2122.0</td></tr><tr><td>4</td><td>300</td><td>13186.4</td><td>17579.4</td><td>18450.8</td><td>19618.0</td><td>19635.0</td></tr><tr><td>5</td><td>300</td><td>5142.0</td><td>10494.4</td><td>11299.5</td><td>11804.9</td><td>12343.0</td></tr><tr><td rowspan="6">4</td><td rowspan="6">10</td><td rowspan="6">1000</td><td>3</td><td>200</td><td>5648.8</td><td>8142.9</td><td></td><td>9353.0</td><td>9353.0</td></tr><tr><td>3</td><td>300</td><td>9143.5</td><td>13687.6</td><td></td><td>17853.0</td><td>17853.0</td></tr><tr><td>4</td><td>200</td><td>-104.4</td><td>440.0</td><td></td><td>642.4</td><td>919.0</td></tr><tr><td>4</td><td>300</td><td>11029.5</td><td>15854.7</td><td></td><td>18508.0</td><td>18432.0</td></tr><tr><td>5</td><td>300</td><td>724.9</td><td>6125.5</td><td></td><td>6740.1</td><td>7368.0</td></tr><tr><td rowspan="6">4</td><td rowspan="6">15</td><td rowspan="6">3375</td><td>3</td><td>200</td><td>5111.3</td><td>7072.0</td><td></td><td>7646.6</td><td>8222.0</td></tr><tr><td>3</td><td>300</td><td>9030.9</td><td>13040.7</td><td></td><td>17615.0</td><td>17615.0</td></tr><tr><td>2</td><td>150</td><td>2490.9</td><td>2673.5</td><td>2763.4</td><td>2763.3</td><td>2975.4</td></tr><tr><td>2</td><td>200</td><td>8888.4</td><td>11192.5</td><td>11313.7</td><td>11525.6</td><td>11525.6</td></tr><tr><td>3</td><td>250</td><td>7809.0</td><td>8800.3</td><td>9065.4</td><td>9104.8</td><td>9382.3</td></tr><tr><td>3</td><td>300</td><td>14702.3</td><td>16898.8</td><td>17214.3</td><td>17308.7</td><td>18156.1</td></tr><tr><td>5</td><td>4</td><td>256</td><td>3</td><td>300</td><td>12367.7</td><td>15591.0</td><td>15989.0</td><td>16511.1</td><td>17192.6</td></tr><tr><td>5</td><td>5</td><td>625</td><td>3</td><td>300</td><td>11134.5</td><td>15053.4</td><td></td><td>16191.6</td><td>16680.0</td></tr><tr><td>5</td><td>6</td><td>1296</td><td>3</td><td>400</td><td>15013.4</td><td>20271.7</td><td></td><td>25157.0</td><td>25157.0</td></tr><tr><td rowspan="3">6</td><td rowspan="3">4</td><td rowspan="3">1024</td><td>3</td><td>400</td><td>15492.3</td><td>21457.9</td><td></td><td>28163.0</td><td>28163.0</td></tr><tr><td>4</td><td>400</td><td>7124.3</td><td>14805.2</td><td></td><td>15400.6</td><td>15473.0</td></tr><tr><td>4</td><td>500</td><td>14445.4</td><td>24887.7</td><td></td><td>32973.0</td><td>32973.0</td></tr><tr><td rowspan="3">7</td><td rowspan="3">3</td><td rowspan="3">729</td><td>3</td><td>300</td><td>24.3</td><td>1495.9</td><td></td><td>2137.3</td><td>2383.0</td></tr><tr><td>3</td><td>400</td><td>12994.7</td><td>17774.1</td><td></td><td>19554.5</td><td>19983.0</td></tr><tr><td>4</td><td>400</td><td>3267.5</td><td>4303.1</td><td></td><td>5388.1</td><td>5427.0</td></tr><tr><td rowspan="2">8</td><td rowspan="2">3</td><td rowspan="2">2187</td><td>3</td><td>500</td><td>16149.1</td><td>25321.4</td><td></td><td>30259.0</td><td>30259.0</td></tr><tr><td>4</td><td>600</td><td>16892.3</td><td>27608.1</td><td></td><td>28717.7</td><td>28747.0</td></tr></tbody></table>
<table><caption>**Table 5** Running times for the larger-size newsvendor problem instances</caption><thead><tr><th rowspan="2">T</th><th rowspan="2">BR</th><th rowspan="2">|Ξ<sup>T</sup>|</th><th rowspan="2">I</th><th rowspan="2">B</th><th colspan="4">Time (s)</th></tr><tr><th>ν<sup>LDR</sup></th><th>ν<sup>2S-LDR</sup></th><th>ν&#773;<sup>NA-DO</sup></th><th>PI</th></tr></thead><tbody><tr><td rowspan="10">4</td><td rowspan="5">10</td><td rowspan="5">1000</td><td>3</td><td>200</td><td>38</td><td>31</td><td>89</td><td>1</td></tr><tr><td>3</td><td>300</td><td>50</td><td>31</td><td>20</td><td>1</td></tr><tr><td>4</td><td>200</td><td>47</td><td>28</td><td>637</td><td>1</td></tr><tr><td>4</td><td>300</td><td>58</td><td>30</td><td>2712</td><td>1</td></tr><tr><td>5</td><td>300</td><td>59</td><td>33</td><td>3088</td><td>2</td></tr><tr><td>15</td><td>3375</td><td>3</td><td>200</td><td>158</td><td>140</td><td>558</td><td>3</td></tr><tr><td>20</td><td>8000</td><td>3</td><td>300</td><td>726</td><td>675</td><td>2313</td><td>7</td></tr><tr><td rowspan="6">5</td><td rowspan="3">5</td><td rowspan="3">625</td><td>3</td><td>300</td><td>16</td><td>15</td><td>1423</td><td>1</td></tr><tr><td>4</td><td>300</td><td>22</td><td>19</td><td>&gt; 1h</td><td>1</td></tr><tr><td>6</td><td>1296</td><td>3</td><td>400</td><td>54</td><td>54</td><td>64</td><td>1</td></tr><tr><td rowspan="3">6</td><td rowspan="3">4</td><td rowspan="3">1024</td><td>3</td><td>400</td><td>57</td><td>53</td><td>99</td><td>1</td></tr><tr><td>4</td><td>400</td><td>70</td><td>67</td><td>12</td><td>2</td></tr><tr><td>4</td><td>500</td><td>74</td><td>69</td><td>3405</td><td>2</td></tr><tr><td rowspan="3">7</td><td rowspan="3">3</td><td rowspan="3">729</td><td>3</td><td>300</td><td>37</td><td>38</td><td>280</td><td>1</td></tr><tr><td>3</td><td>400</td><td>37</td><td>36</td><td>&gt; 1h</td><td>1</td></tr><tr><td>4</td><td>400</td><td>50</td><td>50</td><td>&gt; 1h</td><td>1</td></tr><tr><td rowspan="2">8</td><td rowspan="2">3</td><td rowspan="2">2187</td><td>3</td><td>500</td><td>364</td><td>335</td><td>408</td><td>4</td></tr><tr><td>4</td><td>600</td><td>462</td><td>464</td><td>&gt; 1h</td><td>5</td></tr></tbody></table>
## D.2. Location-Transportation Problem

<table><caption>**Table 6** Detailed results for the location-transportation problem over a budgeted uncertainty set</caption><thead><tr><th>(T, I, J, α<sup>d</sup>)</th><th>α<sup>u</sup></th><th>ν<sup>2S-LDR</sup></th><th>ν&#773;<sup>NA-DO</sup><sub>R</sub></th><th>(T, I, J, α<sup>d</sup>)</th><th>α<sup>u</sup></th><th>ν<sup>2S-LDR</sup></th><th>ν&#773;<sup>NA-DO</sup><sub>R</sub></th></tr></thead><tbody><tr><td rowspan="4">(3, 10, 10, 0.1)</td><td>0.1</td><td>738969.9</td><td>684460.5</td><td rowspan="4">(4, 5, 7, 0.5)</td><td>0.1</td><td>1052392.2</td><td>826648.8</td></tr><tr><td>0.4</td><td>738967.8</td><td>647069.1</td><td>0.4</td><td>1052393.0</td><td>767446.4</td></tr><tr><td>0.7</td><td>738967.0</td><td>573052.3</td><td>0.7</td><td>1052392.4</td><td>863507.3</td></tr><tr><td>1</td><td>738969.3</td><td>574580.2</td><td>1</td><td>1052392.9</td><td>806706.2</td></tr><tr><td rowspan="4">(3, 10, 10, 0.3)</td><td>0.1</td><td>1657362.2</td><td>1457789.5</td><td rowspan="4">(4, 5, 10, 0.1)</td><td>0.1</td><td>2414351.2</td><td>2008281.8</td></tr><tr><td>0.4</td><td>1637485.0</td><td>1364705.6</td><td>0.4</td><td>2414351.2</td><td>1736011.9</td></tr><tr><td>0.7</td><td>1535360.3</td><td>1205974.5</td><td>0.7</td><td>2414351.2</td><td>2168851.3</td></tr><tr><td>1</td><td>1554797.6</td><td>1223643.1</td><td>1</td><td>2414351.2</td><td>1954758.7</td></tr><tr><td rowspan="4">(3, 10, 10, 0.5)</td><td>0.1</td><td>1641883.1</td><td>1430838.7</td><td rowspan="4">(4, 5, 10, 0.3)</td><td>0.1</td><td>2414351.2</td><td>1912871.0</td></tr><tr><td>0.4</td><td>1605346.2</td><td>1363127.9</td><td>0.4</td><td>2414351.2</td><td>1675507.4</td></tr><tr><td>0.7</td><td>1646129.3</td><td>1178537.7</td><td>0.7</td><td>2414351.2</td><td>1747647.4</td></tr><tr><td>1</td><td>1592763.4</td><td>1474506.6</td><td>1</td><td>2414351.5</td><td>2071113.7</td></tr><tr><td rowspan="4">(3, 10, 15, 0.1)</td><td>0.1</td><td>1133251.5</td><td>857198.3</td><td rowspan="4">(4, 5, 10, 0.5)</td><td>0.1</td><td>2414351.1</td><td>1745320.2</td></tr><tr><td>0.4</td><td>1133251.5</td><td>825580.3</td><td>0.4</td><td>2342846.9</td><td>2076574.6</td></tr><tr><td>0.7</td><td>1157049.6</td><td>1073536.8</td><td>0.7</td><td>2386394.3</td><td>1959412.9</td></tr><tr><td>1</td><td>1156385.8</td><td>956570.9</td><td>1</td><td>2386394.3</td><td>1824441.4</td></tr><tr><td rowspan="4">(3, 10, 10, 0.3)</td><td>0.1</td><td>1157948.4</td><td>1062129.6</td><td rowspan="4">(4, 10, 10, 0.1)</td><td>0.1</td><td>2502997.3</td><td>2064097.4</td></tr><tr><td>0.4</td><td>1157948.4</td><td>861882.4</td><td>0.4</td><td>2508960.6</td><td>1842514.1</td></tr><tr><td>0.7</td><td>1157948.4</td><td>1092108.1</td><td>0.7</td><td>2508960.5</td><td>1688369.1</td></tr><tr><td>1</td><td>1157948.4</td><td>1000064.7</td><td>1</td><td>2508960.8</td><td>2237510.8</td></tr><tr><td rowspan="4">(3, 10, 10, 0.5)</td><td>0.1</td><td>1157948.4</td><td>831435.5</td><td rowspan="4">(4, 10, 10, 0.3)</td><td>0.1</td><td>2344765.1</td><td>2189919.0</td></tr><tr><td>0.4</td><td>1157948.4</td><td>981328.1</td><td>0.4</td><td>2508960.6</td><td>2095408.1</td></tr><tr><td>0.7</td><td>1156079.7</td><td>946148.4</td><td>0.7</td><td>2508961.0</td><td>2071323.3</td></tr><tr><td>1</td><td>1116748.8</td><td>885303.1</td><td>1</td><td>2401193.3</td><td>1728094.5</td></tr><tr><td rowspan="4">(4, 5, 5, 0.1)</td><td>0.1</td><td>770943.7</td><td>724852.8</td><td rowspan="4">(4, 10, 10, 0.5)</td><td>0.1</td><td>2432431.8</td><td>2130923.4</td></tr><tr><td>0.4</td><td>770943.7</td><td>573157.6</td><td>0.4</td><td>2474962.8</td><td>2203068.7</td></tr><tr><td>0.7</td><td>770943.7</td><td>687320.0</td><td>0.7</td><td>2347843.6</td><td>1921134.8</td></tr><tr><td>1</td><td>770943.7</td><td>727980.6</td><td>1</td><td>2347843.6</td><td>1743911.8</td></tr><tr><td rowspan="4">(4, 5, 5, 0.3)</td><td>0.1</td><td>770943.7</td><td>710778.0</td><td rowspan="4">(5, 5, 10, 0.1)</td><td>0.1</td><td>3562840.8</td><td>3075701.1</td></tr><tr><td>0.4</td><td>770943.7</td><td>631020.7</td><td>0.4</td><td>3595989.4</td><td>2855492.7</td></tr><tr><td>0.7</td><td>770943.7</td><td>664643.9</td><td>0.7</td><td>3563956.9</td><td>2788612.9</td></tr><tr><td>1</td><td>770943.7</td><td>696703.2</td><td>1</td><td>3563956.4</td><td>3256824.0</td></tr><tr><td rowspan="4">(4, 5, 5, 0.5)</td><td>0.1</td><td>770943.7</td><td>562210.5</td><td rowspan="4">(5, 5, 10, 0.3)</td><td>0.1</td><td>3623768.3</td><td>2694805.6</td></tr><tr><td>0.4</td><td>770943.7</td><td>662298.1</td><td>0.4</td><td>3501499.4</td><td>2826482.5</td></tr><tr><td>0.7</td><td>770943.7</td><td>620073.7</td><td>0.7</td><td>3649157.4</td><td>2483339.2</td></tr><tr><td>1</td><td>770943.7</td><td>729544.4</td><td>1</td><td>3470084.5</td><td>2450441.3</td></tr><tr><td rowspan="4">(4, 5, 7, 0.1)</td><td>0.1</td><td>1052392.2</td><td>908927.4</td><td rowspan="4">(5, 5, 10, 0.5)</td><td>0.1</td><td>3630095.7</td><td>2474291.5</td></tr><tr><td>0.4</td><td>1052392.2</td><td>847845.0</td><td>0.4</td><td>3573873.7</td><td>2622461.9</td></tr><tr><td>1</td><td>1052392.2</td><td>794593.6</td><td>0.7</td><td>3610736.5</td><td>2973602.4</td></tr><tr><td>1</td><td>1052392.2</td><td>848889.1</td><td>1</td><td>3489038.0</td><td>2543037.8</td></tr><tr><td rowspan="4">(4, 5, 7, 0.3)</td><td>0.1</td><td>1052392.2</td><td>824873.8</td><td rowspan="4"></td><td></td><td></td><td></td></tr><tr><td>0.4</td><td>1052392.2</td><td>984627.8</td><td></td><td></td><td></td><td></td></tr><tr><td>0.7</td><td>1052392.2</td><td>855154.4</td><td></td><td></td><td></td><td></td></tr><tr><td>1</td><td>1052392.7</td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>
<table><caption>Table 7 Algorithmic details for the location-transportation problem over a budgeted uncertainty set</caption><thead><tr><th rowspan="2">(T, I, J, α<sup>d</sup>)</th><th rowspan="2">α<sup>u</sup></th><th colspan="3">ν<sup>2S-LDR</sup>, C&CG</th><th colspan="3">ν<sup>Ω(LDR)</sup></th><th colspan="3">ν<sup>Ω(2S-LDR)</sup></th><th colspan="3">ν̄<sup>NA-DO</sup></th></tr><tr><th>Time (s)</th><th>Time (s)</th><th>#iterations</th><th>Gap</th><th>Time (s)</th><th>Time (s)</th><th>Time (s)</th><th>Time (s)</th><th>#iterations</th><th>Gap</th><th>Time (s)</th><th>Time (s)</th><th>#iterations</th><th>Gap</th></tr></thead><tbody><tr><td rowspan="4">(3, 10, 10, 0.1)</td><td>0.1</td><td>0.5</td><td>168.3</td><td>2</td><td>0.0%</td><td>2.4</td><td>1.4</td><td>1376.9</td><td>617</td><td>4.9%</td><td>1376.9</td><td>617</td><td>4.9%</td><td>1376.9</td><td>617</td></tr><tr><td>0.4</td><td>6.5</td><td>59.6</td><td>2</td><td>0.0%</td><td>9.2</td><td>11.5</td><td>1113.5</td><td>809</td><td>2.6%</td><td>1113.5</td><td>809</td><td>2.6%</td><td>1113.5</td><td>809</td></tr><tr><td>0.7</td><td>10.8</td><td>87.0</td><td>2</td><td>0.0%</td><td>9.3</td><td>9.8</td><td>1088.4</td><td>726</td><td>4.3%</td><td>1088.4</td><td>726</td><td>4.3%</td><td>1088.4</td><td>726</td></tr><tr><td>1</td><td>1.3</td><td>141.8</td><td>2</td><td>0.0%</td><td>16.7</td><td>10.9</td><td>&gt; 10h</td><td>1315</td><td>7.9%</td><td>1315</td><td>1315</td><td>7.9%</td><td>1315</td><td>1315</td></tr><tr><td rowspan="4">(3, 10, 10, 0.3)</td><td>0.1</td><td>14.6</td><td>867.2</td><td>12</td><td>4.8%</td><td>2.2</td><td>0.1</td><td>&gt; 10h</td><td>1013</td><td>8.3%</td><td>1013</td><td>1013</td><td>8.3%</td><td>1013</td><td>1013</td></tr><tr><td>0.4</td><td>4.7</td><td>51.9</td><td>2</td><td>4.9%</td><td>8.1</td><td>16.5</td><td>1082.4</td><td>242</td><td>4.1%</td><td>1082.4</td><td>242</td><td>4.1%</td><td>1082.4</td><td>242</td></tr><tr><td>0.7</td><td>12.6</td><td>66.1</td><td>2</td><td>1.2%</td><td>3.1</td><td>1.6</td><td>&gt; 10h</td><td>730</td><td>7.1%</td><td>730</td><td>730</td><td>7.1%</td><td>730</td><td>730</td></tr><tr><td>1</td><td>8.4</td><td>71.6</td><td>2</td><td>4.9%</td><td>2.6</td><td>0.9</td><td>588.0</td><td>984</td><td>3.0%</td><td>984</td><td>984</td><td>3.0%</td><td>984</td><td>984</td></tr><tr><td rowspan="4">(3, 10, 10, 0.5)</td><td>0.1</td><td>12.9</td><td>124.6</td><td>2</td><td>4.7%</td><td>7.9</td><td>10.5</td><td>&gt; 10h</td><td>271</td><td>7.0%</td><td>271</td><td>271</td><td>7.0%</td><td>271</td><td>271</td></tr><tr><td>0.4</td><td>16.3</td><td>111.8</td><td>3</td><td>4.9%</td><td>1.0</td><td>0.7</td><td>&gt; 10h</td><td>665</td><td>9.1%</td><td>665</td><td>665</td><td>9.1%</td><td>665</td><td>665</td></tr><tr><td>0.7</td><td>1.8</td><td>83.2</td><td>3</td><td>4.3%</td><td>1.7</td><td>0.7</td><td>1132.4</td><td>1064</td><td>3.2%</td><td>1064</td><td>1064</td><td>3.2%</td><td>1064</td><td>1064</td></tr><tr><td>1</td><td>25.2</td><td>96.5</td><td>3</td><td>3.9%</td><td>2.1</td><td>2.2</td><td>&gt; 10h</td><td>1405</td><td>4.1%</td><td>1405</td><td>1405</td><td>4.1%</td><td>1405</td><td>1405</td></tr><tr><td rowspan="4">(3, 10, 15, 0.1)</td><td>0.1</td><td>10.2</td><td>150.1</td><td>3</td><td>1.2%</td><td>2.1</td><td>1.3</td><td>2634.8</td><td>258</td><td>2.9%</td><td>2634.8</td><td>258</td><td>2.9%</td><td>2634.8</td><td>258</td></tr><tr><td>0.4</td><td>3.5</td><td>112.6</td><td>2</td><td>1.2%</td><td>2.3</td><td>1.1</td><td>&gt; 10h</td><td>337</td><td>6.9%</td><td>337</td><td>337</td><td>6.9%</td><td>337</td><td>337</td></tr><tr><td>0.7</td><td>29.3</td><td>96.6</td><td>2</td><td>0.0%</td><td>1.7</td><td>2.0</td><td>&gt; 10h</td><td>432</td><td>5.5%</td><td>432</td><td>432</td><td>5.5%</td><td>432</td><td>432</td></tr><tr><td>1</td><td>8.3</td><td>29.3</td><td>2</td><td>0.0%</td><td>2.2</td><td>2.7</td><td>&gt; 10h</td><td>824</td><td>9.9%</td><td>824</td><td>824</td><td>9.9%</td><td>824</td><td>824</td></tr><tr><td rowspan="4">(3, 10, 10, 0.3)</td><td>0.1</td><td>2.5</td><td>169.7</td><td>2</td><td>1.1%</td><td>3.0</td><td>2.4</td><td>4193.8</td><td>771</td><td>2.1%</td><td>4193.8</td><td>771</td><td>2.1%</td><td>4193.8</td><td>771</td></tr><tr><td>0.4</td><td>13.4</td><td>95.6</td><td>2</td><td>1.2%</td><td>3.8</td><td>0.4</td><td>4706.4</td><td>974</td><td>4.9%</td><td>4706.4</td><td>974</td><td>4.9%</td><td>4706.4</td><td>974</td></tr><tr><td>0.7</td><td>13.0</td><td>94.6</td><td>2</td><td>1.1%</td><td>2.3</td><td>1.2</td><td>5043.6</td><td>475</td><td>2.0%</td><td>5043.6</td><td>475</td><td>2.0%</td><td>5043.6</td><td>475</td></tr><tr><td>1</td><td>10.2</td><td>119.9</td><td>2</td><td>1.1%</td><td>1.6</td><td>2.0</td><td>5015.7</td><td>1187</td><td>4.4%</td><td>5015.7</td><td>1187</td><td>4.4%</td><td>5015.7</td><td>1187</td></tr><tr><td rowspan="4">(3, 10, 10, 0.5)</td><td>0.1</td><td>5.7</td><td>97.1</td><td>6</td><td>1.3%</td><td>2.0</td><td>1.9</td><td>3757.0</td><td>1253</td><td>1.1%</td><td>3757.0</td><td>1253</td><td>1.1%</td><td>3757.0</td><td>1253</td></tr><tr><td>0.4</td><td>2.8</td><td>155.3</td><td>2</td><td>1.2%</td><td>3.0</td><td>1.5</td><td>2902.3</td><td>803</td><td>4.5%</td><td>2902.3</td><td>803</td><td>4.5%</td><td>2902.3</td><td>803</td></tr><tr><td>0.7</td><td>0.7</td><td>112.0</td><td>2</td><td>0.0%</td><td>8.5</td><td>1.6</td><td>&gt; 10h</td><td>884</td><td>9.7%</td><td>884</td><td>884</td><td>9.7%</td><td>884</td><td>884</td></tr><tr><td>1</td><td>2.3</td><td>89.9</td><td>2</td><td>1.2%</td><td>2.3</td><td>4.5</td><td>3898.8</td><td>1120</td><td>4.6%</td><td>3898.8</td><td>1120</td><td>4.6%</td><td>3898.8</td><td>1120</td></tr><tr><td rowspan="4">(4, 5, 5, 0.1)</td><td>0.1</td><td>0.2</td><td>0.8</td><td>2</td><td>0.0%</td><td>7.3</td><td>16.5</td><td>1198.6</td><td>951</td><td>4.8%</td><td>1198.6</td><td>951</td><td>4.8%</td><td>1198.6</td><td>951</td></tr><tr><td>0.4</td><td>0.3</td><td>1.2</td><td>2</td><td>0.0%</td><td>7.1</td><td>21.8</td><td>&gt; 10h</td><td>1483</td><td>7.2%</td><td>1483</td><td>1483</td><td>7.2%</td><td>1483</td><td>1483</td></tr><tr><td>0.7</td><td>0.6</td><td>1.1</td><td>2</td><td>0.0%</td><td>2.5</td><td>5.4</td><td>1558.8</td><td>629</td><td>2.1%</td><td>1558.8</td><td>629</td><td>2.1%</td><td>1558.8</td><td>629</td></tr><tr><td>1</td><td>0.5</td><td>1.2</td><td>2</td><td>0.0%</td><td>1.4</td><td>2.6</td><td>&gt; 10h</td><td>1267</td><td>7.3%</td><td>1267</td><td>1267</td><td>7.3%</td><td>1267</td><td>1267</td></tr><tr><td rowspan="4">(4, 5, 5, 0.3)</td><td>0.1</td><td>0.7</td><td>1.3</td><td>2</td><td>0.0%</td><td>5.6</td><td>0.9</td><td>1649.9</td><td>1399</td><td>0.6%</td><td>1649.9</td><td>1399</td><td>0.6%</td><td>1649.9</td><td>1399</td></tr><tr><td>0.4</td><td>0.4</td><td>0.9</td><td>2</td><td>0.0%</td><td>8.2</td><td>3.0</td><td>&gt; 10h</td><td>569</td><td>9.0%</td><td>569</td><td>569</td><td>9.0%</td><td>569</td><td>569</td></tr><tr><td>0.7</td><td>0.3</td><td>0.4</td><td>2</td><td>0.0%</td><td>3.5</td><td>1.1</td><td>905.6</td><td>1016</td><td>3.3%</td><td>1016</td><td>1016</td><td>3.3%</td><td>1016</td><td>1016</td></tr><tr><td>1</td><td>0.9</td><td>0.4</td><td>2</td><td>0.0%</td><td>1.5</td><td>3.3</td><td>&gt; 10h</td><td>548</td><td>8.4%</td><td>548</td><td>548</td><td>8.4%</td><td>548</td><td>548</td></tr><tr><td rowspan="4">(4, 5, 5, 0.5)</td><td>0.1</td><td>0.4</td><td>0.9</td><td>2</td><td>0.0%</td><td>2.2</td><td>0.0</td><td>&gt; 10h</td><td>550</td><td>7.0%</td><td>550</td><td>550</td><td>7.0%</td><td>550</td><td>550</td></tr><tr><td>0.4</td><td>0.5</td><td>1.3</td><td>2</td><td>0.0%</td><td>9.5</td><td>14.1</td><td>&gt; 10h</td><td>284</td><td>7.0%</td><td>284</td><td>284</td><td>7.0%</td><td>284</td><td>284</td></tr><tr><td>0.7</td><td>0.2</td><td>1.4</td><td>2</td><td>0.0%</td><td>3.9</td><td>2.0</td><td>1064.2</td><td>920</td><td>2.6%</td><td>920</td><td>920</td><td>2.6%</td><td>920</td><td>920</td></tr><tr><td>1</td><td>0.2</td><td>1.2</td><td>2</td><td>0.0%</td><td>3.9</td><td>2.5</td><td>&gt; 10h</td><td>912</td><td>5.6%</td><td>912</td><td>912</td><td>5.6%</td><td>912</td><td>912</td></tr><tr><td rowspan="4">(4, 5, 7, 0.1)</td><td>0.1</td><td>4.7</td><td>77.2</td><td>2</td><td>0.0%</td><td>1.4</td><td>0.9</td><td>1207.0</td><td>332</td><td>4.8%</td><td>1207.0</td><td>332</td><td>4.8%</td><td>1207.0</td><td>332</td></tr><tr><td>0.4</td><td>9.7</td><td>133.8</td><td>2</td><td>0.0%</td><td>4.8</td><td>10.1</td><td>&gt; 10h</td><td>433</td><td>8.7%</td><td>433</td><td>433</td><td>8.7%</td><td>433</td><td>433</td></tr><tr><td>1</td><td>1.1</td><td>159.8</td><td>2</td><td>0.0%</td><td>2.7</td><td>5.5</td><td>&gt; 10h</td><td>1073</td><td>6.8%</td><td>1073</td><td>1073</td><td>6.8%</td><td>1073</td><td>1073</td></tr><tr><td>1</td><td>2.4</td><td>123.2</td><td>2</td><td>0.0%</td><td>2.9</td><td>4.6</td><td>886.7</td><td>864</td><td>4.9%</td><td>886.7</td><td>864</td><td>4.9%</td><td>886.7</td><td>864</td></tr><tr><td rowspan="4">(4, 5, 7, 0.3)</td><td>0.4</td><td>6.6</td><td>197.4</td><td>2</td><td>0.0%</td><td>4.9</td><td>2.1</td><td>1364.0</td><td>1361</td><td>4.4%</td><td>1364.0</td><td>1361</td><td>4.4%</td><td>1364.0</td><td>1361</td></tr><tr><td>0.7</td><td>5.3</td><td>179.7</td><td>2</td><td>0.0%</td><td>2.2</td><td>2.1</td><td>&gt; 10h</td><td>787</td><td>6.9%</td><td>787</td><td>787</td><td>6.9%</td><td>787</td><td>787</td></tr><tr><td>1</td><td>3.2</td><td>156.8</td><td>2</td><td>0.0%</td><td>3.7</td><td>0.8</td><td>&gt; 10h</td><td>966</td><td>9.3%</td><td>966</td><td>966</td><td>9.3%</td><td>966</td><td>966</td></tr><tr><td>1</td><td>6.2</td><td>121.9</td><td>2</td><td>0.0%</td><td>1.5</td><td>1.2</td><td>1047.0</td><td>1296</td><td>4.5%</td><td>1047.0</td><td>1296</td><td>4.5%</td><td>1047.0</td><td>1296</td></tr><tr><td rowspan="4">(4, 5, 7, 0.5)</td><td>0.4</td><td>3.9</td><td>101.3</td><td>2</td><td>0.0%</td><td>10.3</td><td>14.3</td><td>988.4</td><td>392</td><td>4.7%</td><td>988.4</td><td>392</td><td>4.7%</td><td>988.4</td><td>392</td></tr><tr><td>0.7</td><td>25.5</td><td>135.8</td><td>2</td><td>0.0%</td><td>2.2</td><td>6.6</td><td>970.9</td><td>562</td><td>4.8%</td><td>970.9</td><td>562</td><td>4.8%</td><td>970.9</td><td>562</td></tr><tr><td>1</td><td>4.6</td><td>204.7</td><td>2</td><td>0.0%</td><td>4.1</td><td>3.1</td><td>&gt; 10h</td><td>591</td><td>8.9%</td><td>591</td><td>591</td><td>8.9%</td><td>591</td><td>591</td></tr><tr><td>1</td><td>4.8</td><td>83.7</td><td>2</td><td>1.7%</td><td>3.0</td><td>7.2</td><td>&gt; 10h</td><td>1508</td><td>9.4%</td><td>1508</td><td>1508</td><td>9.4%</td><td>1508</td><td>1508</td></tr><tr><td rowspan="4">(4, 5, 10, 0.1)</td><td>0.4</td><td>13.6</td><td>88.8</td><td>2</td><td>1.6%</td><td>1.1</td><td>1.0</td><td>&gt; 10h</td><td>723</td><td>6.4%</td><td>723</td><td>723</td><td>6.4%</td><td>723</td><td>723</td></tr><tr><td>0.7</td><td>6.8</td><td>102.2</td><td>2</td><td>1.6%</td><td>2.1</td><td>5.0</td><td>4502.0</td><td>479</td><td>4.9%</td><td>479</td><td>479</td><td>4.9%</td><td>479</td><td>479</td></tr><tr><td>1</td><td>6.4</td><td>118.4</td><td>2</td><td>1.8%</td><td>12.6</td><td>4.2</td><td>&gt; 10h</td><td>566</td><td>7.4%</td><td>566</td><td>566</td><td>7.4%</td><td>566</td><td>566</td></tr><tr><td>1</td><td>17.9</td><td>199.3</td><td>2</td><td>1.8%</td><td>3.8</td><td>8.2</td><td>7782.2</td><td>1059</td><td>4.2%</td><td>7782.2</td><td>1059</td><td>4.2%</td><td>7782.2</td><td>1059</td></tr><tr><td rowspan="4">(4, 5, 10, 0.3)</td><td>0.4</td><td>25.3</td><td>103.5</td><td>2</td><td>1.8%</td><td>6.7</td><td>6.3</td><td>12134.1</td><td>667</td><td>2.6%</td><td>667</td><td>667</td><td>2.6%</td><td>667</td><td>667</td></tr><tr><td>0.7</td><td>13.8</td><td>50.2</td><td>2</td><td>1.7%</td><td>9.0</td><td>10.3</td><td>11770.6</td><td>586</td><td>4.5%</td><td>11770.6</td><td>586</td><td>4.5%</td><td>11770.6</td><td>586</td></tr><tr><td>1</td><td>8.6</td><td>107.1</td><td>2</td><td>1.7%</td><td>2.0</td><td>3.8</td><td>9143.7</td><td>421</td><td>2.2%</td><td>9143.7</td><td>421</td><td>2.2%</td><td>9143.7</td><td>421</td></tr><tr><td>1</td><td>10.4</td><td>189.7</td><td>2</td><td>1.7%</td><td>1.2</td><td>0.6</td><td>&gt; 10h</td><td>366</td><td>9.0%</td><td>366</td><td>366</td><td>9.0%</td><td>366</td><td>366</td></tr><tr><td rowspan="4">(4, 5, 10, 0.5)</td><td>0.4</td><td>6.2</td><td>38.2</td><td>2</td><td>1.8%</td><td>5.7</td><td>5.5</td><td>&gt; 10h</td><td>1429</td><td>9.3%</td><td>1429</td><td>1429</td><td>9.3%</td><td>1429</td><td>1429</td></tr><tr><td>0.7</td><td>4.0</td><td>72.6</td><td>2</td><td>1.8%</td><td>3.1</td><td>3.1</td><td>12449.6</td><td>617</td><td>4.9%</td><td>12449.6</td><td>617</td><td>4.9%</td><td>12449.6</td><td>617</td></tr><tr><td>1</td><td>1.0</td><td>41.1</td><td>2</td><td>1.8%</td><td>17.4</td><td>40.7</td><td>11130.6</td><td>739</td><td>4.3%</td><td>11130.6</td><td>739</td><td>4.3%</td><td>11130.6</td><td>739</td></tr></tbody></table>
<table><caption>**Table 7** Algorithmic details for the location-transportation problem over a budgeted uncertainty set (continued)</caption><thead><tr><th rowspan="2">(T, I, J, α<sup>d</sup>)</th><th rowspan="2">α<sup>u</sup></th><th colspan="2">ν<sup>LDR</sup></th><th colspan="2">ν<sup>2S-LDR</sup>, C&CG</th><th colspan="2">ν<sup>Ω(LDR)</sup></th><th colspan="2">ν<sup>Ω(2S-LDR)</sup></th><th colspan="2">ν̄<sup>NA-DO</sup></th></tr><tr><th>Time (s)</th><th>Time (s)</th><th>#iterations</th><th>Gap</th><th>Time (s)</th><th>Time (s)</th><th>Time (s)</th><th>#iterations</th><th>Time (s)</th><th>Gap</th></tr></thead><tbody><tr><td rowspan="4">(4, 10, 10, 0.1)</td><td>0.1</td><td>10.7</td><td>13318.8</td><td>1</td><td>4.1%</td><td>21.0</td><td>43.0</td><td>&gt; 10h</td><td>1880</td><td>5.2%</td></tr><tr><td>0.4</td><td>18.9</td><td>13171.6</td><td>1</td><td>4.8%</td><td>14.8</td><td>9.2</td><td>&gt; 10h</td><td>2154</td><td>8.7%</td></tr><tr><td>0.7</td><td>33.7</td><td>4419.5</td><td>2</td><td>4.6%</td><td>18.0</td><td>49.5</td><td>&gt; 10h</td><td>776</td><td>8.2%</td></tr><tr><td>1</td><td>3.4</td><td>7159.0</td><td>2</td><td>4.8%</td><td>14.6</td><td>6.9</td><td>14098.5</td><td>1275</td><td>4.1%</td></tr><tr><td rowspan="4">(4, 10, 10, 0.3)</td><td>0.1</td><td>35.8</td><td>10763.5</td><td>1</td><td>2.4%</td><td>4.6</td><td>9.1</td><td>9702.9</td><td>885</td><td>4.8%</td></tr><tr><td>0.4</td><td>12.7</td><td>7688.3</td><td>2</td><td>4.8%</td><td>9.3</td><td>8.2</td><td>&gt; 10h</td><td>755</td><td>8.6%</td></tr><tr><td>0.7</td><td>24.5</td><td>12108.4</td><td>1</td><td>4.6%</td><td>8.5</td><td>5.2</td><td>&gt; 10h</td><td>1343</td><td>13.4%</td></tr><tr><td>1</td><td>33.8</td><td>7580.7</td><td>2</td><td>3.2%</td><td>8.8</td><td>1.2</td><td>13045.5</td><td>862</td><td>4.9%</td></tr><tr><td rowspan="4">(4, 10, 10, 0.5)</td><td>0.1</td><td>23.1</td><td>8361.0</td><td>2</td><td>2.4%</td><td>36.8</td><td>7.7</td><td>10972.8</td><td>765</td><td>1.5%</td></tr><tr><td>0.4</td><td>17.5</td><td>9441.9</td><td>2</td><td>4.4%</td><td>36.9</td><td>69.7</td><td>&gt; 10h</td><td>791</td><td>5.7%</td></tr><tr><td>0.7</td><td>16.8</td><td>5638.9</td><td>2</td><td>2.3%</td><td>38.5</td><td>50.6</td><td>14347.7</td><td>1511</td><td>4.4%</td></tr><tr><td>1</td><td>26.7</td><td>3497.6</td><td>2</td><td>2.4%</td><td>29.5</td><td>20.6</td><td>&gt; 10h</td><td>696</td><td>8.7%</td></tr><tr><td rowspan="4">(5, 5, 10, 0.1)</td><td>0.1</td><td>2.3</td><td>372.6</td><td>4</td><td>1.6%</td><td>34.3</td><td>17.0</td><td>&gt; 10h</td><td>1439</td><td>7.5%</td></tr><tr><td>0.4</td><td>5.8</td><td>989.5</td><td>4</td><td>2.8%</td><td>28.3</td><td>38.4</td><td>&gt; 10h</td><td>267</td><td>8.2%</td></tr><tr><td>0.7</td><td>5.4</td><td>657.6</td><td>2</td><td>1.9%</td><td>21.4</td><td>31.2</td><td>15167.1</td><td>605</td><td>4.6%</td></tr><tr><td>1</td><td>3.6</td><td>784.6</td><td>3</td><td>1.7%</td><td>31.3</td><td>52.0</td><td>&gt; 10h</td><td>904</td><td>7.0%</td></tr><tr><td rowspan="4">(5, 5, 10, 0.3)</td><td>0.1</td><td>1.8</td><td>883.5</td><td>2</td><td>2.8%</td><td>39.3</td><td>84.0</td><td>&gt; 10h</td><td>789</td><td>5.2%</td></tr><tr><td>0.4</td><td>5.5</td><td>607.0</td><td>2</td><td>1.9%</td><td>24.5</td><td>23.9</td><td>16175.2</td><td>1053</td><td>4.8%</td></tr><tr><td>0.7</td><td>4.5</td><td>414.9</td><td>10</td><td>2.8%</td><td>27.4</td><td>33.6</td><td>13106.8</td><td>762</td><td>4.8%</td></tr><tr><td>1</td><td>3.5</td><td>959.7</td><td>2</td><td>1.3%</td><td>19.5</td><td>32.5</td><td>14376.6</td><td>1016</td><td>2.0%</td></tr><tr><td rowspan="4">(5, 5, 10, 0.5)</td><td>0.1</td><td>5.0</td><td>1009.4</td><td>2</td><td>1.8%</td><td>35.5</td><td>94.5</td><td>&gt; 10h</td><td>418</td><td>5.6%</td></tr><tr><td>0.4</td><td>6.2</td><td>723.7</td><td>2</td><td>3.0%</td><td>38.5</td><td>52.0</td><td>13067.1</td><td>1329</td><td>4.5%</td></tr><tr><td>0.7</td><td>7.9</td><td>383.6</td><td>2</td><td>2.5%</td><td>39.5</td><td>53.6</td><td>15368.2</td><td>911</td><td>4.0%</td></tr><tr><td>1</td><td>9.9</td><td>724.3</td><td>2</td><td>1.6%</td><td>39.5</td><td>66.5</td><td>14150.1</td><td>521</td><td>4.7%</td></tr></tbody></table>

### D.3. Capital Budgeting with Loan

<table><caption>**Table 8** Results for the capital budgeting problem with unrestricted loans</caption><thead><tr><th rowspan="2">Instance</th><th rowspan="2">(T, I)</th><th rowspan="2">B</th><th rowspan="2">ν<sup>x</sup></th><th colspan="3">UB</th><th colspan="3">Optimality Gap (%)</th></tr><tr><th>ν<sup>Ω</sup></th><th>ν̄<sup>NA-DO</sup></th><th>ν̄<sup>DNA-DO</sup></th><th>(ν<sup>Ω</sup> - ν<sup>x</sup>) / (ν<sup>Ω</sup>)</th><th>(ν̄<sup>NA-DO</sup> - ν<sup>x</sup>) / (ν<sup>Ω</sup>)</th><th>(ν̄<sup>DNA-DO</sup> - ν<sup>x</sup>) / (ν<sup>Ω</sup>)</th></tr></thead><tbody><tr><td>1</td><td></td><td>0</td><td>1.8</td><td>2.2</td><td>2.1</td><td>2.1</td><td>19.7%</td><td>16.1%</td><td>14.5%</td></tr><tr><td>2</td><td></td><td>50</td><td>6.7</td><td>7.9</td><td>7.6</td><td>7.5</td><td>18.0%</td><td>14.1%</td><td>12.6%</td></tr><tr><td>3</td><td>(3,5)</td><td>100</td><td>7.6</td><td>8.8</td><td>8.5</td><td>8.4</td><td>16.2%</td><td>12.8%</td><td>10.4%</td></tr><tr><td>4</td><td></td><td>150</td><td>7.8</td><td>8.8</td><td>8.5</td><td>8.3</td><td>13.1%</td><td>9.1%</td><td>7.0%</td></tr><tr><td>5</td><td></td><td>0</td><td>6.7</td><td>7.9</td><td>7.6</td><td>7.5</td><td>18.7%</td><td>13.8%</td><td>12.3%</td></tr><tr><td>6</td><td></td><td>50</td><td>11.7</td><td>13.9</td><td>13.4</td><td>13.1</td><td>18.9%</td><td>14.5%</td><td>11.8%</td></tr><tr><td>7</td><td>(3,10)</td><td>100</td><td>16.7</td><td>19.7</td><td>19.2</td><td>18.8</td><td>18.5%</td><td>15.2%</td><td>12.6%</td></tr><tr><td>8</td><td></td><td>150</td><td>16.7</td><td>19.7</td><td>19.1</td><td>18.7</td><td>18.0%</td><td>14.1%</td><td>11.8%</td></tr><tr><td>9</td><td></td><td>0</td><td>8.9</td><td>10.4</td><td>10.1</td><td>9.9</td><td>16.5%</td><td>13.3%</td><td>11.0%</td></tr><tr><td>10</td><td></td><td>50</td><td>13.7</td><td>16.4</td><td>15.9</td><td>15.5</td><td>19.4%</td><td>15.8%</td><td>13.0%</td></tr><tr><td>11</td><td>(3,15)</td><td>100</td><td>18.9</td><td>22.4</td><td>21.6</td><td>21.1</td><td>18.7%</td><td>14.3%</td><td>11.8%</td></tr><tr><td>12</td><td></td><td>150</td><td>21.6</td><td>26.0</td><td>25.1</td><td>24.6</td><td>20.4%</td><td>16.3%</td><td>13.9%</td></tr><tr><td>13</td><td></td><td>0</td><td>11.9</td><td>13.8</td><td>13.2</td><td>13.0</td><td>16.0%</td><td>11.2%</td><td>8.9%</td></tr><tr><td>14</td><td></td><td>50</td><td>16.4</td><td>19.8</td><td>19.2</td><td>18.9</td><td>21.1%</td><td>17.6%</td><td>15.6%</td></tr><tr><td>15</td><td>(3,20)</td><td>100</td><td>21.7</td><td>25.8</td><td>25.1</td><td>24.7</td><td>19.0%</td><td>15.8%</td><td>13.8%</td></tr><tr><td>16</td><td></td><td>150</td><td>27.1</td><td>31.8</td><td>31.0</td><td>30.4</td><td>17.5%</td><td>14.4%</td><td>12.4%</td></tr><tr><td>17</td><td></td><td>0</td><td>17.7</td><td>20.2</td><td>19.1</td><td>18.8</td><td>14.2%</td><td>8.3%</td><td>6.7%</td></tr><tr><td>18</td><td></td><td>50</td><td>22.5</td><td>26.2</td><td>25.3</td><td>24.9</td><td>16.5%</td><td>12.7%</td><td>10.9%</td></tr><tr><td>19</td><td>(3,25)</td><td>100</td><td>26.5</td><td>32.2</td><td>31.2</td><td>30.8</td><td>21.6%</td><td>17.8%</td><td>16.4%</td></tr><tr><td>20</td><td></td><td>150</td><td>31.7</td><td>38.2</td><td>36.8</td><td>36.2</td><td>20.5%</td><td>16.1%</td><td>14.3%</td></tr><tr><td>21</td><td></td><td>0</td><td>22.2</td><td>26.4</td><td>25.6</td><td>25.3</td><td>18.9%</td><td>15.4%</td><td>13.9%</td></tr><tr><td>22</td><td></td><td>50</td><td>26.5</td><td>32.4</td><td>31.2</td><td>30.8</td><td>22.3%</td><td>17.9%</td><td>16.5%</td></tr><tr><td>23</td><td>(3,30)</td><td>100</td><td>31.4</td><td>38.4</td><td>37.4</td><td>36.9</td><td>22.3%</td><td>19.1%</td><td>17.6%</td></tr><tr><td>24</td><td></td><td>150</td><td>37.1</td><td>44.4</td><td>42.9</td><td>42.4</td><td>19.7%</td><td>15.8%</td><td>14.3%</td></tr></tbody></table>
<table><caption>**Table 8** Results for the capital budgeting problem with unrestricted loans (continued)</caption><thead><tr><th rowspan="2">Instance</th><th rowspan="2">(T, I)</th><th rowspan="2">B</th><th rowspan="2">ν<sup>K</sup></th><th colspan="3">UB</th><th colspan="3">Optimality Gap (%)</th></tr><tr><th>ν<sup>Ω</sup></th><th>ν̃<sub>R</sub><sup>NA-DO</sup></th><th>ν̃<sub>R</sub><sup>DNA-DO</sup></th><th>(ν<sup>Ω</sup> - ν<sup>K</sup>)/ν<sup>K</sup></th><th>(ν̃<sub>R</sub><sup>NA-DO</sup> - ν<sup>K</sup>)/ν<sup>K</sup></th><th>(ν̃<sub>R</sub><sup>DNA-DO</sup> - ν<sup>K</sup>)/ν<sup>K</sup></th></tr></thead><tbody><tr><td>25</td><td></td><td>0</td><td>3.3</td><td>3.8</td><td>3.6</td><td>3.5</td><td>17.1%</td><td>10.7%</td><td>8.0%</td></tr><tr><td>26</td><td></td><td>50</td><td>8.3</td><td>9.9</td><td>9.4</td><td>9.2</td><td>19.2%</td><td>13.7%</td><td>11.1%</td></tr><tr><td>27</td><td>(4,5)</td><td>100</td><td>8.7</td><td>10.2</td><td>9.7</td><td>9.5</td><td>17.2%</td><td>12.0%</td><td>9.5%</td></tr><tr><td>28</td><td></td><td>150</td><td>7.6</td><td>9.1</td><td>8.6</td><td>8.4</td><td>18.8%</td><td>13.1%</td><td>10.3%</td></tr><tr><td>29</td><td></td><td>200</td><td>8.7</td><td>10.2</td><td>9.9</td><td>9.6</td><td>17.4%</td><td>13.4%</td><td>10.7%</td></tr><tr><td>30</td><td></td><td>250</td><td>8.7</td><td>10.2</td><td>9.6</td><td>9.3</td><td>16.7%</td><td>9.7%</td><td>6.9%</td></tr><tr><td>31</td><td></td><td>0</td><td>8.4</td><td>9.9</td><td>9.5</td><td>9.2</td><td>18.3%</td><td>12.7%</td><td>9.9%</td></tr><tr><td>32</td><td></td><td>50</td><td>13.4</td><td>16.0</td><td>15.1</td><td>14.7</td><td>19.2%</td><td>12.3%</td><td>9.4%</td></tr><tr><td>33</td><td>(4,10)</td><td>100</td><td>18.0</td><td>22.0</td><td>21.2</td><td>20.8</td><td>22.1%</td><td>17.7%</td><td>15.4%</td></tr><tr><td>34</td><td></td><td>150</td><td>20.8</td><td>25.0</td><td>23.9</td><td>23.3</td><td>19.7%</td><td>14.7%</td><td>11.7%</td></tr><tr><td>35</td><td></td><td>200</td><td>20.4</td><td>25.0</td><td>24.1</td><td>23.5</td><td>22.2%</td><td>18.0%</td><td>15.2%</td></tr><tr><td>36</td><td></td><td>250</td><td>21.1</td><td>25.0</td><td>24.0</td><td>23.4</td><td>18.1%</td><td>13.5%</td><td>10.7%</td></tr><tr><td>37</td><td></td><td>0</td><td>12.9</td><td>15.3</td><td>14.5</td><td>14.2</td><td>18.0%</td><td>11.7%</td><td>9.3%</td></tr><tr><td>38</td><td></td><td>50</td><td>18.0</td><td>21.3</td><td>20.0</td><td>19.5</td><td>18.2%</td><td>11.0%</td><td>8.2%</td></tr><tr><td>39</td><td>(4,15)</td><td>100</td><td>23.1</td><td>27.3</td><td>26.3</td><td>25.8</td><td>18.2%</td><td>14.0%</td><td>11.9%</td></tr><tr><td>40</td><td></td><td>150</td><td>28.3</td><td>33.3</td><td>32.1</td><td>31.4</td><td>17.8%</td><td>13.7%</td><td>11.3%</td></tr><tr><td>41</td><td></td><td>200</td><td>31.6</td><td>38.2</td><td>36.9</td><td>36.0</td><td>20.9%</td><td>16.6%</td><td>13.8%</td></tr><tr><td>42</td><td></td><td>250</td><td>32.3</td><td>38.2</td><td>36.2</td><td>35.5</td><td>18.1%</td><td>12.0%</td><td>9.7%</td></tr><tr><td>43</td><td></td><td>0</td><td>16.0</td><td>19.3</td><td>18.8</td><td>18.4</td><td>20.6%</td><td>17.0%</td><td>15.1%</td></tr><tr><td>44</td><td></td><td>50</td><td>21.3</td><td>25.3</td><td>23.9</td><td>23.4</td><td>18.9%</td><td>12.0%</td><td>9.6%</td></tr><tr><td>45</td><td>(4,20)</td><td>100</td><td>25.8</td><td>31.3</td><td>30.4</td><td>29.7</td><td>21.6%</td><td>17.8%</td><td>15.4%</td></tr><tr><td>46</td><td></td><td>150</td><td>31.3</td><td>37.3</td><td>35.2</td><td>34.5</td><td>19.2%</td><td>12.4%</td><td>10.1%</td></tr><tr><td>47</td><td></td><td>200</td><td>36.6</td><td>43.3</td><td>41.3</td><td>40.6</td><td>18.3%</td><td>12.9%</td><td>11.0%</td></tr><tr><td>48</td><td></td><td>250</td><td>39.7</td><td>48.3</td><td>46.8</td><td>45.9</td><td>21.7%</td><td>17.8%</td><td>15.6%</td></tr><tr><td>49</td><td></td><td>0</td><td>22.7</td><td>27.7</td><td>26.8</td><td>26.3</td><td>22.2%</td><td>18.2%</td><td>16.0%</td></tr><tr><td>50</td><td></td><td>50</td><td>26.8</td><td>33.7</td><td>31.8</td><td>31.3</td><td>25.6%</td><td>18.5%</td><td>16.7%</td></tr><tr><td>51</td><td>(4,25)</td><td>100</td><td>32.4</td><td>39.7</td><td>37.7</td><td>37.1</td><td>22.5%</td><td>16.4%</td><td>14.5%</td></tr><tr><td>52</td><td></td><td>150</td><td>37.9</td><td>45.7</td><td>43.7</td><td>42.9</td><td>20.5%</td><td>15.3%</td><td>13.1%</td></tr><tr><td>53</td><td></td><td>200</td><td>42.2</td><td>51.7</td><td>50.1</td><td>49.3</td><td>22.6%</td><td>18.7%</td><td>16.9%</td></tr><tr><td>54</td><td></td><td>250</td><td>46.8</td><td>57.7</td><td>55.6</td><td>54.6</td><td>23.3%</td><td>18.8%</td><td>16.7%</td></tr><tr><td>55</td><td></td><td>0</td><td>23.0</td><td>27.9</td><td>26.3</td><td>25.9</td><td>21.6%</td><td>14.6%</td><td>12.6%</td></tr><tr><td>56</td><td></td><td>50</td><td>26.1</td><td>33.9</td><td>32.8</td><td>32.5</td><td>29.7%</td><td>25.5%</td><td>24.2%</td></tr><tr><td>57</td><td>(4,30)</td><td>100</td><td>32.6</td><td>39.9</td><td>38.0</td><td>37.5</td><td>22.3%</td><td>16.4%</td><td>14.9%</td></tr><tr><td>58</td><td></td><td>150</td><td>36.4</td><td>45.9</td><td>43.7</td><td>43.1</td><td>26.1%</td><td>20.1%</td><td>18.4%</td></tr><tr><td>59</td><td></td><td>200</td><td>41.5</td><td>51.9</td><td>49.6</td><td>48.9</td><td>25.1%</td><td>19.6%</td><td>17.8%</td></tr><tr><td>60</td><td></td><td>250</td><td>46.1</td><td>57.9</td><td>55.5</td><td>54.8</td><td>25.7%</td><td>20.5%</td><td>19.0%</td></tr><tr><td>61</td><td></td><td>0</td><td>4.9</td><td>6.1</td><td>5.8</td><td>5.6</td><td>23.6%</td><td>17.4%</td><td>14.5%</td></tr><tr><td>62</td><td></td><td>50</td><td>10.5</td><td>13.0</td><td>12.3</td><td>12.0</td><td>23.5%</td><td>16.9%</td><td>13.3%</td></tr><tr><td>63</td><td></td><td>100</td><td>14.4</td><td>17.6</td><td>16.7</td><td>16.2</td><td>22.4%</td><td>16.4%</td><td>13.0%</td></tr><tr><td>64</td><td>(5,5)</td><td>150</td><td>14.1</td><td>17.6</td><td>17.0</td><td>16.5</td><td>24.2%</td><td>20.0%</td><td>16.6%</td></tr><tr><td>65</td><td></td><td>200</td><td>14.4</td><td>17.6</td><td>16.7</td><td>16.2</td><td>21.7%</td><td>15.6%</td><td>12.1%</td></tr><tr><td>66</td><td></td><td>250</td><td>14.0</td><td>17.6</td><td>16.7</td><td>16.3</td><td>25.6%</td><td>19.3%</td><td>16.5%</td></tr><tr><td>67</td><td></td><td>300</td><td>14.3</td><td>17.6</td><td>16.9</td><td>16.4</td><td>22.6%</td><td>17.7%</td><td>14.5%</td></tr><tr><td>68</td><td></td><td>350</td><td>14.4</td><td>17.6</td><td>16.6</td><td>16.2</td><td>22.0%</td><td>15.6%</td><td>12.3%</td></tr><tr><td>69</td><td></td><td>0</td><td>10.0</td><td>12.0</td><td>11.3</td><td>11.0</td><td>19.5%</td><td>12.6%</td><td>9.6%</td></tr><tr><td>70</td><td></td><td>50</td><td>14.0</td><td>18.0</td><td>17.2</td><td>16.9</td><td>29.0%</td><td>23.3%</td><td>20.9%</td></tr><tr><td>71</td><td></td><td>100</td><td>19.7</td><td>24.0</td><td>23.1</td><td>22.6</td><td>21.9%</td><td>17.6%</td><td>14.6%</td></tr><tr><td>72</td><td>(5,10)</td><td>150</td><td>24.0</td><td>30.0</td><td>28.5</td><td>27.7</td><td>24.9%</td><td>18.8%</td><td>15.4%</td></tr><tr><td>73</td><td></td><td>200</td><td>24.6</td><td>30.0</td><td>28.8</td><td>28.0</td><td>21.9%</td><td>17.1%</td><td>13.8%</td></tr><tr><td>74</td><td></td><td>250</td><td>23.1</td><td>30.0</td><td>28.8</td><td>28.2</td><td>29.9%</td><td>24.8%</td><td>22.3%</td></tr><tr><td>75</td><td></td><td>300</td><td>23.5</td><td>30.0</td><td>28.7</td><td>28.0</td><td>27.7%</td><td>22.4%</td><td>19.3%</td></tr><tr><td>76</td><td></td><td>350</td><td>22.7</td><td>30.0</td><td>28.4</td><td>27.8</td><td>31.9%</td><td>25.1%</td><td>22.4%</td></tr></tbody></table>
<table><caption>**Table 8** Results for the capital budgeting problem with unrestricted loans (continued)</caption><thead><tr><th rowspan="2">Instance</th><th rowspan="2">(T, I)</th><th rowspan="2">B</th><th rowspan="2">ν<sup>K</sup></th><th colspan="3">UB</th><th colspan="3">Optimality Gap (%)</th></tr><tr><th>ν<sup>Ω</sup></th><th>ν̃<sub>R</sub><sup>NA-DO</sup></th><th>ν̃<sub>R</sub><sup>DNA-DO</sup></th><th>(ν<sup>Ω</sup> - ν<sup>K</sup>)/ν<sup>K</sup></th><th>(ν̃<sub>R</sub><sup>NA-DO</sup> - ν<sup>K</sup>)/ν<sup>K</sup></th><th>(ν̃<sub>R</sub><sup>DNA-DO</sup> - ν<sup>K</sup>)/ν<sup>K</sup></th></tr></thead><tbody><tr><td>77</td><td></td><td>0</td><td>12.7</td><td>16.2</td><td>15.3</td><td>14.9</td><td>28.1%</td><td>20.8%</td><td>17.5%</td></tr><tr><td>78</td><td></td><td>50</td><td>17.6</td><td>22.2</td><td>21.5</td><td>21.1</td><td>26.2%</td><td>22.3%</td><td>19.6%</td></tr><tr><td>79</td><td></td><td>100</td><td>21.7</td><td>28.2</td><td>26.6</td><td>25.8</td><td>29.9%</td><td>22.5%</td><td>18.9%</td></tr><tr><td>80</td><td></td><td>150</td><td>26.2</td><td>34.2</td><td>32.4</td><td>31.5</td><td>30.5%</td><td>23.7%</td><td>20.0%</td></tr><tr><td>81</td><td>(5,15)</td><td>200</td><td>30.8</td><td>40.2</td><td>38.9</td><td>38.1</td><td>30.7%</td><td>26.3%</td><td>23.8%</td></tr><tr><td>82</td><td></td><td>250</td><td>31.6</td><td>40.5</td><td>39.0</td><td>38.2</td><td>28.1%</td><td>23.3%</td><td>20.8%</td></tr><tr><td>83</td><td></td><td>300</td><td>30.7</td><td>40.5</td><td>39.2</td><td>38.4</td><td>32.2%</td><td>27.8%</td><td>25.4%</td></tr><tr><td>84</td><td></td><td>350</td><td>31.3</td><td>40.5</td><td>38.8</td><td>38.0</td><td>29.6%</td><td>24.0%</td><td>21.6%</td></tr><tr><td>85</td><td></td><td>0</td><td>19.1</td><td>23.6</td><td>22.9</td><td>22.3</td><td>23.1%</td><td>19.6%</td><td>16.4%</td></tr><tr><td>86</td><td></td><td>50</td><td>23.1</td><td>29.6</td><td>28.6</td><td>27.9</td><td>27.9%</td><td>23.7%</td><td>20.7%</td></tr><tr><td>87</td><td></td><td>100</td><td>28.1</td><td>35.6</td><td>34.4</td><td>33.7</td><td>26.6%</td><td>22.6%</td><td>20.0%</td></tr><tr><td>88</td><td></td><td>150</td><td>33.8</td><td>41.6</td><td>39.7</td><td>38.9</td><td>23.1%</td><td>17.7%</td><td>15.3%</td></tr><tr><td>89</td><td>(5,20)</td><td>200</td><td>38.2</td><td>47.6</td><td>46.5</td><td>45.4</td><td>24.5%</td><td>21.8%</td><td>18.9%</td></tr><tr><td>90</td><td></td><td>250</td><td>39.9</td><td>53.6</td><td>52.4</td><td>51.1</td><td>34.2%</td><td>31.2%</td><td>28.0%</td></tr><tr><td>91</td><td></td><td>300</td><td>46.8</td><td>58.9</td><td>57.0</td><td>55.7</td><td>25.9%</td><td>21.7%</td><td>19.1%</td></tr><tr><td>92</td><td></td><td>350</td><td>45.9</td><td>58.9</td><td>57.7</td><td>56.6</td><td>28.4%</td><td>25.7%</td><td>23.3%</td></tr><tr><td>93</td><td></td><td>0</td><td>23.5</td><td>29.6</td><td>29.0</td><td>28.4</td><td>25.9%</td><td>23.4%</td><td>20.9%</td></tr><tr><td>94</td><td></td><td>50</td><td>28.6</td><td>35.6</td><td>34.7</td><td>33.9</td><td>24.3%</td><td>21.3%</td><td>18.3%</td></tr><tr><td>95</td><td></td><td>100</td><td>33.4</td><td>41.6</td><td>39.9</td><td>39.3</td><td>24.6%</td><td>19.7%</td><td>17.6%</td></tr><tr><td>96</td><td></td><td>150</td><td>36.5</td><td>47.6</td><td>45.9</td><td>44.9</td><td>30.5%</td><td>25.9%</td><td>23.1%</td></tr><tr><td>97</td><td>(5,25)</td><td>200</td><td>42.6</td><td>53.6</td><td>51.7</td><td>50.6</td><td>25.9%</td><td>21.4%</td><td>19.0%</td></tr><tr><td>98</td><td></td><td>250</td><td>47.1</td><td>59.6</td><td>58.2</td><td>56.9</td><td>26.4%</td><td>23.4%</td><td>20.6%</td></tr><tr><td>99</td><td></td><td>300</td><td>51.6</td><td>65.6</td><td>63.5</td><td>62.6</td><td>27.0%</td><td>23.0%</td><td>21.2%</td></tr><tr><td>100</td><td></td><td>350</td><td>56.1</td><td>71.6</td><td>69.7</td><td>68.3</td><td>27.6%</td><td>24.3%</td><td>21.8%</td></tr><tr><td>101</td><td></td><td>0</td><td>25.6</td><td>32.2</td><td>31.1</td><td>30.6</td><td>26.1%</td><td>21.8%</td><td>19.8%</td></tr><tr><td>102</td><td></td><td>50</td><td>29.1</td><td>38.2</td><td>37.1</td><td>36.5</td><td>31.3%</td><td>27.4%</td><td>25.3%</td></tr><tr><td>103</td><td></td><td>100</td><td>34.4</td><td>44.2</td><td>43.3</td><td>42.6</td><td>28.7%</td><td>26.0%</td><td>24.1%</td></tr><tr><td>104</td><td></td><td>150</td><td>37.8</td><td>50.2</td><td>48.7</td><td>48.1</td><td>32.8%</td><td>28.7%</td><td>27.2%</td></tr><tr><td>105</td><td>(5,30)</td><td>200</td><td>42.0</td><td>56.2</td><td>54.7</td><td>53.5</td><td>33.9%</td><td>30.2%</td><td>27.4%</td></tr><tr><td>106</td><td></td><td>250</td><td>47.2</td><td>62.2</td><td>60.3</td><td>59.5</td><td>31.9%</td><td>27.8%</td><td>26.2%</td></tr><tr><td>107</td><td></td><td>300</td><td>54.4</td><td>68.2</td><td>66.4</td><td>65.3</td><td>25.5%</td><td>22.1%</td><td>20.0%</td></tr><tr><td>108</td><td></td><td>350</td><td>60.6</td><td>74.2</td><td>72.1</td><td>70.7</td><td>22.4%</td><td>18.9%</td><td>16.5%</td></tr><tr><td>109</td><td></td><td>0</td><td>5.6</td><td>6.8</td><td>6.6</td><td>6.4</td><td>20.2%</td><td>18.0%</td><td>14.1%</td></tr><tr><td>110</td><td></td><td>50</td><td>10.6</td><td>12.8</td><td>12.5</td><td>12.1</td><td>20.5%</td><td>18.0%</td><td>14.2%</td></tr><tr><td>111</td><td></td><td>100</td><td>13.6</td><td>16.9</td><td>16.3</td><td>15.8</td><td>24.0%</td><td>19.8%</td><td>16.4%</td></tr><tr><td>112</td><td></td><td>150</td><td>12.8</td><td>16.9</td><td>16.5</td><td>16.0</td><td>31.4%</td><td>28.4%</td><td>24.8%</td></tr><tr><td>113</td><td>(6,5)</td><td>200</td><td>13.2</td><td>16.9</td><td>16.5</td><td>16.0</td><td>27.7%</td><td>24.9%</td><td>21.2%</td></tr><tr><td>114</td><td></td><td>250</td><td>12.9</td><td>16.9</td><td>16.5</td><td>15.9</td><td>30.5%</td><td>27.7%</td><td>23.1%</td></tr><tr><td>115</td><td></td><td>300</td><td>13.0</td><td>16.9</td><td>16.5</td><td>15.9</td><td>30.1%</td><td>27.2%</td><td>22.8%</td></tr><tr><td>116</td><td></td><td>350</td><td>13.3</td><td>16.9</td><td>16.5</td><td>16.0</td><td>27.2%</td><td>24.2%</td><td>20.3%</td></tr><tr><td>117</td><td></td><td>400</td><td>12.9</td><td>16.9</td><td>16.4</td><td>15.9</td><td>30.7%</td><td>26.9%</td><td>23.1%</td></tr><tr><td>118</td><td></td><td>0</td><td>10.9</td><td>14.0</td><td>13.8</td><td>13.4</td><td>27.8%</td><td>26.0%</td><td>22.5%</td></tr><tr><td>119</td><td></td><td>50</td><td>15.9</td><td>20.0</td><td>19.6</td><td>18.9</td><td>25.8%</td><td>23.5%</td><td>19.0%</td></tr><tr><td>120</td><td></td><td>100</td><td>20.7</td><td>26.0</td><td>25.4</td><td>24.5</td><td>25.4%</td><td>22.4%</td><td>18.2%</td></tr><tr><td>121</td><td></td><td>150</td><td>24.9</td><td>32.0</td><td>31.4</td><td>30.3</td><td>28.3%</td><td>25.8%</td><td>21.5%</td></tr><tr><td>122</td><td>(6,10)</td><td>200</td><td>27.1</td><td>35.0</td><td>34.2</td><td>33.3</td><td>29.2%</td><td>26.5%</td><td>23.1%</td></tr><tr><td>123</td><td></td><td>250</td><td>26.6</td><td>35.0</td><td>34.2</td><td>33.2</td><td>31.5%</td><td>28.7%</td><td>24.7%</td></tr><tr><td>124</td><td></td><td>300</td><td>26.8</td><td>35.0</td><td>34.4</td><td>33.3</td><td>30.4%</td><td>28.1%</td><td>24.0%</td></tr><tr><td>125</td><td></td><td>350</td><td>27.3</td><td>35.0</td><td>34.4</td><td>33.3</td><td>28.3%</td><td>26.3%</td><td>22.2%</td></tr><tr><td>126</td><td></td><td>400</td><td>26.9</td><td>35.0</td><td>34.2</td><td>33.2</td><td>29.9%</td><td>27.0%</td><td>23.3%</td></tr></tbody></table>
<table><caption>**Table 8** Results for the capital budgeting problem with unrestricted loans (continued)</caption><thead><tr><th rowspan="2">Instance</th><th rowspan="2">(T, I)</th><th rowspan="2">B</th><th rowspan="2">ν<sup>k</sup></th><th colspan="3">UB</th><th colspan="3">Optimality Gap (%)</th></tr><tr><th>ν<sup>Ω</sup></th><th>ν̃<sub>R</sub><sup>NA-DO</sup></th><th>ν̃<sub>R</sub><sup>DNA-DO</sup></th><th>(ν<sup>Ω</sup> - ν<sup>k</sup>)/ν<sup>k</sup></th><th>(ν̃<sub>R</sub><sup>NA-DO</sup> - ν<sup>k</sup>)/ν<sup>k</sup></th><th>(ν̃<sub>R</sub><sup>DNA-DO</sup> - ν<sup>k</sup>)/ν<sup>k</sup></th></tr></thead><tbody><tr><td>127</td><td></td><td>0</td><td>15.9</td><td>19.9</td><td>19.5</td><td>19.0</td><td>24.7%</td><td>22.5%</td><td>19.3%</td></tr><tr><td>128</td><td></td><td>50</td><td>19.8</td><td>25.9</td><td>25.5</td><td>24.9</td><td>30.7%</td><td>28.7%</td><td>25.8%</td></tr><tr><td>129</td><td></td><td>100</td><td>25.3</td><td>31.9</td><td>31.4</td><td>30.6</td><td>25.8%</td><td>23.9%</td><td>20.9%</td></tr><tr><td>130</td><td></td><td>150</td><td>28.9</td><td>37.9</td><td>37.3</td><td>36.4</td><td>31.2%</td><td>29.3%</td><td>26.1%</td></tr><tr><td>131</td><td>(6,15)</td><td>200</td><td>33.5</td><td>43.9</td><td>43.1</td><td>42.1</td><td>30.8%</td><td>28.3%</td><td>25.5%</td></tr><tr><td>132</td><td></td><td>250</td><td>37.0</td><td>49.7</td><td>48.7</td><td>47.6</td><td>34.3%</td><td>31.7%</td><td>28.7%</td></tr><tr><td>133</td><td></td><td>300</td><td>36.9</td><td>49.7</td><td>49.0</td><td>47.8</td><td>34.8%</td><td>32.9%</td><td>29.7%</td></tr><tr><td>134</td><td></td><td>350</td><td>38.5</td><td>49.7</td><td>48.8</td><td>47.6</td><td>29.1%</td><td>26.8%</td><td>23.8%</td></tr><tr><td>135</td><td></td><td>400</td><td>37.9</td><td>49.7</td><td>48.9</td><td>47.7</td><td>31.2%</td><td>29.0%</td><td>25.9%</td></tr><tr><td>136</td><td></td><td>0</td><td>22.4</td><td>28.6</td><td>28.3</td><td>27.8</td><td>27.6%</td><td>26.5%</td><td>23.9%</td></tr><tr><td>137</td><td></td><td>50</td><td>27.6</td><td>34.6</td><td>34.3</td><td>33.5</td><td>25.4%</td><td>24.3%</td><td>21.6%</td></tr><tr><td>138</td><td></td><td>100</td><td>31.7</td><td>40.6</td><td>40.1</td><td>39.3</td><td>27.9%</td><td>26.3%</td><td>23.7%</td></tr><tr><td>139</td><td></td><td>150</td><td>36.9</td><td>46.6</td><td>46.1</td><td>45.3</td><td>26.4%</td><td>25.2%</td><td>22.8%</td></tr><tr><td>140</td><td>(6,20)</td><td>200</td><td>41.0</td><td>52.6</td><td>52.1</td><td>50.9</td><td>28.4%</td><td>27.2%</td><td>24.4%</td></tr><tr><td>141</td><td></td><td>250</td><td>45.1</td><td>58.6</td><td>58.2</td><td>57.1</td><td>29.8%</td><td>28.8%</td><td>26.4%</td></tr><tr><td>142</td><td></td><td>300</td><td>49.7</td><td>64.6</td><td>63.8</td><td>62.6</td><td>30.0%</td><td>28.3%</td><td>26.0%</td></tr><tr><td>143</td><td></td><td>350</td><td>53.2</td><td>70.6</td><td>70.0</td><td>68.8</td><td>32.6%</td><td>31.5%</td><td>29.3%</td></tr><tr><td>144</td><td></td><td>400</td><td>54.3</td><td>71.5</td><td>70.9</td><td>69.6</td><td>31.6%</td><td>30.6%</td><td>28.1%</td></tr><tr><td>145</td><td></td><td>0</td><td>29.0</td><td>37.2</td><td>36.9</td><td>36.2</td><td>28.2%</td><td>27.2%</td><td>24.9%</td></tr><tr><td>146</td><td></td><td>50</td><td>32.8</td><td>43.2</td><td>42.8</td><td>42.1</td><td>31.5%</td><td>30.3%</td><td>28.2%</td></tr><tr><td>147</td><td></td><td>100</td><td>37.5</td><td>49.2</td><td>48.7</td><td>48.0</td><td>31.3%</td><td>30.0%</td><td>28.0%</td></tr><tr><td>148</td><td></td><td>150</td><td>42.5</td><td>55.2</td><td>54.6</td><td>53.7</td><td>29.9%</td><td>28.5%</td><td>26.3%</td></tr><tr><td>149</td><td>(6,25)</td><td>200</td><td>46.2</td><td>61.2</td><td>60.5</td><td>59.7</td><td>32.5%</td><td>31.0%</td><td>29.2%</td></tr><tr><td>150</td><td></td><td>250</td><td>51.9</td><td>67.2</td><td>66.7</td><td>65.7</td><td>29.4%</td><td>28.5%</td><td>26.6%</td></tr><tr><td>151</td><td></td><td>300</td><td>56.1</td><td>73.2</td><td>72.3</td><td>71.2</td><td>30.4%</td><td>28.9%</td><td>26.8%</td></tr><tr><td>152</td><td></td><td>350</td><td>61.1</td><td>79.2</td><td>78.6</td><td>77.4</td><td>29.6%</td><td>28.6%</td><td>26.7%</td></tr><tr><td>153</td><td></td><td>400</td><td>64.5</td><td>85.2</td><td>84.5</td><td>83.0</td><td>32.0%</td><td>30.9%</td><td>28.7%</td></tr><tr><td>154</td><td></td><td>0</td><td>31.5</td><td>42.6</td><td>42.2</td><td>41.6</td><td>35.2%</td><td>34.1%</td><td>32.2%</td></tr><tr><td>155</td><td></td><td>50</td><td>37.5</td><td>48.6</td><td>48.1</td><td>47.5</td><td>29.5%</td><td>28.3%</td><td>26.6%</td></tr><tr><td>156</td><td></td><td>100</td><td>40.6</td><td>54.6</td><td>54.1</td><td>53.4</td><td>34.4%</td><td>33.1%</td><td>31.5%</td></tr><tr><td>157</td><td></td><td>150</td><td>45.2</td><td>60.6</td><td>60.0</td><td>59.2</td><td>33.9%</td><td>32.6%</td><td>30.8%</td></tr><tr><td>158</td><td>(6,30)</td><td>200</td><td>49.1</td><td>66.6</td><td>66.0</td><td>65.1</td><td>35.5%</td><td>34.4%</td><td>32.5%</td></tr><tr><td>159</td><td></td><td>250</td><td>56.0</td><td>72.6</td><td>71.9</td><td>71.0</td><td>29.7%</td><td>28.5%</td><td>26.9%</td></tr><tr><td>160</td><td></td><td>300</td><td>59.4</td><td>78.6</td><td>78.0</td><td>76.8</td><td>32.4%</td><td>31.4%</td><td>29.4%</td></tr><tr><td>161</td><td></td><td>350</td><td>65.0</td><td>84.6</td><td>83.8</td><td>82.6</td><td>30.2%</td><td>29.0%</td><td>27.1%</td></tr><tr><td>162</td><td></td><td>400</td><td>68.5</td><td>90.6</td><td>89.8</td><td>88.5</td><td>32.2%</td><td>31.1%</td><td>29.2%</td></tr></tbody></table>
D.4. Capital Budgeting with Integer Loan

<figure><img src="image_11.png" /><figcaption>**Figure 9** Optimality gap improvements for capital budgeting instances with general integer loan decisions.</figcaption></figure>
<table><caption>**Table 9** Solution times of different bounding methods in seconds for the capital budgeting problem</caption><thead><tr><th rowspan="2">Instance</th><th rowspan="2">(T, I)</th><th rowspan="2">B</th><th colspan="4">Fractional loan</th><th colspan="4">Integral loan</th></tr><tr><th>ν<sup>K</sup></th><th>ν<sup>Ω</sup></th><th>ν&#773;<sup>NA</sup><sub>R</sub></th><th>ν&#773;<sup>DNA</sup><sub>R</sub></th><th>ν<sup>K</sup></th><th>ν<sup>Ω</sup></th><th>ν&#773;<sup>NA</sup><sub>R</sub></th><th>ν&#773;<sup>DNA</sup><sub>R</sub></th></tr></thead><tbody><tr><td>1</td><td></td><td>0</td><td>27.0</td><td>63.0</td><td>56.1</td><td>62.1</td><td>67.2</td><td>72.7</td><td>112.3</td><td>80.8</td></tr><tr><td>2</td><td></td><td>50</td><td>16.2</td><td>85.2</td><td>83.8</td><td>74.9</td><td>52.6</td><td>99.3</td><td>678.9</td><td>142.2</td></tr><tr><td>3</td><td>(3,5)</td><td>100</td><td>25.8</td><td>60.8</td><td>55.5</td><td>52.2</td><td>43.5</td><td>95.6</td><td>260.8</td><td>287.1</td></tr><tr><td>4</td><td></td><td>150</td><td>25.2</td><td>56.2</td><td>55.3</td><td>35.5</td><td>64.0</td><td>59.3</td><td>259.9</td><td>156.1</td></tr><tr><td>5</td><td></td><td>0</td><td>35.4</td><td>547.4</td><td>547.4</td><td>521.2</td><td>111.3</td><td>840.0</td><td>1133.1</td><td>948.6</td></tr><tr><td>6</td><td></td><td>50</td><td>40.5</td><td>384.5</td><td>254.5</td><td>376.1</td><td>77.5</td><td>771.1</td><td>413.1</td><td>557.9</td></tr><tr><td>7</td><td>(3,10)</td><td>100</td><td>34.5</td><td>340.5</td><td>392.5</td><td>366.2</td><td>96.5</td><td>373.2</td><td>478.1</td><td>432.2</td></tr><tr><td>8</td><td></td><td>150</td><td>44.4</td><td>514.4</td><td>657.4</td><td>471.2</td><td>107.7</td><td>1215.7</td><td>1447.6</td><td>604.0</td></tr><tr><td>9</td><td></td><td>0</td><td>90.9</td><td>1502.9</td><td>1178.9</td><td>1390.9</td><td>218.5</td><td>2029.6</td><td>3812.6</td><td>3569.0</td></tr><tr><td>10</td><td></td><td>50</td><td>90.0</td><td>1169.0</td><td>1007.0</td><td>1061.0</td><td>201.8</td><td>1467.1</td><td>2698.8</td><td>2868.9</td></tr><tr><td>11</td><td>(3,15)</td><td>100</td><td>96.3</td><td>1438.3</td><td>1168.3</td><td>1556.3</td><td>169.1</td><td>2261.3</td><td>2553.9</td><td>5394.1</td></tr><tr><td>12</td><td></td><td>150</td><td>94.8</td><td>1535.8</td><td>1247.8</td><td>1355.8</td><td>200.5</td><td>1974.3</td><td>3441.4</td><td>3313.6</td></tr><tr><td>13</td><td></td><td>0</td><td>611.4</td><td>3138.4</td><td>1896.4</td><td>2568.4</td><td>689.5</td><td>4305.1</td><td>3531.1</td><td>5606.8</td></tr><tr><td>14</td><td></td><td>50</td><td>653.1</td><td>3809.1</td><td>2498.1</td><td>3005.1</td><td>772.2</td><td>5003.3</td><td>3962.0</td><td>7371.5</td></tr><tr><td>15</td><td>(3,20)</td><td>100</td><td>667.2</td><td>3057.2</td><td>2643.2</td><td>4295.2</td><td>743.4</td><td>3777.1</td><td>5651.2</td><td>12202.7</td></tr><tr><td>16</td><td></td><td>150</td><td>609.6</td><td>3539.6</td><td>2918.6</td><td>2557.6</td><td>765.9</td><td>4162.9</td><td>5034.6</td><td>5319.8</td></tr><tr><td>17</td><td></td><td>0</td><td>4292.4</td><td>8689.4</td><td>4392.9</td><td>5521.4</td><td>4561.3</td><td>9868.9</td><td>7116.6</td><td>10188.8</td></tr><tr><td>18</td><td></td><td>50</td><td>3762.0</td><td>6794.0</td><td>3465.3</td><td>5684.5</td><td>3953.5</td><td>7354.5</td><td>4636.6</td><td>11134.0</td></tr><tr><td>19</td><td>(3,25)</td><td>100</td><td>3759.0</td><td>6921.0</td><td>3830.0</td><td>4160.5</td><td>4054.2</td><td>8311.5</td><td>6707.6</td><td>6271.3</td></tr><tr><td>20</td><td></td><td>150</td><td>4012.8</td><td>7998.8</td><td>4716.5</td><td>4608.3</td><td>4336.0</td><td>8751.4</td><td>6883.0</td><td>8159.8</td></tr><tr><td>21</td><td></td><td>0</td><td>1808.1</td><td>5929.1</td><td>3754.7</td><td>8207.1</td><td>2188.2</td><td>6894.2</td><td>6115.2</td><td>20660.0</td></tr><tr><td>22</td><td></td><td>50</td><td>2083.8</td><td>5512.8</td><td>2817.2</td><td>8133.8</td><td>2394.6</td><td>6036.5</td><td>3818.2</td><td>15524.7</td></tr><tr><td>23</td><td>(3,30)</td><td>100</td><td>2027.1</td><td>5422.1</td><td>3152.7</td><td>9087.1</td><td>2307.5</td><td>5763.8</td><td>4756.4</td><td>23493.2</td></tr><tr><td>24</td><td></td><td>150</td><td>1850.7</td><td>6203.7</td><td>4069.8</td><td>7591.7</td><td>2225.7</td><td>7524.8</td><td>5719.4</td><td>15765.4</td></tr><tr><td>25</td><td></td><td>0</td><td>89.2</td><td>1012.2</td><td>626.8</td><td>1185.2</td><td>173.3</td><td>3395.6</td><td>686.6</td><td>1483.1</td></tr><tr><td>26</td><td></td><td>50</td><td>97.2</td><td>1141.2</td><td>592.8</td><td>1236.2</td><td>190.1</td><td>1536.5</td><td>665.1</td><td>1478.5</td></tr><tr><td>27</td><td></td><td>100</td><td>106.0</td><td>1189.0</td><td>624.7</td><td>1259.0</td><td>173.2</td><td>2337.4</td><td>706.7</td><td>1488.1</td></tr><tr><td>28</td><td>(4,5)</td><td>150</td><td>80.8</td><td>849.8</td><td>470.5</td><td>932.8</td><td>170.7</td><td>1740.1</td><td>516.6</td><td>1071.5</td></tr><tr><td>29</td><td></td><td>200</td><td>85.2</td><td>838.2</td><td>390.8</td><td>952.2</td><td>160.4</td><td>3339.3</td><td>423.1</td><td>1060.8</td></tr><tr><td>30</td><td></td><td>250</td><td>107.2</td><td>1039.2</td><td>596.8</td><td>1173.2</td><td>202.7</td><td>2826.5</td><td>665.2</td><td>1332.0</td></tr><tr><td>31</td><td></td><td>0</td><td>590.0</td><td>2420.0</td><td>1277.3</td><td>3150.0</td><td>713.9</td><td>3186.4</td><td>1544.7</td><td>5161.8</td></tr><tr><td>32</td><td></td><td>50</td><td>593.2</td><td>2247.2</td><td>1050.1</td><td>3178.2</td><td>720.7</td><td>5845.5</td><td>1295.2</td><td>4602.0</td></tr><tr><td>33</td><td></td><td>100</td><td>580.8</td><td>2189.8</td><td>1049.2</td><td>2941.8</td><td>779.5</td><td>12174.6</td><td>1236.0</td><td>4424.5</td></tr><tr><td>34</td><td>(4,10)</td><td>150</td><td>470.0</td><td>2230.0</td><td>1449.3</td><td>3485.0</td><td>636.5</td><td>2278.7</td><td>1820.4</td><td>5099.7</td></tr><tr><td>35</td><td></td><td>200</td><td>417.2</td><td>2208.2</td><td>1285.5</td><td>2833.2</td><td>602.3</td><td>5185.9</td><td>1572.6</td><td>4482.1</td></tr><tr><td>36</td><td></td><td>250</td><td>431.6</td><td>2198.6</td><td>1353.7</td><td>3231.6</td><td>583.6</td><td>10838.1</td><td>1576.6</td><td>4862.5</td></tr><tr><td>37</td><td></td><td>0</td><td>1372.8</td><td>5153.8</td><td>3131.9</td><td>5853.8</td><td>1679.6</td><td>7835.0</td><td>4739.6</td><td>11098.8</td></tr><tr><td>38</td><td></td><td>50</td><td>1370.4</td><td>4828.4</td><td>2509.6</td><td>6153.4</td><td>1724.7</td><td>8235.8</td><td>3476.6</td><td>12602.2</td></tr><tr><td>39</td><td></td><td>100</td><td>1268.0</td><td>5458.0</td><td>3486.7</td><td>7861.0</td><td>1525.9</td><td>15446.3</td><td>5541.5</td><td>21591.5</td></tr><tr><td>40</td><td>(4,15)</td><td>150</td><td>1507.6</td><td>5316.6</td><td>2784.4</td><td>7977.6</td><td>1750.2</td><td>17220.9</td><td>3909.3</td><td>21449.1</td></tr><tr><td>41</td><td></td><td>200</td><td>1426.4</td><td>5243.4</td><td>3039.6</td><td>6315.4</td><td>1613.6</td><td>12527.3</td><td>4895.8</td><td>11460.3</td></tr><tr><td>42</td><td></td><td>250</td><td>1589.2</td><td>5702.2</td><td>2990.8</td><td>8046.2</td><td>1911.2</td><td>19692.0</td><td>4633.7</td><td>21338.5</td></tr><tr><td>43</td><td></td><td>0</td><td>4676.4</td><td>10535.4</td><td>5935.6</td><td>17371.4</td><td>5228.1</td><td>20173.7</td><td>9827.4</td><td>&gt; 10h</td></tr><tr><td>44</td><td></td><td>50</td><td>4988.0</td><td>10909.0</td><td>5992.7</td><td>16613.0</td><td>5402.2</td><td>14597.9</td><td>7704.6</td><td>&gt; 10h</td></tr><tr><td>45</td><td></td><td>100</td><td>4942.8</td><td>10226.8</td><td>5601.9</td><td>17168.8</td><td>5356.3</td><td>20126.0</td><td>9011.5</td><td>&gt; 10h</td></tr><tr><td>46</td><td>(4,20)</td><td>150</td><td>4111.2</td><td>9830.2</td><td>5337.5</td><td>13249.2</td><td>4642.0</td><td>27734.9</td><td>7855.0</td><td>&gt; 10h</td></tr><tr><td>47</td><td></td><td>200</td><td>4633.6</td><td>9934.6</td><td>6239.1</td><td>16356.6</td><td>5176.0</td><td>16374.7</td><td>9543.7</td><td>&gt; 10h</td></tr><tr><td>48</td><td></td><td>250</td><td>4344.4</td><td>9147.4</td><td>5522.3</td><td>14321.4</td><td>4749.8</td><td>18371.0</td><td>7313.3</td><td>&gt; 10h</td></tr><tr><td>49</td><td></td><td>0</td><td>8113.2</td><td>16000.2</td><td>10434.8</td><td>25348.2</td><td>8769.9</td><td>&gt; 10h</td><td>13120.7</td><td>&gt; 10h</td></tr><tr><td>50</td><td></td><td>50</td><td>7384.0</td><td>14978.0</td><td>9753.3</td><td>21113.0</td><td>7679.0</td><td>&gt; 10h</td><td>12535.0</td><td>&gt; 10h</td></tr><tr><td>51</td><td></td><td>100</td><td>7619.2</td><td>14613.2</td><td>9046.1</td><td>20856.2</td><td>7865.7</td><td>&gt; 10h</td><td>12054.9</td><td>&gt; 10h</td></tr><tr><td>52</td><td>(4,25)</td><td>150</td><td>7864.4</td><td>15145.4</td><td>9942.3</td><td>22623.4</td><td>8270.7</td><td>&gt; 10h</td><td>12352.3</td><td>&gt; 10h</td></tr><tr><td>53</td><td></td><td>200</td><td>7440.4</td><td>14736.4</td><td>8354.9</td><td>25575.4</td><td>7731.2</td><td>&gt; 10h</td><td>9985.8</td><td>&gt; 10h</td></tr><tr><td>54</td><td></td><td>250</td><td>7986.4</td><td>15734.4</td><td>8246.9</td><td>27221.4</td><td>8265.6</td><td>&gt; 10h</td><td>10005.2</td><td>&gt; 10h</td></tr></tbody></table>
<table><caption>**Table 9** Solution times of different bounding methods in seconds for the capital budgeting problem (continued)</caption><thead><tr><th rowspan="2">Instance</th><th rowspan="2">(T, I)</th><th rowspan="2">B</th><th colspan="4">Fractional loan</th><th colspan="4">Integral loan</th></tr><tr><th>ν<sup>k</sup></th><th>ν<sup>Ω</sup></th><th>ν̄<sup>NA</sup><sub>R</sub></th><th>ν̄<sup>DNA</sup><sub>R</sub></th><th>ν<sup>k</sup></th><th>ν<sup>Ω</sup></th><th>ν̄<sup>NA</sup><sub>R</sub></th><th>ν̄<sup>DNA</sup><sub>R</sub></th></tr></thead><tbody><tr><td>55</td><td></td><td>0</td><td>13898.8</td><td>20608.8</td><td>12197.9</td><td>> 10h</td><td>14146.4</td><td>> 10h</td><td>15337.6</td><td>> 10h</td></tr><tr><td>56</td><td></td><td>50</td><td>13221.6</td><td>20082.6</td><td>10940.4</td><td>33232.6</td><td>13470.7</td><td>> 10h</td><td>14137.2</td><td>> 10h</td></tr><tr><td>57</td><td></td><td>100</td><td>12854.0</td><td>20045.0</td><td>13182.0</td><td>> 10h</td><td>13435.0</td><td>> 10h</td><td>19973.4</td><td>> 10h</td></tr><tr><td>58</td><td>(4,30)</td><td>150</td><td>13757.6</td><td>20898.6</td><td>13569.7</td><td>> 10h</td><td>14320.1</td><td>> 10h</td><td>19708.7</td><td>> 10h</td></tr><tr><td>59</td><td></td><td>200</td><td>12260.8</td><td>19245.8</td><td>10654.5</td><td>> 10h</td><td>12773.9</td><td>> 10h</td><td>14818.3</td><td>> 10h</td></tr><tr><td>60</td><td></td><td>250</td><td>13656.4</td><td>20471.4</td><td>12559.6</td><td>> 10h</td><td>14180.5</td><td>> 10h</td><td>18532.9</td><td>> 10h</td></tr><tr><td>61</td><td></td><td>0</td><td>1188.0</td><td>4539.0</td><td>2692.7</td><td>6018.0</td><td>1302.5</td><td>6041.9</td><td>2993.6</td><td>7706.0</td></tr><tr><td>62</td><td></td><td>50</td><td>1219.0</td><td>4453.0</td><td>2802.0</td><td>5922.0</td><td>1297.0</td><td>4489.5</td><td>3164.9</td><td>6700.7</td></tr><tr><td>63</td><td></td><td>100</td><td>1469.0</td><td>4346.0</td><td>2864.0</td><td>5529.0</td><td>1545.1</td><td>12850.6</td><td>3207.0</td><td>6774.4</td></tr><tr><td>64</td><td></td><td>150</td><td>1315.0</td><td>4093.0</td><td>2428.7</td><td>5442.0</td><td>1443.7</td><td>7996.8</td><td>2647.2</td><td>6284.1</td></tr><tr><td>65</td><td>(5,5)</td><td>200</td><td>1099.0</td><td>4026.0</td><td>2384.0</td><td>5303.0</td><td>1247.0</td><td>12676.3</td><td>2573.5</td><td>6182.0</td></tr><tr><td>66</td><td></td><td>250</td><td>1194.0</td><td>4438.0</td><td>2825.3</td><td>6075.0</td><td>1294.4</td><td>6630.7</td><td>3110.0</td><td>7217.1</td></tr><tr><td>67</td><td></td><td>300</td><td>812.5</td><td>3603.5</td><td>2135.7</td><td>4741.5</td><td>923.0</td><td>12057.9</td><td>2253.7</td><td>5600.9</td></tr><tr><td>68</td><td></td><td>350</td><td>1376.5</td><td>5019.5</td><td>3046.3</td><td>6457.5</td><td>1484.3</td><td>19850.1</td><td>3340.3</td><td>7406.8</td></tr><tr><td>69</td><td></td><td>0</td><td>4275.5</td><td>10708.5</td><td>9053.2</td><td>14882.5</td><td>4512.4</td><td>14699.0</td><td>11601.7</td><td>20366.7</td></tr><tr><td>70</td><td></td><td>50</td><td>3354.0</td><td>11245.0</td><td>9677.3</td><td>13469.0</td><td>3547.8</td><td>13864.4</td><td>11116.0</td><td>16596.1</td></tr><tr><td>71</td><td></td><td>100</td><td>3764.5</td><td>12332.5</td><td>10597.7</td><td>14540.5</td><td>3959.8</td><td>20538.2</td><td>12756.1</td><td>18039.9</td></tr><tr><td>72</td><td></td><td>150</td><td>3769.0</td><td>10903.0</td><td>10864.4</td><td>13991.0</td><td>3991.5</td><td>22267.6</td><td>13768.9</td><td>19160.7</td></tr><tr><td>73</td><td>(5,10)</td><td>200</td><td>3507.5</td><td>10891.5</td><td>10037.7</td><td>13358.5</td><td>3679.3</td><td>21045.9</td><td>12135.6</td><td>18143.1</td></tr><tr><td>74</td><td></td><td>250</td><td>4005.5</td><td>12899.5</td><td>11522.3</td><td>16746.5</td><td>4244.5</td><td>17293.1</td><td>13665.4</td><td>20972.2</td></tr><tr><td>75</td><td></td><td>300</td><td>3507.0</td><td>10998.0</td><td>13136.7</td><td>13005.0</td><td>3724.3</td><td>25089.7</td><td>18343.2</td><td>17303.2</td></tr><tr><td>76</td><td></td><td>350</td><td>3803.0</td><td>9881.0</td><td>8232.7</td><td>12413.0</td><td>4006.9</td><td>21655.3</td><td>9519.8</td><td>17092.7</td></tr><tr><td>77</td><td></td><td>0</td><td>7936.0</td><td>22038.0</td><td>14625.3</td><td>28044.0</td><td>8368.5</td><td>> 10h</td><td>16465.2</td><td>> 10h</td></tr><tr><td>78</td><td></td><td>50</td><td>8516.5</td><td>22334.5</td><td>14089.7</td><td>29802.5</td><td>8893.6</td><td>> 10h</td><td>15924.1</td><td>> 10h</td></tr><tr><td>79</td><td></td><td>100</td><td>9214.0</td><td>21936.0</td><td>14357.3</td><td>29063.0</td><td>9757.7</td><td>> 10h</td><td>15971.1</td><td>> 10h</td></tr><tr><td>80</td><td></td><td>150</td><td>9150.0</td><td>22127.0</td><td>13484.7</td><td>28673.0</td><td>9541.0</td><td>> 10h</td><td>14723.9</td><td>> 10h</td></tr><tr><td>81</td><td>(5,15)</td><td>200</td><td>8487.5</td><td>21143.5</td><td>13829.0</td><td>27331.5</td><td>8850.1</td><td>> 10h</td><td>15523.1</td><td>> 10h</td></tr><tr><td>82</td><td></td><td>250</td><td>7664.5</td><td>21657.5</td><td>13638.3</td><td>28499.5</td><td>8033.2</td><td>> 10h</td><td>14898.5</td><td>> 10h</td></tr><tr><td>83</td><td></td><td>300</td><td>8442.5</td><td>22287.5</td><td>14258.3</td><td>27426.5</td><td>8785.5</td><td>> 10h</td><td>15829.6</td><td>> 10h</td></tr><tr><td>84</td><td></td><td>350</td><td>9361.5</td><td>19241.5</td><td>12027.7</td><td>27496.5</td><td>9747.9</td><td>> 10h</td><td>13072.9</td><td>> 10h</td></tr><tr><td>85</td><td></td><td>0</td><td>12474.0</td><td>31538.0</td><td>24198.3</td><td>> 10h</td><td>13006.9</td><td>> 10h</td><td>28183.8</td><td>> 10h</td></tr><tr><td>86</td><td></td><td>50</td><td>13804.0</td><td>30383.0</td><td>22819.2</td><td>> 10h</td><td>14552.9</td><td>> 10h</td><td>26801.1</td><td>> 10h</td></tr><tr><td>87</td><td></td><td>100</td><td>13531.5</td><td>27682.5</td><td>21402.1</td><td>> 10h</td><td>14425.2</td><td>> 10h</td><td>25016.9</td><td>> 10h</td></tr><tr><td>88</td><td></td><td>150</td><td>12393.5</td><td>30610.5</td><td>23529.6</td><td>> 10h</td><td>13144.1</td><td>> 10h</td><td>28047.3</td><td>> 10h</td></tr><tr><td>89</td><td>(5,20)</td><td>200</td><td>13991.5</td><td>32146.5</td><td>25330.4</td><td>> 10h</td><td>14732.0</td><td>> 10h</td><td>31559.2</td><td>> 10h</td></tr><tr><td>90</td><td></td><td>250</td><td>12486.0</td><td>28563.0</td><td>22135.8</td><td>> 10h</td><td>13412.9</td><td>> 10h</td><td>26835.3</td><td>> 10h</td></tr><tr><td>91</td><td></td><td>300</td><td>14566.5</td><td>32117.5</td><td>24472.9</td><td>> 10h</td><td>15184.6</td><td>> 10h</td><td>28655.3</td><td>> 10h</td></tr><tr><td>92</td><td></td><td>350</td><td>14411.0</td><td>32416.0</td><td>26284.2</td><td>> 10h</td><td>15094.5</td><td>> 10h</td><td>31906.3</td><td>> 10h</td></tr><tr><td>93</td><td></td><td>0</td><td>19659.5</td><td>> 10h</td><td>22510.6</td><td>> 10h</td><td>20733.1</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>94</td><td></td><td>50</td><td>19808.5</td><td>> 10h</td><td>26676.1</td><td>> 10h</td><td>20452.7</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>95</td><td></td><td>100</td><td>21565.0</td><td>> 10h</td><td>27794.4</td><td>> 10h</td><td>22605.3</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>96</td><td></td><td>150</td><td>20473.5</td><td>> 10h</td><td>25248.3</td><td>> 10h</td><td>21390.7</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>97</td><td>(5,25)</td><td>200</td><td>20571.5</td><td>> 10h</td><td>25523.9</td><td>> 10h</td><td>21776.2</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>98</td><td></td><td>250</td><td>21483.0</td><td>> 10h</td><td>24536.7</td><td>> 10h</td><td>22207.9</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>99</td><td></td><td>300</td><td>22307.5</td><td>> 10h</td><td>27119.4</td><td>> 10h</td><td>23392.4</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>100</td><td></td><td>350</td><td>18784.5</td><td>> 10h</td><td>25705.0</td><td>> 10h</td><td>19746.9</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>101</td><td></td><td>0</td><td>16882.5</td><td>> 10h</td><td>21286.1</td><td>> 10h</td><td>17358.9</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>102</td><td></td><td>50</td><td>15126.5</td><td>> 10h</td><td>17585.0</td><td>> 10h</td><td>15856.9</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>103</td><td></td><td>100</td><td>17118.0</td><td>> 10h</td><td>25825.6</td><td>> 10h</td><td>17614.8</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>104</td><td></td><td>150</td><td>16249.5</td><td>> 10h</td><td>20582.8</td><td>> 10h</td><td>16861.9</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>105</td><td>(5,30)</td><td>200</td><td>17442.5</td><td>> 10h</td><td>22880.6</td><td>> 10h</td><td>18291.0</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>106</td><td></td><td>250</td><td>16167.0</td><td>> 10h</td><td>21657.8</td><td>> 10h</td><td>16757.5</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>107</td><td></td><td>300</td><td>16427.5</td><td>> 10h</td><td>19808.3</td><td>> 10h</td><td>17289.3</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>108</td><td></td><td>350</td><td>16505.0</td><td>> 10h</td><td>21061.1</td><td>> 10h</td><td>17336.6</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr></tbody></table>
<table><caption>**Table 9** Solution times of different bounding methods in seconds for the capital budgeting problem (continued)</caption><thead><tr><th rowspan="2">Instance</th><th rowspan="2">(T, I)</th><th rowspan="2">B</th><th colspan="4">Fractional loan</th><th colspan="4">Integral loan</th></tr><tr><th>ν<sup>K</sup></th><th>ν<sup>Ω</sup></th><th>ν̄<sup>NA</sup><sub>R</sub></th><th>ν̄<sup>DNA</sup><sub>R</sub></th><th>ν<sup>K</sup></th><th>ν<sup>Ω</sup></th><th>ν̄<sup>NA</sup><sub>R</sub></th><th>ν̄<sup>DNA</sup><sub>R</sub></th></tr></thead><tbody><tr><td>109</td><td></td><td>0</td><td>3016.8</td><td>7432.8</td><td>5463.7</td><td>9377.8</td><td>3240.6</td><td>13927.0</td><td>6806.2</td><td>13651.4</td></tr><tr><td>110</td><td></td><td>50</td><td>5783.4</td><td>10153.4</td><td>7404.2</td><td>13493.4</td><td>5940.9</td><td>11026.0</td><td>9714.2</td><td>18790.5</td></tr><tr><td>111</td><td></td><td>100</td><td>5794.8</td><td>10968.8</td><td>8336.0</td><td>15323.8</td><td>6040.9</td><td>18911.0</td><td>11222.6</td><td>25385.0</td></tr><tr><td>112</td><td></td><td>150</td><td>4342.2</td><td>9224.2</td><td>6689.4</td><td>11467.2</td><td>4508.2</td><td>13912.7</td><td>8315.9</td><td>16565.2</td></tr><tr><td>113</td><td>(6,5)</td><td>200</td><td>4884.6</td><td>10710.6</td><td>7782.0</td><td>14405.6</td><td>5035.6</td><td>18792.8</td><td>10012.1</td><td>23365.9</td></tr><tr><td>114</td><td></td><td>250</td><td>4446.6</td><td>8944.6</td><td>6575.8</td><td>11688.6</td><td>4707.9</td><td>16686.1</td><td>7770.8</td><td>17262.4</td></tr><tr><td>115</td><td></td><td>300</td><td>5827.2</td><td>11153.2</td><td>8325.5</td><td>14773.2</td><td>6062.7</td><td>17358.7</td><td>11622.5</td><td>25515.4</td></tr><tr><td>116</td><td></td><td>350</td><td>4345.8</td><td>8927.8</td><td>6816.8</td><td>12029.8</td><td>4605.7</td><td>16729.0</td><td>9140.3</td><td>17563.5</td></tr><tr><td>117</td><td></td><td>400</td><td>4180.2</td><td>9836.2</td><td>7160.2</td><td>12836.2</td><td>4367.5</td><td>14595.2</td><td>8522.6</td><td>19430.3</td></tr><tr><td>118</td><td></td><td>0</td><td>10104.6</td><td>> 10h</td><td>10200.6</td><td>24872.6</td><td>10546.4</td><td>27686.4</td><td>13701.4</td><td>> 10h</td></tr><tr><td>119</td><td></td><td>50</td><td>8583.0</td><td>> 10h</td><td>9351.0</td><td>21351.0</td><td>9209.9</td><td>22440.2</td><td>12711.7</td><td>35626.3</td></tr><tr><td>120</td><td></td><td>100</td><td>11280.6</td><td>> 10h</td><td>11376.6</td><td>22130.6</td><td>11747.3</td><td>29362.6</td><td>16320.9</td><td>35307.2</td></tr><tr><td>121</td><td></td><td>150</td><td>9476.4</td><td>> 10h</td><td>9860.4</td><td>23221.4</td><td>9962.1</td><td>31165.4</td><td>12165.8</td><td>> 10h</td></tr><tr><td>122</td><td>(6,10)</td><td>200</td><td>9439.2</td><td>> 10h</td><td>9919.2</td><td>20601.2</td><td>9959.5</td><td>23400.1</td><td>13121.1</td><td>> 10h</td></tr><tr><td>123</td><td></td><td>250</td><td>8892.6</td><td>> 10h</td><td>9084.6</td><td>17979.6</td><td>9535.3</td><td>24697.2</td><td>10903.3</td><td>26120.8</td></tr><tr><td>124</td><td></td><td>300</td><td>9023.4</td><td>> 10h</td><td>9983.4</td><td>18604.4</td><td>9466.8</td><td>30363.2</td><td>11968.1</td><td>26395.9</td></tr><tr><td>125</td><td></td><td>350</td><td>11331.6</td><td>> 10h</td><td>11427.6</td><td>27149.6</td><td>11807.6</td><td>28773.7</td><td>13932.5</td><td>> 10h</td></tr><tr><td>126</td><td></td><td>400</td><td>11216.4</td><td>> 10h</td><td>12080.4</td><td>28663.4</td><td>11832.9</td><td>26791.1</td><td>14588.3</td><td>> 10h</td></tr><tr><td>127</td><td></td><td>0</td><td>18997.2</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>19922.6</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>128</td><td></td><td>50</td><td>18051.6</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>19040.8</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>129</td><td></td><td>100</td><td>16756.2</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>17648.3</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>130</td><td></td><td>150</td><td>20196.0</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>21130.6</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>131</td><td>(6,15)</td><td>200</td><td>19142.4</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>20092.3</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>132</td><td></td><td>250</td><td>19387.2</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>20388.0</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>133</td><td></td><td>300</td><td>16451.4</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>17483.9</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>134</td><td></td><td>350</td><td>16513.2</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>17490.8</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>135</td><td></td><td>400</td><td>15548.4</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>16621.3</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>136</td><td></td><td>0</td><td>26038.2</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>28075.0</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>137</td><td></td><td>50</td><td>25179.0</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>26638.6</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>138</td><td></td><td>100</td><td>27861.0</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>29812.5</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>139</td><td></td><td>150</td><td>27286.2</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>29240.4</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>140</td><td>(6,20)</td><td>200</td><td>26205.0</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>27575.8</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>141</td><td></td><td>250</td><td>26002.2</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>27503.4</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>142</td><td></td><td>300</td><td>28728.0</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>30580.6</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>143</td><td></td><td>350</td><td>24949.8</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>26952.8</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>144</td><td></td><td>400</td><td>29950.2</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>31406.3</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>145</td><td></td><td>0</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>146</td><td></td><td>50</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>147</td><td></td><td>100</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>148</td><td></td><td>150</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>149</td><td>(6,25)</td><td>200</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>150</td><td></td><td>250</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>151</td><td></td><td>300</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>152</td><td></td><td>350</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>153</td><td></td><td>400</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>154</td><td></td><td>0</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>155</td><td></td><td>50</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>156</td><td></td><td>100</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>157</td><td></td><td>150</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>158</td><td>(6,30)</td><td>200</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>159</td><td></td><td>250</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>160</td><td></td><td>300</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>161</td><td></td><td>350</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr><tr><td>162</td><td></td><td>400</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td><td>> 10h</td></tr></tbody></table>