# The impact of the existence of multiple adjustable robust solutions

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## Abstract

In this note we show that multiple solutions exist for the production-inventory example in the seminal paper on adjustable robust optimization in Ben-Tal et al. (Math Program 99(2):351–376, 2004). All these optimal robust solutions have the same worst-case objective value, but the mean objective values differ up to 21.9 % and for individual realizations this difference can be up to 59.4 %. We show via additional experiments that these differences in performance become negligible when using a folding horizon approach. The aim of this paper is to convince users of adjustable robust optimization to check for existence of multiple solutions. Using the production-inventory example and an illustrative toy example we deduce three important implications of the existence of multiple optimal robust solutions. First, if one neglects this existence of multiple solutions, then one can wrongly conclude that the adjustable robust solution does not outperform the nonadjustable robust solution. Second, even when it is a priori known that the adjustable and nonadjustable robust solutions are equivalent on worst-case objective value, they might still differ on the mean objective value. Third, even if it is known that affine decision rules yield (near) optimal performance in the adjustable robust optimization setting, then still nonlinear decision rules can yield much better mean objective values.

## Keywords

Adjustable robust optimization Production-inventory problems Folding horizon Multiple solutions## 1 Introduction

In [2] the Robust Optimization (RO) methodology is extended to multi-stage problems. The proposed Adjustable Robust Optimization (ARO) techniques appeared to be very effective to solve uncertain multi-stage optimization problems. This first paper on ARO has been cited more than 500 times already, and the ARO methodology has been applied to a wide variety of problems (see e.g. the survey papers [3, 6]). Recently, it was shown that (A)RO problems may have multiple optimal solutions, and that not all of these solutions are Pareto robustly optimal [8]. A solution is called Pareto robustly optimal if there is no other robustly feasible solution that has better objective value for at least one scenario, and for all other scenarios in the uncertainty set the objective value is not worse.

In this note we show that the ARO model of the production-inventory problem in [2], which is the seminal work on ARO, also has multiple optimal robust solutions. Although in robust optimization one operates in a distribution-free environment, an often used performance measure is the mean objective value, which is evaluated posteriorly assuming some information on the distribution of the parameters. For the cases considered in [2], we show that among the optimal robust solutions, the difference in mean objective value can be as much as 21.9 % and for individual realizations the difference can be up to 59.4 %. This underlines the importance of the message in [8] that ARO problems may have multiple optimal robust solutions. In such cases one can often find optimal robust solutions that are much better with respect to the mean objective value than solutions that were initially found.

We also extend the experiments performed in [2] by including a folding horizon approach. In a folding horizon approach the model is re-optimized in each period using the available information at that point of time and only the decisions for the current time are implemented. Using this approach we find that there are still multiple optimal robust solutions, but the differences in mean costs diminish. This is mainly due to the fact that the here-and-now decisions are unique in almost all periods. As a last experiment, we also analyze the model and solutions we found when replacing the worst-case objective by an expected value objective. For the expected value objective we find that, for the seminal production-inventory problem considered here, the solution is unique.

In the second part of this note we discuss several important implications for practical ARO. The first implication is that, by ignoring the possibility of multiple solutions, one can incorrectly conclude that the ARO solution is not better than the RO solution, or even incorrectly conclude that ARO is (much) better than RO. The second implication is that even in cases where it is a priori known that RO and ARO are equivalent, i.e., they have the same worst-case optimal objective value, one cannot conclude that there is no value in using ARO. This is because in many cases there are ARO solutions that give much better solutions for the mean costs. The third implication is that even in cases where affine decision rules are (nearly) optimal, i.e., the optimal robust objective value cannot be improved by using nonlinear decision rule, one cannot conclude that there is no value in using nonlinear decision rules. Such a conclusion might be wrong, since nonlinear decision rules may yield much better solutions for the expected objective value. These implications are illustrated by using both the production-inventory example from [2] and two toy examples.

Our aim is to convince users of ARO that one should always check for the existence of multiple solutions. In many papers on ARO it is not reported that one checked for possible existence of multiple solutions. These papers run the risk that much better solutions could have been found, or even that wrong conclusions have been drawn. For example, researchers who use the same production-inventory example as in the seminal work [2] to test new ARO methods, should be aware of the fact that this problem has many optimal robust solutions with big differences in mean costs.

## 2 Multiple adjustable robust solutions

To illustrate the implications of multiple adjustable robust solutions we use three problems. The first problem is the production-inventory problem by [2] in its original setting. The second problem is an illustrative toy example where the existence of multiple solutions is more directly visible. The last toy problem we investigate is a two-stage facility location problem. For all models we study both the impact in a folding and in a non-folding horizon approach.

### 2.1 Production-inventory model by Ben-Tal et al. [2]

We have repeated the experiments for the production-inventory problem by [2]. All solutions are obtained using the commercial solver Gurobi 6.0 [7] programmed in the YALMIP language [10] in MATLAB. All options of Gurobi were left at their default values.

- 1.
Solve the original model from [2], which gives a solution with minimal worst-case costs.

- 2.
Change the objective into minimizing the costs for the nominal demand trajectory. Furthermore, add a constraint that ensures that the worst-case costs do not exceed the costs found in Step 1.

*maximizing*costs for the nominal demand trajectory. This we call the ‘Worst’ solution. Without the two-step approach, and some bad luck, one could have obtained this solution as a ‘First’ solution, i.e. by solving the original problem formulation. The performances of these three optimal robust solutions are given in Table 1. The first column states the uncertainty level, for which we used the same levels as in [2]. If the level of uncertainty is 2.5 %, then this indicates that in each period the realized demand could be up to 2.5 % higher or lower than the nominal demand. The three solutions are all robustly optimal, so they have the same worst-case costs (WC costs). For each of those solutions we have determined the mean costs and the standard deviation. In [2] the mean costs were approximated using 100 simulated demand trajectories drawn from a uniform distribution. The mean costs can also be determined exactly since the objective is linear in the uncertain demand. For the mean costs comparison we assume, as in the original paper, that the mean demand is given by the nominal demand scenario. The standard deviation was derived using the second moment of the uniform distribution, the distribution that was also used in the seminal paper by [2] to sample the scenarios to calculate average costs.

Performance of the Best, First and Worst optimal robust solutions

Uncertainty level (%) | WC costs | Best solution | First solution | Worst solution | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Mean | Std | Mean | Std | Performance gap | Mean | Std | Performance gap | ||||

Mean (%) | Max (%) | Mean (%) | Max (%) | ||||||||

2.5 | 35105 | 33932 | 178 | 35105 | 0 | 3.5 | 7.2 | 35105 | 0 | 3.5 | 7.2 |

5 | 36389 | 34073 | 350 | 35953 | 142 | 5.5 | 11.8 | 36389 | 0 | 6.8 | 14.6 |

10 | 38990 | 34416 | 691 | 38136 | 232 | 10.8 | 24.9 | 38990 | 0 | 13.3 | 30.7 |

20 | 44273 | 35077 | 1373 | 40174 | 696 | 14.5 | 39.4 | 42766 | 315 | 21.9 | 59.4 |

*performance gap*. The mean performance gap is just the percentage increase of the mean costs compared to the mean costs of the ‘Best’ solution. The maximum performance gap is the single demand trajectory that results in the largest difference in costs between the ‘Best’ solution and the ‘Worst’ (or ‘First’) solution. To explain how this gap is calculated, we determine the costs for the ‘Worst’ and the ‘Best’ solution, when trajectory \(\varvec{d}\) realizes, by respectively \(OPT_W(\varvec{d})\) and \(OPT_B(\varvec{d})\). These costs are linear in demand \(\varvec{d}\) because the original objective is linear, fixed recourse and we use linear decision rules. The maximum performance gap for the ‘Worst’ solution is given by

Performance of the best, first obtained and worst optimal robust solutions using the folding horizon approach

Uncertainty level (%) | Best FH solution | First FH solution | Worst FH solution | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Mean | Std | Mean | Std | Performance gap | Mean | Std | Performance gap | |||

Mean (%) | Max (%) | Mean (%) | Max (%) | |||||||

2.5 | 33909 | 179 | 33912 | 179 | 0.0 | 0.1 | 33912 | 178 | 0.0 | 0.1 |

5 | 34057 | 330 | 34061 | 328 | 0.0 | 0.0 | 34059 | 328 | 0.0 | 0.0 |

10 | 34327 | 676 | 34350 | 667 | 0.1 | 0.7 | 34351 | 666 | 0.1 | 0.7 |

20 | 34495 | 1361 | 34517 | 1348 | 0.1 | 0.6 | 34532 | 1339 | 0.1 | 0.6 |

### 2.2 Folding horizon versus non-folding horizon

*t*starting from the first period until the end of the planning horizon. Using this folding horizon approach we again compared solutions that used the two-step approach in each step (Best FH solution), without a two-step approach (First FH solution) and when the two-step approach was used when maximizing for nominal demand in the second step (Worst FH solution). An exact calculation of the mean costs and the standard deviation is not possible for this experiment. Therefore, we draw 100 demand trajectories independently and uniformly distributed in each period. These trajectories are used to approximate the mean costs and the standard deviation when using the folding horizon approach. Simulations were also used in [2] to approximate the mean costs and the standard deviation for the non-folding horizon approach. The results are depicted in Table 2. We stress that this folding horizon approach was not used in [2]. Clearly, using the two-step approach does not yield significantly better results for the folding horizon approach. Often the resulting costs are the same for both approaches, but for one of our simulated realizations the extra costs incurred when not using the two-step approach is 0.7 %. Even stronger, for each simulated demand trajectory, the costs when using the folding horizon approach (Best FH solution) were at most the costs of the “First FH” solution. Finally, note that the mean costs for the folding horizon solutions are not much lower than the mean costs of the ‘Best’ solution given in Table 1, meaning that there is not much additional gain by re-optimizing in each step as is done in the folding horizon approach. It is at a first glance surprising that the effect of having multiple optimal solutions diminishes when using a folding horizon approach. We found that this is mainly because the first stage decisions are unique for almost all time periods and in all simulated scenarios. The question whether or not the first stage decisions are unique can be answered by fixing the worst case costs in the first step, as in the usual two step approach, and then minimize or maximize the order quantity in the current time period. In this way we get, for each time period

*t*, a lower and upper bound on the feasible first stage decisions. In Fig. 1 we depict the differences between the maximum and the minimum for the 20 % uncertainty level for one out of the three factories. The behavior of the solutions depicted was observed for all other cases as well: the vast majority of the first-stage decisions are unique. We only witnessed non-unique optimal here-and-now decisions in time periods 6 and 18, depending on the factory (1, 2 or 3) considered.

Performance of the linear decision rule that minimizes the mean costs

Uncertainty level (%) | WC | Mean | Std |
---|---|---|---|

2.5 | 35108 | 33919 | 178 |

5 | 36412 | 34031 | 357 |

10 | 39040 | 34311 | 708 |

20 | 44298 | 35066 | 1375 |

- 1.
The non-folding horizon solution can be used as a backup solution in case of failure in hardware or software during the re-optimization steps. This is especially important in more critical multi-stage optimization systems such as power systems.

- 2.
Re-optimization might take too much computation time or might not be possible at all. This happens in multi-stage optimization settings when periods follow up close in time, or when the solutions are implemented in low-end software systems. Examples of low-end computer systems are traffic light systems, that are not designed to solve the more computationally demanding optimization models.

### 2.3 Toy examples

*x*and

*y*are nonadjustable. We readily see that the worst-case objective value is 0 and the two solutions, \(RO_1 = (1,0)\) and \(RO_2 = (0,0)\), or any convex combination of these, are worst-case optimal. Without a two-step approach the solver is indifferent between all these optimal robust solutions since they all have optimal worst-case profits. The realized profits as a function of scenario (

*a*,

*b*) are respectively \(p_{RO_1}(a,b) = a\) and \(p_{RO_2}(a,b) = 0\) and the two-step approach yields solution \(RO_1\).

Now suppose that *y* is adjustable and we restrict ourselves to linear decision rules (LDR). Then we find that linear decision rules \(y(b) = -b\) or \(y(b) = -\tfrac{1}{2}b\) are optimal in worst-case sense together with any nonadjustable *x* in [0, 1]. For the first solution \(LDR_1\) we take \((x,y) = (1,-b)\) and for the second solution \(LDR_2 = (0,-\tfrac{1}{2}b)\). The profits of these solutions for scenario (*a*, *b*) are respectively \(p_{LDR_1}(a,b) = a+b\) and \(p_{LDR_2}(a,b) = \tfrac{1}{2}b\). Again, without a two-step approach the solver would be indifferent between these solutions since both have optimal worst-case objective value of 0. The two-step approach yields solution \(LDR_1\).

*perfect hindsight*solution, where parameters

*a*and

*b*are known before deciding upon

*x*and

*y*, equals \((x,y) = (1,-b^2-b)\) for any

*a*,

*b*in [0, 1]. This perfect hindsight solution can also be obtained in the adjustable robust optimization model by allowing for nonlinear decision rules and setting \(NDR_1 = (1,-b^2 - b)\). The profits for this nonlinear decision rule (NDR) are \(p_{NDR_1}(a,b) = a+b^2+b\) for scenario (

*a*,

*b*). Again, there are many more nonlinear decision rules that are optimal in worst-case sense, but have different mean profits. One example is \(NDR_2 = (0,- \frac{1}{2}b^3)\) which yields profit \(p_{NDR_2}(a,b) = \frac{1}{2}b^3\). All these results are summarized in Table 4.

Comparison of the different nonadjustable and adjustable solutions

\({RO}_1\) | \({RO}_2\) | \({LDR}_1\) | \( {LDR}_2\) | \({NDR}_1\) | \({NDR}_2\) | |
---|---|---|---|---|---|---|

Here-and-now | 1 | 0 | 1 | 0 | 1 | 0 |

Wait-and-see\(^\mathrm{a}\) | 0 | 0 | \(-b\) | \(-\tfrac{1}{2}b\) | \(-b^2 - b\) | \(-\tfrac{1}{2}b^3\) |

Profits for scenario ( | | 0 | \(a+b\) | \(\frac{1}{2}b\) | \(a+b^2+b\) | \(\frac{1}{2}b^3\) |

Worst-case profits | 0 | 0 | 0 | 0 | 0 | 0 |

Mean profits (with unif. distr.) | \(\frac{1}{2}\) | 0 | 1 | \(\frac{1}{4}\) | \(\frac{4}{3}\) | \(\frac{1}{8}\) |

*b*lies on the extremes, i.e. \(P(b=0 \text { or } b=1) \ne 1\) and (2) the mean value of

*a*and

*b*is such that \(\mathbb {E}(a) > \tfrac{1}{2}\mathbb {E}(b)\).

Note that for this toy example, contrary to the model from [2], there could be a significant gain from the two-step method in the folding horizon approach. The variable *x* has to be chosen in the first step of the optimization. As we have seen, the optimal robust value is indifferent between any *x* in [0, 1]. In the second step we shall always choose \(y = -b^2 -b\). However, choosing \(x=0\) instead of \(x=1\) gives us a difference of *a* in the objective value. The two-step approach combined with the folding horizon approach returns the optimal (folding horizon) solution, which equals \(NDR_1\).

*N*are both equal to 10. All other customers are (much) closer to both facilities, but unit transportation costs from facility 2 are significantly smaller than from facility 1. This situation is depicted in Fig. 2.

*i*. The facility location problem consists of two types of decisions, namely the decision to open facility 1 (\(x_1=1\)) or facility 2 (\(x_2 = 1\)) and the actual deliveries to the customers from the opened facility. Only one of the facilities may be opened. The total delivery to customer

*i*from facility 1, respectively facility 2, is \(y_{1i}\) and \(y_{2i}\) and has unit costs \(c_{1i}\) and \(c_{2i}\). The goal is to minimize the worst-case transportation costs, which is modeled as:

In the adjustable robust model there are multiple optimal solutions. In the first solution we open facility 1 and transport \(y_{1i} = d_i\) to customer \(i=1,\ldots ,N\). In the second solution we open facility 2 and transport \(y_{2i} = d_i\) to each customer. Clearly, we obtain the same worst-case costs \(c_{1N} = c_{2N}\) as in the perfect hindsight case. However, the costs when \((d_1,d_2,\ldots ,d_N)\) realizes equals \(\sum _{i=1}^Nc_{1i}d_i\) and \(\sum _{i=1}^Nc_{2i}d_i\) respectively. If the expected demand is \(d_i = \tfrac{1}{N}\) for all \(i=1,\ldots ,N\), or any other scenario that does not place all probability mass on the demand realization with \(d_N = 1\), then the two-step approach picks the solution that opens facility 2.

To conclude, in the first toy-example the here-and-now decisions matter in the folding horizon approach for the costs, but there is no impact of the existence of multiple here-and-now decisions on the choice of the optimal wait-and-see decision in the re-optimization step. In the second toy example we do see an impact: once the wrong facility is opened in the first stage, all demand has to be fulfilled from that location at high expected costs in the re-optimization step.

## 3 Implications for robust optimization

The inventory-production problem and the toy examples from the previous section allow us to present some important implications. First, if we analyze and compare the mean objective values of arbitrary optimal robust solutions for RO and ARO, then false conclusions can be drawn regarding the added value of ARO over RO. The mean objective value of an *arbitrary* optimal robust solution, obtained by solving the original RO or ARO problem formulations, might very well be much worse than the solution with best mean objective value among all optimal robust solutions. This best performing solution can be obtained by carrying out the two-step approach. In the production-inventory problem with uncertainty level 2.5 %, the worst-case objective values of the RO and ARO solution are nearly the same: the difference is only 0.5 %. If we compare the RO and ARO solutions on average costs, then the worst ARO solution is also only 0.5 % better than the RO solution. The best ARO solution, however, is 3.5 % better on average, which could be overlooked if the two-step approach is not carried out. For the 20 % uncertainty level, the gap between the average performances of all optimal robust ARO solutions can be as much as 21.9 %. The first toy example illustrates that an arbitrary ARO solution is not guaranteed to do better than a RO solution with respect to average performance. For instance, the average performance of robust solution \(RO_1\) is better than the performance of ARO solution \(LDR_2\). On the other hand, the optimal ARO solution \(LDR_1\) is guaranteed to do better than any RO solution on the average performance. In our small facility location example we have seen that the robust solution results in a much higher objective value, but that it does open the best facility for folding horizon approaches. The linear decision rule on the other hand results in multiple optimal solutions which could lead to undesirable choices for opening the facilities. The two-step approach results in a solution that opens the cheapest facility, mimicking the solution of perfect hindsight.

Second, one might be inclined to jump to the conclusion that ARO can be safely ignored, when it is a priori known that ARO and RO are equivalent with respect to the *worst-case* objective value. One of the situations that we know where ARO is equivalent to RO is the case of constraint-wise uncertainty see [2, Theorem 2.1]. However, the equivalence is not necessarily true for the mean objective value as well. Therefore, one should not ignore ARO for such problems. This is illustrated by the first toy example: the worst-case objective value is zero for both the RO and ARO solutions, but the mean objective values differ significantly.

Third, even if affine decision rules yield (near) optimal worst-case performance, nonlinear decision rules, such as quadratic decision rules, can yield much better performance on the mean objective value. Most applications of ARO restrict decision rules to affine functions, which is referred to as affinely adjustable robust optimization (AARO) [2]. Affine decision rules are known to perform optimal or nearly optimal in many situations [1, 4]. However, once again, this observation is with respect to the worst-case objective value, and not for the mean objective value. This is illustrated by the first toy example. Here, the quadratic decision rule \(NDR_1\) has the same worst-case objective value as any of the other decision rules, but the mean objective value is much better, and, in this particular case, even optimal for each scenario (Bellman optimal).

The encompassing recommendation that follows from these implications is that the two-step approach should always be conducted in any application of robust optimization. The two-step approach enables the optimizer to fully exploit the performance on the mean objective value of the solution, while guaranteeing no deterioration in the worst-case performance. This is especially relevant for ARO, where decision rules can be utilized to enhance the solution’s performance in other than worst-case scenarios. We also recommend the use of the two step approach in folding horizon methods, but we do note that the impact of multiple solutions may be less severe.

## Notes

### Acknowledgments

The authors thank the associate editor and two anonymous referees for their useful comments.

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