# Multi-Objective BiLevel Optimization by Bayesian Optimization
## 4 Experiments
### Noisy Experiments
#### 4.2.3 Test Problems  
The first example is a bi-objective problem that is scalable in terms of the number of follower decision variables. The formulation of the problem is given in Table 5. We choose \(K=14\) and \(K=19\), giving 15 and 20 follower variables, respectively, with 1 leader decision variable. We choose the reference point required to measure hypervolume improvement to be \((1.0,0.5)\). The Pareto-optimal decision sets for this specific bilevel decision-making problem can be found in [45]. Example 2  
The second test problem is the modified test problem with 10 and 20 variable instances. The formulation of 264 the problem is given in Table 5. We choose the required reference point to be \((1.1,1.1)\). The Pareto-optimal front for a given leader is defined as a circle of radius \((1+r)\) with center \(((1+r),(1+r))\). We choose \(K=5\) for our experiments, with parameters \(r=0.1\), \(\tau=1\) and \(\alpha=1\), following [45] so that our results can be compared with those for m-BLEAQ and H-BLEMO. Example 3  
The third test problem is the modified test problem with 10 and 20 variable instances. The formulation of 264 the problem is given in Table 5. We choose the required reference point \((0.8,0.0)\) for measuring the hypervolume improvement during the optimization. Details on the Pareto-optimal solutions are given in [48]. Practical Case: Gold Mining in Kuusamo  
The Kuusamo region is a popular tourist destination known for its natural beauty. There is a lot of interest in this region cause it contains a huge amount of gold deposits and is considered to be a \"highly prospective Paleoproterozoic Kuusamo Schist Belt\" [49]. The expected gold amount in the ore is around 4.9 g per ton according to an Australia-based gold mining company. Even though there is a big potential in terms of providing lots of jobs in the region and leading to a great amount of gold resources, there are concerns about harming the environment. The first of them is that mining operations may cause pollution of the river water in the region. This concerns environmentalists. Second, the ore in the region contains uranium and, if it is mined, it might harm the reputation of nearby tourist resorts. Another is the open pit mines around the area called Ruka, which will affect nearby skiing resorts and hiking routes and reduce tourist interest. The regulating authority, which is the government, acts as a leader, and the mining company is the follower, which reacts rationally to the decisions of the leader in order to maximize its own profit. The leader should find an optimal strategy assuming that he holds the necessary information about the follower. In the situation explained above, the government has a decision-making problem, which is whether to allow mining and to what extent. In the problem above, the leader has two objectives while the follower has one. The first objective is the maximization of revenues coming from the project and the second objective is to minimize the environmental harm, which is a result of the mining. The mining company also aims to maximize its own profit. While the government is optimizing its own taxation strategy, it needs to model how the mining company reacts to any given tax structure. Therefore, the authority makes an environmental regulatory decision instead of solving the problem to optimality. Clearly, the objectives are conflicting, such as the fact that large profits may affect the environment by increasing the damage, which follows with a bad public image. The mathematical formulation of this hierarchical decision-making problem can be found in Table 6. More details can be found about the bilevel modelling of the problem in [45; 46]. Figure 4 presents the Pareto-optimal frontier of the given problem for the government according to the formulation in Table 6. We can observe that increasing the tax revenue decreases the environmental damage.