## 4 Case Study: The Disposal in Finland  
In practice, the scalarized problem (32) with an augmentation term (33) in Step 2 of MITSPA is solved with a branch-and-cut type method for single-objective MINLP problems called BARON [44; 45] in GAMS [46]. The CPU time of solving each problem (32) presented here varies from 9 s to 28000 s while the average CPU time is 3475 s and the median CPU time is 142 s. The weighting vectors used are of the form  
\[\lambda^{U}=\frac{1}{f^{nad}-f^{R}},\quad\lambda^{A}=\frac{1}{f^{R}-f^{id}}\]  
such that \(f^{nad}-f^{R}>0\) and \(f^{R}-f^{id}>0\) as suggested in [27]. The approximation of the nadir vector used is obtained with a pay-off table [31; 32]. We investigate the disposal of the spent nuclear fuel from the European pressurized water reactor (EPR) produced by Olkliuoto 3 in Finland starting to operate in the near future. The length of one disposal period is selected to be 5 years, and the parameters \(N\) and \(Z\) are 19 and 11, respectively. The other parameters used are given in Appendix A, except the cost parameters that are omitted due to their commerce-related nature. This parameter selection yields a multiobjective MINLP problem with 9 objectives, 440 continuous and 475 binary variables, 1144 linear constraints, 20 nonlinear constraints, and three box constraints. Apart from these, we need some auxiliary variables and constraints to overcome the non-smoothness of the problem. Indeed, the two-slope parameterized ASFs are nonsmooth, but due to their min-max structure, the problem (32) can be written in the MINLP form as in [25]. Similarly, this trick can be applied also for the nonsmooth objectives. After that, we have to solve a single-objective problem with 441 continuous and 484 binary variables, 1153 linear constraints, 21-146 nonlinear constraints and 3 box constraints. Before we proceed to the solution process, we discuss the trade-offs of the problem. There are three parts in the final disposal of spent nuclear fuel: the interim storage, the encapsulation facility, and the disposal facility. These three parts interact with each other as is exemplified in the following. * Interim storage versus disposal facility: The interim storage-related goals all imply transferring the spent nuclear fuel from the interim storage as rapidly as possible. However, in order to minimize the disposal facility-related goals, the cooling times should be maximized. * Encapsulation facility versus interim storage: By delaying the start of disposal, it is possible to shorten the operation time of the encapsulation facility, and thus, decrease the operating costs. Again, the delay at the start of the encapsulation can cause an increase of the inventories in the interim storage. * Encapsulation facility versus disposal facility: The disposal should be started and ended as soon as possible. Both of these aims have a tendency to increase the canister heat load, and hence, affect the disposal facility goals. To minimize the operation time of the encapsulation facility, empty assembly positions can be used. However, the price to pay is the increased number of canisters. In addition, a larger number of canisters necessitates an increase in the disposal facility area. In order to investigate these, and other trade-offs, the interactive method MITSPA is employed. In each iteration of MITSPA, some new preference information is asked from the decision-maker reflecting his/her preferences. For each iteration, we compute nine solutions by using the current reference point with different metrics by varying the value of the parameter \(q\) from 1 to 9 in Step 2. In order to exemplify this, the nine solutions computed using reference point 1 are shown in Figure 1. These nine solutions represent different trade-offs between objective function values. The results obtained are scaled to the interval from 0 to 1 such that 0 is the value of the ideal vector and 1 is the value of the nadir vector for the objective under consideration.