## 3 Methodology
### Optimization of Basic Race Strategies
#### 3.4.2 Determination of Optimum Stint Lengths for Each Set  
Optimum stint lengths are those with the minimum race duration for the respective set. When omitting driver interactions, a driver's race duration is simply the sum of his lap times. Looking at Equation (1), we find that there are four lap-dependent elements: \(t_{\text{grid}}\), \(t_{\text{fuel}}\), \(t_{\text{pit,in-lap/out-lap}}\), and \(t_{\text{tire}}\). The grid time loss and time loss due to fuel mass occur independently of the race strategy and can therefore be omitted when determining optimum stint lengths. For a known amount of pit stops per set, pit time losses are also fixed. Thus, under the afore mentioned assumptions, the race duration of a given set depends only on the time loss due to tire degradation and, therefore, on the stint lengths. A set's optimum stint lengths can either be determined by calculating the race durations for all possible combinations of stint lengths (1 lap/69 laps, 2 laps/68 laps, etc. in a race with 70 laps), or, in the case of a linear tire degradation model, by solving a mixed-integer quadratic optimization problem (MIQP). Solving the MIQP is much faster, notably with the 2- and 3-stop strategies. Especially with no very detailed timing data, the linear degradation model is mostly used anyway, as it is the most robust model. The formulation of the MIQP is introduced in the following. With \(l_{\text{tot}}\) being the total number of laps in a race, the objective of the MIQP reads as follows:  
\[\min\,t_{\text{race}}(l_{\text{tot}})\cong\min\sum_{l=1}^{l_{\text{tot}}}t_{ \text{tire}}(l). \tag{4}\]  
Splitting the race into a fixed number of stints \(N\) (pit stops \(=N-1\)) with stint index \(i\), compounds \(c_{i}\) and stint lengths \(\alpha_{i}\) as optimization variables, the problem can be formulated as follows:  
\[\min_{\lfloor\alpha_{1},\alpha_{N}\rfloor} \sum_{i=1}^{N}\sum_{a=1}^{\alpha_{i}}t_{\text{tire}}(a,c_{i})\] (5) subject to \[\sum_{i=1}^{N}\alpha_{i}=l_{\text{tot}}\] \[\alpha_{i}\in\mathbb{N}^{+}\quad\forall\,1\leq i\leq N.\]Next, we introduce the linear tire degradation model. Thus, the time loss \(t_{\text{tire}}\) is determined by tire age \(a\) and two coefficients \(k_{0}\) and \(k_{1}\), which are dependent on the tire compound \(c\):  
\[t_{\text{tire}}(a,c)=k_{0}(c)+k_{1}(c)\cdot a. \tag{6}\]  
In general, a softer tire is faster in the beginning (\(k_{0}\) is smaller than for a harder compound), but also displays a faster degradation (\(k_{1}\) is higher than for a harder compound). Inserting the tire degradation model into Equation (5), we obtain:  
\[\sum_{i=1}^{N}\sum_{a=1}^{a_{i}}t_{\text{tire}}(a,c_{i})=\sum_{i=1}^{N}\left(k _{0}(c_{i})\cdot\alpha_{i}+k_{1}(c_{i})\sum_{a=1}^{a_{i}}a\right). \tag{7}\]  
Using the Gaussian sum for the last part and rewriting \(k_{0}(c_{i})\) and \(k_{1}(c_{i})\) as \(k_{0,i}\) and \(k_{1,i}\), respectively, gives:  
\[\sum_{i=1}^{N}\sum_{a=1}^{a_{i}}t_{\text{tire}}(a,c_{i})=\sum_{i=1}^{N}\left(k _{0,i}\cdot\alpha_{i}+k_{1,i}\left(\frac{1}{2}\alpha_{i}^{2}+\frac{1}{2} \alpha_{i}\right)\right). \tag{8}\]  
For the case that the tires are not new at the start of a stint, for instance for the top ten starters, a starting age \(a_{\text{s},i}\) is introduced into the formulation. Accordingly, \(\alpha_{i}\) is replaced by \(\alpha_{i}+a_{\text{s},i}\) and the offset loss of \(a_{\text{s},i}\) is subtracted:  
\[ \begin{split}\sum_{i=1}^{N}\sum_{a=1}^{\alpha_{i}}t_{\text{tire} }(a,c_{i},a_{\text{s},i})&=\sum_{i=1}^{N}\left(k_{0,i}\left( \alpha_{i}+a_{\text{s},i}\right)+k_{1,i}\left(\frac{1}{2}\left(\alpha_{i}+a_{ \text{s},i}\right)^{2}+\frac{1}{2}\left(\alpha_{i}+a_{\text{s},i}\right) \right)\right.\\ &\qquad\qquad\qquad\qquad\left.-\left(k_{0,i}\cdot a_{\text{s},i }+k_{1,i}\left(\frac{1}{2}a_{\text{s},i}^{2}+\frac{1}{2}a_{\text{s},i}\right) \right)\right).\end{split}  \tag{9}\]  
Switching to vector notation, we obtain the final formulation:  
\[