# Residential Demand Response Scheduling with Consideration of Consumer Preferences
## 4 Computational Experiments  
In this section, we present the results of the performed computational experiments based on the proposed MIP formulation for the problem of interest. The model is implemented using IBM ILOG CPLEX Optimization Studio Version 12.6.1.0 (International Business Machines Corporation, New York, NY, USA) and executed using the default solver settings. The goal of the performed tests is to explore the trade-off between the household preference satisfaction and the production cost. Although we define the model using three objective functions, we investigate the Pareto front (PF) only for the bi-objective optimization problem using objectives \(\mathit{Pro}\mathit{C}\mathit{o}\mathit{c}\mathit{s}\) and \(\mathit{Avg}\mathit{Dissat}\). The third goal \(\mathit{Max}\mathit{Dissat}\) is substituted by a constraint dependent on an input parameter. For the simplified model, we calculate the approximate PF using the \(\epsilon\)-constraint method. In this method, a series of single objective optimization problems are solved. The other objective functions are substituted by constraints in the following way:  
Minimize \[f_{i}(x)\] (17) Subjected to \[x\in S\] (18) \[f_{j}<\epsilon_{j}, i,j=1,2\quad i\
eq j\] (19)  
As is discussed in [28], for bi-objective functions, this method can be used to calculate the PF with the additional effort of removing some dominated points. In our computational experiments, we explore the relation between consumer comfort and production cost represented by two functions \(\mathit{Pro}\mathit{C}\mathit{o}\mathit{s}_{HP}\), \(\mathit{Pro}\mathit{C}\mathit{s}_{LP}\). The experiments are conducted for problems with 50, 150 and 250 households. For each case, we observe that the PF for \(\mathit{Max}\mathit{Dissat}=2,3,4,5\). The computational time for all of the problem sizes is reasonable. On average, it took around 0.3, 1 and 2.5 s to calculate one single objective problem for fixed values of \(\mathit{Max}\mathit{Dissat}\) and \(\mathit{Avg}\mathit{Dissat}\), for problems having 50, 150 and 250 households, respectively. The fixed values for \(\mathit{Avg}\mathit{Dissat}\), corresponding to \(\epsilon\) in Equation (17), used for calculating the PF were taken from the interval \([1,\mathit{Max}\mathit{Dissat}]\) with a step 0.05. In practice, this meant that a maximum of 80 single objective optimization problems would be solved for calculating the PF for a fixed value of \(\mathit{Max}\mathit{Dissat}\). In the case of the problems containing 250 households, the PF is calculated within several minutes. It is noteworthy that, due to the consideration of appliance usage frequencies in the method  
Figure 5: A piecewise linear convex cost function \(f(\hat{l})\) of instantaneous power consumption with seven consumption classes. Cross-over points are used to distinguish the classes. for problem instance generation, a notable number (around 25%) of households (SP, SNP) has only one appliance. In practice, this has made solving the underlying combinatorial optimization problem less computationally expensive. The results of the computational experiments can be seen in Figure 6.  
Figure 6: The relation between average comfort levels and productions costs for a different number of households and the maximal allowed dissatisfaction levels. Production cost is defined as a linear approximation of a convex function (left) and using an hourly pricing scheme (right). Comfort levels vary between 1 and 6; Level 1 represents the maximum satisfaction, while Level 6 is the minimum satisfaction. **(a)**Pareto fronts (PFs) for 5 households using \(\mathit{ProCost}_{LP}\); **(b)** PFs for 50 households using \(\mathit{ProCost}_{HP}\); **(c)** PFs for 150 households using \(\mathit{ProCost}_{LP}\); **(d)** PFs for 150 households using \(\mathit{ProCost}_{HP}\); **(e)** PFs for 250 households using \(\mathit{ProCost}_{LP}\); **(f)** PFs for 250 households using \(\mathit{ProCost}_{HP}\).