I need help with problem, where i have a collection of n triangular functions. We want to select a subset of at most 7 of \(\displaystyle g_i\) functions such that their sum is as constant as possible within the interval [0,1]. As constant as possible means that the difference between the maximum value and the minimum value is as small as possible:
[math]\min_{{{g_1,\ldots,g_m} \subset {f_1,\ldots,f_n}}} \left( \max_{x_i \in B} \sum_{j=1}^n f_j(x_i)v_j - \min_{x_i \in {B}} \sum_{j=1}^n f_j(x_i)v_j \right),[/math]where is
[math]v_j ~=~ \begin{cases} 1\,; ~&f_j \in {g_1,\ldots,g_m}, \\ 0\,; ~&f_j \notin {g_1,\ldots,g_m}. \end{cases}[/math]
[math]{B} = {x_1,\ldots,x_k} \cap [a,b][/math] is set of breakpoints of functions [math]f_i[/math]
I need t use ILP to solve this problem and i don't know how to write it.
Can someone please help me?
[math]\min_{{{g_1,\ldots,g_m} \subset {f_1,\ldots,f_n}}} \left( \max_{x_i \in B} \sum_{j=1}^n f_j(x_i)v_j - \min_{x_i \in {B}} \sum_{j=1}^n f_j(x_i)v_j \right),[/math]where is
[math]v_j ~=~ \begin{cases} 1\,; ~&f_j \in {g_1,\ldots,g_m}, \\ 0\,; ~&f_j \notin {g_1,\ldots,g_m}. \end{cases}[/math]
[math]{B} = {x_1,\ldots,x_k} \cap [a,b][/math] is set of breakpoints of functions [math]f_i[/math]
I need t use ILP to solve this problem and i don't know how to write it.
Can someone please help me?
Last edited: