Start documenting MILP model

README.md

@@ -25,4 +25,44 @@ conda activate trails_through_daybreak_lgc_alignment_choices
 
 # Install library and dev dependencies (ex. linting tools)
 pip install -e '.[dev]'
-```
+```
+
+## MILP model
+WIP: Writing in progress
+
+### Variables
+
+```math
+R_{x} = \text{1 if route ``x'' was chosen in Chapter 5, else 0}
+```
+
+```math
+A_{a_b} = \text{Score of alignment ``a'' at the end of Chapter ``b''}
+```
+
+```math
+O_{c} = \text{1 if choice ``c'' was chosen, else 0}
+```
+
+```math
+H_{e} = \text{1 if the correct route was chosen for event ``e'', else 0}
+```
+
+### Constraints
+Choose exactly one of the routes in Chapter 5.
+
+```math
+R_{law} + R_{grey} + R_{chaos} + R_{fourth} = 1
+```
+
+Choose exactly one of the choices for each dialog prompt.
+
+```math
+\forall_{e \in events} \left(\sum_{c \in e.choices} O_{c}\right) = 1
+```
+
+Model the alignment scores at the end of each chapter.
+
+```math
+\forall_{a \in \{law, grey, chaos\}} \forall_{c \in \{1, 2, 3, 4, 5, 6, 7\}} A_{ac} = \sum_{e \in \{d \in events \mid d.chapter <= c\}} \left(H_{e} * e.completion.a + H_{e} * \left(\sum_{h \in e.choices} O_c * h.a\right)\right)
+```