Setup: Suppose we have a set of cities $C$ which are connected by a set of roads $R$, where traversing each road $(i,j) \in R$ requires paying $m_{i,j}$ dollars for the toll and takes some time $t_{i,j}$. Your goal is to go from city $c_\text{start} \in C$ to city $c_\text{goal} \in C$. For a path $p$, let $\text{PathMoney}(p)$ be the total money paid in traversing $p$, and let $\text{PathTime}(p)$ be the total time spent traversing $p$.
1. [1 point] Suppose such that one unit of time is worth $m$ dollars to you. Define a utility function in terms of $\text{PathMoney}(p)$ and $\text{PathTime}(p)$.
2. [3 point] Construct a state space model and specify an algorithm to maximize the utility function you described above.
3. [1 point] How does PathMoney and PathTime of the optimal path change as you vary $m$?
4. [1 point] Suppose you have a budget of $M$ dollars. You're in a rush and would like to just minimize the amount of time you spend on the road subject to the constraint that you can't spend more than $M$ dollars. Write down your utility function in terms of PathMoney and PathTime.
5. [4 points] Construct a state space model and specify an algorithm to maximize the utility function you described above. Hint: assume that all $m_{i,j}$ are non-negative integers. How big is your state space?
6. [2 points] Describe a non-trivial consistent A* heuristic based on state abstraction. Provide a short and concise justification for the consistency of your heuristic.