Let’s imagine Bob has to plan out a day for his date in Vancouver. He has to figure out how to maximize the most of his day by doing as much activity as possible in a single day based on how important each activity is.
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This is similar to the knapsack problem where instead of maximizing the number of valuable item in a bag, we will maximize the number of activity in a day that is valuable to the person Bob is dating.
$i$ that could be planned or not planned$i$$i$$i$Let’s consider that $x_i$ is an integer $\mathbb Z\in \{0,1\}$, where 0 means is not in the dating plan and 1 meaning is in the dating plan.
Assume Bob would spend $200 for the date
Assume Bob only has 8 hours to spend on the activities
Assume travel time is negligible
Assume the value of the activity is defined by ranked by Bob’s partner from $[n,1]$ where $n$ is the most valuable activity to do and $1$ is the least preferred activity to do.
Activity options in Vancouver:
| Activity Name | Price for two ($) | Time spent Approximately | Links | Person A Value | Person B Value | Person C |
|---|---|---|---|---|---|---|
| Rock Climbing | 60 | 3 | https://hiveclimbing.com/pricing/ | 7 | 2 | 3 |
| Kayak at Deep Cove | 76 | 3 | https://deepcovekayak.com/rental/kayak-rentals/ | 6 | 4 | 4 |
| Rent a Tandem bike & bike around Stanley Park | 40 | 2 | https://spokesbicyclerentals.com/rentals/ | 5 | 3 | 6 |
| Minigolf | 35 | 1 | https://www.westcoastminiputt.com/events/ | 4 | 5 | 2 |
| Escape room | 70 | 1 | https://e-exit.ca/vancouver-west-broadway/3 | 3 | 1 | 1 |
| Capilano Bridge | 133.9 | 2 | https://www.capbridge.com/rates/ | 2 | 6 | 7 |
| VanDusen Botanical Garden | 16.6 | 2 | https://vandusengarden.org/plan-your-visit/hours-admission/ | 1 | 7 | 5 |
$$ \begin{aligned} \text{maximize}\qquad &\sum^n_{i=1}v_ix_i\\ \text{subject to}\qquad & \sum^n_{i=1}\beta_ix_i &\leq \Beta\\ &\sum^n_{i=1}t_ix_i &\leq T\\ \ &x_i \in \{0,1\} \end{aligned} $$