Optimize spacing and headway for a corridor service using the following assumptions and Excel Solver:
Cost per stop - $1000
Cost per mile of transit service - $10,000/mile
Density of passengers - 2000 passengers per mile per hour
Maximum cruising speed - 40 miles per hour
Length of service - 50 miles
Time per stop - 1 minute
Access time - 10 miles per hour
Maximum door - to - door travel time: 1.5 hour

To optimize the spacing and headway for a transit corridor service, given the assumptions, you can model the problem in Excel and use **Excel Solver** to minimize the total cost while satisfying the travel time constraint.

Here's how you can approach the problem:

### Step-by-step process:

1. **Variables to define:**
   - **$ s $**: Stop spacing (miles).
   - **$ h $**: Headway (hours between vehicles).

2. **Travel time components:**
   - **Access time (walking time)**: Passengers walk to the stop at 10 mph.
     \[
     \text{Access time} = \frac{s}{2} \times \frac{1}{10} = \frac{s}{20} \text{ hours}
     \]
     (where $ \frac{s}{2} $ is the average distance walked to the nearest stop).
   
   - **In-vehicle time**: This is the time spent traveling between stops, which depends on the **cruising speed** (40 mph) and the time lost at stops (1 minute/stop). The **average in-vehicle time** is the sum of the cruising time and stopping time:
     \[
     \text{Cruising time} = \frac{\text{Length of service}}{\text{cruising speed}} = \frac{50}{40} = 1.25 \text{ hours}.
     \]
     For stopping time, if the vehicle stops every $ s $ miles:
     \[
     \text{Stopping time} = \frac{50}{s} \times \frac{1}{60} \text{ hours per stop}.
     \]

3. **Total travel time**:
   \[
   \text{Total travel time} = \frac{s}{20} + 1.25 + \frac{50}{s} \times \frac{1}{60}.
   \]
   This must be **less than or equal to 1.5 hours**, the maximum door-to-door travel time constraint.

4. **Cost components**:
   - **Cost of stops**: If the vehicle makes $ \frac{50}{s} $ stops:
     \[
     \text{Cost of stops} = 1000 \times \frac{50}{s}.
     \]
   
   - **Cost of service per mile**: This cost is proportional to the number of miles traveled per hour, which depends on the headway $ h $:


Optimize spacing and headway for a corridor service using the following assumptions and Excel Solver:
Cost per stop - $1000
Cost per mile of transit service - $10,000/mile
Density of passengers - 2000 passengers per mile per hour
Maximum cruising speed - 40 miles per hour
Length of service - 50 miles
Time per stop - 1 minute
Access time - 10 miles per hour
Maximum door-to-door travel time: 1.5 hours.

Learn how to optimize spacing and headway for a transit corridor service using cost and time constraints. This problem incorporates key concepts such as cost minimization, transportation modeling, and constraint optimization using Excel Solver.

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