Optimization of Parking Lot - Linear Programming

[CADHalp]

Hey guys,

We have parking lot, and have chosen a specific area of 160x36ft to redesign its parking spaces. This area will not including marginal space for backing out etc, so we can consider it a packing problem. Assuming parking spots are 9x18ft rectangles, how can I solve for optimization? Of course I need to find out my constraints etc. Disclaimer, I am not whatsoever a mathematician. I am simply a user of AutoCAD that was assigned to compare CAD outputs to linear programming.

Let's assume that we use 90, 45, and 60 degree angles. Obviously 90 degree angles would consist of 17 spots across, for a total of 34 in total. I've tried doing my research, and all thesis' show results, but no code or process to getting their results whether in MATLAB, and LINDO etc. Can anyone point me in the right direction? I know there is no perfect solution, hence optimization.

 

[stapel]

We have parking lot, and have chosen a specific area of 160x36ft to redesign its parking spaces. This area will not including marginal space for backing out etc, so we can consider it a packing problem. Assuming parking spots are 9x18ft rectangles, how can I solve for optimization? Of course I need to find out my constraints etc. Disclaimer, I am not whatsoever a mathematician. I am simply a user of AutoCAD that was assigned to compare CAD outputs to linear programming.

Let's assume that we use 90, 45, and 60 degree angles. Obviously 90 degree angles would consist of 17 spots across, for a total of 34 in total. I've tried doing my research, and all thesis' show results, but no code or process to getting their results whether in MATLAB, and LINDO etc. Can anyone point me in the right direction? I know there is no perfect solution, hence optimization.

What do you mean by "solving for optimization"? How would the various angle measures come into play?

When you reply, please include the full and exact text of the exercise, the complete instructions, any additional necessary information (such as a graphic), and a clear statement of your thoughts and efforts so far.

Note: Most of the volunteers here are math teachers of one sort or another. We likely can't help you with your computer programming, etc. But we'll see if we can assist with the mathematics of this exercise.

Thank you!

 

[JeffM]

Hmm

Not sure what happened to my answer. I apologize if this is a duplicate.

If this is a real life problem, the given rectangle cannot accommodate 36 spaces, but can easily accommodate 34. The 34 spaces solution is trivial. By expanding the rectangle to 36 x 162 you can accommodate 36 spaces.

If this is an exercise, I am not sure that I can prove that 35 spaces are impossible (though I suspect impossiblity due to indivisibility), and, if 35 spaces are achievable in principle, I am quite sure that I cannot give you an algorithm to find such a configuration.

EDIT: I do not currently have time to research this, but I am guessing that you can use calculus to prove that unused space is minimized with 90 degree parking. If so, it should then be relatively easy to prove that 34 is the maximum given the lot's dimensions. This of course is not a proof, but a suggested program for finding a proof.

 

[JeffM]

Hmmmm......whassa going on in your parking area?

So the max number of parking spots is found:
but how do the cars get there?
Isn't some area required for cars to drive in?

Btw, per George Google, the average parking
spot is 8.5 feet by 18 feet....

The problem is not in fact well specified, but I read it as saying that there is sufficient room for access, etc.

 

[stapel]

The problem is not in fact well specified, but I read it as saying that there is sufficient room for access, etc.

If the exercise itself is "not...well specified", then I'm not sure how "optimization" is expected to take place. :shock:

 

[JeffM]

If the exercise itself is "not...well specified", then I'm not sure how "optimization" is expected to take place. :shock:

Ahh that is because you have not lived in the business world. There the problems are never well specified.

 

[JeffM]

Denis

if you divide the area of the small rectangles into the area of the large rectangle, you get a number between 35 and 36. That leads at first blush to thinking you can fit 35 in. I don't think you can because of unusable space left over after fitting in 34 spaces.

Experiment a a bit yourself.