Minimizing a project costs through nonlinear optimization

Question

I have a project and I want to minimize the costs. I am are responsible for the inspection of 1000 miles of sewer grid in Canada. My goal is to provide time high quality inspection reports. I tried to define the problem using optimization but I haven't used for a year. I want to know what would be the optimised number of people I should hire.

The people I have to pay are robots drivers and inspectors.

• I have a $\$90000$ budget, two months on site and two weeks off site before.
• robots drivers take 6 pictures of every pole under different point of views and upload them on my cloud. The drivers upload their pics once a week.
• A well trained driver can inspect 15miles of sewer grid per day. Training lasts for one day, you can train two pilots in parallel.
• A photo inspector performs a quality check of the pictures on the Sterblue cloud. Sterblue asks the drone pilots to reinspect any poles not passing the quality checks. You can assume 10% of the poles have to be re-inspected.
• My AI on the cloud detects the defects. The customer expects 95% accuracy.
• The inspector performs a quality check of the detections found by the AI. They cost $300 a day and can handle 30miles/day for image quality review, 30miles/day to review the work of the AI.
• I have a free operations team that generates the inspection reports. 0,5 day is needed to prepare a report for 100 miles of grid.

$$ \begin{cases} \begin{aligned} \min \ &800 \overbrace{x_1.y_1}^{\text{robot pilots x days}} &+ 300 \overbrace{x_2.y_2}^{\text{inspectors x days}}\\ &800 x_1.y_1 &+ 300 x_2.y_2&\le $90000\\ &y_1&+y_2 &\le 2 months\\ &15 x_1.y_1&&\ge 1000 miles+10\%x_1.y_1 \\ &15 y_1 &&= 30 y_2 \\ \forall i, x_i \end{aligned} \end{cases} $$

Where $x_1$ is the number of robots drivers, $x_2$ the number of inspectors, $y_1$ the time spent by robot drivers, $y_2$ the time spent by inspectors.

I know that inspection can't start before reports, and that I haven't found a way to write the 95% accuracy constraints.

• Can you help me improve the problem so I take into account every constraints ?
• What should $x_1$ and $x_2$, the number of people I should hire, be ?

I know I should do a Gantt diagram as well but I don't know yet where the major steps and dependencies are.

Answer 1

You have multiple things at hand here, none of which seem like an optimization problem to me:

• To understand how many people to hire, consider that you have ca. 620 miles (= 1000km) of grid, and a single driver can do 15 miles of grid per day. Given that you have two months (ca. 60 days), this means that in theory one person could do it all, since $\frac{620}{60} \approx 10$. However, in order to account for off days, vehicle breakdown etc. you should look for 2, possibly 3 drivers. For me, this is not an optimization problem because a lot of qualitative measures need to be taken into account (e.g. worker reliability, actual progress speed) etc.
• The 95% constraint exclusively depends on your "AI" (calling image recognition "AI" is a bit of a stretch for me, sounds more like machine learning to me, but ok), so you need to prove that. The easiest way is to check against some test data (see e.g. here for the split between training and test data) and if that sample is large enough, you can draw some conclusion from that. A step further would be to incorporate the feedback from the inspectors in real time and see whether your prediction quality matches what you assume.
• As for the number of inspectors: you have not specified how many poles there are per mile, but assuming an equidistant distribution, we can say that a total of $620\cdot 0.1 = 62$ miles have a problem. With a speed of 30 miles/day, you basically only need one, since you have a two month period at your disposal. I am not sure what you mean by "AI review", but if you simply want to get test data to validate the predictions from the image recognition software, I would suggest the inspector does that the remaining time he/she is not busy with defect piles.

Of course you could throw some optimization at this, but I don't see the point given on the description you provided.