I have formulated the below optimization problem.

https://book.polandcentral.cloudapp.azure.com/Formulation.JPG

In $(P)$:
- $w_i$ and $p_{ij}$, and $q_i$ are decision variable.
- $\alpha_i,c_{ij},\beta_{i}$ are coefficients.

For the proof of being NP-complete, It is easy to show that problem in is NP. For the reduction part of the proof, I aimed to show that [Capacitated Facility Location Problem](https://en.wikipedia.org/wiki/Facility_location_problem#Capacitated_facility_location) is polynomially reducible to $(P)$.

The main obstacle (so as to prove the instance of CFLP can be solved as instance of our problem) here is that in order to remove the last term in objective function of $(P)$, we can set the decision variable $q_i=0$, however, it yields that $w_i$ should be $1$ for all $i$ (due to existence of third constraints). In CFLP notation, it means that all facilities should be open (which does not parse to me).

Is there any alternative way to for the proof?

On Tuesday, July 4, 2023 at 9:00:19 AM UTC-4, User2121 wrote:
> I have formulated the below optimization problem.
>
> https://book.polandcentral.cloudapp.azure.com/Formulation.JPG
>
> In $(P)$:
> - $w_i$ and $p_{ij}$, and $q_i$ are decision variable.
> - $\alpha_i,c_{ij},\beta_{i}$ are coefficients.
>
> For the proof of being NP-complete, It is easy to show that problem in is NP. For the reduction part of the proof, I aimed to show that [Capacitated Facility Location Problem](https://en.wikipedia.org/wiki/Facility_location_problem#Capacitated_facility_location) is polynomially reducible to $(P)$.
>
> The main obstacle (so as to prove the instance of CFLP can be solved as instance of our problem) here is that in order to remove the last term in objective function of $(P)$, we can set the decision variable $q_i=0$, however, it yields that $w_i$ should be $1$ for all $i$ (due to existence of third constraints). In CFLP notation, it means that all facilities should be open (which does not parse to me).
>
> Is there any alternative way to for the proof?

probably, change the indexing in a non-obvious way.

My thing:

What is needed is a "non-de-recursive-izable forecast," based on access to good data and science, which should be calculated and published. The publication of the forecast could be set up so that the publication of it speeds up the release, but nothing else, provably, can be re-worked to allow input variables to be manipulated in a way that would change the output of the time when this ends and the cure is out.

-Philip White (philipwhite363@gmail.com)