The hacker groups are targeting three different company data centers, each with different vulnerability

levels, as specified in the following table.

Data Center $1$ Data Center $2$ Data Center $3$

Vulnerability

Level $759$

Simultaneously, the Cyber Defenders need to maintain the security of their primary servers, known as

the OmniTech Tower. The protection each member can offer varies depending on the location. The table

below illustrates the maximum security each member can provide:

Protection Value Data Center $1$ Data Center $2$ Data Center $3$ OmniTech Tower

Network Ninja $5345$

Cryptography Captain $6237$

Software Commander $7485$

Data Detective $3254$

Hardware Hawk $2133$

Firewall Titan $6597$

To fully safeguard each data center, the total protection score provided by the assigned members must

meet or surpass the vulnerability level. While doing this, the following conditions should be observed:

$(1)$ Each member of the Cyber Defenders can either stay at OmniTech Tower or be assigned to a

data center.

$(2)$ Each data center must have at least one and no more than three Cyber Defenders deployed.

$(3)$ The total number of Cyber Defenders deployed to the data centers should not exceed five.

$(4)$ At least one of the Cryptography Captain and the Network Ninja should remain at OmniTech Tower.

$(5)$ If the Data Detective is deployed at a data center, the Network Ninja should be deployed there as well.

$(6)$ If the Data Detective and the Hardware Hawk are both deployed at the same data center, they

will end up in disagreements and not contribute to the security, meaning that no more than one of Tower.

$(5)$ If the Data Detective is deployed at a data center, the Network Ninja should be deployed there as

well.

$(6)$ If the Data Detective and the Hardware Hawk are both deployed at the same data center, they

will end up in disagreements and not contribute to the security, meaning that no more than one of

these two can be deployed at any data center.

$(7)$ If the Hardware Hawk is deployed at a data center, at least one of the Cryptography Captain or

Software Commander should also be deployed there to maintain his productivity.

a$)$ Your task is to devise a linear integer program that helps QmniTech Intelligences determine the

optimal deployment plan to maximize QmniTech Tower's protection while ensuring the sufficient

security of all data centers.

b$)$ If the objective is to maximize the total protection value of the QmniTech Tower's and the three

data

centers while ensuring the sufficient security of all data centers, how would you answer differ from

a$).$

these two can be deployed at any data center.

$(7)$ If the Hardware Hawk is deployed at a data center, at least one of the Cryptography Captain or Software Commander should also be deployed there to maintain his productivity.

a$)$ Your task is to devise a linear integer program that helps OmniTech Intelligences determine the

optimal deployment plan to maximize OmniTech Tower's protection while ensuring the sufficient