Let's tackle these questions step$-$by$-$step:

$1.)$ Calculate the company's productivity before and after the change in work rules:

Before the Change:

They made $500$ boxes in $20$ hours $($two $10-$hour shifts$).$

Productivity $($before$)=$ Total boxes produced $/$ Total hours worked

Productivity $($before$)=500$ boxes $/20$ hours $=25$ boxes$/$hour

After the Change:

They made $650$ boxes in $24$ hours $($three $8-$hour shifts$).$

Productivity $($after$)=$ Total boxes produced $/$ Total hours worked

Productivity $($after$)=650$ boxes $/24$ hours $=27\mathrm{.}08$ boxes$/$hour $($rounded to $2$ decimal places$)$

Explanation:

Please refer to the solution in this steps.

Step $2$

$2.)$ Percentage increase in production:

Percentage Increase $=[($New value $-$ Old value$)/$ Old value$]$ x $100$

Percentage Increase $=[(650$ boxes $-500$ boxes$)/500$ boxes$]$ x $100$

Percentage Increase $=150$ boxes $/500$ boxes x $100=30\%$

$3.)$ If production is increased to $700$ boxes per day, what is the new productivity?

Assuming they still work three $8-$hour shifts:

Productivity $($new$)=$ Total boxes produced $/$ Total hours worked

Productivity $($new$)=700$ boxes $/24$ hours $=29\mathrm{.}17$ boxes$/$hour $($rounded to $2$ decimal places$)$

Explanation:

Please refer to the solution in this steps.

Step $3$

$4.)$ Issues with this kind of growth and managing personnel shifts and morale:

Issues:

Sustainability: Increasing production may put a strain on resources, machinery, and employees. Over time, this might lead to more frequent breakdowns, higher maintenance costs, or burnout among the workforce.

Quality Control: Rapid increases in production might lead to oversight in quality control.

Supply Chain Concerns: Ensuring raw materials $($like wood for boxes$)$ can meet the increased production demand.

Employee Fatigue: Shorter but more frequent shifts might mean more days of work for employees leading to fatigue.

Morale Issues: If employees feel they are being overworked, it can affect their morale and overall job satisfaction.

Managing Personnel Shifts and Morale:

Clear Communication: Management should clearly communicate the reasons for change and how it benefits the company and potentially the employees $($e$.$g$.,$ job security, potential for bonuses$).$

Shift Rotation: If possible, rotate shifts among employees so that no one group is perpetually stuck on a less desirable shift.

Incentives: Offer performance incentives or bonuses for meeting production targets.

Feedback: Open channels for employee feedback to ensure any concerns are addressed promptly.

Work$-$Life Balance: Ensure employees have adequate breaks and days off to maintain a healthy work$-$life balance.

Training: Ensure employees are well$-$trained for their tasks to reduce stress and inefficiencies.

Explanation:

All the steps are explained above properly go through it$.$

Answer

Remember, while increased production and growth are often seen as positive for a company, it's crucial to balance growth with the well$-$being of employees, the quality of the product, and long$-$term sustainability.