$1-)$ A bank gives four types of loans to its individual customers. Personal loan types and annual interest rates are $14\%$ for a housing loan, $20\%$ for a vehicle loan, $10\%$ for a mortgage consumer loan, and $20\%$ for a personal consumer loan. The bank's maximum lending capacity is $250$ million currency units. The bank abides by the following policies must: The loan

amount to be allocated to the housing loan must be at least $55\%$ of the total loan amount to be given to the housing and vehicle loans.

The loan amount to be allocated to the housing loan must be at least $25\%$ of the total amount to be given to all loans.

The loan amount allocated for a vehicle loan should not exceed $25\%$ of the total loan amount.

The average interest rate for all loans should not exceed $15\%.$

Set up a linear programming problem that will optimize the bank's total interest income under the given constraints. $($Write by explaining the decision variables and constraints.$)$