My problem is that I$\text{'}$m not sure if the model is well$-$written, coherent, or optimal. Therefore, I need help to ensure that the model is well$-$written, coherent, and optimal, given the context.

Context:

This optimization model has been developed to efficiently manage the vehicle. The primary objective is to minimize the number of vehicles required, including backup vehicles, to meet the daily demand for cigarettes. The model will be used to plan and allocate resources optimally, ensuring that the demand is met efficiently.

Parameters:

$-($C$\_($d$))$: Vehicle capacity per day $(100,000$ units$).$

$-($H$)$: Working hours per day $(8$ hours$).$

$-($T$)$: Time per route in hours $(6$ hours$).$

$-($y$\_($t$))$: Fixed daily number of routes $(15$ routes$).$

$-($I$\_($max$))$: Maximum allowed backup inventory $(7$ vehicles$).$

$-($A$\_($max$))$: Total storage capacity in number of vehicles $(20$ vehicles$).$

$-($Dda$\_($t$))$: Predicted daily demand for month $($t$).$

Decision Variables:

$-($X$\_($t$))$: Number of vehicles required in month $($t$).$

$-($B$\_($t$))$: Number of backup vehicles required in month $($t$).$

$-($I$\_($t$))$: Inventory of backup vehicles in month $($t$).$

Objective Function:

Minimize the number of vehicles and backups:

Minimize Z $=\backslash$sum$\_($t$=1)^$n $($X$\_($t$)+$ B$\_($t$)$Constraints:

$1.$ Daily Demand Satisfaction:

The number of vehicles must be sufficient to meet the daily demand.

X$\_($t$)*$C$\_($d $)>=$ Dda$\_($t$)$ forall t

$2.$ Minimum Backup Vehicles:

At least $30\%$ of the vehicles must be maintained as backup.

B$\_($t$)>=0\mathrm{.}3*$ X$\_($t$)$ forall t

$3.$ Minimum and Maximum Capacity per Vehicle:

Each vehicle must carry between $40\%$ and $80\%$ of its capacity.

$0\mathrm{.}4*$ X$\_($t$)*$C$\_($d$)<=$Dda$\_($t$)<=0\mathrm{.}8*$X$\_($t$)*$C$\_($d$)$ forall t

$4.$ Route Time Constraint:

Working hours per day must not exceed $8$ hours.

Y$\_($t$)*$ T$<=$ H $*$ X$\_($t$)$ forall t

$5.$ Fixed Daily Routes:

The number of daily routes is fixed at $15.$

Y$\_($t$)=15$ forall t

$6.$ Maximum Routes per Vehicle:

A vehicle can make a maximum of $2$ routes per day$6.$ Maximum Routes per Vehicle:

A vehicle can make a maximum of $2$ routes per day.

Y$\_($t$)<=2*$X$\_($t$)$ forall t

$7.$ Maximum Allowed Inventory:

The inventory of backup vehicles must not exceed the maximum allowed.

I$\_($t$)<=$ I$\_($max$)$ forall t

$8.$ Inventory Balance:

The inventory balance must be maintained from one month to another.

I$\_($t$)=$ I$\_($t$-1)+$ B$\_($t$)-(($Dda$\_($t$))/($C$\_($d$)))$ forall t$>1$

$9.$ Total Storage Capacity:

The total storage capacity must be respected.

X$\_($t$)+$ B$\_($t$)<=$ A$\_($max$)$ forall t

This model should provide a framework for the planning and optimization of the vehicle fleet, ensuring that daily demand is met efficiently while maintaining an adequate number of backup vehicles for contingencies.