Suppose that you are the logistics manager of a retailer store. You are currently planning to transport a specific product from the manufacturer to your retailer store. You pay $$5$ for purchasing one unit of the product from the manufacturer and you will sell each unit for $$15.$ You can ship at most $500$ units of the product from the manufacturer. To ship the product, you will use truckload carriers and you have $3$ different truckload carriers that you can use. To use trucks from any truckload carrier, you need to pay an upfront fee to the truckload carrier. Also, for each truck you use, there is a truck fee; and trucks of different carriers have varying capacities. The table below gives the information about the carriers$$ upfront fees, per truck fees, and per truck capacity.

Truckload Carrier Upfront Fee Per Truck Fee Per Truck Capacity

$1$ $$130$ $$2065$ units

$2$ $$80$ $$3570$ units

$3$ $$100$ $$2575$ units

You will be transporting only full truck loads. For instance, if you choose to use $3$ trucks from Truckload Carrier $2,$ you must pay the upfront fee of $$80,$ and $$35$ for each truck $($so$,$ $$105$ for the trucks you are using from Carrier $2),$ and you will be transporting $70$ units with each truck $($so$,210$ units will be transported with Carrier $2)$ to your retailer store.

As the logistics manager, you want to determine which truckload carriers to use and how many trucks to use from each truckload carrier. Note that, if you do not pay the upfront fee to a truckload carrier, you cannot use any truck from that carrier. In your problem, you want to maximize the net profit assuming that everything you transport to the retailer store will be sold. The net profit is equal to revenues from sales minus purchase cost paid to the manufacturer minus transportation costs $($transportation costs include upfront fees and truck fees$).$

a$)$ Mathematically formulate an integer$/$binary linear optimization model for the above transportation planning problem. To do so$,$ clearly define the decision variables $($for binary variables, define what $0$ means and what $1$ means$),$ and then formulate the objective function and the constraints using the decision variables.