A company must decide which of its factories to keep operational and how to distribute its products from these factories to various customers. The goal is to minimize the total costs, which include the fixed costs of operating each factory and the transportation costs, while ensuring that the demand of each customer is met.

Given Data:

Factories $($with production capacities and fixed operating costs$)$:

Factory $1($F$1)$: Capacity $=500$ units, Fixed Cost $=$ $$10,000$

Factory $2($F$2)$: Capacity $=400$ units, Fixed Cost $=$ $$8,000$

Factory $3($F$3)$: Capacity $=600$ units, Fixed Cost $=$ $$12,000$

Customers $($with demand$)$:

Customer A: Demand $=300$ units

Customer B: Demand $=350$ units

Customer C: Demand $=250$ units

Customer D: Demand $=200$ units

Transportation Costs $($per unit$)$ from Factories to Customers:

Customer A Customer B Customer C Customer D

F$1$ $$2$ $$3$ $$2\mathrm{.}5$ $$4$

F$2$ $$3$ $$2\mathrm{.}5$ $$3$ $$2$

F$3$ $$2\mathrm{.}5$ $$3\mathrm{.}5$ $$2$ $$3$

Objective:

Determine which factories to operate and the optimal way to distribute products from these factories to customers in order to minimize the total costs, comprising both fixed operational costs and variable transportation costs.

$1-$ Carefully write the mathematical formulations including the constraints.

$2-$ Using Excel Solver, Minimize the total cost of operating factories, the distribution center, and transporting goods from factories to the DC and then to customers.