A manufacturing company needs to decide whether to open or close each of its three factories based on demand from six customers, production capacities, and fixed operating costs. The total demand from customers exceeds the company's maximum production capacity if all factories are operational. Each factory has a fixed cost associated with opening it and variable transportation costs for shipping products directly to customers. Additionally, there's a penalty cost for every unit of unmet demand. The goal is to minimize the total costs, which include the fixed costs of operating factories, transportation costs, and penalty costs for any unmet demand.

Given Data:

Factories $($with maximum production capacity and fixed costs$)$:

F$1$: $200$ units, Fixed Cost $=$ $$10,000$

F$2$: $250$ units, Fixed Cost $=$ $$12,000$

F$3$: $300$ units, Fixed Cost $=$ $$15,000$

Customers $($with demand$)$:

C$1$: $150$ units

C$2$: $200$ units

C$3$: $180$ units

C$4$: $160$ units

C$5$: $190$ units

C$6$: $220$ units

Penalty Cost per Unit of Unmet Demand:

$$5$ per unit

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

C$1$ C$2$ C$3$ C$4$ C$5$ C$6$

F$1$ $$3$ $$2$ $$4$ $$3$ $$2$ $$5$

F$2$ $$4$ $$3$ $$2$ $$2$ $$3$ $$4$

F$3$ $$2$ $$3$ $$3$ $$4$ $$1$ $$2$

$3-$ Carefully write the mathematical formulations including the constraints.

$4-$ 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.