Problem $1.(45$ points$)$ on Inteser ProrrmminrSuppose that you are the production manager of a furniture factory. Your company Sahikea can produce couches in four different production facilities. To use a facility, you need to pay a fixed net up cost that is paid upfront. Also, for each couch you produce in a facility you need to pay a manufacturing cost; andeach facility has a production capacity, The following table summarizes the information about thefacilities' set up cost$,$ unit manufacturing cost, and capacity.Production FacilitySet$-$Up Cost$$2500$Manufacturing CostFacility Capacity$2$$$2200$$$160$$$175500750$$$1775$$$1504004$$$1500$$$250800$You can produce couches in a facility only after you pay the set$-$up cost$.$ For instance, if you choose to produce couches at facility $2,$ you have to pay the set$-$up cost of $$2200,$ and $$175$ for each couch you produce there. Note that you cannot produce any couches in a facility if you do not pay the set$-$up cost$.$Now, suppose that Sahlkea wants to have $1800$ couches in their inventory for the next selling season. As the production manager, you want to determine which facilities to use and how many couches to produce in each facility while minimizing the total production cost $($this includes set up costs and manufacturing costs$).$a$)(25$ points$)$ Mathematically formulate an integer$/$binary optimization model for the above production 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.