Suppose 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 set up cost that is

paid upfront. Also, for each couch you produce in a facility you need to pay a manufacturing cost; and

each facility has a production capacity. The following table summarizes the information about the

facilities' set up cost$,$ unit manufacturing cost, and capacity.

Production Facility Set$-$up cost Manufacturing cost Facility cost

$12500$ $$160500$

$22000$ $$175750$

$31775$ $$150400$

$41500$ $$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 SahIkea 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.