$24.$ Clean$-$Sheet Manufacturing Location. In this problem, we revisit the Martin$-$Beck plant location problem described in Problem $10.$ The management of Martin$-$Beck has decided to do a clean$-$sheet analysis. Rather than assume that the St$.$ Louis plant is fixed as open, management wants to run a model that allows for any plant or set of plants to be open so that total cost is minimized. The annual fixed cost and the capacities of the proposed plants have been estimated $($see Problem $10).$ The variable costs for the proposed plant locations are estimated to be the following: Detroit $($$$4\mathrm{.}26),$ Toledo $($$$4\mathrm{.}19),$ Denver $($$$4\mathrm{.}69),$ and Kansas City $($$$4\mathrm{.}20).$

We need an estimate of the fixed cost and variable cost at the current St$.$ Louis plant. The file StLouisMB contains $15$ observations from previous years that will allow us to estimate these.

a$.$ Use simple linear regression, with total cost as the dependent variable $($y$)$ and volume as the independent variable $($x$).$ The model y $=$ b$0+$ b$1$x will give you estimates of b$1=$ the per unit variable cost and b$0=$ annual fixed cost. Round the variable cost estimate to the nearest cent.

b$.$ Using the data from Problem $10,$ the values of variable cost given above for the proposed locations and the fixed and variable costs for St$.$ Louis estimated in part $($a$),$ build an optimization model to minimize total cost to meet demand. Which plants are open and what is the total cost to meet demand?