$[9$:$22$ pm$,23/4/2024]$ Vanmathi S P: $15.$ The ACC company has developed a voice$-$activated Internet assistant for home. This product is manufactured at four plants, located in Atlanta, Kansas City, Aberdeen, and Austin. After production, the units are shipped to three warehouses, located in Nashville, San Jose, and Houston. ACC sells through the retail channel. In particular, five different retailers currently sell the product$-$Sears, Best Buy, Fry$\text{'}$s$,$ Walmart, and Office Depot. ACC makes weekly shipments to the main warehouses of these five retailers. The shipping cost from each plant to each ware$-$house, along with the production cost and weekly production capacity at each plant, are given in table

$[9$:$23$ pm$,23/4/2024]$ Vanmathi S P: Table $1$

Shipping Cost $($$$/$unit$)$

San Jose

Plant

Nashville

Houston

Production Cost $($$$/$unit$)$

Capacity $($units$/$week$)$

Atlanta

$$30$

$$40$

$$50$

$$208$

$200$

Kansas City

$$25$

$$45$

$$40$

$5214$

$300$

Aberdeen

$$45$

$$30$

$$55$

$5215$

$300$

Austin

$530$

$$50$

$$30$

$5210$

$400$

Shipping Cost $($$$/$unit$)$

Table $2$

Variable Cost $($unit$)$

Best Buy

Comp USA

Office Max

Warehouse

Sears

Fry$\text{'}$s

Capacity $[$units$/$week$)$

Now

$140$

$$45$

$530$

$575$

$530$

$54$

$300$

$$50$

$525$

$$15$

$540$

$300$

Haalar

$$50$

$$25$

$515$

$540$

$150$

$300$

Cantomer somand

$100$

$10$

$15$

$300$

$150$

Table $3$

Plant

A

Kansas City

Warehouse

Fixed Cost $(5/$week$)$

Fixed Cost $($$$/$week$)$

$19\mathrm{.}500$

Nas

$55\mathrm{.}000$

San Jose

$$10\mathrm{.}000$

Houston

$54\mathrm{.}500$

$55,000$

$$1000$

The shipping cost from each warehouse to each customer, the variable cost $($cost per unit moved through the warehouse$),$ the capacity $($maximum number of units that can be moved through the warehouse per week$)$ for each warehouse, and the weekly demand given in table a$.$ Formulate a linear programming model to determine the plan for weekly for each customer are $2.$ production and distribution of the product from the various plants, through the warehouses, to the customers that will minimize total costs.

b$.$ Now suppose that ACC is considering saving money by closing some of its production facilities and$/$or warehouses. Suppose there is a fixed cost to operate each plant and each warehouse as indicated in table $3.$ Add binary variables to your model to incorporate the decision of which plants and warehouses to keep open so as to minimize total cost $($including the fixed costs for any plant or ware$-$house that is operated$).$