A company that makes a special laptop battery must meet the following monthly demand for its major distributors. The batteries are currently produced in its own South Korea facility with the following maximum capacity and cost estimates. Month (t), 2024 Demand (Dt) Capacity Cost (South Korea) Cost (External firm) January February March 35,500 42,700 52,000 26,500 48,000 22,000 35,000 32,500 $32/unit $42/unit $31/unit $38/unit $36/unit $35/unit $39/unit $28/unit $33/unit $41/unit April May 64,200 35,000 49,000 36,500 $33/unit $29/unit June The company will have a backlog of 3,300 units at the beginning of January 2024. To meet the demand, several options may be considered. a). Use the inventory of its own South Korea facility at a holding cost of $4/unit charged against the ending inventory of each month. b). Subcontract part of the production to an external South Korea manufacturer. Due to the capacity limit, the maximum amount of quantity subcontracted to that external company cannot exceed 35,000 units per month. c). Including a candidate Mexican subcontractor into this planning process. The Mexican subcontractor charges $30/unit (great price!) and has a maximum capacity 26,000/month. However, due to lack of experience, it is preferred that the total quantity outsourced to Mexico over the planning horizon is no more than 35% of the total demand. d) Consider a planned shortage policy so that a backlog of orders can be delivered in the following month at a penalty cost of $8/unit (i.e., the rush order handling). The company needs to have an ending inventory of 11,000 units, but not any backlog, by the end of June 2024. Task 1: Write down a linear programming (LP) mathematical model to minimize the total production, inventory, subcontracting, and planned shortage cost over the given planning horizon. You need to define the variables, write down the objective function, and write down all the necessary constraints (c.g., the inventory balance constraints, the capacity constraints, the initial and ending constraints, and other constraints). Task 2: Implement the LP model in Excel and use Solver to solve it to optimality. Please show all your calculations in Excel for this task.