A company makes tools. The demand forecast for the company's tools are shown in the table below. The company needs to meet the demand and end December with at least 700 tools (i.e.. no stockouts at the end of December and at least 700 tools in inventory). At the beginning of January, the company has a workforce of 50 employees and 1000 tools in inventory. The material cost for each tool is $10. The company has a total of 20 working days in each month, and each employee earns $4 per hour regular time and $5 per hour overtime. Each employee works eight hours per day on regular time and the rest on overtime. No employee works more than 10 hours of overtime each month. Each employee can produce 40 tools per month on regular time and one tool for every four hours of overtime. At the beginning of each month, the company decides how many additional employees are hired or how many current employees are laid off. The cost of hiring an employee is $300 and the cost of laying off an employee is $400. To meet the demand, the company can build up inventory during the slow months (i.c., months in which the demand is low) or building up a backlog of orders (stockouts) that will be delivered late to customers. The cost of carrying inventory is $2 per tool per month, and the cost of stocking out is $3 per tool per month. In addition to manufacturing tools in its facility, the company can subcontract out some of the work. The subcontracting cost is $29/tool. (i.e., the company can purchase one tool at $29 from suppliers to meet the demand.) Please formulate an LP model to help the company minimize the total cost. Please write a GAMS input file using subscripted equations and symbolic parameters and find an optimal solution and the optimal value. Month May June January 1000 February 2000 March 1500 April 800 2500 2000 Demand Forecast (Number of Tools) Month July August December September 900 October 3500 November 4000 2900 Demand Forecast (Number 1750 2000 of Tools)