A company has forecasted its monthly demand of its product for the next year as below. The company currently has 1250 full-time regular workers with hourly salary of $20. The production plant operates for 8 hours per day, 20 days per month. Overtime is allowed but limited to a maximum 20 hours per full-time worker. Overtime is also paid hourly, at 50\% higher than regular hourly payment. Making each unit of product requires 10 minutes labor time and $20 of materials. The inventory holding cost is $3 per unit. Currently, the company has 50000 units as inventory. It wants to have an aggregated plan for the coming 12 months, that ends with the same level of inventory. For the scenario, write the mathematical models with constraints. Keeping the overtime restriction as no more than 20 hours per regular workers per month at 50\% extra charge. To be more realistic, newly hired workers required training before they can produce as efficient and effective as the experience workers. The hiring cost at $500 /worker and layoff cost of $750 /worker. Assume that the newly hired workers' productivity is only 75% for their first month and they are not allowed to work any overtime. In addition, the company CANNOT layoff more than 20% of its existing workforce. Backlog is allowed at penalty charge of $5/ unit. The company can also subcontract the production at $26/ unit. For t=1,,12 Wt= Workforce size for month t Ht= Number of employees hired at the beginning of month t Lt= Number of employees laid off at the beginning of month t Pt= Production in month t It= Inventory at the end of month t St= Number of units backlogged at the end of month t Ct= Number of units subcontracted for month t Ot= Number of overtime hours worked in month t