Beta Co$.$ manufactures air conditioning compressors and markets them to ASEAN markets. The company receives the following demand forecasts in a $6-$month period:

Month Jul Aug Sep Oct Nov Dec

Demand $($units$)8509601060950540760$

Working Days $212321222221$

The "demand" refers to the quantities of product ordered by customers in a month, and "days" refer to the number of working days $($exclusive of public holidays$)$ in a month.

The company's production department currently hires $8$ employees in its manufacturing site. On average, one employee can produce $5$ units of product per working day. The company's HR policy prohibits employees in the production department to work overtime but allows them to work undertime. For example, there are $21$ working days in July. If an employee works $18$ days and another employee works $19$ days in July, then these two employees have a total of $3$ days of undertime in July. According to HR policy, the maximum allowable undertime is $3$ days per month per employee.

The company has the option to subcontract its production to a contractor in any month. However, due to the nature of batch processing, the contractor only allows three $(3)$ options of subcontracting, i$.$e$.,$ batch sizes of $100$ units, $150$ units and $300$ units. The lump sum payments for the three batch sizes of subcontracting are:

Batch Size Lump Sum Payment

$100$ units $$65,500$

$150$ units $$94,300$

$300$ units $$182,120$

On average, one unit of product generates a revenue of $$1,700.$ The average material cost of one unit of product is $$456.$ Employees are paid on fixed salary and the average salary for an employee is $$5,000/$month$.$ An order that is unfulfilled $($in the month when it was received$)$ is considered as backorder, which would incur a backorder $($delay$)$ cost of $$245$ per unit per month. Unsold finished products are stored in warehouse and incurs inventory holding cost of $$98$ per unit per month.

Assume that the initial inventory at the beginning of July is $130$ units. State the decision variables, objective and constraints. Use proper notations to denote decision variables and related parameters.