The following data is the gathered data from a manufacturing plant that fabricates three sub assemblies, S$1,$ S$2,$ and S$3,$ in a pre$-$planned manufacturing sequence in four workshops. The output rates of the four workshops, in units of products per hour, are given in this data. W$1$ outputs $20$ S$1$ per hour. W$1$ outputs $30$ S$2$ per hour. W$1$ outputs $60$ S$3$ per hour. available working hours in W$1$ is $150.$ W$2$ outputs $40$ S$1$ per hour. W$2$ outputs $25$ S$2$ per hour. available working hours in W$2$ is $160.$ W$3$ outputs $10$ S$2$ per hour. W$3$ outputs $20$ S$3$ per hour. available working hours in W$3$ is $130.$ W$4$ outputs $22$ S$2$ per hour. W$4$ outputs $5$ S$3$ per hour. available working hours in W$4$ is $100.$ For instance, S$1$ only needs to be processed in Workshops W$1$ and W$2$ and Workshop W$1$ outputs $20$ S$1$ per hour. The company owner wants to make maximum profit by manufacturing the best possible product mix, in the coming month.

As the company moves forward with their manufacturing plant operations, it is important to consider some critical information. to ensure optimal production and maximise profit, the company must schedule W$1$ and W$2$ for at least $100$ hours. it is also crucial that they meet the minimum demand of $1000$ units for S$1$ in the upcoming month. to achieve this, they must male sure that they have enough of the unique assembly components required for both S$1$ and S$3.$ unfortunately, they only have $42000$ units of this component available for the coming month. s$1$ requires one unit of this assembly component, while s$3$ requires three units. the main goal is to ensure that they meet their production goals and satisfy the customer demand in the most profitable manner.

$($a$)$ develop a linear programming model that helps the company achieve such a goal. clearly present all decision variables, the objective function and the constraints.

$($b$)$ solve the model in $($a$)$ using microsoft solver