EXERCISE $7\mathrm{.}13$ from DMD textbook $$textbook $($Data$,$ Models, and Decisions: The Fundamentals of Management Science by Dimitris

Bertsimas and Robert M$.$ Freund, Dynamic Ideas LLC$,2004$

A manufacturing company is anticipating an increase in demand for its products. However, management is concerned about the adequacy of their employees to meet the increased demand. This inadequacy, combined with high workforce turnover $(5\%$ of employees leave the company at the end of each month and must be replaced$),$ will threaten the company's ability to meet the increased demand. Rather than simply hiring new workers, management is contemplating enrolling some or all of their employees in a month$-$long intensive training program. After the successful completion of the training program, an employee would receive an increase in salary; furthermore, the employee would sign a contract not to leave the company for at least $6$ months. The management believes that successful completion of the program would increase an employee's productivity by $20\%.$ However, only $90\%$ of the employees are estimated to be able to complete the training program successfully. $($We assume that if the employee enrolls in the training program but does not successfully complete the program, then he or she will return to the workforce at the pre$-$training skill level, and that he or she can enroll in the program again at any time$).$

The monthly demand for untrained employees for the next six months is shown in Table $7\mathrm{.}50.$ Note that if there are trained employees available, their higher productivity would allow management to satisfy demand with fewer employees. For example, the demand in month $1$ can be satisfied with $100$ untrained employees, or with $82$ untrained and $15$ trained employees $($since $100=82+(1\mathrm{.}20)\backslash$times $15).$

An employee cannot be engaged in production and be enrolled in the training program during the same month. At the beginning of January, there are $145($un$-$trained$)$ employees on the workforce. Monthly payroll costs to the company are $$3,000$ per untrained employee $($engaged in either production or in the training program$)$ and $$3,300$ per trained employee.

The company would like to design a training schedule for its employees that would minimize their total six$-$month payroll cost and allow them to satisfy the workforce demand throughout the six months from January through June.

Month Number of $($untrained$)$ employees required

January $100$

February $100$

March $115$

April $125$

May $140$

June $150$

Table $7\mathrm{.}50$ The demand for $($untrained$)$ employees at a manufacturing company.

$($a$)(6$ points$)$ Construct and then solve a linear optimization model to design such a schedule. What is the optimal training schedule?

Hint: The decisions that have to be made in this problem are how many untrained employees to allow to enroll in the training program each month. There will be similar conditions that have to be satisfied every month, and so as an illustration, let us consider the month of April. Let us rename the months January through June as months $1-6.$ Then April is month $4,$ for example.

The decision variables are as follows:

$=$ the number of trained employees available in month i$,$ for i $=1,...,6.$

$=$ the number of untrained employees engaged in production in month i$,$ for i $=1,...,6.$

$=$ the number of untrained employees enrolled in the training program in month i$,$ for i $=1,...,6.$

Then for i $=4,$ assuming that the $5\%$ turnover rate among untrained employees who are currently engaged in production prevails, we would have the following constraints:

$4+1\mathrm{.}24>=125($the demand for employees engaged in production has to be satisfied$)$

$4=3+0\mathrm{.}93($the trained employees in month $4$ consist of those in month $3$ plus those who successfully complete their training in month $3)$

$4+4=0\mathrm{.}953+0\mathrm{.}13($the untrained employees in month $4$ consist of those who remain from month $3($less the $5\%$ who leave the company every month$)$ plus those who did not successfully complete the training program in month $3).$