Question

Workco must have the following number of workers available during the next three months: month 1, 20; month 2, 16; month 3, 25. At the beginning of month 1, Workco has no workers. It costs Workco $100 to hire a worker and $50 to fire a worker. Each worker is paid a salary of $140/month. Formulate the problem of determining a hiring and firing strategy that minimizes the total cost incurred during the next three months as a MCNF problem.

Answer 1

Let us assume that Workco does not have the option to have less number of workers than 20, 16 & 25 in the first, second and third months respectively.

Month 1 : Since there are zero workers at the start, the cost of First month can easily be calculated as cost of hiring 20 workers and salary of 20 workers i.e (20*100)+(20*140) = $4800

Month 2 : Since number of workers required is less than month 1, workco has the option of firing maximum of 4 workers to bring down number of workers to 16. The decision of whether to fire the workers depend on 2 factors :

a) Cost of firing + Cost of hiring - If we see, cost of firing + hiring is ($50 + $100) = $150 which is greater than the salary of worker of $140/month

b) Number of workers required in 3rd Month - It is important to note that the number of workers needed in 3rd month is the highest i.e more than the 1st month also which effectively means that we will have to rehire all the workers fired in month two and then hire 5 more workers (Difference between number of workers in 1st and 3rd month).

Since cost of hiring + firing is more than the salary, it makes sense to not fire any worker in month 2.

Considering the above points, cost in month two is 20*140 = $2800

Month 3 : Nom of workers at the end of month two is 20 and requirement is 25. So Workco will have to hire 5 more workers costing him (5*100)=$500 and then pay salary to 25 workers (25*140)= $3500 taking the total cost in month 3 to $500 + $3500 = $4000

So, total least possibe cost in three months is $4800 + $2800 + $4000 = $11600