# [Math] the objective function and constraints of this problem

linear programming

How to solve this problem? This is what I know so far.

Let A be the no. of drivers at the beginning of the year.
Let B be the no. of drivers fired.
Let C be the no. of drivers recruited

Is the objective function 12000A + 2000B + 1000C

Am I on the correct track? What is next?

SBS bus company just won the tender to run premium bus services from the east of New York. The company predicts that they need to have at least the following numbers of bus drivers in coming 5 years: Year 1: 100 drivers; Year 2: 90 drivers; Year 3: 110 drivers; Year 4: 120 drivers and Year 5: 90 drivers.

Recruiting one driver costs the company 1,000 USD which includes pre-deployment training. Firing one driver costs the company 2,000 USD including the severance pay. Each driver on payroll costs the company 12,000 USD per year. The company currently has 60 bus drivers on payroll. Apply linear programming model to develop a formulation to minimize the total recruiting, firing and payroll cost for the bus company. List assumptions you made in the formulation. Note that you are not required to solve the problem.

It´s not enough. You have to regard each period (year). The most important thing in (linear) programming is to define the variables:

$h_i$: Number of drivers, who are hired at the beginning of year $i \in \{1,2,3,4,5\}$

$f_i$: Number of drivers, who are fired at the end of year $i-1 \in \{1,2,3,4,5\}$

$d_i=h_i-f_i:$ Difference of hired and fired drivers in year $i \in \{1,2,3,4,5\}$

$d_1=60$ is the initial number of drivers.

The constraints for the minimum of drivers in year $1$ is

$$\sum\limits_{i=1}^{1}d_i\geq 100$$

The constraints for the minimum of drivers in year $2$ is

$$\sum\limits_{i=1}^{2}d_i\geq 90$$

And so on.

And the objective function is

$$\texttt{min}\quad 1000\cdot \sum_{i=1}^5 h_i+2000\cdot \sum_{i=1}^5 f_i+12000\cdot( \sum_{k=1}^5 \sum_{i=1}^k d_i)$$

$h_i,f_i \in \mathbb N$

If you have any questions feel free to ask.