Question $2)(25$ points$)$ Consider the following ride matching problem where at a given time of the day in a region, we have $3$ customers and $4$ drivers. We want to match drivers to customers to maximize the total profit of matching. The profit of matching each customer and each driver is given in the table below.

$\backslash$table$[[,$Customer A$,$Customer B$,$Customer C$,,,],[$Driver $1,$$$,20,$$$,22,$$$,24],[$Driver $2,$$$,21,$$$,20,$$$,30],[$Driver $3,$$$,27,$$$,32,$$$,29],[$Driver $4,$$$,37,$$$,38,$$$,21]]$

A driver can be assigned to at most one customer, and a customer can be assigned to at most one driver.

a$)$ Formulate a mathematical optimization model to maximize the total profit for this problem. Clearly specify decision variables, objective function, and constraints.

Answer:

Notation and definition of the decision variables:

Objective function:

Constraints:

b$)$ Use Excel solver to solve this problem. What is the maximum total profit? Which Driver is not assigned to any customer?

Answer:

Maximum total profit:

Driver that is not assigned: