This problem requires you to set up a standard minimization problem like you did in Section 4.4.

**Problem 7** The Marshall County trash incinerator in Norton burns 10 tons of trash per hour and co-generates 6 kilowatts of electricity, while the Wiseburg incinerator burns 5 tons per hour and co-generates 4 kilowatts of electricity. The county needs to burn at least 70 tons of trash and co-generate at least 48 kilowatts of electricity every day. If the Norton incinerator costs $80 per hour to operate and the Wiseburg incinerator costs $50 per hour.

a. Set up the linear programming problem (but do not solve) to determine how many hours the incinerators should operate each day with the least cost.

**Solution** Let *y*_{1} be number of hours the Norton incinerator operates and *y*_{2} be the number of hours the Wiseburg incinerator operates.

Objective Function: *C* = 80 *y*_{1} + 50 *y*_{2}

Constraints: 10 *y*_{1} + 5 *y*_{2} __>__ 70 (trash burning requirement)

6 *y*_{1} + 4 *y*_{2} __>__ 48 (electricity generation requirement)

* y*_{1} __>__ 0, *y*_{2} __>__ 0

b. Construct the dual problem to the linear programming problem in a. Make sure you find both the objective function and all constraints.

**Solution **Write out the coefficients of the standard linear programming problem and write out the transpose by interchanging the rows and columns.

Now reconstruct the corresponding standard minimization problem.

Objective Function: *z* = 70 *x*_{1} + 48 *x*_{2}

Constraints: 10 *x*_{1} + 6 *x*_{2} __<__ 80

5 *x*_{1} + 4 *x*_{2} __<__ 50

*x*_{1} __>__ 0, *x*_{2} __>__ 0