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 y1 be number of hours the Norton incinerator operates and y2 be the number of hours the Wiseburg incinerator operates.
Objective Function: C = 80 y1 + 50 y2
Constraints: 10 y1 + 5 y2 > 70 (trash burning requirement)
6 y1 + 4 y2 > 48 (electricity generation requirement)
y1 > 0, y2 > 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 x1 + 48 x2
Constraints: 10 x1 + 6 x2 < 80
5 x1 + 4 x2 < 50
x1 > 0, x2 > 0