(“A Gentle Introduction to Optimization”, by B. Guenin, J. K ̈onemann and L. Tun ̧cel, Cam- bridge University Press, 2014) A chemical plant produces a noxious byproduct, called Chem- ical X, that is highly toxic and needs to be disposed of properly. The chemical plant is connected by a pipe system to a recycling plant that can safely process Chemical X. The amount of Chemical X produced in each hour of the day, according to a standard day’s pro- duction schedule, is shown in Table 1 (Chemical X is not produced in any hour omitted in the table). Table 1: Amount of Chemical X produced during each of the day. Time interval 9-10am 10-11am 11am-12pm 12-1pm 1-2pm 2-3pm
(“A Gentle Introduction to Optimization”, by B. Guenin, J. K ̈onemann and L. Tun ̧cel, Cam-
bridge University Press, 2014) A chemical plant produces a noxious byproduct, called Chem-
ical X, that is highly toxic and needs to be disposed of properly. The chemical plant is
connected by a pipe system to a recycling plant that can safely process Chemical X. The
amount of Chemical X produced in each hour of the day, according to a standard day’s pro-
duction schedule, is shown in Table 1 (Chemical X is not produced in any hour omitted in
the table).
Table 1: Amount of Chemical X produced during each of the day.
Time interval 9-10am 10-11am 11am-12pm 12-1pm 1-2pm 2-3pm
Chemical X (litres) 300 240 600 200 300 900
The chemical plant has a storage capacity of 1000 litres for Chemical X, and, for environmental
safety reasons, no Chemical X can be kept, unprocessed, overnight at the chemical plant.
The cost for the recycling plant to process Chemical X varies throughout the day, as given in
Table 2.
Table 2: Price for processing Chemical X at the recycling plant, at each hour of the day.
Time 10am 11am 12pm 1pm 2pm 3pm
Price ($ per litre) 30 40 35 45 38 50
Formulate an LP model to assist the chemical plant manager to minimize the cost of safely
disposing of Chemical X. Write out your full model in words and then using the problem
data.
PLEASE provide steps and comments explaining how you worked towards the answer
The LP model for this problem can be formulated as follows:
Step 1: Define the decision variables Let x_i be the amount of Chemical X processed at the recycling plant during the i-th hour.
Step 2: Define the objective function Minimize the cost of processing Chemical X, which is given by the sum of the product of the price per litre and the amount of Chemical X processed during each hour: minimize 30x_10 + 40x_{11} + 35x_{12} + 45x_{1} + 38x_{2} + 50x_{3}
Step 3: Define the constraints
-
The amount of Chemical X produced in each hour must be equal to or less than the amount processed at the recycling plant, since no Chemical X can be kept unprocessed overnight: 300x_10 <= 300 240x_{11} <= 240 600x_{12} <= 600 200x_1 <= 200 300x_2 <= 300 900x_3 <= 900
-
The amount of Chemical X processed at the recycling plant cannot exceed the storage capacity of 1000 litres: x_10 + x_{11} + x_{12} + x_1 + x_2 + x_3 <= 1000
Step 4: Write the LP model in mathematical form minimize 30x_10 + 40x_{11} + 35x_{12} + 45x_{1} + 38x_{2} + 50x_{3} subject to 300x_10 <= 300 240x_{11} <= 240 600x_{12} <= 600 200x_1 <= 200 300x_2 <= 300 900x_3 <= 900 x_10 + x_{11} + x_{12} + x_1 + x_2 + x_3 <= 1000 x_10, x_{11}, x_{12}, x_1, x_2, x_3 >= 0
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
