Water Treatment

You work for a regional water quality authority and must suggest how much wastewater treatment must be performed by each of two cities (Lundville and Bellido City) on the Cohen River. Each city discharges a certain flow into the river with a given concentration of a pollutant. The pollutant then mixes with the water already in the river and decays biologically at a known rate while it is in the river. At Lundville, the concentration of pollutants must be below 3 units per mgd. At Bellido City, the concentration of pollutants must be below 4 units per mgd.

Lundville's wastewater stream would add 145 units of pollution daily and the cleaner Bellido City would add 130 units daily. The cost of removing a unit of pollutants from Lundville's wastewater stream is $1900 and for Bellido City, the unit removal cost from the wastewater stream is $1850. The flow in the Cohen River is 6 mgd at Lundville (Q1). A tributary (Q2) adds 2 mgd between Lundville and Bellido City. Thirty percent of the pollutants entering the waste stream at Lundville decay biologically before arriving at Bellido City.

a) Formulate the problem in terms of the objective function and constraints and solve using the Lagrange Multipliers method by hand (For simplicity, it is okay to assume non-negativity constraints as non-binding).

b) How many units of pollutants should be removed by each city to meet the water quality requirement and keep the Cohen River clean?

c) What are Lagrange Multiplier values for constraints?