Question

Consider a wastewater treatment plant in which it is possible to include five different treatment processes in series. These treatment processes must together remove at least 90% of the 100 units of influent waste. Assuming that R

i
​
is the amount of waste removed by process i, the following conditions must hold: 20≤R
1
​
≤30,0≤R
2
​
≤30,0≤R
3
​
≤10,0≤R
4
​
≤20,0≤R
5
​
≤30 The costs of the various discrete sizes of each unit process i are dependent upon the waste entering the process i as well as the amount of waste removed, as indicated in the Table. Write an optimization model (objective function and constraints) for finding the least-cost combinations of removals R
1
​
that will remove 90% of the influent waste. Solve the problem using dynamic programming Show that this problem could be described by a network in which the nodes are the state variables (the influent quantity I
1
​
) at each stage (treatment process i ) and the links are the decision variables (the quantities of waste removed R
1
​
) and the associated costs, C
1
​
(I
1
​
,R
1
​
). Locate the least-cost path on the network. (Adapted from problem 2.18, Loucks, Haith and Stedinger, Water Resource Systems Planning and Analysis)

Answer 1

An optimization model can be used to find the least-cost combinations of removals in a wastewater treatment plant using dynamic programming.

Step-by-step explanation:

The optimization model for finding the least-cost combination of removals can be solved using dynamic programming. The objective function is to minimize the total cost of waste removal, while the constraints are that the total amount of waste removed should be at least 90 units and the individual removal amounts for each process should satisfy the given ranges.

To solve this problem using dynamic programming, we can create a network where the nodes represent the state variables (influent quantity at each stage) and the links represent the decision variables (quantities of waste removed) and their associated costs. Each node represents a treatment process, and the links connect the nodes to show the flow of waste and associated costs.

We can use dynamic programming to find the least-cost path on this network by considering all possible combinations of waste removals and calculating the total cost for each combination. By systematically iterating through the treatment processes and considering the costs of all possible removal amounts, we can find the path with the minimum cost.