Assignment # 10 (2)

Article Reference:
Tze Chin Pan, Jehng Jung Kao, GA-QP Model to Optimize Sewer System Design, January 2009, Journal of Environmental Engineering

Summary:
The paper explains the importance of optimizing the sewer network (Pipe diameter, pipe slopes, pipe buried depths etc) in reducing the total cost of establishing the sewer system. Optimizing sewer network is difficult because of few typical constraints associated in the design. Typical constraints are – maintaining minimum velocity (for self cleansing), preventing maximum velocity (which causes scouring), upstream elevation greater than the downstream elevation giving sufficient slope for the flow, designing the depth to accommodate the flow of the design capacity, commercially available pipe diameters, and ensuring the diameter of the downstream pipes to be greater or equal than the upstream pipes thereby letting the downstream pipes to carry cumulative flow. Many approaches have been developed to design sewer network. One such approach was using discrete differential dynamic programming (DDDP), but this approach restricts the search space which reduces the opportunities locate global optimum. Pan and Kao used combined GA and QP as QP is a better way of representing a non linear cost function than LP where you need to convert the nonlinear functions to linear by piecewise linearization. As few of the design factors such as geology, traffic impact, people’s preference, land availability etc are not considered in the study as they are difficult to model. Authors mention that the solution obtained from this model would not be feasible when un-modeled factors are evaluated. This paper explains modeling a different Modeling for Generating Alternative (MGA) function to evaluate the difference between the results obtained from GA-QP and DDDP models.

GA Model components-
Genes: Pipe and pumping station locations in binary code
Chromosomes: Pipe diameters and pumping locations which represent a design layout of the sewer system.
Fitness function: Reciprocal of the cost function (1/Cost of chromosome)
Selection Process: Based on greater the area on the wheel higher the probability being chosen (Simpson et.al. 1994)
Crossover: Random mother chromosomes binary (gene) information is exchanged to generate two child chromosomes.
Mutation: Changes the binary information randomly

The values of the genes are randomly produced. All the unacceptable chromosomes are discarded than repairing as it is too complex, thus producing new set of chromosomes. The two major constraints in deciding the acceptability of the chromosomes are – pipe having required flow capacity (i.e dia of pipe should be capable to carry the flow) and maintain flow continuity (i.e d/s dia is greater than the u/s). Fitness function used in the GA model to evaluate the fitness of the chromosomes generated is defined as the reciprocal of the cost function. Therefore chromosomes with high fitness value are superior to the chromosome with low fitness value. Selection, mutation and crossover are used to create new set of chromosomes. As the cost decreases, the fitness value increases thereby occupying greater area on the wheel which increases the probability of being selected for crossover. The Crossover process exchanges pipe diameters and pumping locations of randomly generated chromosomes to generated new set of chromosomes. Mutation randomly changes the binary information in the chromosomes so that they are not stuck with the local optimum.

Quadratic Programming-
This study has used QP to evaluate the fitness value of the chromosome. The objective function includes – construction cost of pipe based on the diameter, depth and the length, construction cost of manhole and construction cost of pumping station. The authors have constrained the flow velocities and depth by limiting the pipe slope. All the sewer pipes need to be buried at a specific depth from the ground so as to be able to connect the household waste flow. Therefore the decision variables associated in this model are pipe slopes and the buried depths of the downstream end of the pipes.

MGA identifies maximally different solutions which could still be regarded as good and different alternatives.

Discussion:
The authors have explained in detail how each component of GA is taken into consideration. Implementation of GA in the real time practical models is clearer to me after reading this paper. It is true as mentioned by Author about the solution not being feasible without taking into account few factors like land availability, geology etc, which is considered to be of major component in the designing procedure. This paper carries out a practical case explanation of importance of having generating alternative solutions as discussed in the earlier article by Brill.The comparison of different results/solutions DDDP, GA, MGA1, and MGA2 clearly explains the importance of having a feasible alternative which could be considered when all the non considered factors are evaluated into these solutions. Every individual sewer system has specific concern which could not be modeled mathematically into the model, this approach of generating alternatives would be helpful to see and evaluate the feasibility of these factors along with the optimization solution obtained from the GA model. This would be more of a practical approach than just optimizing without considering these factors which are vital in the decisions of these systems.