2nd International Conference on Engineering Optimization September 6 - 9, 2010, Lisbon, Portugal
Waste Load Allocation Using Non-dominated Archiving Multi-colony Ant Algorithm Seyyed Asghar Mostafavi1 and Abbas Afshar2 1
Research Assistant, Civil Engineering Department, Iran University of Science and Technology, Tehran, Iran,
[email protected] 2 Professor, Civil Engineering Department, Iran University of Science and Technology, Tehran, Iran,
[email protected]
Abstract The optimal Waste Load Allocation (WLA) problem, in general, is a mathematical model incorporating a water quality simulator within the framework of multi objective optimization. The goals are defined as to improve the water quality and minimize the total treatment cost of dischargers. The fraction removal levels of the dischargers are the decision variables. This paper explores the capabilities of Ant Colony algorithms to solve the optimal Waste Load Allocation as a multi-objective optimization problem. We use QUAL2K as a water quality simulator and a powerful ant colony algorithm known as Non-dominated Archiving Multi-colony Ant Algorithm (NA-ACO). The applicability of the model is demonstrated through a hypothetical case example. Results' location near global optima demonstrates the suitable performance of applied algorithm. Keywords: Waste load allocation, Water pollution, Multi Objective Optimization, Ant Colony Optimization. 1. Introduction Rivers are the most important sources supplying water for municipal, agricultural and industrial uses. Considerable increasing of the waste discharge leads to increasing pollution of this water body. The ability of river to clean itself due to its assimilative capacity makes necessary determination of waste water treatment levels for the polluters. Waste Load Allocation (WLA) as a well known water quality control strategy is used to determine the optimal pollutant removal at a number of point sources along the river. In fact, the WLA model proposed the discharge levels to achieve the pollution control agency and other stakeholders’ goals. This model, in general, consists of a quality simulator to model the pollutant transport in river system and an optimization model to find the optimal removal fraction levels of the dischargers. The waste load allocation model can be formulated as a multi objective optimization model considering conflicting goals. Typical multi objective waste load allocation problems address, for example, minimization of the total treatment cost and also the inequity among the polluters, subject to constraints on satisfaction of a specified Dissolved Oxygen (DO) standard at all of the check points along river (Brill et al., 1984; Burn and Yulianti, 2001; Yandamuri et al., 2006). Moreover, in some previous proposed models, the violation from the standards at the checkpoints is considered as an objective of the WLA model (Yandamuri et al., 2006; Saadatpour and Afshar, 2007). Recently, Metaheuristic algorithms such as Genetic Algorithm (GA), Simulated Annealing (SA) and Ant Colony (AC) algorithms have been employed to deal with multi criteria problems. For example, Burn and Yulianti (2001), Yandamuri et al. (2006), Saadatpur and Afshar (2007) used genetic algorithm to solve the multi objective WLA problems. Ant Colony Optimization (ACO) algorithms are the most successful and widely recognized algorithmic techniques based on real ant behaviors (Dorigo and Stutzle, 2004). Multi-objective Ant Colony Optimization algorithms have recently been applied to solve multi-objective problems (Mariano and Morales, 1999; Gambardella et al., 1999; Doerner, et. al., 2001, 2003; Baran and Schaerer, 2003; Doerner et al., 2004). present work focuses on the application of powerful and recently developed multi objective ACO technique known as “Non-dominated Archiving ACO” (NA-ACO) proposed by Afshar et al. (2008) to solve the waste load allocation problem. The goals are defined as to minimize the total treatment costs and also the violation index. The index is a weighted sum of three components that reflects DO violation characteristics: number of DO violations, maximum and total DO violations at the checkpoints along the river. We use QUAL2K as a simulator to predict the water quality response at specified checkpoints, for various possible combinations of discharging. From the trade-off curve (surfaces) generated, the decision makers can select the appropriate treatment strategy under consideration.
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2. Violation index A violation index for the water quality of a system is proposed in this study. The proposed index is expressed as a weighted sum of three individual ratios including the number of violations, magnitude of maximum violation and magnitude of total violations from specified DO standard over all the checkpoints along the river. The three ratios are expressed as follows:
Na N0 M IM = a M0 T IT = a T0 IN =
(1) (2) (3)
Where I N , I M and I T was the violation indices in terms of Number of violations, Maximum violation and Total violations, respectively. Subscripts 0 and a indicate the level of treatment corresponding to no treatment and actual treatment, respectively. As mentioned above, the overall violation index of a waste load allocation policy ( I WLA ) is expressed as a weighted sum of three violation indices, I N , I M , and I T , already defined. That is
IWLA= (wNIN + wMIM + wT IT )
(4)
Where wN , wM and wT was weights associated with the violation indices corresponding to the number of violations, the magnitude of maximum violation, and the magnitude of total violations of the DO standard, respectively. These weights are to be estimated by the decision maker for the particular case being studied. 3. Model Formulation The proposed cost-violation waste load allocation model developed to minimize the total treatment cost and also the overall violation index. The model identifies the trade-off relationship between the wastewater treatment cost and the overall DO standard violation in the river system. The decision variables, the fraction removal, are selected from a set of possible removal level ( xs j ) available for any polluters. The model consists of NS dischargers, and the DO standard violations will be checked in NR checkpoints along the river. This model formulated as: NS
Minimize : ∑ C j ( x j )
(5)
j =1
NR
Minimize : ∑ IWLA
(6)
x j ∈ xs j
(7)
r =1
Subject to:
Where Cj (xj ) is the cost of treatment at source j,
x j is the removal fraction for source j and r is the number of
checkpoints. In the optimization model, at each iteration, the related treatment cost is calculated after selecting the removal fractions for each discharger from their possible options. The water quality simulator used to evaluate the water quality response and the magnitude of overall violation index ( I WLA ) at checkpoint r. at the end, a set of optimal solutions known as non-dominated solutions which enables Decision maker to select a solution from the optimal discharge scenarios under considerations. 4. Non-dominated Archiving Ant Colony Optimization Algorithm As mentioned previously, the optimization algorithm that we use in this paper to solve the multi objective WLA problem is the Non-dominated Archiving ACO (NA-ACO) which is proposed by Afshar et al., (2008). By considering the general framework of ACO algorithms, the present multi objective optimization algorithm was developed based on archiving the nondominated solutions. In this algorithm, a colony of agents with same population is assigned for each objective of the problem. The ants in each colony simultaneously explore a solution according to the colony’s objective. The solutions constructed in one colony are not evaluated in the same colony. Then, these trial solutions are transferred to the next colony to be evaluated according to the objective of that colony. The new solutions found on the basis of the updated pheromone trail in the second colony are transferred to the first colony (in the case of two-objective problem). This cycle is repeated for a predefined number of iterations
2
F2
known as Cycle Iteration. At the end of any cycle iteration, the non-dominated solutions among all the solutions transferred to an offline archive for further pheromone updating. Then, the pheromone levels of both colonies are updated according to the non-dominated solutions in the archive and the algorithms return to the starting point. This process continues up to a predefined number of iterations (total iterations) or a desired number of archived Pareto solutions. At the end of running this algorithm, the present nondominated solutions in the archive are the optimal solutions of the multi objective problem. More detail about this algorithm can be found in Afshar et al. (2008) study. In order to demonstrate the performance of this algorithm, two well known unconstrained convex and non-convex multi objective optimization problems, ZDT1 and ZDT2 (introduced by Zitzler et al. 2000) are used. Since the actual solutions of these problems are known (Zitzler et al. 2000), testing the algorithm performance is easy. Figures 1 and 2 illustrate the results of applying NA-ACO algorithm to these test problems. The curves show the Pareto-optimal fronts (Actual solutions). Moreover, Afshar et al. (2008) compared the NA-ACO algorithm with previously developed algorithms (such as SPEA, PAES and NSGA-II algorithms). 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
Actual solutions
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
NA-ACO solutions
0.8
0.9
1.0
F1
F2
Figure 1: ZDT1 Pareto front
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
Actual solutions
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
NA-ACO solutions
0.8
0.9
1.0
F1
Figure 1: ZDT2 Pareto front 5. Water Quality Simulator One of the basic components of each WLA problem is the water quality simulation model. In fact, the quality simulator used to predict the water quality response in terms of DO concentration at specified checkpoint along the river, for various combination of waste discharge by polluters. Most of WLA models have employed the Streeter-Phelps equation (Streeter and Phelps 1925; Burn and McBean 1985; Vasquez et al. 2000) as a model to simulate water quality in river systems. This model can estimate the variation of Biochemical Oxygen Demand (BOD) and DO along the river but no for other constituent such as nitrogen, phosphorus and etc. Nowadays, some simulation models like QUAL2E, QUAL2K, and WASP4 are available to simulate the transport and fate of a number of quality constituents (such as temperature, ph, carbonaceous BOD, DO, phytoplankton and several forms of the nutrients phosphorus and nitrogen.) in the river systems. In this paper, the QUAL2K (or Q2K), a river and stream quality simulator that is an improved version of the QUAL2E (or Q2E) model (Brown and Barnwell 1987) is used to predict the effect of BOD discharge on water quality indicator (dissolved oxygen concentration). Detailed descriptions and applications of this model are available in the literature (Park and lee, 2002; Chapra and Pelletier, 2003). The point that must be mentioned here is that, in each generation of new solution, the input data for Q2K model was constant and only the amount of
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waste load was varied according to the selected discharge strategy. 6. Model Application The performance of the proposed multi objective waste load allocation model is examined through a hypothetical river system. For this case study, there are 4 polluters that discharge BOD load along the river. The discharge locations and effluent characteristics are shown in Table 1. Discharging is assumed to be located at the beginning of the reaches. The initial background concentrations of BOD and DO in the main river are assumed to be 3 and 8 mg/l, respectively. For all point sources, the DO concentration is assumed to be 1.0 mg/l and the upstream flow is 6 m3/s. 16 checkpoints are considered along 16 km of the river system, and the O’Connor and Dobbins (1958) equation is used to estimate the reaeration coefficient. The model is run for two DO standards of 6.0 and 6.5 mg/l. The possible range for the removal fraction, for each discharger, is from 0.35 to 0.98 with four options that illustrated with related cost data in Table 2. The non-dominated solutions of the WLA model using NA-ACO approach was obtained when the number of ants in each colony, number of cycle iterations, number of total iterations, and pheromone evaporation rate were 10,10,100, and 0.9, respectively. The value of α and β parameters (see Afshar et al., 2008) were assumed 1 and 0, respectively. Table 1: Effluent Data for the hypothetical river. Point source Discharger 1
Location (km) 16
Flow (m3/l) 0.2
BOD (mg/l) 280
Discharger 2
12
0.45
430
Discharger 3
8
0.15
260
Discharger 4
4
0.3
360
Table 2: Treatment Cost Data. Point source Discharger 1
35% 0.47
$ million/year 67% 1.25
90% 1.67
98% 1.97
Discharger 2
0.92
1.78
2.38
3.2
Discharger 3
0.75
1.53
1.93
2.35
Discharger 4
1.12
2.2
2.82
3.56
Table 3: Cost-Violation Trade-offs DOstd (mg/l) 6.0
6.5
index LC
Solution point (TC, VI) 3.26, 0.608
Removal fraction levels (x1, …, x4) 0.35, 0.35, 0.35, 0.35
MP
4.72, 0.209
0.35, 0.90, 0.35, 0.35
MV
6.74, 0.00
0.90, 0.98, 0.35, 0.35
LC
3.26, 0.678
0.35, 0.35, 0.35, 0.35
MP
5.92, 0.175
0.90, 0.90, 0.35, 0.35
MV
9.3, 0.00
0.98, 0.98, 0.90, 0.67
Note: TC= Total Cost (million $); VI= Violation Index; LC= Least Cost solution; MP= Middle Point; MV= Minimum Violation index.
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0.7 LC
Violation Index
0.6 0.5 0.4 0.3
MP
0.2 0.1
MV
0 3
4
5
6
7
Treatment Cost ($ million)
Figure 1: Cost-violation trade-off (DO=6.0 mg/l) 0.8
Violation Index
0.7
LC
0.6 0.5 0.4 0.3 MP
0.2 0.1 0
MV 3
4
5
6
7
8
9
10
Treatment Cost ($ million)
Figure 1: Cost-violation trade-off (DO=6.5 mg/l) 6. Result of the cost-violation model The cost-violation model examines waste load allocations that vary from the minimum violations (minimum violation index) and corresponding considerably large treatment cost to the solution that results in large violations with a lower cost. Figures 2 and 3 are exhibited the Pareto optimal front for two scenarios of specified DO standards of 6.0 and 6.5 mg/l. Three points (as potential solutions for decision making process) of the solutions are labeled in these figures for further examining the curve. The LC and MV solutions refer to least cost and minimum violation index solutions, respectively. Results of these two solutions with a Middle Point (MP) solution are summarized in Table 3, for two scenarios. From this table, understand that decreasing the violation index (or water quality improvement) lead to increasing the total treatment cost. For example, for the DO standard of 6.0 mg/l, the treatment cost for LC and MV solutions are 3.26 And 6.74 Million dollar and the magnitude of violation index for these points are 0.608 and 0.0, respectively. Moreover, the difference in treatment cost between LC and MV solutions increases from 3.48 to 6.04 million dollar as the DO standard varies from 6.0 to 6.5 mg/l, respectively. Decision maker will be able to select a desirable WLA solution (discharge strategy) from the trade-off curve obtained by the multi-objective model based upon the budget constraint and the acceptable violation index. 7. Conclusions and Recommendations A multi-objective optimization model has been developed to solve the Waste Load Allocation problem considering the total treatment cost and violation from water quality standards as its objectives. The proposed violation index was a weighted sum of three components that reflected to the dissolved oxygen (DO) violation characteristics consist of: number of DO violations, magnitude of maximum DO violation and total magnitude of DO violations at the checkpoints along the river. We use a popular water quality simulator, QUAL2K to predict the water quality response for any combination of discharging the pollutants. The multi-objective problem was solved using a powerful and recently ACO algorithm known as “Non-dominated Archiving Ant Colony Optimization” proposed by Afshar et al. (2008). High efficiency of this algorithm to find most of the non-dominated solutions was tested by two well known unconstrained convex and non-convex multi objective mathematical problems. Future work can explore the application of “NA-ACO” algorithms to the WLA problem considering more than two objectives or considering more than one water quality indicator. In this study, only discharging BOD load considered. In natural systems, a set of different pollutant discharge to the river and the interactions among pollutants should be considered. Therefore, employing a simulator model which considers several pollutants
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discharge and the interactions among pollutants is necessary. The Q2K model can simulate discharging several pollutants along river systems.
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