Research Article Wastewater Treatment Optimization for...

Research ArticleWastewater Treatment Optimization forFish Migration Using Harmony Search

Zong Woo Geem1 and Jin-Hong Kim2

1Department of Energy amp Information Technology Gachon University Seongnam 461-701 Republic of Korea2Department of Civil amp Environmental Engineering Chung-Ang University Seoul 156-756 Republic of Korea

Correspondence should be addressed to Jin-Hong Kim jinhongkimcaugmailcom

Received 3 October 2014 Accepted 20 November 2014 Published 4 December 2014

Academic Editor Youqing Wang

Copyright copy 2014 Z W Geem and J-H Kim This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Certain types of fish migrate between the sea and fresh water to spawn In order for them to swim without any breathing problemriver should contain enough oxygen If fish is passing along the river in municipal area it needs sufficient dissolved oxygen levelwhich is influenced by dumped amount of wastewater into the river If existing treatment methods such as settling and biologicaloxidation are not enough we have to consider additional treatment methods such as microscreening filtration and nitrificationThis study constructed a wastewater treatment optimization model for migratory fish which considers three costs (filtration costnitrification cost and irrigation cost) and two environmental constraints (minimal dissolved oxygen level and maximal nitrate-nitrogen concentration) Results show that the metaheuristic technique such as harmony search could find good solutions robustlywhile calculus-based technique such as generalized reduced gradient method was trapped in local optima or even divergent

1 Introduction

Various species of fish migrate on time periods ranging fromdaily to annually and over long distances up to thousands ofkilometers [1] Fish normallymigrate because of reproductiveor diet purposes

In order for anadromous fishes such as salmon whichmigrate from the sea into fresh water to spawn to swimupstream without any breathing problem the river shouldcontain enough oxygen or the level of dissolved oxygen (DO)should be more than certain criterion

To control the DO level in a river wastewater dumpedinto the river should be well treated Actually there are severalwastewater treatment techniques such as settling biologicaloxidation microscreening filtration and nitrification [2]

This study intends to find the optimal wastewater treat-ment portfolio which suggests overall minimal cost whilesatisfying minimal DO level over the river reach in order formigratory fishes to swim upstream well

Actually optimal DO control issue for water quality whilelimiting wastewater load has been researched for variousrivers all over the world such asWillamette River in Oregon

USA [3] Schuylkill River in Pennsylvania USA [4] NitraRiver in Slovakia [5] Yasu River in Japan [6] and YamunaRiver in India [7] However real-world problems sometimesrequire model simplification [8] or experience local optimaentrapment [2] This study tries to adopt a more realisticmodel considering nonlinearity of the problem and to findgood results without being entrapped in premature solutions

2 Optimization Formulation

The basic structure of wastewater treatment model in thisstudy came from Haith [2] and can be visualized in Figure 1

A city (population = 100000) dumps wastewater into ariver The wastewater (40000m3day) undergoes secondarytreatment which consists of settling and biological oxidationto remove organic material from the wastewater Howeverthis treatment is not enough to satisfy the water qualitystandard for migratory fishes An environmental regulatoryagency set this standard that the outflow of wastewater treat-ment should be more than 5mgliter (or 5 ppm) of DO insummertime to preserve aquatic life Therefore the city is

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 313157 5 pageshttpdxdoiorg1011552014313157

2 Mathematical Problems in Engineering

Boundary40000m3day

Secondary

Filtration

Boundary

Aquifer

River

Percolation

110000m3day

Nitrification

Q2 + Q4Q3 minus Q4

Q4

Q2

Q1

Q3

Q5

Storage

Irrigation A r

Figure 1 Schematic diagram of wastewater treatment portfolio

now forced to consider additional treatment processes thatenhance the water quality

Because the secondary treatment (settling and biologicaloxidation) is not enough the city should consider extra pro-cesses such as filtration and nitrification as shown in Figure 1Also certain amount of wastewater can be diverted to reducethe river discharge This diverted one can be used for cropirrigation

So we can consider total five decision variables for thiswastewater treatment optimization as follows

(1) 1198761 wastewater dumped without additional treatment

(unit 103m3day)(2) 1198762 wastewater undergoing additional filtration (unit

103m3day)(3) 1198763 wastewater undergoing additional nitrification

(unit 103m3day)(4) 1198764 wastewater undergoing additional nitrification

and filtration (unit 103m3day)(5) 1198765 wastewater diverted for irrigation (unit

103m3day)

Also there is onemore decision variable 119903which denotes irri-gation rate (cmweek) and the relationship between 119876

5and

119903 is 119903 = 701198765119860 Here119860 (hectare or 104m2) denotes irrigated

area and 70 is unit conversion factorWith the above decision variables we can construct the

objective function as follows

Minimize 119911 = 119891 (1198761 1198762 1198763 1198764 1198765 119903) = 119862ft + 119862nt + 119862ir

(1)

Here the cost for filtration 119862ft ($103yr) is represented asfollows

119862ft = 3 (1198762 + 1198764)093

+ 67 (1198762+ 1198764)055

(2)

where the first term in right-hand side of (2) represents thecapital cost of filtration treatment and the second termrepresents operation and maintenance cost

The cost for nitrification 119862nt ($103yr) is represented as

follows

119862nt = 1381198763068

+ 1061198763

042 (3)

where the first term in right-hand side of (3) represents thecapital cost of nitrification treatment and the second termrepresents operation and maintenance cost

The cost for irrigation 119862ir ($103yr) is represented asfollows

119862ir = 219119876028

5+ 12119876

078

5+ 02119876

054

5

+ (131 +48

119903)119876(074+032119903)

5

+ (51 +19

119903)119876(079+028119903)

5minus 068119860

(4)

where the first term in right-hand side of (4) represents thecapital cost of transmission line the second term representsthe capital cost of storage system (lagoon) the third term rep-resents the operation andmaintenance cost of storage systemthe fourth term represents the capital cost of irrigation sys-tem the fifth term represents the operation and maintenancecost of irrigation system and the sixth term represents thenet benefit (negative cost) of cropping (crop sales minus landrent)

Constraints for this optimization problem can be

1198761+ 1198762+ 1198763+ 1198765= 40 (5)

1198763ge 1198764 (6)

0 le 119876119894le 40 119894 = 1 5 (7)

Mathematical Problems in Engineering 3

8 (1 minus 119890minus0063119909

) + 1198620119890minus0063119909

minus 2331198610(119890minus0044119909

minus 119890minus0063119909

)

minus 0671198730(119890minus0025119909

minus 119890minus0063119909

) ge 5

119909 = 5 10 15 20 25 30 40 50

(8)

1198620=8 (110) + 2 (119876

1+ 1198762+ 1198763)

110 + 1198761+ 1198762+ 1198763

(9)

1198610=2 (110) + 25119876

1+ 13119876

2+ 13 (119876

3minus 1198764) + 7119876

4

110 + 1198761+ 1198762+ 1198763

(10)

1198730=5 (110) + 54119876

1+ 50119876

2+ 10 (119876

3minus 1198764) + 10119876

4

110 + 1198761+ 1198762+ 1198763

(11)

654 le 119903 le 1307 (12)

Equations (8) to (11) stand for DO level constraint and (12)stands for nitrate-nitrogen level constraint More detailsabout these constraints are explained in the next section

3 Mathematical Model of Water Quality

Dissolved oxygen 119862(119909) in ppm at a distance (119909 km) down-stream of a point wastewater discharge can be represented asthe following differential equation

119906119889119862

119889119909= 1198962(119862119904minus 119862) minus 119896

1119861 minus 119896119899119873 (13)

where 119906 is river flow velocity (79 kmday in this study) 1198962

is reaeration rate (05day in this study) 119862119904is saturation DO

(80 ppm in this study) 119861 and119873 are remaining carbonaceousbiochemical oxygen demand (CBOD) and nitrogenous bio-chemical oxygen demand (NBOD) (ppm) at distance 119909 and1198961and 119896119899are rate constants (035day and 02day resp) The

first term in the right-hand side of (13) denotes oxygenincrease due to reaeration and the second and third termsdenote oxygen decrease due to oxidation of carbonaceous andnitrogenous material respectively

The above differential equation has the analytic solutionas follows

119862 (119909) = 119862119904(1 minus 119890

minus1198962119909119906) + 1198620119890minus1198962119909119906

minus11986101198961

1198962minus 1198961

(119890minus1198961119909119906minus 119890minus1198962119909119906)

minus1198730119896119899

1198962minus 119896119899


(14)

Equation (14) is identical to (8) where 1198620 1198610 and 119873

0are

respectively river DO CBOD and NBOD right after dis-charge If river water and wastewater are completely mixed atthe discharge point initial 119862

0 1198610 and 119873

0can be calculated

using weighted average as expressed in (9) to (11) where riverflow is 110000m3day river DO is 80 ppm river CBOD is20 ppm and river NBOD is 50 ppm Table 1 shows effluentwater quality after wastewater treatment

Table 1 Effluent water quality after wastewater treatment

Treatment type Effluent quality (ppm)DO CBOD NBOD

Secondary(settling + biological oxidation) 119876

1

2 25 54

Secondary+ filtration (microscreening) 119876

2

2 13 50

Secondary+ nitrification 119876

3minus 1198764

2 13 10

Secondary+ nitrification + filtration 119876

4

2 7 10

For the diverted amount 1198765 we can consider the follow-

ing mass balance equation

Percolation = 119903119879 + 119875 minus ET (15)

where 119879 is irrigation duration (13 weeks in this study) 119875 isprecipitation (cm) during the irrigation season and ET isevapotranspiration (cm) Here 119875 minus ET is zero in this study

A nitrogen balance is used for estimating the nitrogen lossin percolation into groundwater If the nitrogen concentra-tion of119876

5is 119899 (20 ppm in this study) total nitrogen amount to

the irrigation area is 01119903119879119899 (kgha) where 01 is unit conver-sion factor If NC is crop nitrogen uptake (170 kgha in thisstudy) the unused nitrogen of119876

5becomes 01119903119879119899minusNC and

the nitrate-nitrogen concentration 119888119899(ppm) in the percola-

tion becomes as follows

119888119899=nitrogen losspercolation

=119903119879119899 minus 10NC119903119879 + 119875 minus ET

(16)

In this study 119888119899should be less than or equal to 10 ppm for

public health purpose which makes the following constraint

119903119879119899 minus 10NC119903119879 + 119875 minus ET

le 10 (17)

Also the nitrogen contained in1198765should be enough to satisfy

the croprsquos nitrogen requirement as follows

119903119879119899 ge 10NC (18)

From (17) and (18) we can obtain (12) for nitrate-nitrogenconstraint

4 Optimization Using Harmony Search

Thewastewater treatment optimizationmodel for fishmigra-tion constructed in previous sections has a complex structurewhich is difficult to devise a proper search strategy [2] Sothe problem was tackled by commercial software namedMicrosoft Excel Solver which uses the generalized reducedgradient (GRG2) technique [9]

When Solver was applied to the model with five initialsolution vectors it was trapped in local optima or even diver-gent instead of convergent Table 2 shows details

When GRG2 started with the first solution vector (0 0 00 0 0) it was trapped in one of local optima (3349) with


Table 2 Results by generalized reduced gradient technique

Initial vector Final vector1198761

1198762

1198763

1198764

1198765

119903 Cost 1198761

1198762

1198763

1198764

1198765

119903 Cost0 0 0 0 0 0 0 0 0 40 3033 0 654 33490 0 0 0 40 10 3458 0 0 0 0 40 10 34580 0 40 25 0 10 3187 0 0 40 3033 0 10 33498 0 0 0 32 10 3038lowast Divergent10 10 10 5 10 10 3149 Divergent

the final solution vector (0 0 40 3033 0 654) when GRG2started using the second solution vector (0 0 0 0 40 10)with the cost of 3458 it did not move any further from itsstarting point when GRG2 started using the third solutionvector (0 0 40 25 0 10) it was trapped in one of local optima(3349) with the final solution vector (0 0 40 3033 0 10)which is identical to the final solution vector of the first case interms of cost but the value of 119903 is different Because119876

5is zero

different 119903 values do not affect the objective function valuewhen GRG2 started using the fourth solution vector (8 0 00 32 10) it was even divergent and disabled to compute anyfurther Although the fourth vector has a good cost (3038)it also slightly violated minimal DO condition at the distanceof 20 km downstream and whenGRG2 started using the fifthsolution vector (10 10 10 5 10 10) with the cost of 3149 it wasalso divergent

On top of those five trials additional trials with differentstarting vectors have been tried But mostly it reached diver-gence instead of convergence because the model structure iscomplicated Thus a metaheuristic approach instead ofcalculus-based approach using the harmony search algo-rithm was introduced to this study

The harmony search (HS) algorithm was inspired bymusic improvisation [10] and applied to various optimizationproblems [11] It has its own unique human-experience-basedderivative [12] For this DO control problem the followingoptimization formulation for HS was used

Minimize 119911 = 119891 (1198761 1198762 1198763 1198764 1198765 119903)

= 119862ft (1198762 1198764) + 119862nt (1198763) + 119862ir (1198765 119903) (19)

subject to

119862 (119909 1198761 1198762 1198763 1198764) ge 5 ppm

119909 = 5 10 15 20 25 30 40 50(20)

1198761+ 1198762+ 1198763+ 1198765= 40 (21)

1198763ge 1198764 (22)

0 le 119876119894le 40 119894 = 1 5 (23)

654 le 119903 le 1307 (24)

For the above optimization model HS found the optimalsolution using the following steps

Step 1 HS constructs initial memory place named harmonymemory (HM) as in (25) and fills HM with initial solutionvectors as many as harmony memory size (HMS this is 10 inthis study) The initial vectors should satisfy the problemconstraints in (20) to (24)

HM

=

[[[[[

[

1198761

11198761

21198761

31198761

41198761

51199031

119891 (x1)1198762

11198762

21198762

31198762

41198762

51199032

119891 (x2)

119876HMS1

119876HMS2

119876HMS3

119876HMS4

119876HMS5

119903HMS

119891 (xHMS)

]]]]]

]

(25)

Step 2 A new harmony is generated using the followingequation

119909New119894

larr997888

119909119894isin [119909

Lower119894

119909Upper119894

] wp (1 minusHMCR)

119909119894isin HM

= 1199091

119894 1199092

119894 119909

HMS119894

wp HMCR sdot (1 minus PAR)

119909119894+ Δ 119909

119894isin HM wp HMCR sdot PAR

119894 = 1 6

(26)

where HMCR is harmony memory considering rate (095in this study) and PAR is pitch adjustment rate (03 in thisstudy)

Step 3 If the generated vector xNew is better than the worstone xWorst in HM in terms of objective function value thelatter is replaced with the former as follows

xNew isin HM and xWorstnotin HM (27)

Step 4 If termination criterion is satisfied the computation isended Otherwise Step 2 is performed again with updatedHM

When the HS approach was applied to the DO controloptimization model it could successfully find good resultswithout any divergence When ten different runs were per-formed HS found solutions ranging from 3030 to 3117 with


Table 3 Results by harmony search algorithm

Run Final vector119891(x) DO at 119909 km (ppm)

1198761

1198762

1198763

1198764

1198765

119903 5 10 15 20 25 30 40 501 647 0 0 0 3353 728 3099 641 570 535 523 527 540 579 6212 756 0 0 0 3244 727 3043 629 555 518 506 511 525 566 6103 766 0 0 0 3234 725 3038 628 553 516 505 509 523 565 6094 626 0 0 0 3374 713 3110 643 573 538 527 530 543 581 6235 781 0 0 0 3219 740 3030 627 551 514 503 507 521 563 6086 661 0 0 0 3339 697 3092 640 568 533 521 525 538 577 6197 684 0 0 0 3316 714 3080 637 565 529 518 522 535 574 6178 646 0 0 0 3354 728 3100 641 570 535 523 527 540 579 6219 767 0 0 0 3233 726 3037 628 553 516 505 509 523 564 60910 612 0 0 0 3388 688 3117 645 575 540 529 533 545 583 624

the average of 3075 Table 3 shows more details about thecomputation results

Furthermore this study tackled the above optimizationmodel using genetic algorithm (GA) which is another pop-ular metaheuristic algorithm When GA was applied to thismodel it could find solutions without any divergence Whenten different runs were performed GA found solutions rang-ing from 3033 to 3413 with the average of 3357 AlthoughGA found good solution (3033) only once mostly it foundpremature solutions

5 Conclusions

Thewastewater treatment optimizationmodel for fishmigra-tion was constructed and solved using HS The optimizationmodel considered three costs such as filtration cost nitri-fication cost and irrigation cost and two environmentalconstraints such as minimal DO requirement over riverreach and maximal nitrate-nitrogen concentration for publichealth

While the existing mathematical approach such as GRG2had hard time to identify solutions HS could find bettersolutions without divergence Also HS did not require initialsolution vectors which are very sensitive to final solutionquality When compared with GA HS could find bettersolutions in terms of minimal and average costs

For future study more realistic problems in wastewatertreatment field are expected to be considered and moreupdated techniques for these problems are expected to bedeveloped

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was supported by a grant (12-TI-C02) fromAdvanced Water Management Research Program funded byMinistry of Land Infrastructure and Transport of Koreangovernment

References

[1] httpenwikipediaorgwikiFish migration[2] D A Haith Environmental Systems Optimization Wiley New

York NY USA 1982[3] D H Burn and B J Lence ldquoComparison of optimization for-

mulations for waste-load allocationsrdquo Journal of EnvironmentalEngineering vol 118 no 4 pp 597ndash613 1992

[4] H Cardwell and H Ellis ldquoStochastic dynamic program-ming models for water quality managementrdquo Water ResourcesResearch vol 29 no 4 pp 803ndash813 1993

[5] L Somlyody M Kularathna and I Masliev ldquoDevelopment ofleast-cost water quality control policies for theNitra River Basinin SlovakiardquoWater Science andTechnology vol 30 no 5 pp 69ndash78 1994

[6] T Kawachi and SMaeda ldquoDiagnostic appraisal of water qualityand pollution control realities in Yasu River using GIS-aidedepsilon robust optimization modelrdquo Proceedings of the JapanAcademy Series B Physical and Biological Sciences vol 80 no8 pp 399ndash405 2004

[7] A P Singh S K Ghosh and P Sharma ldquoWater quality manage-ment of a stretch of river Yamuna an interactive fuzzy multi-objective approachrdquo Water Resources Management vol 21 no2 pp 515ndash532 2007

[8] D P Loucks C S ReVelle and W R Lynn ldquoLinear program-ming for water pollution controlrdquoManagement Science vol 14no 4 pp B166ndashB181 1967

[9] httpwwwsolvercomcontentbasic-solver-algorithms-and-methods-used

[10] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[11] ZWGeemMusic-InspiredHarmony SearchAlgorithmsTheoryand Applications Springer Berlin Germany 2009

[12] ZW Geem ldquoNovel derivative of harmony search algorithm fordiscrete design variablesrdquo Applied Mathematics and Computa-tion vol 199 no 1 pp 223ndash230 2008

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Boundary40000m3day

Secondary

Filtration

Boundary

Aquifer

River

Percolation

110000m3day

Nitrification

Q2 + Q4Q3 minus Q4

Q4

Q2

Q1

Q3

Q5

Storage

Irrigation A r

Figure 1 Schematic diagram of wastewater treatment portfolio

now forced to consider additional treatment processes thatenhance the water quality

Because the secondary treatment (settling and biologicaloxidation) is not enough the city should consider extra pro-cesses such as filtration and nitrification as shown in Figure 1Also certain amount of wastewater can be diverted to reducethe river discharge This diverted one can be used for cropirrigation

So we can consider total five decision variables for thiswastewater treatment optimization as follows

(1) 1198761 wastewater dumped without additional treatment

(unit 103m3day)(2) 1198762 wastewater undergoing additional filtration (unit

103m3day)(3) 1198763 wastewater undergoing additional nitrification

(unit 103m3day)(4) 1198764 wastewater undergoing additional nitrification

and filtration (unit 103m3day)(5) 1198765 wastewater diverted for irrigation (unit

103m3day)

Also there is onemore decision variable 119903which denotes irri-gation rate (cmweek) and the relationship between 119876

5and

119903 is 119903 = 701198765119860 Here119860 (hectare or 104m2) denotes irrigated

area and 70 is unit conversion factorWith the above decision variables we can construct the

objective function as follows

Minimize 119911 = 119891 (1198761 1198762 1198763 1198764 1198765 119903) = 119862ft + 119862nt + 119862ir

(1)

Here the cost for filtration 119862ft ($103yr) is represented asfollows

119862ft = 3 (1198762 + 1198764)093

+ 67 (1198762+ 1198764)055

(2)

where the first term in right-hand side of (2) represents thecapital cost of filtration treatment and the second termrepresents operation and maintenance cost

The cost for nitrification 119862nt ($103yr) is represented as

follows

119862nt = 1381198763068

+ 1061198763

042 (3)

where the first term in right-hand side of (3) represents thecapital cost of nitrification treatment and the second termrepresents operation and maintenance cost

The cost for irrigation 119862ir ($103yr) is represented asfollows

119862ir = 219119876028

5+ 12119876

078

5+ 02119876

054

5

+ (131 +48

119903)119876(074+032119903)

5

+ (51 +19

119903)119876(079+028119903)

5minus 068119860

(4)

where the first term in right-hand side of (4) represents thecapital cost of transmission line the second term representsthe capital cost of storage system (lagoon) the third term rep-resents the operation andmaintenance cost of storage systemthe fourth term represents the capital cost of irrigation sys-tem the fifth term represents the operation and maintenancecost of irrigation system and the sixth term represents thenet benefit (negative cost) of cropping (crop sales minus landrent)

Constraints for this optimization problem can be

1198761+ 1198762+ 1198763+ 1198765= 40 (5)

1198763ge 1198764 (6)

0 le 119876119894le 40 119894 = 1 5 (7)


8 (1 minus 119890minus0063119909

) + 1198620119890minus0063119909

minus 2331198610(119890minus0044119909

minus 119890minus0063119909

)

minus 0671198730(119890minus0025119909

minus 119890minus0063119909

) ge 5

119909 = 5 10 15 20 25 30 40 50

(8)

1198620=8 (110) + 2 (119876

1+ 1198762+ 1198763)

110 + 1198761+ 1198762+ 1198763

(9)

1198610=2 (110) + 25119876

1+ 13119876

2+ 13 (119876

3minus 1198764) + 7119876

4

110 + 1198761+ 1198762+ 1198763

(10)

1198730=5 (110) + 54119876

1+ 50119876

2+ 10 (119876

3minus 1198764) + 10119876

4

110 + 1198761+ 1198762+ 1198763

(11)

654 le 119903 le 1307 (12)




119906119889119862

119889119909= 1198962(119862119904minus 119862) minus 119896

1119861 minus 119896119899119873 (13)






119862 (119909) = 119862119904(1 minus 119890

minus1198962119909119906) + 1198620119890minus1198962119909119906

minus11986101198961

1198962minus 1198961


minus1198730119896119899

1198962minus 119896119899


(14)


0are


0 1198610 and 119873

0can be calculated





1

2 25 54


2

2 13 50


3minus 1198764

2 13 10


4

2 7 10












=119903119879119899 minus 10NC119903119879 + 119875 minus ET

(16)



119903119879119899 minus 10NC119903119879 + 119875 minus ET

le 10 (17)



119903119879119899 ge 10NC (18)









1198762

1198763

1198764

1198765

119903 Cost 1198761

1198762

1198763

1198764

1198765



5is zero




Minimize 119911 = 119891 (1198761 1198762 1198763 1198764 1198765 119903)

= 119862ft (1198762 1198764) + 119862nt (1198763) + 119862ir (1198765 119903) (19)

subject to

119862 (119909 1198761 1198762 1198763 1198764) ge 5 ppm

119909 = 5 10 15 20 25 30 40 50(20)

1198761+ 1198762+ 1198763+ 1198765= 40 (21)

1198763ge 1198764 (22)

0 le 119876119894le 40 119894 = 1 5 (23)

654 le 119903 le 1307 (24)



HM

=

[[[[[

[

1198761

11198761

21198761

31198761

41198761

51199031

119891 (x1)1198762

11198762

21198762

31198762

41198762

51199032

119891 (x2)

119876HMS1

119876HMS2

119876HMS3

119876HMS4

119876HMS5

119903HMS

119891 (xHMS)

]]]]]

]

(25)


119909New119894

larr997888

119909119894isin [119909

Lower119894

119909Upper119894

] wp (1 minusHMCR)

119909119894isin HM

= 1199091

119894 1199092

119894 119909

HMS119894


119909119894+ Δ 119909


119894 = 1 6

(26)









1198761

1198762

1198763

1198764

1198765

119903 5 10 15 20 25 30 40 501 647 0 0 0 3353 728 3099 641 570 535 523 527 540 579 6212 756 0 0 0 3244 727 3043 629 555 518 506 511 525 566 6103 766 0 0 0 3234 725 3038 628 553 516 505 509 523 565 6094 626 0 0 0 3374 713 3110 643 573 538 527 530 543 581 6235 781 0 0 0 3219 740 3030 627 551 514 503 507 521 563 6086 661 0 0 0 3339 697 3092 640 568 533 521 525 538 577 6197 684 0 0 0 3316 714 3080 637 565 529 518 522 535 574 6178 646 0 0 0 3354 728 3100 641 570 535 523 527 540 579 6219 767 0 0 0 3233 726 3037 628 553 516 505 509 523 564 60910 612 0 0 0 3388 688 3117 645 575 540 529 533 545 583 624



5 Conclusions






Acknowledgment


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










8 (1 minus 119890minus0063119909

) + 1198620119890minus0063119909

minus 2331198610(119890minus0044119909

minus 119890minus0063119909

)

minus 0671198730(119890minus0025119909

minus 119890minus0063119909

) ge 5

119909 = 5 10 15 20 25 30 40 50

(8)

1198620=8 (110) + 2 (119876

1+ 1198762+ 1198763)

110 + 1198761+ 1198762+ 1198763

(9)

1198610=2 (110) + 25119876

1+ 13119876

2+ 13 (119876

3minus 1198764) + 7119876

4

110 + 1198761+ 1198762+ 1198763

(10)

1198730=5 (110) + 54119876

1+ 50119876

2+ 10 (119876

3minus 1198764) + 10119876

4

110 + 1198761+ 1198762+ 1198763

(11)

654 le 119903 le 1307 (12)




119906119889119862

119889119909= 1198962(119862119904minus 119862) minus 119896

1119861 minus 119896119899119873 (13)






119862 (119909) = 119862119904(1 minus 119890

minus1198962119909119906) + 1198620119890minus1198962119909119906

minus11986101198961

1198962minus 1198961


minus1198730119896119899

1198962minus 119896119899


(14)


0are


0 1198610 and 119873

0can be calculated





1

2 25 54


2

2 13 50


3minus 1198764

2 13 10


4

2 7 10












=119903119879119899 minus 10NC119903119879 + 119875 minus ET

(16)



119903119879119899 minus 10NC119903119879 + 119875 minus ET

le 10 (17)



119903119879119899 ge 10NC (18)









1198762

1198763

1198764

1198765

119903 Cost 1198761

1198762

1198763

1198764

1198765



5is zero




Minimize 119911 = 119891 (1198761 1198762 1198763 1198764 1198765 119903)

= 119862ft (1198762 1198764) + 119862nt (1198763) + 119862ir (1198765 119903) (19)

subject to

119862 (119909 1198761 1198762 1198763 1198764) ge 5 ppm

119909 = 5 10 15 20 25 30 40 50(20)

1198761+ 1198762+ 1198763+ 1198765= 40 (21)

1198763ge 1198764 (22)

0 le 119876119894le 40 119894 = 1 5 (23)

654 le 119903 le 1307 (24)



HM

=

[[[[[

[

1198761

11198761

21198761

31198761

41198761

51199031

119891 (x1)1198762

11198762

21198762

31198762

41198762

51199032

119891 (x2)

119876HMS1

119876HMS2

119876HMS3

119876HMS4

119876HMS5

119903HMS

119891 (xHMS)

]]]]]

]

(25)


119909New119894

larr997888

119909119894isin [119909

Lower119894

119909Upper119894

] wp (1 minusHMCR)

119909119894isin HM

= 1199091

119894 1199092

119894 119909

HMS119894


119909119894+ Δ 119909


119894 = 1 6

(26)









1198761

1198762

1198763

1198764

1198765

119903 5 10 15 20 25 30 40 501 647 0 0 0 3353 728 3099 641 570 535 523 527 540 579 6212 756 0 0 0 3244 727 3043 629 555 518 506 511 525 566 6103 766 0 0 0 3234 725 3038 628 553 516 505 509 523 565 6094 626 0 0 0 3374 713 3110 643 573 538 527 530 543 581 6235 781 0 0 0 3219 740 3030 627 551 514 503 507 521 563 6086 661 0 0 0 3339 697 3092 640 568 533 521 525 538 577 6197 684 0 0 0 3316 714 3080 637 565 529 518 522 535 574 6178 646 0 0 0 3354 728 3100 641 570 535 523 527 540 579 6219 767 0 0 0 3233 726 3037 628 553 516 505 509 523 564 60910 612 0 0 0 3388 688 3117 645 575 540 529 533 545 583 624



5 Conclusions






Acknowledgment


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198762

1198763

1198764

1198765

119903 Cost 1198761

1198762

1198763

1198764

1198765



5is zero




Minimize 119911 = 119891 (1198761 1198762 1198763 1198764 1198765 119903)

= 119862ft (1198762 1198764) + 119862nt (1198763) + 119862ir (1198765 119903) (19)

subject to

119862 (119909 1198761 1198762 1198763 1198764) ge 5 ppm

119909 = 5 10 15 20 25 30 40 50(20)

1198761+ 1198762+ 1198763+ 1198765= 40 (21)

1198763ge 1198764 (22)

0 le 119876119894le 40 119894 = 1 5 (23)

654 le 119903 le 1307 (24)



HM

=

[[[[[

[

1198761

11198761

21198761

31198761

41198761

51199031

119891 (x1)1198762

11198762

21198762

31198762

41198762

51199032

119891 (x2)

119876HMS1

119876HMS2

119876HMS3

119876HMS4

119876HMS5

119903HMS

119891 (xHMS)

]]]]]

]

(25)


119909New119894

larr997888

119909119894isin [119909

Lower119894

119909Upper119894

] wp (1 minusHMCR)

119909119894isin HM

= 1199091

119894 1199092

119894 119909

HMS119894


119909119894+ Δ 119909


119894 = 1 6

(26)









1198761

1198762

1198763

1198764

1198765

119903 5 10 15 20 25 30 40 501 647 0 0 0 3353 728 3099 641 570 535 523 527 540 579 6212 756 0 0 0 3244 727 3043 629 555 518 506 511 525 566 6103 766 0 0 0 3234 725 3038 628 553 516 505 509 523 565 6094 626 0 0 0 3374 713 3110 643 573 538 527 530 543 581 6235 781 0 0 0 3219 740 3030 627 551 514 503 507 521 563 6086 661 0 0 0 3339 697 3092 640 568 533 521 525 538 577 6197 684 0 0 0 3316 714 3080 637 565 529 518 522 535 574 6178 646 0 0 0 3354 728 3100 641 570 535 523 527 540 579 6219 767 0 0 0 3233 726 3037 628 553 516 505 509 523 564 60910 612 0 0 0 3388 688 3117 645 575 540 529 533 545 583 624



5 Conclusions






Acknowledgment


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198761

1198762

1198763

1198764

1198765

119903 5 10 15 20 25 30 40 501 647 0 0 0 3353 728 3099 641 570 535 523 527 540 579 6212 756 0 0 0 3244 727 3043 629 555 518 506 511 525 566 6103 766 0 0 0 3234 725 3038 628 553 516 505 509 523 565 6094 626 0 0 0 3374 713 3110 643 573 538 527 530 543 581 6235 781 0 0 0 3219 740 3030 627 551 514 503 507 521 563 6086 661 0 0 0 3339 697 3092 640 568 533 521 525 538 577 6197 684 0 0 0 3316 714 3080 637 565 529 518 522 535 574 6178 646 0 0 0 3354 728 3100 641 570 535 523 527 540 579 6219 767 0 0 0 3233 726 3037 628 553 516 505 509 523 564 60910 612 0 0 0 3388 688 3117 645 575 540 529 533 545 583 624



5 Conclusions






Acknowledgment


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Wastewater Treatment Optimization for...

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