Dynamic-chance-constrained-based Fuzzy Programming Approach for

Optimizing Wastewater Facultative Ponds for Multi-period Case

K. KARTONO1,, S. SUTRISNO1,*, S. SUNARSIH1, W. WIDOWATI1,

TOSPORN ARREERAS2, MUHAMMAD SYUKUR2

1Department of Mathematics, Diponegoro University,

Tembalang 50275 Semarang,

INDONESIA

2School of Management, Mae Fah Luang University,

Chiang Rai 57100,

THAILAND

*Corresponding Author

Abstract: - In this article, a novel optimization model that was specifically designed as a dynamic-chance-

constrained fuzzy uncertain programming framework is introduced. This model serves the purpose of optimizing

the efficiency of facultative ponds utilized in domestic wastewater treatment. The primary focus of this study was

maximizing the amount of the wastewater treated in the facility subject to quality requirements via the assessment

of wastewater quality through the measurement of Biological Oxygen Demand (BOD). The model's development

was grounded in a real-world scenario, where decision-makers encountered uncertainties in various parameters,

such as the rate of BOD degradation and the incoming wastewater load, both characterized by fuzzy membership

functions. In light of this uncertainty, the decision-maker aimed to maximize the wastewater treatment capacity

while maintaining a suitable safety margin for both objective and constraint functions, employing policies founded

on probability and chance. A case study was carried out at the Bantul domestic wastewater treatment plant, situated

in Yogyakarta, Indonesia. The study successfully identified optimal decisions regarding wastewater flow rates and

processing times. As a result, it can be concluded that the proposed model effectively resolved the problem at hand,

making it a valuable tool for decision-makers in similar contexts.

Key-Words: - Biological oxygen demand, chance-constrained optimization, domestic wastewater, dynamic fuzzy

programming, dynamic optimization, facultative ponds, wastewater treatment

Received: October 19, 2023. Revised: November 15, 2023. Accepted: November 30, 2023. Available online: December 13, 2023.

Nomenclature

Decision variables on the observation day j:

0()jQ

: The rate of the wastewater volume

inflow at the inlet (m3)

()

ijQ

: The rate of the wastewater volume at

the facultative pond i (m3)

()tj

: Average detention time (day)

Fuzzy parameters:

: The daily rate of the waste load at the

facility inlet (kg)

: The daily BOD degradation rate

Semi-decision variables:

()

L j

: Waste load at the inlet of the facultative

pond i (kg) on the observation day j

()

: Waste load processed in the facultative

pond i (kg) on the observation day j

Crisp or deterministic parameters:

()

: The BOD concentration at pond i

(mg/L) estimated on the

observation day j

()

: The BOD degradation efficiency

index at pond i (in percentage)

estimated on the observation day j

()

: Target or reference point for the

BOD degradation efficiency index

at pond i estimated on the

observation day j

BM : Wastewater quality standard

, 1,2

pi

: Percentage of waste load processed

in pond i

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2024.23.3

K. Kartono, S. Sutrisno, S. Sunarsih,

W. Widowati, Tosporn Arreeras, Muhammad Syukur

E-ISSN: 2224-2678

Volume 23, 2024

1 Introduction

Before disposal, wastewater needs to undergo a

stabilization process to uphold water and

environmental sustainability objectives. However, the

availability of wastewater treatment plants remains

limited, especially in developing countries,

necessitating the optimization of their performance to

handle wastewater to the fullest extent possible.

In most wastewater treatment plants,

microorganisms such as algae, bacteria, and

zooplankton are commonly harnessed to reduce

pollutant concentration, [1]. Among the parameters

employed to assess the quality of treated water, the

focus in this study was on Biological Oxygen

Demand (BOD). The types of wastewater typically

encountered are of domestic origin, originating from

households, hotels, and general industries. Facultative

ponds are chosen for use in wastewater treatment

plants due to their straightforwardness in degrading

pollutants in domestic wastewater until they meet

specified concentration standards, often measured

through BOD levels, [2].

Mathematical optimization models have been

integrated into wastewater treatment management to

enhance the capacity and efficiency of facultative

ponds. Numerous models have been devised for

wastewater management, each tailored to address

specific challenges faced by decision-makers, see e.g.,

[3], [4], [5]. These models vary in complexity and

purpose, encompassing simple models like the one

proposed in, [6], to manage pollutant concentrations

based on quantitative prototypes. Other models cater

to distinct scenarios, including linear models with

deterministic parameters, [7], models focusing on

adsorbents for wastewater treatment analysis, [8],

quantitative models for sewage treatment, [9], and

models assessing the construction costs of wastewater

plants, [10].

Beyond wastewater treatment optimization,

additional models serve various objectives, such as

sewage management, [10], [11], energy analysis, [12],

effluent and sludge analysis, [13], microplastics

removal, [14], and (bio)energy generation from

wastewater, [15], [16], [17], [18], [19], [20].

However, none of these models have been formulated

in a chance-constrained framework, enabling

decision-makers to impose chance-based constraints

on uncertainty-containing parameters.

To address this gap, a new model was developed

to optimize the performance of facultative ponds in

wastewater treatment, accommodating uncertain

parameters like pollutant concentration at the inlet,

and in a dynamic manner over time. This allows

decision-makers to introduce additional chance-based

constraints to the model, such as the probability of

violating uncertain constraints under predefined

values. These uncertainties are treated as fuzzy

parameters with membership functions determined

based on the observations of the decision-maker.

Among the parameters investigated in this study,

BOD levels were monitored, using the Bantul

facultative ponds in Yogyakarta, Indonesia as a case

study to develop the model and compute optimal

decisions based on the proposed framework.

Fig. 1: The layout view of the Sewon wastewater

treatment plant

2 Mathematical Model

2.1 Problem Setting and Assumptions

This study focuses on the degradation of Biological

Oxygen Demand (BOD) in facultative ponds,

specifically designed within the layout of the Bantul

wastewater treatment plant situated in Yogyakarta,

Indonesia. The facility processes domestic wastewater

originating from households, industries, offices, and

hotels through a multi-step treatment process, as

illustrated in Figure 1. The primary objective is to

maximize the processing capacity of all facultative

ponds to ensure that the BOD concentration meets the

quality standard, considering certain uncertain

parameters. Furthermore, the problem is solved

dynamically in terms of observation time periods,

meaning that the model should be able to provide

optimal decisions for multiple periods of

implementation in one calculation. To be precise and

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2024.23.3

K. Kartono, S. Sutrisno, S. Sunarsih,

W. Widowati, Tosporn Arreeras, Muhammad Syukur

E-ISSN: 2224-2678

Volume 23, 2024

to provide a clearer understanding, the specifications

and assumptions used in this research are elaborated

as follows.

The parameter for assessing wastewater quality

was BOD. Data related to BOD were collected from

wastewater samples at specific grid points within each

facultative pond. The BOD degradation rate was

treated as a fuzzy parameter, with the decision-maker

developing its membership function. Furthermore, An

index value was employed to regulate the BOD

degradation process, as described in the mathematical

model.

The source of domestic wastewater entering the

facility was exclusively from the Yogyakarta

province. The uncertain inflow waste load was

monitored at the inlet per day, incorporating fuzzy

uncertainty. The decision-maker established the

membership function for the inflow waste load based

on observations and historical data. Secondary

historical data and observations informed the

formulation of this membership function.

Optimizations were conducted over days, and they

covered multiple days of observations in one model

and one calculation. Moreover, all fuzzy parameters

were assumed to have discrete membership functions.

The methodology adopted in this study can be

summarized as follows: Initially, the decision-maker

constructs membership functions for the fuzzy

parameters and assesses the likelihood of not

violating the lower bounds of the chance-based

constraints within the constraint functions.

Subsequently, the objective function, representing the

wastewater inflow rate, is formulated. Additionally,

the BOD efficiency index control term is defined as

the quadratic difference between a reference point and

the actual efficiency index. The reference point is

determined based on the decision-maker's intuition

and experience with managing the facultative ponds'

performance. Furthermore, constraint functions are

formulated and expressed in a mathematical model,

taking into account the structure of the wastewater

treatment facility and the necessary conditions that

must be met.

The formulated mathematical model is then solved

using a computer, with the model being translated

into a programming language using LINGO 19.0 and

subsequently solved with the embedded solver in the

software. Chance-constrained-based programming is

employed to calculate the optimal decision, as

detailed in, [21]. Finally, the generated solution is

applied to the wastewater treatment facility.

2.2 Chance-constrained-based Fuzzy

Optimization Model

In this optimization challenge, the following two

primary objectives were considered: 1) maximizing

the influx of wastewater and 2) minimizing the

quadratic expression representing the disparity

between the BOD degradation efficiency index and

the reference value stipulated by the decision-maker.

The setup of these two goals led to the formulation of

the following optimization problem subject to

constraints functions that were formulated following

the specifications and assumptions of the problem

described in the previous section (explanations for

each will follow afterward):

1 1 1

min ( ) [ ( ) ( )]

JJ r

j j i

Z Q j E j E j

  

   

 

(1)

subject to,

1,2,...,jJ

 

( ) ( )

( ) , 1, 2,3, 4;

1000

j C j

L j i



  

(2)

1( ) ( ;) , ,2,3,4

EBjjC M i 

(3)

1 2 0

( ) ( ) ;L j L j L

(4)

 

3 1 1 1

( ) (1 ( )) ( ) ;

Cr L j p j L j



   

(5)

 

4 2 2 2

( ) (1 ( )) ( ) ;

Cr L j p j L j



   

(6)

1 2 0

{ } , 1, 2,3,4;( ) ( ) i

Cr L j L L ij



   

(7)

1 2 0

( ) ( ) ( );

Q j Q j Q j

(8)

3 1 4 2

( ) 0.5 ( ) and ( ) 0.5 ( );

e e e e

Q j Q j Q j Q j   

(9)

()

() () , 1, 2, 3, 4

Cr E i







  







(10)

In the preceding minimization problem, we

consider the value of

Q

the objective function since

the original problem aims to maximize it. The

constraint function (2) denotes that the waste load is

determined by both the inflow rate and organic matter

while (3)ensuring that the BOD concentration remains

below an upper threshold. Additionally, equality (4)

signifies that the total waste load entering ponds I and

II equals that at the inlet. Inequalities (5) (6) govern

the waste load transfer from pond I to pond III and

from pond II to pond IV, respectively, where

, 1,2





they represent the confidence levels used

for pond I and II provided as an appropriate safety

margin by the manager/decision-maker for the

corresponding constraint functions to hold. The

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2024.23.3

K. Kartono, S. Sutrisno, S. Sunarsih,

W. Widowati, Tosporn Arreeras, Muhammad Syukur

E-ISSN: 2224-2678

Volume 23, 2024

probabilistic inequality (7) is employed to restrict the

treated wastewater load in facultative pond i, ensuring

it does not exceed its pre-treatment value.

Concerning Figure 1, the combined inflow rate

into ponds I and II equals that of the inlet, while the

inflow rates into ponds III and IV are half that of

ponds I and II, denoted by inequalities (8) and (9)

respectively. Formula (10) outlines the computation

of the required biological oxygen degradation

efficiency index, where k represents the BOD

degradation rate for the detention time spans one day,

and where

, 1,2,3, 4





to represent the confidence

levels used for pond i provided as an appropriate

safety margin by the manager/decision-maker for the

corresponding constraint functions to hold.

Combining both objective functions and all

constraint functions yields a probabilistic

optimization problem. To address this problem, the

chance-constraint programming algorithm is required

to compute the optimal decision. Furthermore, it's

worth noting that all constraint functions are closed

and bounded, ensuring that this optimization problem

always possesses an optimal solution as long as its

feasible region is not empty.

The chance-constrained optimization problem (1)

was solved by using the chance-constrained

programming method introduced in, [22].

Furthermore, to calculate the optimal decision, the

uncertain programming method based on the

deterministic equivalent approach provided in, [21],

was utilized. To calculate the expectation of fuzzy

numbers with discrete membership functions, the

fuzzy number theory in, [22], was utilized.

3 Case Study

The case study was carried out at the Bantul

wastewater treatment facility, and Figure 1 illustrates

the treatment process flow. The subsequent

subsection presents the parameters and outcomes of

the chance-constrained fuzzy programming model.

3.1 Parameter Setting

The membership functions for the fuzzy parameters

were generated randomly, centered around the mean

of the data provided in, [7]. Figure 2 displays both

their membership values and weights. In compliance

with the Yogyakarta Province's local government

policy, the BOD concentration in treated wastewater

should not surpass 50 mg/L. Simultaneously, the

decision-maker aimed for an efficiency index of 0.5

for each pond. The calculations were executed using

the LINGO 19.0 optimization software, employing

the generalized reduced gradient algorithm as outlined

in references, [21], [22].

(a) (b)

Fig. 2: Graphs of the membership functions of the fuzzy parameters (a) waste load for ponds I and II (b) waste load

for ponds III and IV (c) BOD degradation rate for ponds I and II (d) BOD degradation rate for ponds III and IV

0,00

1,00

2,00

0,470 0,480 0,490 0,500 0,510 0,520 0,530

Amount

Membership Weight

0,00

2,00

4840 4850 4860 4870 4880 4890 4900

Amount

Membership Weight

0,00

2,00

0,490 0,500 0,510 0,520 0,530 0,540 0,550

Amount

Membership Weight

0,00

1,00

2,00

1,00 1,05 1,10 1,15 1,20

Membership Weight

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K. Kartono, S. Sutrisno, S. Sunarsih,

W. Widowati, Tosporn Arreeras, Muhammad Syukur

E-ISSN: 2224-2678

Volume 23, 2024

(a) (b)

Fig. 3: The optimal decisions (a)

()

ijQ

: The rate of the wastewater volume at the facultative pond i (m3) on the

observation day j (b)

()tj

: Average detention time (day) on observation day j

3.2 Results and Discussion

Figure 3 displays the optimal outcome derived from

the proposed model. Meanwhile, the optimal inflow

rate at the inlet stands at 26950 m3 per day. The

optimized allocation for wastewater treatment across

the facultative ponds consists of 13475 m3/day for

both Ponds I and II, along with 6737.5 m3/day for

Ponds III and IV. According to this calculation, the

anticipated BOD concentration post-treatment is 50

mg/L, and it is not imperative for the detention time to

span the entire day. Should the decision-maker opt for

a full-day detention time, the expected BOD

concentration would be lower than 50 mg/L.

It is worth noting that the efficiency index values

varied among the four ponds due to parameter

fluctuations, but their average remained at 31%. This

implies that overall performance in the facultative

ponds needs enhancement, primarily through sludge

removal. Notably, Pond II exhibited the highest

efficiency index value and should be maintained,

while Pond I, with the lowest value, requires

improved treatment, such as the addition of an aerator.

From the results, several managerial insights

emerge regarding the management of facultative

ponds, those are explained as follows. First, decision-

makers are likely to consider varying confidence

levels for each constraint function in the mathematical

model, allowing adjustments based on their

experience and intuition. Second, some parameters

had unknown actual values during computation,

indicating decisions were made under uncertainty.

This suggests that achieved goals may differ from the

mathematical model's expected values. Therefore,

when dealing with multiple probability values, actual

results can be either better or worse. Third, it is

possible to perform multiple optimizations with

different parameter values, such as various

membership functions, until the decision-maker gains

sufficient confidence to execute a decision. However,

computational time should be considered when the

decision-maker has the time and expects improved

results.

4 Conclusion

In this study, a novel dynamic-chance-constrained

fuzzy programming model has been introduced, aimed

at improving the efficiency of facultative wastewater

stabilization ponds with multiple time periods of

observation. An empirical investigation was

conducted at the Bantul wastewater treatment plant,

and the outcomes indicated the effective optimization

of the facility through the proposed method.

Looking ahead, there are several forthcoming

challenges. These encompass the development of

more intricate models to address complex scenarios,

including the management of pollutant degradation

processes within maturation ponds. Additionally,

exploring the impact of sludge analysis on pond

performance is an intriguing avenue for future

research.

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5000

10000

15000

12345

Observation day

Pond I Pond II Pond III Pond IV

0,5

1 2 3 4 5

day

Observation day

Pond I Pond II Pond III Pond IV

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K. Kartono, S. Sutrisno, S. Sunarsih,

W. Widowati, Tosporn Arreeras, Muhammad Syukur

E-ISSN: 2224-2678

Volume 23, 2024

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W. Widowati, Tosporn Arreeras, Muhammad Syukur

E-ISSN: 2224-2678

Volume 23, 2024

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Contribution of Individual Authors to the Creation

of a Scientific Article (Ghostwriting Policy)

- K. Kartono managed and supervised the research

activities.

- K. Kartono, S. Sutrisno, S. Sunarsih, W. Widowati,

Tosporn Arreeras, Muhammad Syukur modelled

and verified the programming.

- S Sutrisno carried out the computational simulation.

- K. Kartono, S. Sutrisno, S. Sunarsih, W. Widowati

prepared the draft of the manuscript.

- Tosporn Arreeras, Muhammad Syukur validated the

results, reviewed, and edited the manuscript.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

This work is funded by the Faculty of Science and

Mathematics, Universitas Diponegoro via Riset

Utama 2023 Research Grant contract no.

22/UN7.F8/PP/II/2023.

Conflict of Interest

The authors have no conflicts of interest to declare.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en_

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DOI: 10.37394/23202.2024.23.3

K. Kartono, S. Sutrisno, S. Sunarsih,

W. Widowati, Tosporn Arreeras, Muhammad Syukur

E-ISSN: 2224-2678

Volume 23, 2024