Application of Fuzzy Multi-Objective Programming to Regional Sewer System Planning

Abstract

Planning of sewer systems typically involves limitations and problems, regardless of whether traditional planning methods or optimization models are used. Such problems include non-quantifiability, fuzzy objectives, and uncertainties in decision-making variables which are commonly applied in the planning of any process. Particularly, uncertainties have prevented the inclusion of these variables in models. Consequently, the theoretical optional solution of the mathematical models is not the true optimum solution to practical problems. In this study, to solve the above problems for regional sewer system planning, multi-objective programming (MOP), nonlinear programming, mixed-integer programming, and compromise fuzzy programming were used. The objectives of this study were two-fold: (1) determination of the necessary decision-making variables or parameters, such as the optimum number of plants, piping layout, size of the plant, and extent of treatment; (2) establishment of a framework and methodology for optimal planning for designing a regional sewer system, matching demanded targets with the lowest cost, which would achieve the aim of lower space and energy requirements as well as consumption and high treatment efficiency for the purpose of meeting effluent standards. The findings of this study revealed that individual regional sewage treatment plants could be merged to form a centralized system. Land acquisition was difficult; thus, reducing the number of plants was required. Therefore, the compromise-fuzzy-based MOP method could effectively be used to build a regional sewer system plan, and the amount of in-plant establishment reached its maximized value with a minimized cost.

1. Introduction

Regional sewer system planning is a complex and difficult task, because a thorough understanding of the concerned region is needed, including information about the life expectancy of the designed system, the population of the service area, sources of wastewater and water quality, resources required, plant size, and extent of treatment. Furthermore, other additional information is required, for example: the control of the wastewater quantity, selection of plant location, consideration of cost selection of the treatment process, determination of automation systems, consideration of resource reuse, and assessment of related nontechnical factors (such as the public acceptance level and related environmental impact).

There are several problems associated with the application of an optimization method to wastewater treatment system planning [1], and these are mentioned as follows. Many objectives with different characteristics with no transferred or substituted conditions should be subjected to an optimization analysis. The amount of nonquantifiable or fuzzy objectives could not be included in the mathematical optimization models. Various decision-making variables contained uncertainties. The optimal solution of the models did not match that of the true problem.

Multi-objective programming (MOP) is considered one of the common problems in practical processes. The basic theory of MOP has been established, but its practical implementation continues to pose a challenge [2,3,4]. The theory of MOP was first presented by Koopmans [5], who introduced the concept of the efficient vector. Kuhn derived optimality conditions for vector optimization, thereby establishing the foundation of MOP [6]. Zadeh presented the fuzzy set theory [7], and Zimmerman used the max–min operator theory of Bellman [8] and the fuzzy set theory proposed by Zadeh to establish Zimmermann fuzzy programming [6]. The results were possibly nonlinear using the fuzzy programming, because membership functions depended on a decision maker’s operation. Hence, Hannan, Leberling, and Sakawa applied the max–min operator to linear, exponential, hyperbolic, and intermittent linear regression functions [9,10,11].

Moreover, compromise fuzzy programming (CFP) is a proper and multi-objective programming method which is available for application to several objectives, in those situations where the simultaneous optimization of all the considered objectives may not be matched, with the presence of a high level of conflict between rules [12]. Many fuzzy multi-objective programming methods united defuzzification operations that illustrated fuzzy values into crisp ones. A fuzzy value might decrease into one or more single crisp values because of the effect regarding defuzzification, which is described as a process of synthesis, so the entire concept of defuzzification was opposite to the principle of fuzzy sets theory. As a result, the last steps of the resolution process were a desirable feature due to the addition of maintenance of fuzziness [12]. Furthermore, Hsu and Tzeng applied CFP to design the regional wastewater treatment system, where CFP involved combining all used objective functions, so CFP application was suitable to be used to pursue optimization of all the considered multi-objectives [13]. However, few available studies have utilized the CFP method in the application of the actual practical problems. MOP determined discrete, continuous, and continuous-discrete solutions using a combination of integer, linear, and mixed-integer forms. Therefore, the MOP model used in this study could evaluate the feasibility of achieving these multi-objective goals in regional sewer system planning.

2. Methodology

Tarafder used the MOP method to optimize the operation and design of a styrene manufacturing process [14]. Tan used robust interval fuzzy programming, and Chae developed a mixed-integer linear programming (MILP) model [15,16]. To optimize water networks in industrial processes, Lovelady and Halwagi developed a nonlinear programming (NLP) model [17]. Most optimization studies have been based on process simulation and single-objective optimization [4,18,19,20,21]. Consequently, the optimal solutions of MOP problems resulted in solution sets, also referred to as Pareto sets.

Multi-objective genetic algorithms, non-dominated sorting genetic algorithms, and other modern multi-objective algorithms have involved Pareto multi-objective evolutionary algorithms [4,22]. The decision factors in the algorithm strongly affect the Pareto sets. However, accurately determining decision factors is challenging because of the fuzzy relationships between decision factors and optimal objectives in practical systems. When the optimization objectives are in conflict with each other, this is true basis of the problem of accurate determination [4,23,24,25].

Furthermore, the MOP method has been applied in various areas, including economics, energy conservation, engineering, logistics, and production planning [26,27]. There are many real practical cases in this field, including the following: Wu et al. (2020), who proposed a means of urban traffic signal control based on multi-objective joint optimization [28]; Huang et al. (2020), who presented a multi-objective optimization strategy for a distribution network considering V2G-enabled electric vehicles [29]; Ma et al. (2020), who demonstrated multi-objective optimization of traffic signals based on vehicle trajectory data [30]; Bagheri et al. (2020), who depicted Fuzzy arithmetic DEA approach for the fuzzy multi-objective transportation problem [31]; Yang et al. (2020), who published ensemble fuzzy radial basis function neural networks architecture driven with the aid of multi-optimization [32], Da Silva et al. (2021), who submitted multi-objective optimization design and control of plug-in hybrid electric vehicle powertrain [33]; Yue et al. (2021), who demonstrated fuzzy multi-objective modeling for managing water-food-energy-climate change-land nexus towards sustainability [34], Jafarian-Moghaddam (2021), who wrote Economical Speed for Optimizing the Travel Time and Energy Consumption in Train Scheduling using a Fuzzy Multi-Objective Model [35]; Ozdemir et al. (2021), who proposed a fuzzy multi-objective model for assembly line balancing with ergonomic risk consideration [36]. This study used NLP, mixed-integer programming (MIP), and CFP to derive an MOP method that would provide the optimal solution for a regional wastewater treatment system in terms of the lowest number of treatment plants, lowest cost, and lowest environmental impact. The mathematical models used are depicted as follows.

2.1. Mixed-Integer Programming

In many practical cases, the number of persons or automobile variables shall be integers to let variables have meaningful value. In this study, only a limited number of variables are integers, and the integer value is presented as either 0 or 1. This practice is called MIP, and it is resolved using the following mathematical model (Equations (1) and (2)).

Let I_i = 1, i = 1, 2, …, m, when selected

I_i = 0, when not selected

Hence,

$$\mathrm{Max}. Z={\displaystyle \sum _{i=1}^n{I}_i}{f}_i(Q_i)$$

(1)

subject to

$${\displaystyle \sum _{i=1}^n{I}_i=1}$$

(2)

Here, f_i(Q_i) is the construction, operational, and maintenance cost function of the j class treatment plant, and Q is the wastewater treatment capacity of the plant at point i.

2.2. Fuzzy Programming

The fundamentals of the fuzzy programming model developed by Zimmermann are described as follows [8].

Min. Z = CX

subject to

AX ≤ b

X ≥ 0

where Z is a k × 1 vector matrix (k is the number of objectives), A is an m × n matrix of the coefficients of constraints, b is an m × 1 matrix, C is a k × n vector matrix, and X is an n × 1 matrix of decision variables.

If, eventually, Z fuzzy objectives exist, then the equation can be rewritten as

CX ≤ Z

AX ≤ b

X ≥ 0

According to fuzzy set theory, to determine the membership function (Equation (3)), the decision maker can select a constant $d_i$ such that

$$U_i({(AX)}_i)=\left\{\begin{array}{cc}\hfill 1\hfill & {(AX)}_i\le b_i\\ 1-\frac{{(AX)}_i-b_i}{d_i}\hfill & b_i<{(AX)}_i<b_i+d_i\\ 0\hfill & {(AX)}_i\ge b_i+d_i\end{array}\right.$$

(3)

There are three conditions for range selection of AX, while AX is situated in the range between combinations of b_i and d_i, and the combinations of b_i and d_i are available to present various choices, after the addition of d_i to b_i for the selection of the decision maker on its demand. The above statements match the definition of fuzzy set theory, because various conditions are considered in Equation (3) for the optimal choices of the final decision according to the requests of the optimal choices. According to the algorithmic definition of a fuzzy optimal decision, the compromise solution can be obtained by solving Equation (4).

$$\underset{x\ge 0}{\max }.\underset{i}{\min }. [U_i (AX\left){}_i\right]$$

(4)

The fuzzy multi-objective programming model can be transformed into a mathematical model as follows (Equation (5)):

Max. λ

$$\mathrm{subject} \mathrm{to} \mathsf{\lambda } \le b_i-\left(\mathrm{AX}\right)$$

(5)

X, λ ≥ 0

2.3. Compromise Fuzzy Programming

Hsu and Tzeng developed a mathematical model in which a minimum operator is used to link objective functions, written as Equations (6) and (7) [13].

Min. β

$$\mathrm{subject} \mathrm{to} U_i \left(Z_i\right) \ge f_i-w_i\times \beta , i=1,2,\dots \dots .n$$

(6)

X ∈ S

$$\underset{i}{\max }\frac{f_i-1}{w_i}\le \beta \le \underset{i}{\min }\frac{f_i}{w_i}$$

(7)

Here, $U_i$ is the objective membership function including linear and nonlinear functions, $f_i$ is the approximate membership specified by the decision maker, $w_i$ is the compromise weight provided by the decision maker, and β is the compromise factor.

2.4. Analysis of Model Scheme

The scheme of a general mathematical model includes objective functions, decision variables, constant parameters, and constraints.

2.4.1. Objective Function

Three objective functions are used in this study: the total cost is intended to be minimized; biochemical oxygen demand (BOD) removal is required to be maximized; and the number of required treatment plants is intended to be minimized. These objectives are linked after using CFP.

A.

Objective 1: The total cost, including the construction cost of plants and the pipeline cost, shall be the minimum, as written as Equation (8).

$$Min[{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^3{I}_{ij}}}¡E{f}_i(Q_i)+{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^n{l}_{ij}CP(\left|q_{ij}-e_{ji}\right|}})]$$

(8)

where f_i(Q_i) is the construction, operational, and maintenance cost function of the j class treatment plant; Q is the wastewater treatment capacity of the plant at point i; and CP is the unit length pipeline construction cost function. Furthermore, q_ij is the wastewater quantity flowing from point j to point i, and e_ij is the wastewater quantity flowing out from point i.

B.

Objective 2: Maximize the total removal of BOD, as written as Equation (9).

$$Max[{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^3{I}_{ij}}\cdot Q\cdot E_j/}{\displaystyle \sum _{i=1}^n{Q}_i}]$$

(9)

where E_j is the BOD removal at the treatment plant at point j, L_i is the quantity of wastewater produced at point i, and l_ij is the distance between point i and point j.

C.

Objective 3: Minimize the number of treatment plants required.

Min. [number of treatment plants]

The number of plants shall be minimized because of the limited and available land for construction. Fewer plants means less environmental impact, and this also renders management of the plants easy. The objective of controlling the number of plants can be achieved by placing a lower limit on plant size. Therefore, the objective function can be rewritten as Equation (10):

$$\begin{array}{l}Max[R]\\ 0\le R\le {\displaystyle \sum _{i=1}^n{Q}_i}\end{array}$$

(10)

$R$ = [limit on the minimum capacity of a wastewater treatment plant that is to be constructed]

2.4.2. Decision Variables

This study selected 46 decision variables, including the flow rate in the pipeline, treatment capacity of the plant, operator of the treatment degree decision variable, compromise factors, and a lower limit on plant size.

2.4.3. Constant Parameters

The constant parameters include the lengths of the pipelines, wastewater quantity produced at each point source, and the constant coefficients and exponents of cost functions.

2.4.4. Constraints

The constraints include the flow rates and minimum capacity of the plant treatment degree, both of which are controlled.

A.

Constraint of constant flow rate

This constraint is given by Equation (11):

$${{\displaystyle L}}_i+{\displaystyle \sum _{j=1}^n{{\displaystyle q}}_{ji}-Q_i-{\displaystyle \sum _{j=1}^n{{\displaystyle e}}_{ij}}=0 (i=1,2,\dots ,n)}$$

(11)

The first two terms represent the daily wastewater quantity produced at point i and the wastewater quantity flowing into the plant at point i, respectively. The final two terms depict the treatment capacity and output quantity at point i, respectively.

B.

Limit on minimum capacity of treatment plant

As operational and economic factors considerably influence the establishment of a wastewater treatment plant, a limit is placed on its treatment capacity. The capacity should always be above a certain limit, as shown by Equation (12). The following equation can be used for this purpose.

$$\left|Q_i-\frac{R}{2}\right|\le \frac{R}{2} (i=1,\dots ,n)$$

(12)

where R is the lower limit of the capacity.

C.

Constraint for treatment level selection

In design of this study, the treatment plants could choose a treatment level, ranging from primary to tertiary treatment. The final decision was made by the operator of the treatment degree decision variable I_ix, where X was situated in the range of 1–3 (I_i1, I_i2, I_i3)$\in${0,1}, and (I_i1, I_i2, I_i3) could have only one variable with a value of 1 as Equation (13) (the others had a value of 0). The following expression could be used for this constraint:

(I_i1 + I_i2 + I_i3) = 1

(13)

D.

Constraint for compromise fuzzy programming

The following constraint was considered for CFP as Equation (14):

$$\underset{i}{\max }\frac{f_i-1}{w_i}\le \beta \le \underset{i}{\min }\frac{f_i}{w_i}$$

(14)

E.

Non-negative constraint

All decision variables, except for the compromise factor, were greater than or equal to zero.

3. Results and Discussion

Considering practical application for real cases by MOP, Roy et al. proposed neutrosophic linear programming derived from fuzzy theory to solve the decision-making problem. For example, the concept of a trapezoidal neutrosophic number aimed to solve the decision-making problem, including water resources, medical diagnosis, and supply chain management [37]. Neutrosophic programming was employed to obtain multi-optimal solutions for the fixed-charge solid transportation problem regarding carbon emissions and budget constraints [38]. Moreover, Maiti and Roy presented multi-resolutions for the Stackelberg game via intuitionistic fuzzy programming that integrated analysis of interval and multi-choice bi-level programming [39]. Many practical programming methods were derived from Fuzzy theory for various resolutions of real problems, since Fuzzy theory was broadly used to solve the decision-making problem. Each method acquired its advantages and benefits for optimal solutions from real problems, depending on the purpose of the study.

An actual example, obtained from the study of Downey Brill and Nakamura [40], was considered for comparison purposes. The example is shown in Figure 1, indicating that it consists of 7 nodes (wastewater sources or plant locations) and 11 possible paths for the pipeline, while nodes 2 and 6 are not allowed to build treatment plants and 5 paths are available to have water flow in both directions. The production of wastewater is estimated to be 0.08–2.30 million gallons per day (MGD), while the total wastewater quantity is estimated to be 5.0 MGD, and the distance between nodes is estimated to be 2.3–10.0 miles. Figure 2 shows the flow chart of experimental processes.

Before the model computations, it was necessary to determine the fuzzy bounds and fuzzy interval of each objective to establish the membership function of the objective.

Table 1. Fuzzy bounds and fuzzy intervals of each objective.

	Fuzzy Bounds	Fuzzy Intervals dj
Total cost of system BOD Removal Lower limit of Plant Capacity	[6401978, 12722172] [85%, 95%] [0, 2.51]	6320194 10% 2.51

shows the results, indicating that the fuzzy interval is obtained by bifurcating the lower limit from the upper limit.

A.

Membership function of total cost.

Table 1 shows that the minimum total cost b_i of the system is USD 6,401,978 and the fuzzy interval d_i is 6,320,194, while its membership function is illustrated in Figure 1.

B.

The membership function is shown in Figure 3, and it is obtained by assuming that the BOD removal values of the secondary and the tertiary treatment systems are 85% and 95%, respectively.

C.

Membership function of minimum plant capacity:

The treated total volume of wastewater in this system was 5.0 MGD. Therefore, if the minimum plant capacity was 2.51 MGD, then only one treatment plant would suffice (2.51 × 2 > 5.0). Its membership function is shown in Figure 4.

The optimization model was based on FORTRAN programming, and it could be used for solving linear and NLP problems. FORTRAN programming facilitated the use of various computation methods. The resulting combination could be divided into three categories: (1) strategies, (2) optimizer, and (3) one-dimensional search. Each category contained numerous resolving methods. This method of solving a problem could choose one method and group it with another.

To obtain an effective resolving method, a test should be conducted first. In this study, the test was conducted with 50 combinations of resolving methods and 100 random initial values (0–5). The results are presented as a scatter diagram in Figure 5. Computations were repeated a total of 5000 times for this combination arrangement. From this diagram, the combinations beneficial to the system were selected and used as the ADS method. To determine whether this predicted resolution affected the optimization results, we divided the 50 combinations into three distribution sections: (0–2.5), (2.5–5.0), and (0–5.0). A cost–number accumulation diagram was then obtained (Figure 6). As shown in Figure 6, the resolution quality was higher when the range of random numbers for the predicted values was 0–2.5. Hence, a lower construction cost could be achieved if the initial prediction values are set within the range 0–2.5, as shown in Figure 7.

Based on the CFP scheme, while the value of the compromise weight W_i was chosen by the decision maker and the f_i value was presented as the approximate membership, this paper presented eight preferred schemes (the first to fifth schemes involved three objectives, whereas the sixth to eighth contained two objectives) to cover decision makers’ preferences. The optimal and compromised solution of this model, obtained through computations with its scheme, is shown in

Table 2. Results for five schemes involving the consideration of three or two objectives.

Scheme	Variables Considered	Total Cost (Z1)	BOD Removal (Z2)	Min Number of Plants	Total Cost	BOD Removal (%)	Required Min. Capacity of Plant (MGD)	No. of Plants
Scheme 1	The compromise weighted w	0.33	0.33	0.33	10,362,600	92.6%	0.894	4
	The approximate membership value f	0.80	0.80	0.80
Scheme 2	The compromise weighted w	0.10	0.45	0.45	9,395,210	89.6%	0.346	5
	The approximate membership value f	0.90	0.60	0.60
Scheme 3	The compromise weighted w	0.45	0.10	0.45	9,628,680	94.0%	0.500	4
	The approximate membership value f	0.60	0.90	0.60
Scheme 4	The compromise weighted w	0.45	0.45	0.10	9,822,130	85.0%	2.444	2
	The approximate membership value f	0.60	0.60	0.90
Scheme 5	The compromise weighted w	0.20	0.20	0.60	9,822,130	87.9%	0.914	4
	The approximate membership value f	0.90	0.90	0.60
Scheme 6	The compromise weighted w	0.50	-	0.50	8,076,930	85.0%	0.998	3
	The approximate membership value f	0.80	-	0.80
Scheme 7	The compromise weighted w	0.20	-	0.80	7,517,060	85.0%	0.50	5
	The approximate membership value f	0.80	-	0.20
Scheme 8	The compromise weighted w	0.80	-	0.20	9,468,140	85.0%	1.905	1

. Clearly, when the compromise weight was equal to the approximate membership, the system tended to have four wastewater treatment plants, matching the result of Scheme 1, Figure 8. If the decision maker preferred to emphasize the construction cost, then the entire system tended to become decentralized, and the system then required five treatment plants. Therefore, the corresponding cost was the lowest of the costs of all schemes (Scheme 2, Figure 8). If emphasis was placed on environmental protection, then the system selected the scheme with four treatment plants, which provided a BOD removal of 94% (Scheme 3, Figure 8). Furthermore, the decision-maker might consider the difficulty of land acquisition for plant construction and consequently demand a reduction in the number of plants required. In this case, Scheme 4 (Figure 8) could be used, and this scheme involves two secondary treatment plants. If land acquisition is not a priority concern, then the total cost and environmental protection can be equally emphasized, and Scheme 5 (Figure 8) can be preferred. This scheme involves four secondary treatment plants.

When considering the two objectives, namely minimizing the total cost and minimizing the number of constructed plants, it means that both objectives are equally important. Thus, the system tends to divide the results into two sections, as is matched by the result of Scheme 6 (Figure 8), which requires three treatment plants. If the decision maker emphasizes cost, the system requires more plants and the cost is the minimum, as achieved by the result of Scheme 7, Figure 8. However, if land acquisition is assumed to be difficult, only one treatment plant exists, and the total cost is the maximum, as the result of Scheme 8 (Figure 8) showed.

Table 3. Comparison of available studies related to sewer system planning.

Titles	Methods	Objective	Objective Function	Variables and Constant Parameters
Robust optimization approach to regional wastewater system planning [42]	Robust optimization	Multi-objective	1. The minimum total cost; 2. the minimum cost and dissolved oxygen; 3. the maximum predicted dissolved oxygen of rivers	1. Production of wastewater; 2. flow of pipeline; 3. length of pipeline; 4. flow of rivers; 5. dissolved oxygen of stream segment; 6. length of stream segment; 7. possibility of any scenario’s occurrence; 8. the maximum treatment capacity of wastewater plants; 9. cost of discount for wastewater systems; 10. forfeit for disobeying the standard of water quality parameters.
Optimal Design of Regional Wastewater Pipelines and Treatment Plant Systems [43]	Genetic algorithm	Single-objective	1. The minimum total cost	1. The minimum depth of pipeline; 2. cost of energy; 3. coefficients of pipeline cost; 4. coefficients of pump cost; 5. coefficients of wastewater plants; 6. the minimum depth of the digging hole; 7. operational hours of water pumping; 8. interest of total cost; 9. gradient; 10. width of pipeline excavation; 11. length of pipeline connection; 12. production of wastewater at nodes; 13. life-span of pump; 14. life-span of wastewater plants; 15. the minimum and maximum velocity of pipeline.
Optimal Expansion Strategy for a Sewer System under Uncertainty [44]	Grey mixed integer programming	Single-objective	1. The minimum total cost	1. The maintained and operational cost of facilities; 2. recovery cost for maintaining sewer facilities; 3. cost of construction of the new facilities; 4. cost of wastewater transportation; 5. cost of facilities expansion; 6. income of population; 7. size of population.
A composite indicator approach to assess the sustainability and resilience of wastewater management alternatives [45]	Analytic hierarchy process	Multi-objective	1. The total cost; 2. the carbon emission amount; 3. the values of chemical oxygen demand, and total nitrogen content, total phosphate; 4. recovery ability after disasters; 5. the consecutively comprehensive indexes	1. Flow of untreated wastewater; 2. concentration of untreated wastewater; 3. operational cost; 4. capital cost; 5. size of population
A multi-objective optimization model for decision support in water reclamation system planning [45]	Triangle Splitting Method	Multi-objective	1.The minimum cost of wastewater treatment; 2. the minimum greenhouse gas emission of the sewer system	1. The capacity of wastewater plants; 2. the maintained and operational cost of facilities; 3. transportation cost of reclaimed water; 4. transportation cost of wastewater; 5. greenhouse gas emissions of the sewer system; 6. greenhouse gas emissions of reclaimed water; 7. greenhouse gas emissions of wastewater transportation; 8. the maximum capacity of greenhouse gas emissions for the sewer system; 9. life-span of facilities; 10. price of reclaimed water; 11. usage ratio of reclaimed water; 12. distance between citizen center location and wastewater plant; 13. size of population in the citizen center; 14. amount of recycled wastewater produced from customers.

shows most of available studies related to sewer system planning which considered multiple factors and objectives [41,42,43,44,45]. For example, Zeferino et al. (2012) considered several parameters, including production of wastewater, flow of the pipeline, length of the pipeline, flow of rivers, dissolved oxygen of the stream segment, length of the stream segment, and possibility of any scenarios occurrence, etc., for wastewater system planning [41]. Brand and Ostfeld (2011) regarded diversified parameters including the minimum depth of the pipeline, cost of energy, coefficients of pipeline cost, coefficients of pump cost, coefficients of wastewater plants, the minimum depth of the digging hole, operational hours of water pumping, etc., for wastewater system planning [42]. Sun et al. (2020) considered parameters including flow of untreated wastewater, concentration of untreated wastewater, operational cost, capital cost, and population size for wastewater management [44]. Although those above studies presented useful and great results due to detailed consideration of several parameters, the decision maker might not understand the meaning of all consideration parameters, because the decision maker might be not an expert or a specialist. This study assumed eight conditions (Schemes 1–8, Figure 8) with simple meaning parameters, including BOD removal, length of the pipelines, required minimal capacity of the plant, and total cost for the selection of the decision maker. Thus, the results of this study made it easy for the decision maker to easily and quickly propose his/her decision, even if he/she did not have the relevant expertise or professional engineering background.

4. Conclusions

This study successfully combined integer programming and CFP in a multi-objective programming model for the purpose of solving fuzzy characteristics and constraints, although fuzzy models are non-quantifiable in a wastewater treatment system. Decision makers can, therefore, use the model to obtain numerous decision options. This study focused on regional sewer system planning. There were several major variables for regional sewer system planning, including BOD removal, length of the pipelines, required minimal capacity of the plant, and total cost, and the above variables were represented by consecutive real numbers. However, the variable of a minimal number of plants was represented by integer. There were three levels indicating the number of plants being built: the values of 0, 1, and 2 represented no plants existing, a secondary treatment, and tertiary treatment, respectively. Therefore, the model of MOP was suitable to deal with the problem of combinations of real numbers and integers for system planning. Then, the study reviewed a real case regarding a regional sewer system via the MOP method (integrations of NLP, MIP, and CFP), and the findings of this study can be summarized as follows. Limiting plant capacity is emphasized when land acquisition is rather difficult. The results of this study reveal that all regional sewage treatment plants can form a centralized treatment system, thereby reducing the number of plants required. In consideration of the number of treatment plants, establishment of four wastewater treatment plants is the optimal choice, as shown in the results of Scheme 1, Figure 8. If construction cost is the most critical consideration, then the number of required plants should be maximized with the lowest cost, regardless of the type of objective. The optimal choice of construction cost occurs in the results of Scheme 7, Figure 8. Regarding inspection of environmental protection, the results of Scheme 3, Figure 8 depicts a BOD removal of 94% of four treatment plants. Furthermore, in careful consideration of the real problem (land acquisition for plant construction), there are two secondary treatment plants in the results of Scheme 4, Figure 8.

The choice of the initial predicted values and the different combinations of resolving methods strongly affect the solution which can be obtained when a highly nonlinear optimization problem is resolved. Therefore, it is essential to choose initial predicted values with the appropriate range, and to use the appropriate combination of methods to achieve high reproducibility and a low total cost. This study demonstrates that the compromise fuzzy programming method can be effectively applied to the planning of a regional wastewater treatment system. The method will help in achieving high BOD removal and reducing the number of required wastewater treatment plants.