A stochastic equilibrium chance-constrained programming ... - NSFC

Qual Quant DOI 10.1007/s11135-015-0301-2

A stochastic equilibrium chance-constrained programming model for municipal solid waste management of the City of Dalian, China Min Zhou1 • Shasha Lu2 • Shukui Tan1 • Danping Yan1 • Guoliang Ou3 • Dianfeng Liu4 • Xiang Luo5 • Yanan Li1 • Lu Zhang1 • Zuo Zhang1,6 • Xiangbo Zhu1

Springer Science+Business Media Dordrecht 2015

Abstract In this paper, a stochastic equilibrium chance-constrained programming (SECCP) model was developed for tackling the municipal waste management issue under uncertainty. The main advantage of this model is that it effectively reflected the dualrandom characteristics of uncertain parameters through incorporating the opinions and judgments from various respondents into the parameter identification processes. This will lead to birandom variables, where their mean values and standard deviations are allowed to be the random variables, instead of the fixed values. The generation of birandom variables

& Shasha Lu [email protected] Min Zhou [email protected] Shukui Tan [email protected] Danping Yan [email protected] Guoliang Ou [email protected] Dianfeng Liu [email protected] Xiang Luo [email protected] Yanan Li [email protected] Lu Zhang [email protected] Zuo Zhang [email protected] Xiangbo Zhu [email protected]

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will enrich the stochastic optimization theory and improve the accuracy and rationality of parameters design and estimation. The equilibrium chance-constrained programming algorithm was used to solve the SECCP model, which is capable of tackling birandom variables and is overcoming limitations of traditional stochastic chance-constrained programming while parameters with normal distribution are required strictly. Currently, the application of SECCP model in the environmental management fields was limited. As the first attempt, the regional waste management of the City of Dalian, China, was used as a study case for demonstration. A variety of solutions are beneficial in providing decision space to the local managers through designing and adjusting the constraints-violation levels. This solution process also reflected trade-off between system economy and reliability. The successful application in regional waste management system is expected to be a good example for tackling other similar problems. Keywords

Birandom variable Equilibrium chance Waste management Dalian

1 Introduction With rapid development of social economy, acceleration of urbanization process and continuous growth of population size, the quantity of municipal solid waste (MSW) remains steady growth. According to the ‘‘Twelfth Five-Year Plan on the Construction of Harmless-Disposal Facility for Municipal Solid Wastes in China’’, by the end of 2014, the annual delivering quantity of MSW in China is about 221 million tons with average growth rate of 9.5 % per year; the annual waste-production amounts per capita is 440 kg and the harmless-disposal rate of MSW reaches 63.5 %. There are planning to increase the disposal capacity of 580,000 tons/day; the collection and transfer capacity of 457,000 tons/day, and transportation capability of 457,000 tons/day in the urban areas. It is expected to launch 1882 pollution control projects with a total investment of 263.6 billion RMB, which will provide strong support for solving MSW management issue. However, the problem caused by the large amount of MSWs has not been adequately addressed, although many investment activities and treatment facilities had made their contributions in the MSW disposal and management. This is mainly due to vast system components and complicated system structure associated with management system, leading to difficulties in designing and executing rational and effective waste management strategies. Based on deep understanding and analysis of management system, uncertain optimization model will be crucial for generating appropriate facility-construction schemes and waste-allocation strategies.

1

College of Public Administration, Huazhong University of Science and Technology, Wuhan 430074, China

2

School of Economics and Management, Beijing Forestry University, Beijing 100083, China

3

School of Construction and Environmental Engineering, Shenzhen Polytechnic, Shenzhen, China

4

School of Resource and Environment Sciences, Wuhan University, Wuhan, China

5

Key Laboratory for Geographical Process Analysis & Simulation, Central China Normal University, Wuhan, China

6

School of Business, Hubei University, Youyi Road, Wuhan 430062, China

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A stochastic equilibrium chance-constrained programming…

As mentioned above, inherent complexities and interactive relationships in the MSW management system force some system parameters should be treated as the uncertain variables for ensuring the management model is more closing the real situation. For example, the waste-generation amounts are affected by many factors, including the economic output, population amounts and legal regulation; meanwhile, it also exerts a significant influence on the generations of allocation schemes, thus, they need to be expressed as the uncertain forms through data collection and analysis, and human judgments. This fact leads to a large number of inexact optimization techniques were developed to deal with the MSW management problems under uncertainty (Chang and Wang 1997; Chang and Davila 2007, 2008; Cheng et al. 2009; Fan et al. 2009, 2012, 2014a, b; Greene and Tonjes 2014; Huang et al. 1992, 1993, 1995; Li et al. 2009; Maqsood and Huang 2013; Nie et al. 2006; Pariatamby and Tanaka 2014; Ravindra et al. 2015; Rigamonti et al. 2016; Terazono et al. 2015; Ulfik and Nowak 2014; Xu et al. 2009a, 2009b, 2010a, b, 2012a, 2014a, b; Zou et al. 2000). Currently, existing uncertain optimization techniques were divided to the three types: stochastic mathematical programming (SMP), fuzzy mathematical programming (FMP) and interval linear programming (ILP) (Cheng 2013; Ellis et al. 1985, 1986; Li 2012; Li et al. 2012; Zhou et al. 2015). Among them, the stochastic chance-constrained programming (SCCP) approach, which was firstly proposed by Charnes et al. (1972), was extensively applied in many management fields due to its low requirement in the constraints satisfaction and less computational burden in solving model (Charnes and Cooper 1983; Morgan et al. 1993; Huang 1998; Liu et al. 2003; Li et al. 2007; Xu et al. 2012b). For example, Huang (1998) proposed an integrated optimization model of ILP and SCCP methods for supporting the water quality management within an agricultural system, where interval solutions under various constraints-violation levels were obtained. Xu et al. (2012b) developed an intervalparameter stochastic chance constrained programming (IPSCCP) model and applied it to handle the urban water supply problem. The results generated by the IPSCCP model were effective in helping decision makers establish rational water supply patterns under complex uncertainties. However, SCCP also has some limitations in handling the random variables, where they must be expressed as traditional random variables with normal distributions. In fact, in order to better describe the uncertain parameters, comprehensive consideration in many estimations and opinions from a variety of respondents with different backgrounds is necessary. For example, it is firstly assumed that the waste-generation amounts n are expressed as the random variables with the normal distributions, i.e. n * N (m, d2), where m and d denotes the mean value and standard deviation, respectively. Due to various survey or estimation results from n group of respondents, n groups of random variables could be obtained, i.e. (m1, d1) (m2, d2),…,(mn, dn), such that the m and d values are more suitable to be random variables based on above collected data information, rather than the fixed values as the traditional random variables (Xu et al. 2014a). Referring to the Peng and Liu (2007), the waste-generation amounts presenting as the dual-random characteristics are defined as the birandom variables. Birandom variable, which is firstly developed by Peng and Liu (2007), is mainly used to reflect the parameters with dual-random nature. The reason for describing those parameters as the ‘‘birandom’’ is mainly due to the fact that the parameters themselves not only are assumed as the random variables, but also their characteristic values for reflecting the probabilistic distribution also follow the random distribution (Peng and Liu 2007; Xu and Tao 2012). Currently, the applications of birandom programming were relatively limited. For example, Peng and Liu (2007) mainly introduced the definition of birandom variable and described the solution procedure of birandom programming. Finally, some numerical examples were established for illustrating the feasibility and effectiveness of solution

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algorithm. Xu and Tao (2012) developed a multi-objective decision-making model with birandom parameters and applied it to solve a hydropower station operation problem. It is concluded that developed model owns the potential in tackling more types of practical problems. The major advantage owned by birandom variable is that it is capable of utilizing data information sufficiently and reflecting estimation opinions from various respondents well. However, the application in of birandom variables in MSW management field is lack. Therefore, as the first attempt, a stochastic equilibrium chance-constrained programming (SECCP) is developed for supporting the solid waste management in the Dalian Development Zone of the City of Dalian, China. The rest of this paper is organized as follows. The Sect. 2 introduces current situation and existing problems of the MSW management system in studied region. The regional waste management model, some definitions and concepts of birandom variables and solution algorithm for the SECCP model are included in the Sect. 3. In the Sect. 4, the variation trend in obtained solutions and potential improvement are analyzed and discussed. Finally, a conclusion is given in the last section.

2 Case study 2.1 Overview of the regional MSW system Dalian Development Zone (DDZ) was established in 1984 and is located on the southern part of Liaodong Peninsula in China bordering Jinzhou District to the west and north part and Yellow sea to the east and south part with a geographic coordinates of 38560 4100 – 39120 3000 N and 121420 3000 –12290 49.600 E. DDZ is about 392 km2 with a length of 38 km from east to west and width of 27 km from south to north. The population amounts are nearly 500 million. The Gross domestic product (GDP) in this district reached 170.30 billion RMB by 2014 approximately. Because of its unique geographical advantages, sound industrial base and government’s policy support, DDZ is the first national development zone approved to be established by the State Council in September 1984. After the development and construction in the past 30 years, its economical scale has reached the development level of middle-size cities in China, which ranks the fourth in Province Liaoning and brings great impetus to the economic development of Province Liaoning. With rapid expansion and development of DDZ, recently, the population amount and economical scales are experiencing a continuous increase which has resulted in increasing production of the MSWs in DDZ. Conversely, the treated capacities provided by existing facilities in the studied region are limited and there is a small-scale incinerator with the full-load operation. A large amount of untreated wastes occur, which lead to the severe pollution imposed to atmosphere, water body and soil and has become an important barrier affecting the urban development, environmental protection and residents’ life. Therefore, how to understand and analyze system structure and components comprehensively and thoroughly, formulate and solve the MSW management model accurately are critical for a sustainable development of DDZ. After a series of survey and analysis, the MSW management system of DDZ is introduced and summarized as follows:

2.1.1 Waste generation and composition The MSW management system of DDZ is composed of ten sub-districts, which included Maqiaozi, Dagushan, Haiqingdao, Wanli, Dongjiagou, Jinshitan, Desheng, Dalijia,

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Baosuiqu and Dengsahe, respectively. As shown in survey results, the daily average generation amount in DDZ was 733.4 ton in 2014 and the total generation rate of the wastes are almost 0.268 million ton, which will reach 0.595 million ton by 2020 based on some analytical and predicted results. The waste generated in DDZ can be divided into recyclables, compostable waste, hazardous waste, mixed waste and other waste. In detail, the compositions of wastes mainly include plastics, rubber, textiles, glasses, metals, papers, organics and battery, respectively.

2.1.2 Waste collection, transportation and treatment There are three processes of the waste management system in the studied district, including collection, transfer and treatment. The collection system is a process when the sanitation workers transport the garbage generated from the specified districts to regulated transfer stations. Recently, the common ways is the classified collection. During the collection process, some factors, such as the waste features, climatic characteristics and economic conditions, should be taken into the consideration. Transfer system is a process when wastes from each collection center are transported by the garbage trucks to transfer stations for the storage and pretreatment. Currently, there are seven transfer stations have been used, including three stations in Maqiaozi, two stations in Wanli and two stations in Dagushan, respectively. Six stations are under construction in order to satisfy treatment requirements, including two stations in Maqiaozi, two stations in Haiqingdao, two stations in Jinshitan, respectively. The treatment system is a process when the garbage from the transfer stations is disposed through physical, chemical or biological techniques in order realize the three main treatment principles, i.e. reduction, reuse and recycle. Now, approximately 90 % of the wastes would be buried in the landfill facility located in outside the DDZ, and 10 % of which would be transported to existing small-scale incinerator in the DDZ.

2.1.3 Analysis and estimation of the complexity and uncertainty As mentioned in above sections, the whole MSW system involved many processes, such as waste generation, collection, transportation and treatment. Moreover, these processes are associated with many factors, such as waste-generation rate, treatment capacities, as well as transportation and operational cost. Their interrelationships and interaction may lead to some difficulties in designing and determining the parameter values; meanwhile, the possible bias and error in the data collection and calculation processes also would bring about inevitable uncertainty.

2.2 System parameters and data information In this MSW management system, as described in the Sect. 1, the waste-generation rates in ten districts are assumed to present the dual-random natures because they are affected by many natural, social, economical and environmental factors. In fact, this uncertaintyexpression way is useful for identifying the parameters more accurately through taking into account of survey or estimation results from respondents with various backgrounds. Similarly, the treatment capacities of existing incinerator and designed safety factors for assuring the waste flows can be treated completely also are subjected to human’s judgments, which also are expressed as the birandom variables. Table 1 describes their

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M. Zhou et al. Table 1 The birandom variables included in this MSW management system Transfer stations

Planning period K=1

K=1

K=1

The waste-generation amounts of ten districts (ton/day) Maqiaozi

N (l, 11.22), l * N (157, 10.33)

N (l, 12.66), l * N (254, 11.65)

N (l, 14.07), l * N (361, 12.95)

Dagushan

N (l, 5.66), l * N (90.78, 5.21)

N (l, 7.79), l * N (172.04, 7.17)

N (l, 9.50), l * N (255.51, 8.74)

Haiqingdao

N (l, 5.90), l * N (98.6, 5.43)

N (l, 6.77), l * N (129.88, 6.23)

N (l, 7.55), l * N (161.5, 6.95)

Wanli

N (l, 7.35), l * N (153, 6.76)

N (l, 8.73), l * N (215.9, 8.03)

N (l, 9.95), l * N (280.5, 9.15)

Dongjiagou

N (l, 3.87), l * N (42.5, 3.56)

N (l, 6.10), l * N (105.4, 5.61)

N (l, 7.75), l * N (170, 7.13)

Jinshitan

N (l, 3.87), l * N (42.5, 3.56)

N (l, 6.00), l * N (102, 5.52)

N (l, 7.43), l * N (156.4, 6.84)

Desheng

N (l, 2.18), l * N (13.43, 2.00)

N (l, 2.66), l * N (20.06, 2.45)

N (l, 3.12), l * N (27.54, 2.87)

Dalijia

N (l, 2.32), l * N (15.30, 2.14)

N (l, 2.76), l * N (21.59, 2.54)

N (l, 3.19), l * N (28.90, 2.94)

Baosuiqu

N (l, 4.24), l * N (51.00, 3.90)

N (l, 5.08), l * N (73.10, 4.67)

N (l, 5.74), l * N (93.50, 5.28)

Dengsahe

N (l, 1.45), l * N (5.95, 1.33)

N (l, 1.87), l * N (9.86, 1.72)

N (l, 2.22), l * N (13.94, 2.04)

The treatment capacities of the existing incinerator (ton/day) Incinerator

N (l, 110), l * N (1440, 95)

N (l, 110), l * N (1440, 95)

N (l, 110), l * N (1440, 95)

The safety coefficient for realizing the full treatment of the generated wastes Safety coefficient

N (l, 0.03), l * N (0.97, 0.01)

The variables included in this table are the birandom variables

probability distribution information. Compared with rapid increase of the waste-generation amounts, the treated capacities of existing facilities are insufficient. Therefore, existing small-scale incinerator in the studied district would be expanded; meanwhile, new landfill and composting plant should be constructed and are located in studied region. The economical parameters, including the transportation costs of wastes, operational costs, treatment revenues, expansion and construction costs of the facilities, are fluctuating in a small range and are thus assumed as the fixed values. The detailed parameter information is listed in Table 2. In fact, according to their uncertain characteristics, the parameters included in Table 2 are allowable to be expressed as discrete intervals. In order to reflect the applicability and feasibility of SECCP model, the interval uncertainty is not taken into consideration. Therefore, developed SECCP model in this study will be specifically suitable for tackling the MSW management problem.

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A stochastic equilibrium chance-constrained programming… Table 2 Economical parameters related to the system facilities

Facilities

Type

Costs and benefits (RMB) K=1

Landfill

Incinerator

Composting plant

Transfer station Facilities

Incinerator

Composting plant TC transportation cost, OC operational cost, OB operational benefit

K=3

TC

1.00

1.19

1.45

OC

44.40

48.00

52.80

OB

12.00

14.40

17.40

TC

1.00

1.19

1.45

OC

108.00

115.20

123.60

OB

70.00

75.00

81.00

TC

1.00

1.19

1.45

OC

66.00

72.00

80.40

OB

13.00

15.00

18.00

TC

1.00

1.19

1.45

OC

24.00

30.00

37.20

Option

Expansion or construction cost (9104 RMB) K=1

Landfill

K=2

K=2

K=3

1

16.80

14.40

11.40

2

19.20

16.80

13.80

3

22.80

20.40

17.40

1

144.00

120.00

90.00

2

168.00

144.00

120.00

3

204.00

180.00

156.00

1

600.00

360.00

144.00

2

960.00

600.00

252.00

3

1440.00

960.00

432.00

3 Formulation of solid waste management model In order to generate a cost-effective management strategy of complex MSW system, comprehensive understanding, in-depth analysis and appropriate simplification are necessary. With regard to the MSW system in this area, planning period (2015–2030) is divided into three periods with each period has 5 years; the wastes generated by ten districts are collected, transported and treated by existing incinerator, proposed new landfill and composting plant via the corresponding transfer stations. The system objective is to minimize the economical costs caused by the system operation while the waste-treatment demand and the treatment-facilities limitations are met. Moreover, some system parameters, including waste-generation amounts, treatment capacities and safety coefficient, are associated with random uncertainties and other parameters are expressed as the fixed values. Therefore, the stochastic optimization management model with birandom variables for supporting MSW management is developed (Qin et al. 2011; Xu et al. 2014b).

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3.1 The stochastic optimization model for waste management in studied region 3.1.1 Objective function

Min f ¼

J X I X K X

LEk XIjik ðDIji TRk þ TSk þ OIik Þ

j¼1 i¼1 k¼1

þ

J X C X K X

LEk XCjck ðDCjc TRk þ TSk þ OCck Þ

j¼1 c¼1 k¼1

þ

I X L X K X i¼1 l¼1

þ

C X L X K X c¼1 l¼1

þ

LEk YCclk ðDRCcl TRk þ OLlk RLlk Þ

k

L X U X K X l¼1 u¼1 k¼1

LEk YIilk ðDRIil TRk þ OLlk RLlk Þ

k

J X I X K X j¼1 i¼1 k¼1

ELluk BUluk þ

I X V X K X

EIivk BVivk þ

i¼1 v¼1 k¼1

LEk XIjik RIik

J X C X K X

C X W X K X

ECcwk BWcwk

c¼1 w¼1 k¼1

LEk XCjck RCck

j¼1 c¼1 k¼1

ð1aÞ where f is the total system cost ($); k is the index of time periods; LEk is the length of time period k (d); l, i, and c is the indexes of specific landfills, incinerators, and composting plants, respectively; j is the index of transfer stations; u is the index of construction options for landfill; v is the index of expansion options for incinerator; w is the index of construction options for composting plant; XIjik, and XCjck is the waste amounts from transfer station j to incinerator i and composting plant c (ton/day), respectively, which are decision variables; YIilk and YCclk is the residue flows sourced from incinerator i and composting plant c to landfill l (ton/day), respectively, which also are decision variables; BUluk, BVivk, BWcwk is the options whether construct the facilities under which scales for landfill l, incinerator i, and composting plant c, respectively, which are binary decision variables; DIji and DCjc is the distances from transfer station j to incinerator i and composting plant c (km), respectively; DRIil and DRCcl is the distances from incinerator i and composting plant c to landfill l (km), respectively; ELluk, EIivk, ECcwk is the expansion or construction costs for landfill l, incinerator i, and composting plant c ($/t), respectively; OLlk, OIik, and OCck is the operational costs for landfill l, incinerator i, and composting plant c ($/t), respectively; RLlk, RIik and RCck is the revenues for landfill l, incinerator i and composting plant c ($/t), respectively; TRk is the transportation cost ($/tkm); TSk is the operational cost for transfer stations ($/t). All parameters included in objective function are expressed as the fixed values. The objective function (1a) in model (1) is to minimize total system costs, which is equal to total costs minus revenue. The system costs include the transportation and operational costs for the transfer stations and the operational costs for the treatment facilities; the revenues are sourced from treatment process of the landfill, incinerator and composting facilities, where total costs and revenue generated by incinerator are PJ PI PK PJ PI PK and j¼1 i¼1 k¼1 LEk XIjik ðDIji TRk þ TSk þ OIik Þ j¼1 i¼1 k¼1 LEk XIjik RIik ,

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respectively; the costs and revenue of wastes allocated to the composting plant are PJ PC PK PJ PC PK j¼1 c¼1 k¼1 LEk XCjck ðDCjc TRk þ TSk þ OCck Þ and j¼1 c¼1 k¼1 LEk XCjck RCck , respectively; the difference between the costs and revenues of wastes transported from PC PL PK incinerator to landfill is c¼1 l¼1 k¼1 LEk YCclk ðDRCcl TRk þ OLlk RLlk Þ; the difference between the costs and revenues of wastes from composting plant to landfill is PI PL PK described as i¼1 l¼1 k¼1 LEk YIilk ðDRIil TRk þ OLlk RLlk Þ; the construction and expansion costs of the landfill, incinerator and composting plant are expressed as PL PU PK PI PV l¼1 u¼1 k¼1 ELluk BUluk þ PK PC PW PKi¼1 v¼1 k¼1 EIivk BVivk þ c¼1 w¼1 k¼1 ECcwk BWcwk .

3.1.2 Constraints of treatment/disposal capacity K X k¼1

LEk

X I X L

YIilk þ

C X L X

i¼1 l¼1

X L X U X K YCclk DCLluk BUluk ;

c¼1 l¼1

J X I I I X V X k X X X e g e g ð xÞ þ b ðxÞ XIjik DCIivk0 BVivk0 CAI i j¼1 i¼1

XCjck

8k;

ð1cÞ

i¼1 v¼1 k0 ¼1

i¼1

J X C X

ð1bÞ

l¼1 u¼1 k¼1

C X W X k X

DCCcwk0 BWcwk0

8k;

ð1dÞ

c¼1 w¼1 k0 ¼1

j¼1 c¼1

where CAIi is the existing capacities of incinerator i (ton/day), which are assumed as the birandom variables; DCLluk, DCIivk and DCCcwk is the capacity-construction amounts for landfill l and composting plant c, capacity-expansion amounts for incinerator i (ton/day), e e ðxÞ is the safety coefficient for respectively, which are described as the fixed values; b ensuring the wastes allocated to the incinerator is treated completely, which is a birandom variable. The constraints (1b) to (1d) require the allocated waste amounts to the three facilities must be less than or equal to the treatment capacities, respectively.

3.1.3 Constraints for treatment demand I X

XIjik þ

i¼1

C X

g gjk ; XCjck GW

8j; k;

ð1eÞ

c¼1

where GWjk is waste amount generated in various blocks covered by transfer station j (ton/day), which is expressed as the birandom variable. The constraints (1e) are used to P P ensure the treated amounts of two facilities Ii¼1 XIjik þ Cc¼1 XCjck are more than or equal g gjk . to the generated amounts of ten sub-districts GW

3.1.4 Mass balance equations L X l¼1

YIilk ¼

J X

XIjik FIk ;

8i; k;

ð1fÞ

j¼1

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M. Zhou et al. L X

YCclk ¼

J X

XCjck FCk ;

8c; k;

ð1gÞ

j¼1

l¼1

where FIk and FCk are residue rates from incinerator and composting plant to landfill (%), respectively. The constraints (1f) and (1g) show the relationships between the waste amounts and the residues from incinerators and composting plants, respectively.

3.1.5 Constraints for capacity-expansion options and non-negativity variables

0

U X K X

BUluk 1; 0

u¼1 k¼1

V X

BVivk 1; 0

v¼1

W X K X

BWcwk 1;

8l; u; i; v; c; w; k;

w¼1 k¼1

ð1hÞ BUluk , BVivk , and BWcwk are integers; XIjik 0; XCjck 0;

8i; j; c; l; k;

ð1iÞ

where the constraints (1h) regulate the expansion can only happen once during the planning period. The constraints (1i) required the decision variables must be greater or equal to zero.

3.2 Methodology 3.2.1 Birandom variable with the normal distributions To better reflect random variable with twofold random characteristics, the birandom variable is firstly developed by Peng and Liu (2007). Considering traditional random variables own many expression forms, including the normal, mean and exponential distributions, similarly, the birandom variables also should be expressed as various forms based on their random natures. Because the normal distribution is a usual form for reflecting the random features and is applied in many fields extensively (Huang 1998; Liu et al. 2003; Xu et al. 2012b), the birandom variables with the normal distributions are thus selected to be representative. For any x, n(x) is a birandom variable with normal distribution and is expressed as N ðlðxÞ; r2 ðxÞÞ, where l(x) and r(x) also are the random variables, respectively.

3.2.2 Stochastic equilibrium chance-constrained programming Stochastic equilibrium chance-constrained programming (SECCP) model was firstly proposed by Peng and Liu (2007), which was used to solve the SMP model involving birandom variables. The main focus of the SECCP model is to find the quantitative measures of occurred chance of a birandom event for comparing the occurred degree of two birandom events. Through the comparison of existing chance measures, the equilibrium chance measure is considered as representative since it is a real number and is helpful in assisting decision makers to rank the decisions through utilizing the natural order of the real numbers (Tao and Xu 2013). Referring to Peng and Liu (2007) and Tao and Xu (2013), a SECCP model could be written as:

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Minimize f ¼ CX

ð2aÞ

e e e ðxÞX B e ðxÞ A

ð2bÞ

e e e ðxÞ e ðxÞX E D

ð2cÞ

X0

ð2dÞ

e e ðxÞ 6¼ 0 e e D C; AðxÞ;

ð2eÞ

Subject to:

where the objective function f, decision variable X and coefficient C are expressed as the e e e e e ðxÞ, B e ðxÞ are e ðxÞ and E e ðxÞ, D fixed values. With respect to other auxiliary coefficients, A presented as the birandom variables, where they follow the normal distributions, i.e. e e e e e ðxÞ N A e ðxÞ; r2 , B e ðxÞ N B e ðxÞ; r2B , D e ðxÞ N D e ðxÞ; r2D e ðxÞ A and E A e ðxÞ; r2E , respectively. The reason called them as the birandom variables is that their N E mean values are the random variables with normal distributions, which are expressed as e ðxÞ N lA ; r2 0 , B e ðxÞ N lB ; r2B0 , D e ðxÞ N lD ; r2D0 and E e ðxÞ N lE ; r2E0 , A A respectively. The critical step of solving the model (2) is to convert the random constraints (2b) and (2c) into their fixed equivalents. Based on the equilibrium chance measure, constraints (2b) and (2c) can be reformulated as follows (Peng and Liu 2007; Tao and Xu 2013): n o e e e e e ðxÞX B e ðxÞX B e ðxÞ , Che A e ðxÞ 1 ar A n o n o e e e ðxÞX B e ðxÞ 1 ar 1 ar , Pr x 2 XPr A ð3Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , lA X þ U1 ðar Þ ð X ÞT rA X þ ðrB Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ U1 ðar Þ ðX ÞT rA0 X þ ðrB0 Þ2 lB ; 8r e e e ðxÞX E e ðxÞ D n o e e e ðxÞ 1 ar e ðxÞX E , Che D n o n o e e e ðxÞ 1 ar 1 ar e ðxÞX E , Pr x 2 XPr D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , lD X þ U1 ðar Þ ð X ÞT rD X þ ðrE Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ U1 ðar Þ ð X ÞT rD0 X þ ðrE0 Þ2 lE ; 8r

ð4Þ

where Che is the chance of the event {}; ar are allowable probability-violation levels, where r is the type of probability-violation level. The term U1 ðar Þ is the inverse function of cumulative distribution function of standard normally distributed variable. Based on the constraints (3) and (4), the model (2) is converted into the deterministic model and is rewritten as follows: Minimize f ¼ CX

ð5aÞ

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Subject to: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lA X þ U1 ðar Þ ð X ÞT rA X þ ðrB Þ2 þ U1 ðar Þ ðX ÞT rA0 X þ ðrB0 Þ2 lB ;

8r

ð5bÞ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lD X þ U1 ðar Þ ð X ÞT rD X þ ðrE Þ2 þ U1 ðar Þ ð X ÞT rD0 X þ ðrE0 Þ2 lE ;

8r

ð5cÞ

X0

ð5dÞ

e e e ðxÞ; D e ðxÞ 6¼ 0 C; A

ð5eÞ

Finally, the solutions of the objective values and decision variables under different ar values are obtained, which effectively the trade-off between system economy and reliability, where the term ‘‘system economy’’ is mainly used to evaluate the model performance in the economical aspect. Generally, it is defined as the high ‘‘system economy’’ while the solutions generated from SECCP model lead to the low system cost or high system benefit; conversely, the solutions with the high cost or low benefit are provided by SECCP model and is called as the low ‘‘system economy’’. Figure 1 shows the general framework of formulating and solving SECCP model, where the commercial optimization software LINGO 12.0 is used to solve the proposed model. The SECCP model runs very quickly with the support of software LINGO 12.0, where the computational time of solving optimization model is within a few seconds. Such a short operation time will be favorable in generating a variety of solutions under various probability-violation levels and helping decision makers identify final decision schemes. The procedures of formulating and solving a SECCP model are summarized as follows:

Waste-generation rates of ten blocks

The safety coefficients for treatment facilities

The options whether construct the facilities

The economical and distance parameters

Birandom variables with normal distribution

Birandom variables with normal distribution

Binary variables (0-1)

Fixed values

Stochastic birandom Programming

Mixed Integer Programming

Stochastic equilibrium change-constrained programming model Equilibrium change mesure

Trade-off between system economy and reliability

Optimal solutions under various probability levels by SECCP model

Generation of decision alternatives

Fig. 1 General framework of the SECCP model

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Step 1

Step 2

Step 3 Step 4 Step 5

Investigate, collect and analyze the data information associated with the MSW management system and design the system parameters as the birandom variables and the deterministic values. Accomplish comprehensive understanding, in-depth analysis and appropriate simplification about this management system and determine the objective function and constraints in the MSW management model. Formulate SECCP model for the MSW management based on steps 1 and 2. Convert the random constraints with birandom variables into their respective fixed equivalents based on equilibrium chance-constrained algorithm. Solve deterministic model and generate final solutions of objective function and decision variables under various probability-violation levels, respectively.

4 Result analysis and discussion To better reflect the properties and characteristics of proposed SECCP model, the comparison of generated solutions under various constraints-violation levels (i.e. ar values) and their variation-trend analysis are necessary, where the selection and determination of ar value are mainly depending on the attitude of decision makers to the trade-off between system economy and reliability. The decision schemes under the high ar values are useful in ensuring the random constraints are satisfied; meanwhile, the high system costs are unavoidable. Conversely, the low system costs and system reliability are associated with the low ar value. Therefore, the range of the ar value should be sufficiently wide for providing more decision spaces to local managers. Tables 3 and 4 listed the solutions of the objective function values and decision variable under ar values at 0.01, 0.05 and 0.1, which are expressed as the integer and numeric types, respectively. As shown in Table 3, the variation in the constraints-violation levels would lead to the various facilities-expansion alternatives, where the expansion amounts and expansion periods are distinct. For example, the incinerator is expanded with the increment of 240 ton/day at ar = 0.1 in the period 3; correspondingly, when ar value equals to 0.05, the expansion amounts of the incinerator in the period 3 reach 480 ton/day. As for the ar of 0.01, the incinerator would be expanded twice at periods 2 and 3, with each having the same increment of 360 ton/day, respectively. Similar difference also is happening in the construction schedule of the composting plant. For instance, at two violation levels of 0.05 and 0.1, the designed amounts of the composting plant in the period 1 are the same, being 660 ton/day, respectively; correspondingly, under the ar = 0.1, the composting plant owns the capacities of 828 ton/day in the period 1. The major reasons leading to the variation are that the increase of the ar value means the constraints with the birandom variables would be strict, such that the waste-generation amounts would increase and the existing capacities of incinerator become smaller. Thus, under the low violation level, the expansion or construction period of treatment facilities is designed as the earlier one and the expansion or construction amounts are the largest. In fact, the earlier expansion or construction schedule with the large expansion or construction amounts is more safe and reliable for the waste treatment, although the high expenses are inevitable. Conversely, the time-lagged and the low-increment alternatives are advantageous in the low cost; nevertheless, the system-failure risk would increase. This reflects the trade-off between system economy and reliability.

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M. Zhou et al. Table 3 Integer solutions of the SECCP model under various ar levels Facilities

Probability level

Option

Planning period K=1

Landfill

Incinerator

Composting plant

K=2

K=3

p = 0.01

u=2

1

0

0

p = 0.05

u=2

1

0

0

p = 0.1

u=2

1

0

0

p = 0.01

u=2

0

1

1

p = 0.05

u=3

0

0

1

p = 0.1

u=1

0

0

1

p = 0.01

u=2

1

0

0

p = 0.05

u=1

1

0

0

p = 0.1

u=1

1

0

0

The construction or expansion options are expressed as 1 or 0, representing yes or no answers. The options which are not listed in this table are 0

Table 4 Numerical solutions of the SECCP model under various ar levels Districts

Facilities

Solutions from SECCP model (103 tonne/day) ar = 0.01 K=1

i=1 i=2 i=3 i=4 i=5 i=6 i=7 i=8 i=9 i = 10

ar = 0.05

K=2

ar = 0.1

K=3

K=1

K=2

K=3

K=1

K=2

K=3 462.38

j=1

180.12

269.67

423.67

196.89

291.30

436.69

218.98

346.46

j=2

77.35

127.71

142.81

33.49

66.51

71.66

0

0

20.23

j=1

85.21

152.98

248.04

83.05

148.16

227.84

151.85

215.18

j=2

58.07

109.43

140.01

43.39

84.84

115.86

33.07

68.00

107.70

j=1

96.08

122.99

169.46

96.10

122.38

159.66

102.06

129.17

154.03

j=2

59.26

78.34

81.61

41.49

56.39

62.28

27.81

39.79

54.04

j=1

149.66

205.93

299.04

152.90

209.35

288.30

166.63

226.72

283.65

j=2

86.63

122.83

131.62

57.94

84.59

94.59

33.81

52.93

89.06

j=1

40.46

90.47

159.98

37.94

84.43

140.32

37.88

83.57

126.29

j=2

30.83

73.68

100.56

24.02

60.31

89.38

19.72

52.23

89.06

j=1

33.06

73.19

124.64

27.27

60.98

97.96

24.00

53.93

76.85

j=2

38.09

85.87

113.80

34.50

79.15

112.93

33.35

77.45

122.06

85.80

j=1

12.27

16.74

26.38

9.73

13.23

19.35

8.30

11.16

13.54

j=2

14.24

20.28

22.18

12.58

18.33

22.35

11.98

17.78

25.07

j=1

10.92

14.30

22.98

6.84

8.84

13.64

3.88

4.73

4.55

j=2

18.80

25.44

27.67

18.42

25.32

30.36

19.28

26.88

36.93

j=1

49.31

67.21

95.11

47.27

64.04

85.18

48.14

64.74

78.26

j=2

34.83

49.66

53.23

26.22

38.62

44.84

20.44

31.39

43.06

j=1

5.16

7.54

12.83

3.05

4.52

7.44

1.59

2.35

2.46

j=2

8.87

13.26

14.50

8.48

12.91

15.76

8.74

13.47

19.03

Total system costs

1199.64 (9 106 $)

1046.52 (9 106 $)

974.22 (9 106 $)

The symbol i represents the transfer station; the symbol j represents the treatment facilities, where j is the incinerator and composting plant under j = 1 and j = 2, respectively

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The change in violation level not only exerts the influence on the decision variables expressing as the integer forms, but also on the numerical type of decision variables. From Table 4, as the increase of violation level, the waste amounts allocated to the incinerator would be unstable; conversely, the treatment amounts of the composting plant would remain decrease. For example, as for the Dalijia district, the treatment amounts of the incinerator in the period 2 are 14.30, 8.84 and 4.73 ton/day, respectively. The waste amounts allocated to the incinerator in the Baoshui district at the period 3 are 95.11, 85.18 and 78.26 ton/day, respectively. Conversely, the disposal amounts in the Maqiaozi district over three periods would increase. In the period 1, they are 180.12, 196.89 and 218.98 ton/day, respectively; the amounts in the period 2 are 269.67, 291.30 and 346.46 ton/day, respectively; the amounts in the period 3 are 423.67, 436.69 and 462.38 ton/day, respectively. As for the Wanli district, the treatment amounts in the periods 1 and 2 would increase, being 149.66, 152.90 and 166.63 ton/day, and 205.93, 209.35 and 226.72 ton/day, respectively. The treatment amounts in the period 3 would decrease, being 299.04, 288.30 and 283.65 ton/day, respectively. This is due to the fact that the increase of violation level leads to existing capacities of incinerator would increase; nevertheless, the waste-generation amounts would decrease. Thus, the variation trend of treatment amounts in the incinerator would be unstable. Conversely, the allocated amounts to composting plant become smaller due to the decrease of waste-generation amounts. For instance, over three periods, the disposal waste amounts of composting plant in the Maqiaozi district would decrease where three groups of solutions are 77.35, 33.49 and 0 ton/day; 127.71, 66.51 and 0 ton/day; 142.81, 71.66 and 20.23 ton/day, respectively. Similarly, three groups of management patterns for the Dagushan district are 58.07, 43.39 and 33.07 ton/day; 109.43, 84.84 and 68.00 ton/day; 140.01, 115.86 and 107.70 ton/day, respectively. In order to further reflect the change of allocated amounts under various violation levels, the summation amounts of all districts at three planning periods are calculated and their variation trends are shown in Fig. 2. As demonstrated in Fig. 2, as the increase of violation level, the amounts allocated to the incinerator are still unstable due to the opposite influences of increased treated capacities of the incinerator and decreased waste-generation amounts. In detail, they would increase in the first two periods, being 662.24, 661.07 and 697.27 ton/day in period 1, respectively; being 1021.02, 1007.21 and 1074.68 ton/day in period 2, respectively. Conversely, the treatment amounts in the period 3 would decrease, being 1582.12, 1476.38 and 1417.19 ton/day, respectively. As for the treatment amounts of composting plant, they would decrease stably under the influence of the decrease in the waste-generation amounts. In the period 1, the treatment amounts are 426.97, 300.54 and 208.21 ton/day, respectively; in the period 2, they reach 706.52, 526.98 and 379.92 ton/ day, respectively; in the period 3, they are 828, 660 and 593.69 ton/day, respectively. Apart from the decision variables, the impacts on obtained solutions caused by the determination of the probability-violation level are also reflected in the objective function value. As shown in Fig. 3, total system costs under ten probability levels from 0.01 to 0.1 are 1199.64, 1136.21, 1097.11, 1068.48, 1046.52, 1027.97, 1012.32, 998.15, 985.68 and 974.22 9 106$, respectively. The reason leading to the decrease of total costs is because that the increase of violation level means the waste-generation amounts would decrease; meanwhile, the treated capacities of the incinerator would increase, such that the total waste-treated amounts would decrease; meanwhile, the treatment patterns would be optimal due to the increase of disposal capacities, where more wastes are allocated to the incinerator with the low treatment, operational and expansion costs. Therefore, the total system costs would decrease as the increase of violation level. The trade-off between

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The waste amounts allocated to the two facilities under probability-violation level of 0.01 The waste amounts allocated to the two facilities under probability-violation level of 0.05

Waste-treatment amounts ( t/d)

The waste amounts allocated to the two facilities under probability-violation level of 0.1

0.01

1800 1600

Composting plant

1400 1200

0.1

1000

0.05

800

Incinerator

600 400 200 k=1

k=2

k=3

k=1

k=2

k=3

Planning period

Fig. 2 The summation of optimized waste-treated amounts in the various districts

7& 7&

αU

7& αU

αU

7& αU

7&

αU αU

7&

αU αU αU αU

7&

7&

7&

7&

αU probability-violation level TC = total cost

Fig. 3 System cost under various ar levels

system economy and reliability is evaluated through the adjustment and selection of violation level. The high violation level is corresponding to the low total waste-treatment amounts, high system economy and high system-failure risk and vice versa. Generally, the decision makers are preferable in the compromise alternative for ensuring the balance between the system economy and system-failure risk as possible. Therefore, the schemes under the medium levels (i.e. 0.04, 0.05 and 0.06) are more appropriate. In this study, two critical problems in the MSW management system are solved: (i) how to expand the capacities of existing facility or construct new facilities for meeting increasingly stringent treatment requirement; (ii) how to allocate the wastes to various facilities for realizing the minimization of total system costs. As shown in above

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comparison and analysis of obtained solutions, it can be seen that developed SECCP model is advantageous in both aspects of theory and practical application. On the one hand, it improved traditional SMP theory through designing and expressing the system parameters with dual-random characteristics as birandom variables, where the estimations and opinions from various respondents are incorporated into the uncertainties’ identification sufficiently. The equilibrium chance-constrained programming algorithm is used to solve the SECCP model, which is capable of tackling birandom variables and is overcoming limitations of traditional SCCP while random parameters with normal distribution are required strictly. This will ensure SECCP model has a clear practical background. On the other hand, the application of SECCP model in the MSW management system demonstrated model practicability, where the solutions under various violation levels are considered as the decision base, which are effective in making the decision-making process is more flexible, providing wide decision spaces to the local managers and reflecting the trade-off between system economy and reliability. However, SECCP model still needs to be further improved in some aspects for tackling more-complex management problems in the future. Firstly, as a common probabilistic distribution form, the normal distribution always plays an important role in describing and reflecting the random-uncertain characteristics; nevertheless, it also may be invalid and may affect the accuracy and feasibility of obtained solutions when random variables in the optimization model exhibit other uncertain characteristics. For example, the frequent occurrence of earthquake disaster should be expressed as the gamma distribution. As for the rainfall amounts, they follow the log-normal distribution. Therefore, it is necessary that other types of random distribution forms should be incorporated into SECCP framework in order to better express the random uncertainties of system parameters Secondly, as major braches of uncertain analysis approaches, FMP and IMP are suitable in handling the uncertain parameters with limited data information. The integration of other types of uncertain optimization methods and SECCP is beneficial in avoiding the over-high parameters requirements and over-complex computational process and reflecting the diversity of the uncertain parameters. Thirdly, nonlinear programming (NP), as a type of important optimization technique, has potential to be incorporated into SECCP for tackling more-complex management problem. Within MSW management system, NP is useful in reflecting the economy-of-scale feature of treatment cost through designing nonlinear objective function. Moreover, effective solution algorithms proposed by Huang and Chen (2001) and Qin et al. (2007) also provided well demonstration for the integration of SECCP and NP in the future. Finally, the design and variation in ar value enrich the solution space and complicates the decision making process simultaneously. The Multi-criteria decision analysis (i.e. MCDA) tools could potentially be integrated in the SECCP framework for tackling such a difficulty (Xu et al. 2010b).

5 Conclusion In this study, a stochastic equilibrium chance-constrained programming model was formulated for supporting the regional waste management of the City of Dalian, China. Because the system parameters, including waste-generation amounts, safety coefficient and treatment capacities, exhibit obvious dual-random characteristics in the parameters-identification process, the birandom variables are thus used to describe them. This expression way improved traditional SMP theory and the algorithm based on equilibrium chance

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measure overcomes the limitations of SCCP which is only capable of handling the parameters with normal distribution. A variety of solutions are obtained through adjusting the probability-violation levels, where the solutions under the medium level are recommended as potential decision alternative since it realized the balance between system economy and system-failure risk. The proposed model still needs to be improved for tackling more-complex issue in the future, such that the introduction of new probabilistic distribution forms for describing system parameters accurately, the incorporation of other types of uncertain optimization methods (i.e. FMP and IMP), optimization techniques (i.e. NP) and MCDA techniques. This study made the first attempt in applying SECCP model to the MSW management problem. The study results demonstrated that SECCP model is also applicable to many other environmental problems. Acknowledgments This research was supported by National Natural Science Foundation of China (No. 41401631, 41401192, 41201164), National 985 Project of Non-traditional Security at Huazhong University of Science and Technology, Wuhan Social Science Fund (No. 14007) and Humanity and Social Science Foundation of Ministry of Education of China (No. 13YJC630115). The authors deeply appreciate the anonymous reviewers for their insightful comments and suggestions which contributed much to improving the manuscript.

References Chang, N.B., Wang, S.F.: A fuzzy goal programming approach for the optimal planning of metropolitan solid waste management systems. Eur. J. Oper. Res. 32(4), 303–321 (1997) Chang, N.B., Davila, E.: Minimax regret optimization analysis for a regional solid waste management system. Waste Manag 27, 820–832 (2007) Chang, N.B., Davila, E.: Municipal solid waste characterizations and management strategies for the Lower Rio Grande Valley. Texas. Waste Manag 28, 776–794 (2008) Charnes, A., Cooper, W.W., Kirby, P.: Chance constrained programming: an extension of statistical method. In: Rustagi, J.S. (ed.) Optimizing Methods in Statistics, pp. 391–402. Academic Press, New York (1972) Charnes, A., Cooper, W.W.: Response to decision problems under risk and chance constrained programming: dilemmas in the transitions. Manag. Sci. 29, 750–753 (1983) Cheng, G.H., Huang, G.H., Li, Y.P., Cao, M.F., Fan, Y.R.: Planning of municipal solid waste management systems under dual uncertainties: a hybrid interval stochastic programming approach. Stoch. Environ. Res. Risk. A 23, 707–720 (2009) Cheng, H.W.: A satisficing method for fuzzy goal programming problems with different importance and priorities. Qual. Quant. 47(1), 485–498 (2013) Ellis, J.H., McBean, E.A., Farquhar, G.J.: Chance-constrained stochastic linear programming model for acid rain abatement-I. Complete and noncolinearity. Atmos. Environ. 19, 925–937 (1985) Ellis, J.H., McBean, E.A., Farquhar, G.J.: Chance-constrained stochastic linear programming model for acid rain abatement-II. Limited colinearity. Atmos. Environ. 20, 501–511 (1986) Fan, Y.R., Huang, G.H., Liang, Y.X., Nie, X.H.: Optimum management model of municipal solid waste management under dual uncertainty (in Chinese). Environ. Eng. 27, 371–374 (2009) Fan, Y.R., Huang, G.H., Li, Y.P.: Robust interval linear programming for environmental decision making under uncertainty. Eng. Optim. 44(11), 1321–1336 (2012) Fan, Y.R., Huang, G.H., Huang, K., Jin, L., Suo, M.Q.: A generalized fuzzy integer programming approach for environmental management under uncertainty. Math. Probl. Eng. 62(1), 72–86 (2014a) Fan, Y.R., Huang, G.H., Jin, L., Suo, M.Q.: Solid waste management under uncertainty: a generalized fuzzy linear programming approach. Civ. Eng. Environ. Syst. 31(4), 331–346 (2014b) Greene, K.L., Tonjes, D.J.: Quantitative assessments of municipal waste management systems: using different indicators to compare and rank programs in New York State. Waste Manag 34, 825–836 (2014) Huang, G.H., Baetz, B.W., Patry, G.G.: A grey linear programming approach for municipal solid waste management planning under uncertainty. Civ. Eng. Syst. 9, 319–335 (1992) Huang, G.H., Baetz, B.W., Patry, G.G.: A grey fuzzy linear programming approach for municipal solid waste management planning under uncertainty. Civ. Eng. Syst. 10, 123–146 (1993)

123

A stochastic equilibrium chance-constrained programming… Huang, G.H., Baetz, B.W., Patry, G.G.: Grey integer programming: an application to waste management planning under uncertainty. Eur. J. Oper. Res. 83, 594–620 (1995) Huang, G.H.: A hybrid inexact-stochastic water management model. Eur. J. Oper. Res. 107, 137–158 (1998) Huang, G.H., Chen, M.J.: A derivative algorithm for inexact quadratic program- application to environmental decision-making under uncertainty. Eur. J. Oper. Res. 128, 570–586 (2001) Li, T.F., Huang, T.S., Yang, M.F.: Application of fuzzy mathematical programming to imprecise project management decisions. Qual. Quant. 46(5), 1451–1470 (2012) Li, T.F.: Integrated manufacturing/distribution planning decisions with multiple imprecise goals in an uncertain environment. Qual. Quant. 46(1), 137–153 (2012) Li, Y.P., Huang, G.H., Nie, S.L., Qin, X.S.: ITCLP: an inexact two-stage chance-constrained program for planning waste management systems. Resour. Conserv. Recy. 49, 284–307 (2007) Li, Y.P., Huang, G.H., Yang, Z.F., Chen, X.: Inexact fuzzy-stochastic constraint-softened programming: a case study for waste management. Waste Manag 29, 2165–2177 (2009) Liu, L., Huang, G.H., Liu, Y., Fuller, G.A.: A fuzzy-stochastic robust optimization model for regional air quality management under uncertainty. Eng. Optim. 35(2), 177–199 (2003) Morgan, D.R., Eheart, J.W., Valocchi, A.J.: Aquifer remediation design under uncertainty using a new chance constrained programming technique. Water Resour. Res. 29, 551–568 (1993) Maqsood, I., Huang, G.H.: A two-stage interval-stochastic programming model for waste management under uncertainty. J. Air Waste Manag. 53(5), 540–552 (2013) Nie, X.H., Huang, G.H., Li, Y.P., Liu, L.: A hybrid interval-parameter fuzzy robust programming approach for waste management planning under uncertainty. J. Environ. Manag. 84, 1–11 (2006) Pariatamby, A., Tanaka, M.: Municipal Solid Waste Management in Asia and the Pacific Islands. Springer, Singapore (2014) Peng, J., Liu, B.: Birandom variables and birandom programming. Comput. Ind. Eng. 53, 433–453 (2007) Qin, X.S., Huang, G.H., Zeng, G.M., Chakma, A., Huang, Y.F.: An interval-parameter fuzzy nonlinear optimization model for stream water quality management under uncertainty. Eur. J. Oper. Res. 180(3), 1331–1357 (2007) Qin, X.S., Xu, Y., Su, J.: Optimizing trash-flow allocation through chance-constrained programming with trapezoidal-shaped fuzzy parameters. Environ. Eng. Sci. 28(8), 573–584 (2011) Ravindra, K., Kaur, K., Mor, S.: System analysis of municipal solid waste management in Chandigarh and minimization practices for cleaner emissions. J. Cleaner Prod. 89, 251–256 (2015) Rigamonti, L., Sterpi, I., Grosso, M.: Integrated municipal waste management systems: an indicator to assess their environmental and economic sustainability. Ecol. Indic. 60, 1–7 (2016) Tao, Z.M., Xu, J.P.: An equilibrium chance-constrained multiobjective programming model with birandom parameters and its application to inventory problem. J. Appl. Math. 2013, 1–12 (2013) Terazono, A., Oguchi, M., Iino, S., Mogi, S.: Battery collection in municipal waste management in Japan: challenges for hazardous substance control and safety. Waste Manag 39, 246–257 (2015) Ulfik, A., Nowak, S.: Determinants of municipal waste management in suistainable development of regions in Poland. Pol. J. Environ. Stud. 23(3), 1039–1044 (2014) Xu, J.P., Tao, Z.M.: A class of multi-objective equilibrium chance maximization model with twofold random phenomennon and its application to hydropower station operation. Math. Comput. Simul. 85, 11–33 (2012) Xu, Y., Huang, G.H., Qin, X.S., Cao, M.F.: SRCCP: a stochastic robust chance-constrained programming model for municipal solid waste management under uncertainty. Resour. Conserv. Recycl. 53, 352–363 (2009a) Xu, Y., Huang, G.H., Qin, X.S., Huang, Y.: SRFILP: a stochastic robust fuzzy interval linear programming model for municipal solid waste management under uncertainty. J. Environ. Inf. 14(2), 74–82 (2009b) Xu, Y., Huang, G.H., Qin, X.S., Cao, M.F., Sun, Y.: An interval-parameter stochastic robust optimization model for supporting municipal solid waste management under uncertainty. Waste Manag 30(2), 316–327 (2010a) Xu, Y., Huang, G.H., Qin, X.S.: An inexact fuzzy-chance-constrained air quality management model. J. Air. Waste Manag. 60(7), 805–819 (2010b) Xu, Y., Huang, G.H., Cao, M.F., Dong, C., Li, Y.F.: A hybrid waste-flow allocation model considering multiple stage and interval-fuzzy chance constraints. Civ. Eng. Environ. Syst. 29(1), 59–78 (2012a) Xu, Y., Huang, G.H., Xu, T.Y.: Inexact management modeling for urban water supply systems. J. Environ. Inf. 20(1), 34–43 (2012b) Xu, Y., Huang, G.H., Shao, L.G.: A solid waste management model with fuzzy random parameters. Civ. Eng. Environ. Syst. 31(1), 64–78 (2014a) Xu, Y., Huang, G.H., Xu, L.: A fuzzy robust optimization model for waste allocation planning under uncertainty. Environ. Eng. Sci. 31(10), 1–15 (2014b)

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M. Zhou et al. Zhou, M., Tan, S.K., Tao, L.Z.: An interval fuzzy land-use allocation model (IFLAM) for Beijing in association with environmental and ecological consideration under uncertainty. Qual. Quant. 49(6), 2269–2290 (2015) Zou, R., Lung, W.S., Guo, H.C., Huang, G.H.: Independent variable controlled grey fuzzy linear programming approach for waste flow allocation planning. Eng. Optim. 33(1), 87–111 (2000)

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