A review of inexact optimization modeling and its application to integrated water resources management

Frontiers of Earth Science

Volumn 9, Issue 1, 2015, Pages 51-64

  Wang, Ran ^a   Li, Yin ^b   Tan, Qian ^c,d  

^a LANZHOU UNIVERSITY   (China)

^b RESEARCH INSTITUTE OF HIGHWAY MINISTRY OF TRANSPORT   (China)

^c NORTH CHINA ELECTRIC POWER UNIVERSITY   (China)

^d UNIVERSITY OF REGINA   (Canada)

Author keywords

Fuzzy sets; Inexact optimization; Integrated water resources management; Stochastic; Uncertainty

Indexed keywords

COMPUTER SIMULATION; INTEGRATED APPROACH; NUMERICAL MODEL; OPTIMIZATION; RESOURCE MANAGEMENT; SOCIOECONOMIC CONDITIONS; WATER MANAGEMENT; WATER RESOURCE;

EID: 84939871471     PISSN: 20950195     EISSN: 20950209     Source Type: Journal    
DOI: 10.1007/s11707-014-0449-4     Document Type: Review

References (193)

1. Abu-Taleb M, Mareschal B (1995). Water resources planning in the Middle East: application of the PROMETHEE V multicriteria method. Eur J Oper Res, 81(3): 500–511
2. Ahmed S, King A J, Parija G (2003). A multi-stage stochastic integer programming approach for capacity expansion under uncertainty. J Glob Optim, 26(1): 3–24
3. Akter T, Simonovic S P (2005). Aggregation of fuzzy views of a large number of stakeholders for multi-objective flood management decision-making. J Environ Manage, 77(2): 133–143
4. Albornoz V M, Benario P, Rojas M E (2004). A two-stage stochastic integer programming model for a thermal power system expansion. Int Trans Oper Res, 11(3): 243–257
5. Alefeld G, Herzberger J (1983). Introductions to Interval Computations. New York: Academic Press, 67
6. Almadizar F, Ghazanfari M, Fatemi Ghomi S M T (2009). Application of chance-constrained programming for stochastic group shop scheduling problem. Int J Adv Manuf Technol, 42(3–4): 321–334
7. Arora S R, Gupta R (2009). Interactive fuzzy goal programming approach for bilevel programming problem. Eur J Oper Res, 194(2): 368–376
8. Aviso K B, Tan R R, Culaba A B, Cruz J B Jr (2010). Bi-level fuzzy optimization approach for water exchange in eco-industrial parks. Process Saf Environ Prot, 88(1): 31–40
9. Bao C, Fang C L (2007). Water resources constraint force on urbanization in water deficient regions: a case study of the Hexi Corridor, arid area of NW China. Ecol Econ, 62(3–4): 508–517
10. Bass B, Huang G H, Russo J (1997). Incorporation climate change into risk assessment using grey mathematical programming. J Environ Manage, 49(1): 107–123
11. Beale E M L (1955). On minimizing a convex function subject to linear inequalities. J R Stat Soc, B, 17(2): 173–184
12. Bellman R, Zadeh L A (1970). Decision-making in a fuzzy environment. Manage Sci, 17(4): 141–164
13. Ben Abdelaziz F, Masri H (2009). Multistage stochastic programming with fuzzy probability distribution. Fuzzy Sets Syst, 160(22): 3239–3249
14. Bender M J, Simonovic S P (2000). A fuzzy compromise approach to water resource systems planning under uncertainty. Fuzzy Sets Syst, 115(1): 35–44
15. Beraldi P, Musmanno R, Triki C (2000). Solving stochastic linear programs with restricted recourse using interior point methods. Comput Optim Appl, 15(3): 215–234
16. Birge J R, Louveaux F (1997). Introduction to Stochastic Programming. New York: Springer, 24
17. Birge J R, Louveaux F V (1988). A multicut algorithm for two-stage stochastic linear programs. Eur J Oper Res, 34(3): 384–392
18. Bosch P, Jofré A, Schultz R (2007). Two-stage stochastic programs with mixed probabilities. SIAM J Optim, 18(3): 778–788
19. Buckley J J, Feuring T (2000). Evolutionary algorithm solution to fuzzy problems: fuzzy linear programming. Fuzzy Sets Syst, 109(1): 35–53
20. Cai Y P, Huang G H, Lin Q G, Nie X H, Tan Q (2009c). An optimization-model-based interactive decision support system for regional energy management systems planning under uncertainty. Expert Syst Appl, 36(2): 3470–3482
21. Cai Y P, Huang G H, Lu H W, Yang Z F, Tan Q (2009d). I-VFRP: an interval-valued fuzzy robust programming approach for municipal waste management planning under uncertainty. Eng Optim, 41(5): 399–418
22. Cai Y P, Huang G H, Nie X H, Li Y P, Tan Q (2007). Municipal solid waste management under uncertainty: a mixed interval parameter fuzzy-stochastic robust programming approach. Environ Eng Sci, 24(3): 338–352
23. Cai Y P, Huang G H, Tan Q (2009b). An inexact optimization model for regional energy systems planning in the mixed stochastic and fuzzy environment. Int J Energy Res, 33(5): 443–468
24. Cai Y P, Huang G H, Tan Q, Chen B (2011). Identification of optimal strategies for improving eco-resilience to floods in ecologically vulnerable regions of a wetland. Ecol Modell, 222(2): 360–369
25. Cai Y P, Huang G H, Tan Q, Yang Z F (2009e). Planning of community-scale renewable energy management systems in a mixed stochastic and fuzzy environment. Renew Energy, 34(7): 1833–1847
26. Cai Y P, Huang G H, Yang Z F, Lin Q G, Tan Q (2009a). Community-scale renewable energy systems planning under uncertainty — An interval chance-constrained programming approach. Renew Sustain Energy Rev, 13(4): 721–735
27. Cai Y P, Huang G H, Yang Z F, Sun W, Chen B (2009f). Investigation of public’s perception towards rural sustainable development based on a two-level expert system. Expert Syst Appl, 36(5): 8910–8924
28. Cai Y P, Huang G H, Yang Z F, Tan Q (2009g). Identification of optimal strategies for energy management systems planning under multiple uncertainties. Appl Energy, 86(4): 480–495
29. Campos L, Verdegay J L (1989). Linear programming problems and ranking of fuzzy numbers. Fuzzy Sets Syst, 32(1): 1–11
30. Cao C W, Gu X S, Xin Z (2009). Chance constrained programming models for refinery short-term crude oil scheduling problem. Appl Math Model, 33(3): 1696–1707
31. Carter N, Kreutzwiser R D, de Loë R C (2005). Closing the circle: linking land use planning and water management. Land Use Policy, 22(2): 115–127
32. Castelletti A, Pianosi F, Soncini-Sessa R (2008). Integration, participation and optimal control in water resources planning and management. Appl Math Comput, 206(1): 21–33
33. Chanas S, Kuchta D (1996). Multiobjective programming in optimization of interval objective functions — A generalized approach. Eur J Oper Res, 94(3): 594–598
34. Chang N B, Hernandez E A (2008). Optimal expansion strategy for a sewer system under uncertainty. Environ Model Assess, 13(1): 93–113
35. Chang N B, Wang S F (1995). A grey nonlinear programming approach for planning coastal wastewater treatment and disposal systems. Water Sci Technol, 32(2): 19–29
36. Chang N B, Wen C G, Chen Y L, Yong Y C (1996). A grey fuzzy multiobjective programming approach for the optimal planning of a reservoir watershed, Part A: theoretical development. Water Res, 30(10): 2329–2334
37. Charnes A, Cooper W W, Symonds G H (1958). Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil. Manage Sci, 4(3): 235–263
38. Chen M J, Huang G H (2001). A derivative algorithm for inexact quadratic program — Application to environmental decision-making under uncertainty. Eur J Oper Res, 128(3): 570–586
39. Chen S H, Wu J (2004). Interval optimization of dynamic response for structure with interval parameters. Comput Struc, 82(1): 1–11
40. Chen W Q, Sim M, Sun J, Teo C P (2010). From CVaR to uncertainty set: implications in joint chance-constrained optimization. Oper Res, 58(2): 470–485
41. Chen Y, Marc Kilgour D, Hipe K W (2006). Multiple criteria classification with an application in water resources planning. Comput Oper Res, 33(11): 3301–3323
42. Cheng S, Chan C W, Huang G H (2003). An integrated multi-criteria decision analysis and inexact mixed integer linear programming approach for solid waste management. Eng Appl Artif Intell, 16(5–6): 543–554
43. Cheung R K, Chen C Y (1998). A two-stage stochastic network model and solution methods for the dynamic empty container allocation problem. Transport Sci, 32(2): 142–162
44. Chi G F (1997). Integrated Planning of a Solid Waste Management System in the City of Regina. MASc Thesis, University of Regina, Regina, Saskatchewan, Canada, 23, 25, and 31
45. Cho G M (2005). Log-barrier method for two-stage quadratic stochastic programming. Appl Math Comput, 164(1): 45–69
46. Dai C, Cai X H, Cai Y P, Huo Q, Lv Y, Huang G H (2014). An interval-parameter mean-CVaR two-stage stochastic programming approach for waste management under uncertainty. Stochastic Environ Res Risk Assess, 28(2): 167–187
47. Darby-Downman K, Barker S, Audsley E, Parsons D (2002). A two-stage stochastic programming with recourse model for determining robust planting plans in horticulture. J Oper Res Soc, 51(1): 83–89
48. Davila E, Chang N B (2005). Sustainable pattern analysis of a publicly owned recycling recovery facility in a fast-growing urban setting under uncertainty. J Environ Manage, 75(4): 337–351
49. Davila E, Chang N B, Diwakaruni S (2005). Landfill space consumption dynamics in the Lower Rio Grande Valley by grey integer programming-based games. J Environ Manage, 75(4): 353–365
50. Dong C, Huang G H, Cai Y P, Liu Y (2012). An inexact optimization modeling approach for supporting energy systems planning and air pollution mitigation in Beijing city. Energy, 37(1): 673–688
51. Dong C, Huang G H, Cai Y P, Liu Y (2013). Robust planning of energy management systems with environmental and constraint-conservative considerations under multiple uncertainties. Energy Convers Manage, 65: 471–486
52. Du X P, Sudjianto A, Huang B Q (2005). Reliability-based design with the mixture of random and interval variables. J Mech Des, 127(6): 1068–1076
53. Dubois D, Prade H (1988). Possibility Theory: An Approach to Computerized Processing of Uncertainty. New York: Plenum Press, 102
54. Dubois D, Prade H, Sabbadin R (2001). Decision-theoretic foundations of qualitative possibility theory. Eur J Oper Res, 128(3): 459–478
55. Edirisinghe N C P, Ziemba W T (1994). Bounds for two-stage stochastic programs with fixed recourse. Math Oper Res, 19(2): 292–313
56. Ellis J H (1991). Stochastic programs for identifying critical structural collapse mechanisms. Appl Math Model, 15(7): 367–373
57. Ellis J H, McBean E A, Farquhar G J (1985). Chance-constrained/stochastic linear programming model for acid rain abatement-I. Complete and noncolinearity. Atmos Environ, 19(6): 925–937
58. Ellis J H, McBean E A, Farquhar G J (1986). Chance-constrained stochastic linear programming model for acid rain abatement-II. Limited colinearity. Atmos Environ, 20(3): 501–511
59. Fortemps P, Roubens M (1996). Ranking and defuzzification methods based on area compensation. Fuzzy Sets Syst, 82(3): 319–330
60. Fortin M, McBean E A (1983). A management model for acid rain abatement. Atmos Environ, 17(11): 2331–2336
61. Ganji A, Khalili D, Karamouz M, Ponnambalam K, Javan M (2008). A fuzzy stochastic dynamic nash game analysis of policies for managing water allocation in a reservoir system. Water Resour Manage, 22(1): 51–66
62. Guldmann J M (1986). Interactions between weather stochasticity and the locations of pollution sources and receptors in air quality planning: a chance-constrained approach. Geogr Anal, 18(3): 198–214
63. Guo P, Huang G H, He L, Sun B W (2008). ITSSIP: interval-parameter two-stage stochastic semi-infinite programming for environmental management under uncertainty. Environ Model Softw, 23(12): 1422–1437
64. Guo P, Huang G H, Li Y P (2010). An inexact fuzzy-chance-constrained two-stage mixed-integer linear programming approach for flood diversion planning under multiple uncertainties. Adv Water Resour, 33(1): 81–91
65. Gurgur C Z, Luxhoj J T (2003). Application of chance-constrained programming to capital rationing problems with asymmetrically distributed cash flows and available budget. Eng Economist, 48(3): 241–258
66. Higle J L, Sen S (1991). Stochastic decomposition: an algorithm for two-stage linear programs with recourse. Math Oper Res, 16(3): 650–669
67. Huang G H, Baetz B W, Patry G G (1992). A grey linear programming approach for municipal solid waste management planning under uncertainty. Civ Eng Syst, 9(4): 319–335
68. Huang G H, Baetz B W, Patry G G (1993). A grey fuzzy linear programming approach for municipal solid waste management planning under uncertainty. Civ Eng Syst, 10(2): 123–146
69. Huang G H, Baetz B W, Patry G G (1995a). Grey fuzzy integer programming: an application to regional waste management planning under uncertainty. Socioecon Plann Sci, 29(1): 17–38
70. Huang G H, Baetz B W, Patry G G (1995b). Grey integer programming: an application to waste management planning under uncertainty. Eur J Oper Res, 83(3): 594–620
71. Huang G H, Baetz B W, Patry G G (1995c). Grey quadratic programming and its application to municipal waste management planning under uncertainty. Eng Optim, 23(3): 201–223
72. Huang G H, Baetz B W, Patry G G (1996a). A grey hop, skip, and jump approach: generating alternatives for expansion planning of waste management facilities. Can J Civ Eng, 23(6): 1207–1219
73. Huang G H, Chi G F, Li Y P (2005b). Long-term planning of an integrated solid waste management system under uncertainty-II. A North American case study. Environ Eng Sci, 22(6): 835–853
74. Huang G H, Linton J D, Yeomans J S, Yoogalingam R (2005a). Policy planning under uncertainty: efficient starting populations for simulation-optimization methods applied to municipal solid waste management. J Environ Manage, 77(1): 22–34
75. Huang G H, Loucks D P (2000). An inexact two-stage stochastic programming model for water resources management under uncertainty. Civ Eng Environ Syst, 17(2): 95–118
76. Huang G H, Yin Y Y, Cohen S J, Bass B (1995d). Interval parameter modelling to generate alternatives: a software for environmental decision-making under certainty. In: Brebbia C A, eds. Computer Techniques in Environmental Studies. Southampton: Computational Mechanics Publications, 213–223
77. Huang X X (2007). Chance-constrained programming models for capital budgeting with NPV as fuzzy parameters. J Comput Appl Math, 198(1): 149–159
78. Huang Y F, Huang G H, Hu Z Y, Maqsood I, Chakma A (2005c). Development of an expert system for tackling the public’s perception to climate-change impacts on petroleum industry. Expert Syst Appl, 29(4): 817–829
79. Hulsurkar S, Biswal M P, Sinha S B (1997). Fuzzy programming approach to multi-objective stochastic linear programming problems. Fuzzy Sets Syst, 88(2): 173–181
80. Huo H, Wei Z P (2008). Selection of suppliers under multi-product purchase based on fuzzy multi-objective integer program model. In: Proceeding of the International Conference on Logistics Engineering and Supply Chain. Changsha, China, August 20–22, 40–46
81. Hwang J, Singh M R (1998). Optimal production policies for multi-stage systems with setup costs and uncertain capacities. Manage Sci, 44(9): 1279–1294
82. Inuiguchi M, Ichihashi H, Tanaka H (1990). Fuzzy programming: a survey of recent developments. In: Slowinski R, Teghem J, eds. Stochastic versus Fuzzy Approaches to Multiobjective Mathematical Programming under Uncertainty. Dordrecht: Kluwer Academic Publishers, 45–70
83. Inuiguchi M, Ramík J (2000). Possibilistic linear programming: a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem. Fuzzy Sets Syst, 111(1): 3–28
84. Inuiguchi M, Sakawa M (1998). Robust optimization under softness in a fuzzy linear programming problem. Int J Approx Reason, 18(1–2): 21–34
85. Inuiguchi M, Sakawa M, Kume Y (1994). The usefulness of possibilistic programming in production planning problems. Int J Prod Econ, 33(1–3): 45–52
86. Jamison K D, Lodwick W A (2001). Fuzzy linear programming using a penalty method. Fuzzy Sets Syst, 119(1): 97–110
87. Jansson C (1988). A self-validating method for solving linear programming problems with interval input data. Computing, 6(Supplementum): 33–45
88. Jiang C, Han X, Liu G R, Liu G P (2008). A nonlinear interval number programming method for uncertain optimization problems. Eur J Oper Res, 188(1): 1–13
89. Kall P, Meyer J (2005). Stochastic Linear Programming: Models, Theory, and Computation. New York: Springer, 65–68
90. Kara S S, Onut S (2010). A two-stage stochastic and robust programming approach to strategic planning of a reverse supply network: the case of paper recycling. Expert Syst Appl, 37(9): 6129–6137
91. Karmakar S, Mujumdar P P (2007). A two-phase grey fuzzy optimization approach for water quality management of a river system. Adv Water Resour, 30(5): 1218–1235
92. Karsak E E, Kuzgunkaya O (2002). A fuzzy multiple objective programming approach for the selection of a flexible manufacturing system. Int J Prod Econ, 79(2): 101–111
93. Karuppiah R, Martin M, Grossmann I E (2010). A simple heuristic for reducing the number of scenarios in two-stage stochastic programming. Comput Chem Eng, 34(8): 1246–1255
94. Kataria M, Elofsson K, Hasler B (2010). Distributional assumptions in chance-constrained programming models of stochastic water pollution. Environ Model Assess, 15(4): 273–281
95. Kibzun A I, Nikulin I V (2001). A linear two-stage stochastic programming problem with quantile criterion: its discrete approximation. Autom Remote Control, 62(8): 1339–1348
96. Ko A S, Chang N B (2008). Optimal planning of co-firing alternative fuels with coal in a power plant by grey nonlinear mixed integer programming model. J Environ Manage, 88(1): 11–27
97. Kuhn D, Panos P, Berc R (2008). Bound-based decision rules in multistage stochastic programming. Kybernetika, 44(2): 134–150
98. Kumral M (2003). Application of chance-constrained programming based on multi-objective simulated annealing to solve a mineral blending problem. Eng Optim, 35(6): 661–673
99. Lenton R, Muller M (2009). Conclusions: lessons learned and final reflections. In: Lenton R, Muller M, eds. Integrated Water Resource Management in PracticeebetterWater Management for Development. London: Earthscan, 205–219
100. Leung Y (1988). Spatial Analysis and Planning under Imprecision. Amsterdam: North-Holland, 56–60
101. Li S Y, Hu C F (2008). An interactive satisfying method based on alternative tolerance for multiple objective optimization with fuzzy parameters. IEEE Trans Fuzzy Syst, 16(5): 1151–1160
102. Li Y P, Huang G H (2009). Fuzzy-stochastic-based violation analysis method for planning water resources management systems with uncertain information. Inf Sci, 179(24): 4261–4276
103. Li Y P, Huang G H, Veawab A, Nie X H, Liu L (2006). Two-stage fuzzy-stochastic robust programming: a hybrid model for regional air quality management. J Air Waste Manag Assoc, 56(8): 1070–1082
104. Liu B D (1998). Minimax chance constrained programming models for fuzzy decision systems. Inf Sci, 112(1–4): 25–38
105. Liu C Z, Fan Y Y, Ordóñez F (2009). A two-stage stochastic programming model for transportation network protection. Comput Oper Res, 36(5): 1582–1590
106. Liu L, Huang G H, Liu Y, Fuller G A, Zeng G M (2003). A fuzzystochastic robust programming model for regional air quality management under uncertainty. Eng Optim, 35(2): 177–199
107. Liu X W, Sun J (2004). A new decomposition technique in solving multistage stochastic linear programs by infeasible interior point methods. J Glob Optim, 28(2): 197–215
108. Liu Y K, Dai X D (2007). Minimum-risk criteria in two-stage fuzzy random programming. In: Proceeding of the IEEE International Conference on Fuzzy Systems. London, England, July 23–26, vols 1–4, 1007–1011
109. Liu Z F, Huang G H (2009). Dual-interval two-stage optimization for flood management and risk analyses. Water Resour Manage, 23(11): 2141–2162
110. Liu Z F, Huang G H, Li N (2008a). A dynamic optimization approach for power generation planning under uncertainty. Energy Sources, Part A: Recovery, Utilization, and Environmental Effects, 30(14–15): 1413–1431
111. Loucks D P, Stedinger J R, Haith D A (1981). Water Resources Systems Planning and Analysis. Englewood Cliffs: Prentice Hall, 186–193
112. Louveaux F V (1980). A solution method for multistage stochastic programs with recourse with application to an energy investment problem. Oper Res, 28(4): 889–902
113. Lu H W, Huang G H, Zeng G M, Maqsood I, He L (2008). An inexact two-stage fuzzy-stochastic programming model for water resources management. Water Resour Manage, 22(8): 991–1016
114. Luhandjula M K (1996). Fuzziness and randomness in an optimization framework. Fuzzy Sets Syst, 77(3): 291–297
115. Luhandjula M K, Gupta M M (1996). On fuzzy stochastic optimization. Fuzzy Sets Syst, 81(1): 47–55
116. Luo B, Zhou D C (2009). Planning hydroelectric resources with recourse-based multistage interval-stochastic programming. Stochastic Environ Res Risk Assess, 23(1): 65–73
117. Lustig I J, Mulvey J M, Carpenter T J (1991). Formulation two-stage stochastic programs for interior point methods. Oper Res, 39(5): 757–770
118. Maqsood I, Huang G H, Huang Y F, Chen B (2005). ITOM: an interval-parameter two-stage optimization model for stochastic planning of water resources systems. Stochastic Environ Res Risk Assess, 19(2): 125–133
119. Maqsood I, Huang G H, Zeng G M (2004). An inexact two-stage mixed integer linear programming model for waste management under uncertainty. Civ Eng Environ Syst, 21(3): 187–206
120. Matloka M (1992). Some generalization of inexact linear programming. Optimization, 23(1): 1–6
121. Medova E A (1998). Chance-constrained stochastic programming for integrated services network management. Ann Oper Res, 81(10): 213–230
122. Mehrotra S, Ozevin M G (2007). Decomposition-based interior point methods for two-stage stochastic semidefinite programming. SIAM J Optim, 18(1): 206–222
123. Miller B L, Wagner H M (1965). Chance constrained programming with joint constraints. Oper Res, 13(6): 930–945
124. Mohammed W (2000). Chance constrained fuzzy goal programming with right-hand side uniform random variable coefficients. Fuzzy Sets Syst, 109(1): 107–110
125. Moore R E (1979). Method and Application of Interval Analysis. Philadelphia: Society for Industrial and Applied Mathematics (SIAM), 92–95
126. Mula J, Poler R, Garcia J P (2006). MRP with flexible constraints: a fuzzy mathematical programming approach. Fuzzy Sets Syst, 157(1): 74–97
127. Mulvey J M, Vanderbei R J, Zenios S A (1995). Robust optimization of large-scale systems. Oper Res, 43(2): 264–281
128. Ntaimo L (2010). Disjunctive decomposition for two-stage stochastic mixed-binary programs with random recourse. Oper Res, 58(1): 229–243
129. Otto K N, Lewis A D, Antonsson E K (1993). Approximating α-cuts with the vertex method. Fuzzy Sets Syst, 55(1): 43–50
130. Penuel J, Smith J C, Yuan Y (2010). An integer decomposition algorithm for solving a two-stage facility location problem with second-stage activation costs. Naval Research Logistics, 57(5): 391–402
131. Rakes T R, Reeves G R (1985). Selecting tolerances in chanceconstrained programming — A multiple objective linear-programming approach. Oper Res Lett, 4(2): 65–69
132. Razmi J, Songhori MJ, Khakbaz MH (2009). An integrated fuzzy group decision making/fuzzy linear programming (FGDMLP) framework for supplier evaluation and order allocation. Int J Adv Manuf Technol, 43(5–6): 590–607
133. Reddy M J, Adarsh S (2010). Chance constrained optimal design of composite channels using meta-heuristic techniques. Water Resour Manage, 24(10): 2221–2235
134. Rosenberg D E, Lund J R (2009). Modeling integrated decision for a municipal water system with recourse and uncertainties: Amman, Jordan. Water Resour Manage, 23(1): 85–115
135. Ruszczyński A (1993). Parallel decomposition of multistage stochastic programming problems. Math Program, 58(1–3): 201–228
136. Saadatpour M, Afshar A (2007). Waste load allocation modeling with fuzzy goals; simulation-optimization approach. Water Resour Manage, 21(7): 1207–1224
137. Sadegh M, Mahjouri N, Kerachian R (2010). Optimal inter-basin water allocation using crisp and fuzzy Shapley games. Water Resour Manage, 24(10): 2291–2310
138. Sae-Lim N (1999). Long-Term Planning of a Solid Waste Management System under Uncertainty-An Inexact Optimization Approach. MASc Thesis, University of Regina, Regina, Saskatchewan, Canada
139. Safaei N, Saidi-Mehrabad M, Tavakkoli-Moghaddam R, Sassani F (2008). A fuzzy programming approach for a cell formation problem with dynamic and uncertain conditions. Fuzzy Sets Syst, 159(2): 215–236
140. Sakawa M, Katagiri H (2010). Interactive fuzzy programming based on fractile criterion optimization model for two-level stochastic linear programming problems. Cybern Syst, 41(7): 508–521
141. Sakawa M, Kato K (2002). An interactive fuzzy satisficing method for general multiobjective 0–1 programming problems through genetic algorithms with double strings based on a reference solution. Fuzzy Sets Syst, 125(3): 289–300
142. Sakawa M, Yano H (1994). A fuzzy dual decomposition method for large-scale multiobjective nonlinear-programming problems. Fuzzy Sets Syst, 67(1): 19–27
143. Sawyer C S, Lin Y F (1998). Mixed-integer chance-constrained models for ground-water remediation. J Water Resour Plan Manage, 124(5): 285–294
144. Schultz R, Stougie L, van der Vlerk M H (1996). Two-stage stochastic linear programming: a survey. Stat Neerl, 50(3): 404–416
145. Schweickardt G, Miranda V (2009). A two-stage planning and control model toward economically adapted power distribution systems using analytical hierarchy processes and fuzzy optimization. Int J Electr Power Energy Syst, 31(6): 277–284
146. Sengupta A, Pal T K, Chakraborty D (2001). Interpretation of inequality constraints involving interval coefficients and a solution to interval linear programming. Fuzzy Sets Syst, 119(1): 129–138
147. Shapiro A, Dentcheva D, Ruszczyński A (2009). Lectures on Stochastic Programming: Modeling and Theory. MPS-SIAM Series on Optimization, the Society for Industrial and Applied Mathematics (SIAM) and the Mathematical Programming Society
148. Shapiro A, Philpott A (2010). A tutorial on Stochastic Programming. http://stoprog.org/index.html?SPTutorial/SPTutorial.html
149. Shih J S, Frey H C (1995). Coal blending optimization under uncertainty. Eur J Oper Res, 83(3): 452–465
150. SŁowiński R (1986). A multicriteria fuzzy linear programming method for water supply system development planning. Fuzzy Sets Syst, 19(3): 217–237
151. Snellen W B, Schrevel A (2004). IWRM: For Sustainable Use of Water 50 Years of International Experience with the Concept of Integrated Water Management. Ministry of Agriculture, Nature, and Food Quality, Wageningen, the Netherlands
152. Sobral M M, Hipel K W, Fargugar G J (1981). A multicriteria model for solid waste management. J Environ Manage, 12: 97–110
153. Stanciulescu C, Fortemps P, Installe M, Wertz V (2003). Multiobjective fuzzy linear programming problems with fuzzy decision variables. Eur J Oper Res, 149(3): 654–675
154. Sugimoto T, Fukushima M, Ibaraki T (1995). A parallel relaxation method for quadratic programming problems with interval constraints. J Comput Appl Math, 60(1–2): 219–236
155. Sukyirun L (2004). Long-Term Planning of Water Quality Management in the Bang Pakong River Basin under Uncertainty. MASc Thesis, University of Regina, Regina, Saskatchewan, Canada
156. Sun G J, Liu Y K, Lan Y F (2010). Optimizing material procurement planning problem by two-stage fuzzy programming. Comput Ind Eng, 58(1): 97–107
157. Takeuchi K (1986). Chance-constrained model for real-time reservoir operation using drought duration curve. Water Resour Res, 22(4): 551–558
158. Takriti S, Ahmed S (2004). On robust optimization of two-stage systems. Math Program, 99(1): 109–126
159. Takyi A K, Lence B J (1999). Surface water quality management using a multiple-realization chance constraint method. Water Resour Res, 35(5): 1657–1670
160. Tan Q, Huang G H, Cai Y P (2010a). Identification of optimal plans for municipal solid waste management in an environment of fuzziness and two-layer randomness. Stochastic Environ Res Risk Assess, 24(1): 147–164
161. Tan Q, Huang G H, Cai Y P (2010b). Radial-interval linear programming for environmental management under varied protection levels. J Air Waste Manag Assoc, 60(9): 1078–1093
162. Tan Q, Huang G H, Cai Y P (2010c). A superiority-inferiority-based inexact fuzzy stochastic programming approach for solid waste management under uncertainty. Environ Model Assess, 15(5): 381–396
163. Tan Q, Huang G H, Cai Y P (2010d). Waste management with recourse: an inexact dynamic programming model containing fuzzy-boundary intervals in objectives and constraints. J Environ Manage, 91(9): 1898–1913
164. Tan Q, Huang G H, Cai Y P (2011a). Radial interval chance-constrained programming for agricultural non-point source water pollution control under uncertainty. Agric Water Manage, 98(10): 1595–1606
165. Tan Q, Huang G H, Cai Y P (2012). Robust planning of environmental management systems with adjustable conservativeness under compound uncertainty. J Environ Eng, 138(2): 208–222
166. Tan Q, Huang G H, Cai Y P (2013). Multi-source multi-sector sustainable water supply under multiple uncertainties: an inexact fuzzy-stochastic quadratic programming approach. Water Resour Manage, 27(2): 451–473
167. Tan Q, Huang G H, Wu C Z, Cai Y P (2011b). IF-EM: an intervalparameter fuzzy linear programming model for environment-oriented evacuation planning under uncertainty. Journal of Advanced Transportation, 45(4): 286–303
168. Tan Q, Huang G H, Wu C Z, Cai Y P, Yan X P (2009). Development of an inexact fuzzy robust programming model for integrated evacuation management under uncertainty. J Urban Plann Dev, 135(1): 39–49
169. Tanaka H, Okuda T, Asai K (1973). On fuzzy-mathematical programming. Journal of Cybernetics, 3(4): 37–46
170. Teegavarapu R S V (2010). Modeling climate change uncertainties in water resources management models. Environ Model Softw, 25(10): 1261–1265
171. Tometzki T, Engell S (2009). Hybrid evolutionary optimization of twostage stochastic integer programming problems: an empirical investigation. Evol Comput, 17(4): 511–526
172. Tong S C (1994). Interval number and fuzzy number linear programming. Fuzzy Sets Syst, 66(3): 301–306
173. Torabi S A, Hassini E (2008). An interactive possibilistic programming approach for multiple objective supply chain master planning. Fuzzy Sets Syst, 159(2): 193–214
174. Trukhanov S, Ntaimo L, Schaefer A (2010). Adaptive multicut aggregation for two-stage stochastic linear programs with recourse. European Journal of Operational Research, 206(2): 395–406
175. Urli B, Nadeau R (1992). An interactive method to multiobjective linear programming problems with interval coefficients. INFOR: Information Systems and Operational Research, 30(2): 127–137
176. Vladimirou H, Zenios S A (1997). Stochastic linear programs with restricted recourse. Eur J Oper Res, 101(1): 177–192
177. Wang S M, Watada J, Pedrycz W (2009). Value-at-risk-based two-stage fuzzy facility location problems. IEEE Transactions on Industrial Informatics, 5(4): 465–482
178. Wang XW, Cai Y P, Chen J J, Dai C (2013). A grey-forecasting intervalparameter mixed-integer programming approach for integrated electric-environmental management — A case study of Beijing. Energy, 63(15): 334–344
179. Watanabe T, Ellis H (1994). A joint chance-constrained programming model with row dependence. Eur J Oper Res, 77(2): 325–343
180. Weintraub A, Vera J (1991). A cutting plane approach for chance constrained linear programs. Oper Res, 39(5): 776–785
181. Werners B (1987). Interactive multiple objective programming subject to flexible constraints. Eur J Oper Res, 31(3): 342–349
182. Wets R J B (1996). Challenges in stochastic programming. Math Program, 75(2): 115–135
183. Wu S M, Huang G H, Guo H C (1997). An interactive inexact-fuzzy approach for multiobjective planning of water resource systems. Water Sci Technol, 36(5): 235–242
184. Wu X Y, Huang G H, Liu L, Li J B (2006). An interval nonlinear program for the planning of waste management systems with economies-of-scale effects — A case study for the region of Hamilton, Ontario, Canada. Eur J Oper Res, 171(2): 349–372
185. Xu B, Fang W G, Shi R F, Yu J, Liu L (2009a). Three-objective fuzzy chance-constrained programming model for multiproject and multiitem investment combination. Inf Sci, 179(5): 623–641
186. Xu Y, Huang G H, Qin X S (2009b). Inexact two-stage stochastic robust optimization model for water resources management under uncertainty. Environmental Engineering Science, 26(12): 1765–1776
187. Yao J S, Wu K (2000). Ranking fuzzy numbers based on decomposition principle and signed distance. Fuzzy Sets Syst, 116(2): 275–288
188. Yeh S C (1996). Grey Programming and Its Applications to Water Resources Management. Dissertation for PhD degree. New York: Cornell University
189. Zadeh L A (1965). Fuzzy sets. Inf Control, 8(3): 338–353
190. Zadeh L A (1975). The concept of a linguistic variable and its application to approximate reasoning — I. Inf Sci, 8(3): 199–249
191. Zanjani MK, Nourelfath M, Ait-Kadi D (2010). A multi-stage stochastic programming approach for production planning with uncertainty in the quality of raw materials and demand. Int J Prod Res, 48(16): 4701–4723
192. Zhang J D, Rong G (2010). Fuzzy possibilistic modeling and sensitivity analysis for optimal fuel gas scheduling in refinery. Eng Appl Artif Intell, 23(3): 371–385
193. Zimmermann H J (1985). Applications of fuzzy sets theory to mathematical programming. Inf Sci, 36(1–2): 29–58