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

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Abstract

Water is crucial in supporting people’s daily life and the continual quest for socio-economic development. It is also a fundamental resource for ecosystems. Due to the associated complexities and uncertainties, as well as intensive competition over limited water resources between human beings and ecosystems, decision makers are facing increased pressure to respond effectively to various water-related issues and conflicts from an integrated point of view. This quandary requires a focused effort to resolve a wide range of issues related to water resources, as well as the associated economic and environmental implications. Effective systems analysis approaches under uncertainty that successfully address interactions, complexities, uncertainties, and changing conditions associated with water resources, human activities, and ecological conditions are desired, which requires a systematic investigation of the previous studies in relevant areas. Systems analysis and optimization modeling for integrated water resources management under uncertainty is thus comprehensively reviewed in this paper. A number of related methodologies and applications related to stochastic, fuzzy, and interval mathematical optimization modeling are examined. Then, their applications to integrated water resources management are presented. Perspectives of effective management schemes are investigated, demonstrating many demanding areas for enhanced research efforts, which include issues of data availability and reliability, concerns over uncertainty, necessity of post-modeling analysis, and the usefulness of the development of simulation techniques.

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References

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Alefeld G, Herzberger J (1983). Introductions to Interval Computations. New York: Academic Press, 67

  • 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

    Article  Google Scholar 

  • Arora S R, Gupta R (2009). Interactive fuzzy goal programming approach for bilevel programming problem. Eur J Oper Res, 194(2): 368–376

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Beale E M L (1955). On minimizing a convex function subject to linear inequalities. J R Stat Soc, B, 17(2): 173–184

    Google Scholar 

  • Bellman R, Zadeh L A (1970). Decision-making in a fuzzy environment. Manage Sci, 17(4): 141–164

    Article  Google Scholar 

  • Ben Abdelaziz F, Masri H (2009). Multistage stochastic programming with fuzzy probability distribution. Fuzzy Sets Syst, 160(22): 3239–3249

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Birge J R, Louveaux F (1997). Introduction to Stochastic Programming. New York: Springer, 24

    Google Scholar 

  • Birge J R, Louveaux F V (1988). A multicut algorithm for two-stage stochastic linear programs. Eur J Oper Res, 34(3): 384–392

    Article  Google Scholar 

  • Bosch P, Jofré A, Schultz R (2007). Two-stage stochastic programs with mixed probabilities. SIAM J Optim, 18(3): 778–788

    Article  Google Scholar 

  • Buckley J J, Feuring T (2000). Evolutionary algorithm solution to fuzzy problems: fuzzy linear programming. Fuzzy Sets Syst, 109(1): 35–53

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Campos L, Verdegay J L (1989). Linear programming problems and ranking of fuzzy numbers. Fuzzy Sets Syst, 32(1): 1–11

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Chanas S, Kuchta D (1996). Multiobjective programming in optimization of interval objective functions — A generalized approach. Eur J Oper Res, 94(3): 594–598

    Article  Google Scholar 

  • Chang N B, Hernandez E A (2008). Optimal expansion strategy for a sewer system under uncertainty. Environ Model Assess, 13(1): 93–113

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Chen S H, Wu J (2004). Interval optimization of dynamic response for structure with interval parameters. Comput Struc, 82(1): 1–11

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Google Scholar 

  • Cho G M (2005). Log-barrier method for two-stage quadratic stochastic programming. Appl Math Comput, 164(1): 45–69

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Dubois D, Prade H (1988). Possibility Theory: An Approach to Computerized Processing of Uncertainty. New York: Plenum Press, 102

    Book  Google Scholar 

  • Dubois D, Prade H, Sabbadin R (2001). Decision-theoretic foundations of qualitative possibility theory. Eur J Oper Res, 128(3): 459–478

    Article  Google Scholar 

  • Edirisinghe N C P, Ziemba W T (1994). Bounds for two-stage stochastic programs with fixed recourse. Math Oper Res, 19(2): 292–313

    Article  Google Scholar 

  • Ellis J H (1991). Stochastic programs for identifying critical structural collapse mechanisms. Appl Math Model, 15(7): 367–373

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Fortemps P, Roubens M (1996). Ranking and defuzzification methods based on area compensation. Fuzzy Sets Syst, 82(3): 319–330

    Article  Google Scholar 

  • Fortin M, McBean E A (1983). A management model for acid rain abatement. Atmos Environ, 17(11): 2331–2336

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Higle J L, Sen S (1991). Stochastic decomposition: an algorithm for two-stage linear programs with recourse. Math Oper Res, 16(3): 650–669

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Google Scholar 

  • Huang X X (2007). Chance-constrained programming models for capital budgeting with NPV as fuzzy parameters. J Comput Appl Math, 198(1): 149–159

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Chapter  Google Scholar 

  • 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

    Article  Google Scholar 

  • Inuiguchi M, Sakawa M (1998). Robust optimization under softness in a fuzzy linear programming problem. Int J Approx Reason, 18(1–2): 21–34

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Jamison K D, Lodwick W A (2001). Fuzzy linear programming using a penalty method. Fuzzy Sets Syst, 119(1): 97–110

    Article  Google Scholar 

  • Jansson C (1988). A self-validating method for solving linear programming problems with interval input data. Computing, 6(Supplementum): 33–45

    Google Scholar 

  • 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

    Article  Google Scholar 

  • Kall P, Meyer J (2005). Stochastic Linear Programming: Models, Theory, and Computation. New York: Springer, 65–68

    Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Kuhn D, Panos P, Berc R (2008). Bound-based decision rules in multistage stochastic programming. Kybernetika, 44(2): 134–150

    Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Google Scholar 

  • Leung Y (1988). Spatial Analysis and Planning under Imprecision. Amsterdam: North-Holland, 56–60

    Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Liu B D (1998). Minimax chance constrained programming models for fuzzy decision systems. Inf Sci, 112(1–4): 25–38

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Loucks D P, Stedinger J R, Haith D A (1981). Water Resources Systems Planning and Analysis. Englewood Cliffs: Prentice Hall, 186–193

    Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Luhandjula M K (1996). Fuzziness and randomness in an optimization framework. Fuzzy Sets Syst, 77(3): 291–297

    Article  Google Scholar 

  • Luhandjula M K, Gupta M M (1996). On fuzzy stochastic optimization. Fuzzy Sets Syst, 81(1): 47–55

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Matloka M (1992). Some generalization of inexact linear programming. Optimization, 23(1): 1–6

    Article  Google Scholar 

  • Medova E A (1998). Chance-constrained stochastic programming for integrated services network management. Ann Oper Res, 81(10): 213–230

    Article  Google Scholar 

  • Mehrotra S, Ozevin M G (2007). Decomposition-based interior point methods for two-stage stochastic semidefinite programming. SIAM J Optim, 18(1): 206–222

    Article  Google Scholar 

  • Miller B L, Wagner H M (1965). Chance constrained programming with joint constraints. Oper Res, 13(6): 930–945

    Article  Google Scholar 

  • Mohammed W (2000). Chance constrained fuzzy goal programming with right-hand side uniform random variable coefficients. Fuzzy Sets Syst, 109(1): 107–110

    Article  Google Scholar 

  • Moore R E (1979). Method and Application of Interval Analysis. Philadelphia: Society for Industrial and Applied Mathematics (SIAM), 92–95

    Google Scholar 

  • Mula J, Poler R, Garcia J P (2006). MRP with flexible constraints: a fuzzy mathematical programming approach. Fuzzy Sets Syst, 157(1): 74–97

    Article  Google Scholar 

  • Mulvey J M, Vanderbei R J, Zenios S A (1995). Robust optimization of large-scale systems. Oper Res, 43(2): 264–281

    Article  Google Scholar 

  • Ntaimo L (2010). Disjunctive decomposition for two-stage stochastic mixed-binary programs with random recourse. Oper Res, 58(1): 229–243

    Article  Google Scholar 

  • Otto K N, Lewis A D, Antonsson E K (1993). Approximating α-cuts with the vertex method. Fuzzy Sets Syst, 55(1): 43–50

    Article  Google Scholar 

  • 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

    Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Reddy M J, Adarsh S (2010). Chance constrained optimal design of composite channels using meta-heuristic techniques. Water Resour Manage, 24(10): 2221–2235

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Ruszczyński A (1993). Parallel decomposition of multistage stochastic programming problems. Math Program, 58(1–3): 201–228

    Article  Google Scholar 

  • Saadatpour M, Afshar A (2007). Waste load allocation modeling with fuzzy goals; simulation-optimization approach. Water Resour Manage, 21(7): 1207–1224

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Sakawa M, Yano H (1994). A fuzzy dual decomposition method for large-scale multiobjective nonlinear-programming problems. Fuzzy Sets Syst, 67(1): 19–27

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Schultz R, Stougie L, van der Vlerk M H (1996). Two-stage stochastic linear programming: a survey. Stat Neerl, 50(3): 404–416

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Google Scholar 

  • Shapiro A, Philpott A (2010). A tutorial on Stochastic Programming. http://stoprog.org/index.html?SPTutorial/SPTutorial.html

    Google Scholar 

  • Shih J S, Frey H C (1995). Coal blending optimization under uncertainty. Eur J Oper Res, 83(3): 452–465

    Article  Google Scholar 

  • SŁowiński R (1986). A multicriteria fuzzy linear programming method for water supply system development planning. Fuzzy Sets Syst, 19(3): 217–237

    Article  Google Scholar 

  • 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

    Google Scholar 

  • Sobral M M, Hipel K W, Fargugar G J (1981). A multicriteria model for solid waste management. J Environ Manage, 12: 97–110

    Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Google Scholar 

  • 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

    Article  Google Scholar 

  • Takeuchi K (1986). Chance-constrained model for real-time reservoir operation using drought duration curve. Water Resour Res, 22(4): 551–558

    Article  Google Scholar 

  • Takriti S, Ahmed S (2004). On robust optimization of two-stage systems. Math Program, 99(1): 109–126

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Tanaka H, Okuda T, Asai K (1973). On fuzzy-mathematical programming. Journal of Cybernetics, 3(4): 37–46

    Article  Google Scholar 

  • Teegavarapu R S V (2010). Modeling climate change uncertainties in water resources management models. Environ Model Softw, 25(10): 1261–1265

    Article  Google Scholar 

  • Tometzki T, Engell S (2009). Hybrid evolutionary optimization of twostage stochastic integer programming problems: an empirical investigation. Evol Comput, 17(4): 511–526

    Article  Google Scholar 

  • Tong S C (1994). Interval number and fuzzy number linear programming. Fuzzy Sets Syst, 66(3): 301–306

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Google Scholar 

  • Vladimirou H, Zenios S A (1997). Stochastic linear programs with restricted recourse. Eur J Oper Res, 101(1): 177–192

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Watanabe T, Ellis H (1994). A joint chance-constrained programming model with row dependence. Eur J Oper Res, 77(2): 325–343

    Article  Google Scholar 

  • Weintraub A, Vera J (1991). A cutting plane approach for chance constrained linear programs. Oper Res, 39(5): 776–785

    Article  Google Scholar 

  • Werners B (1987). Interactive multiple objective programming subject to flexible constraints. Eur J Oper Res, 31(3): 342–349

    Article  Google Scholar 

  • Wets R J B (1996). Challenges in stochastic programming. Math Program, 75(2): 115–135

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Yao J S, Wu K (2000). Ranking fuzzy numbers based on decomposition principle and signed distance. Fuzzy Sets Syst, 116(2): 275–288

    Article  Google Scholar 

  • Yeh S C (1996). Grey Programming and Its Applications to Water Resources Management. Dissertation for PhD degree. New York: Cornell University

    Google Scholar 

  • Zadeh L A (1965). Fuzzy sets. Inf Control, 8(3): 338–353

    Article  Google Scholar 

  • Zadeh L A (1975). The concept of a linguistic variable and its application to approximate reasoning — I. Inf Sci, 8(3): 199–249

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Zimmermann H J (1985). Applications of fuzzy sets theory to mathematical programming. Inf Sci, 36(1–2): 29–58

    Article  Google Scholar 

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Authors and Affiliations

  1. State Key Laboratory of Grassland Agro-ecosystems, Institute of Arid Agroecology, School of Life Sciences, Lanzhou University, Lanzhou, 730000, China

    Ran Wang

  2. Research Institute of Highway Ministry of Transport, Beijing, 100088, China

    Yin Li

  3. MOE Key Laboratory of Regional Energy and Environmental Systems Optimization, Resources and Environmental Research Academy, North China Electric Power University, Beijing, 102206, China

    Qian Tan

  4. Institute for Energy, Environment and Sustainable Communities, University of Regina, Saskatchewan, S4S 7H9, Canada

    Qian Tan

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  1. Ran Wang
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  2. Yin Li
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Wang, R., Li, Y. & Tan, Q. A review of inexact optimization modeling and its application to integrated water resources management. Front. Earth Sci. 9, 51–64 (2015). https://doi.org/10.1007/s11707-014-0449-4

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