An interval-parameter fuzzy nonlinear optimization model

O.R.ApplicationsAn interval-parameter fuzzy nonlinear optimization model for stream water quality management under uncertaintyX.S.Qina,b,G.H.Huanga,b,*,G.M.Zeng c ,A.Chakma d ,Y.F.HuangeaSino-Canada Center of Energy and Environmental Research,North China Electric Power University,Beijing 102206,ChinabFaculty of Engineering,University of Regina,Regina,Saskatchewan,Canada S4S 0A2cDepartment of Environmental Science and Engineering,Hunan University,Changsha 410082,China dDepartment of Chemical Engineering,University of Waterloo,Waterloo,Ont.,Canada N2L 3G1eInstitute of River and Coastal Engineering,Tsinghua University,Beijing 100084,ChinaReceived 3December 2004;accepted 24March 2006Available online 30June 2006AbstractPlanning for water quality management systems is complicated by a variety of uncertainties and nonlinearities,where difficulties in formulating and solving the resulting inexact nonlinear optimization problems exist.With the purpose of tackling such difficulties,this paper presents the development of an interval-fuzzy nonlinear programming (IFNP)model for water quality management under uncertainty.Methods of interval and fuzzy programming were integrated within a general framework to address uncertainties in the left-and right-hand sides of the nonlinear constraints.Uncertainties in water quality,pollutant loading,and the system objective were reflected through the developed IFNP model.The method of piecewise linearization was developed for dealing with the nonlinearity of the objective function.A case study for water quality management planning in the Changsha section of the Xiangjiang River was then conducted for demon-strating applicability of the developed IFNP model.The results demonstrated that the accuracy of solutions through lin-earized method normally rises positively with the increase of linearization levels.It was also indicated that the proposed linearization method was effective in dealing with IFNP problems;uncertainties can be communicated into optimization process and generate reliable solutions for decision variables and objectives;the decision alternatives can be obtained by adjusting different combinations of the decision variables within their solution intervals.It also suggested that the linear-ized method should be used under detailed error analysis in tackling IFNP problems.Ó2006Elsevier B.V.All rights reserved.Keywords:Environment;Interval programming;Fuzzy programming;Nonlinear optimization;Uncertainty;Linear programming0377-2217/$-see front matter Ó2006Elsevier B.V.All rights reserved.doi:10.1016/j.ejor.2006.03.053*Corresponding author.Address:Faculty of Engineering,University of Regina,Regina,Saskatchewan,Canada S4S 0A2.Tel.:+13065854095;fax:+13065854855.E-mail address:huang@ (G.H.Huang).European Journal of Operational Research 180(2007)1331–13571332X.S.Qin et al./European Journal of Operational Research180(2007)1331–13571.IntroductionEffective planning for water quality management is important for facilitating sustainable socio-economic development in watershed systems.However,such a planning effort is complicated with a variety of uncertain-ties and nonlinearities.For example,dynamic interactions exist between pollutant loadings and receiving waters(Orlob,1983;Lamb and Hull,1985;Hoppe et al.,2004;Huang and Chang,2003),associated with a variety of uncertainties.Moreover,representation of system costs for water quality management involves a number of nonlinear functions for projecting environmental–economic interrelationships.These complexities lead to difficulties in formulating and solving the resulting nonlinear optimization problems.In the past decades,efforts were made in dealing with the uncertainties and nonlinearities in water quality management through interval,stochastic and fuzzy programs(Huang,1998;Huang and Loucks,2000;Huang et al.,2001;Huang et al.,2002).Interval nonlinear optimization methods present uncertainties of real-world systems as intervals.For instance,Zeng et al.(1994)proposed a linearization method to solve interval non-linear programming problems.Xia and Zhang(1993)introduced several methods for solving interval nonlin-ear programming problems based on an extended Kuhn–Tucker theorem,and applied them to water pollution control planning.More recently,Chen and Huang(2001)developed a series of solution algorithms for solving interval quadratic programs and applied them to environmental management under uncertainty.Stochastic programming is another measure to address uncertainties in nonlinear systems.The most com-monly used approach is Monte Carlo simulation(Smith,1994).Burn and Mcbean(1985)studied optimization modeling of water quality management in an uncertain environment through probabilistic analysis.Huang and Chen(1991)developed a stochastic nonlinear optimization method for water quality management,fol-lowed with environmental risk assessment based on the optimization results.In these models,the objective functions were exponential with several modeling parameters being uncertain and presented as probability density functions.By means of Monte Carlo simulation,the problems could be converted into large amounts of deterministic submodels and then solved by traditional nonlinear programming methods.Pereira and Pinto (1991)presented a method for solving multistage stochastic programming problems based on the approxima-tion of the expected-cost-to-go functions by piecewise linear programs obtained from the dual solutions at dif-ferent stages,and applied it to the planning of energy generation within a reservoir system.Another approach for optimization under uncertainty in water quality management is based on fuzzy set theory.It is a method that facilitates the analysis of systems with uncertainties being derived from vagueness or‘‘fuzziness’’rather than randomness alone.It is suitable for situations when the uncertainties cannot be expressed as probability density functions(PDFs),such that adoption of fuzzy membership functions becomes an attractive alternative(Huang et al.,1995).Previously,Huang et al.(1993,1995)proposed several fuzzy-optimization methods and applied them to environmental management problems.More recently,Guo et al.(1999)proposed a fuzzy multiobjective linear programming model and applied it to land-use planning for forestation and agricultural activities in the Erhai River Basin.Research for similar optimization problems under uncertainty and nonlinearity can also be found in other fields such as solid waste management,water resources management,and urban planning(Yeomans and Huang,2003).Taguchi et al.(2003)developed a genetic algorithm(GA)approach for solving nonlinear goal programming problems with interval coefficients.Gen et al.(1997)and Tang et al.(1998)presented several GA approaches to solve fuzzy nonlinear-objective programming problems.More recently,Chen and Huang (2001)developed a derived algorithm for solving interval quadratic program and applied it to environmental decision making under uncertainty.Wu(2000)developed a differentiated algorithm for tackling interval qua-dratic programming,and applied it to solid waste management.Yeomans et al.(2003)developed an evolution-ary algorithm for optimal planning of municipal solid waste management and applied it in the regional municipality of Hamilton–Wentworth.Generally,the above methods have advantages in their effectiveness in dealing with uncertainties.However, the stochastic methods are associated with difficulties in acquiring PDFs for a number of modeling parameters (Marr and Canale,1988;Guo et al.,2003).The GA-based nonlinear optimization methods are ineffective when the parameters are very uncertain;also,they are non-analytical approaches and have high computa-tional requirements(Qin and Zeng,2002;Zeng et al.,2004).Based on Huang et al.(2001),the interval pro-grams can deal with uncertainties that exist in left-hand side coefficients,but have difficulties when the model’sright-hand sides are highly uncertain;Huang et al.(1993)also indicated that fuzzy programming approach would be more applicable when right sides of the constraints have large intervals.The two methods have var-ied strengths and weaknesses,with a potential for compensating each other when they are integrated within a general framework.Therefore,the main objective of this research is to develop an interval-parameter fuzzy nonlinear programming(IFNP)model for water quality management under uncertainty and explore a linear-ization method for model solutions.This objective entails:(a)Integration of interval and fuzzy programming to address uncertainties in the left-and right-hand sidesof the nonlinear models.(b)Linearization of the nonlinear objective function through piecewise conversions.(c)Solution of the nonlinear models through IFLP solution method.(d)A case study for water quality management planning in the Changsha section of the Xiangjiang Riverfor demonstrating applicability of the proposed IFNP model and linearization method.(e)Comparison of the linearization method and a direct nonlinear method to assess effectiveness of the pro-posed methodology.2.Water quality simulationWater quality models were used extensively for supporting environmental management in river basins.Sev-eral water quality models were developed in the past decades,such as the Streeter–Phelps,O’Conner,Dobbins, and Thomas models(Rauch et al.,1998).In this study,the Streeter–Phelps model is used for supporting quan-tification of water quality constraints related to biological oxygen demand(BOD)and dissolved oxygen(DO) discharges as well as their concentrations in river waters(Streeter and Phelps,1925).Afirst-order degradation reaction of BOD can be written as:d Ld t¼ÀK d L;ð1Þwhere L is the ultimate BOD concentration,t is the meanflow time,and k d is thefirst-order deoxygenation rate constant.For dissolved oxygen(DO),a mass balance equation can be written as follows:d Od t¼Àk d Lþk aðO sÀOÞ;ð2Þwhere O is the DO concentration,O s is the saturated DO concentration,and k a is thefirst-order reaeration rate constant.The solutions of Eqs.(1)and(2)are(Thomann and Mueller,1987):L¼L0eÀk d t;ð3aÞO¼O sÀðO sÀO0ÞeÀk d tÀk d L0k aÀk dðeÀk d tÀeÀk a tÞ;ð3bÞwhere L0and O0are BOD and DO concentrations at the starting point of the river system,respectively.Segmentation is necessary since a number of wastewater discharge outlets scatter along the river,with tem-poral and spatial variations of their loadings.Water quality at each section is affected by various sources from the upper stream(Cheng and Cheng,1990).Fig.1shows a conceptual description of stream segmentation that is related to wastewater discharge and water utilization.According to Fig.1,the mass balance,flow continuity,flow rate,and BOD equilibrium can be written as follows:Q2i¼Q1iÀQ3iþQ i;ð4aÞQ1i¼Q2;iÀ1;ð4bÞL2i Q2i ¼L1iðQ1iÀQ3iÞþL i Q i:ð4cÞX.S.Qin et al./European Journal of Operational Research180(2007)1331–13571333From Eq.(3a),the BOD degradation process can be written as:L1i¼L2;iÀ1eÀk d;iÀ1t iÀ1:ð5ÞDefine a i¼eÀk d i t i.Substituting the corresponding item in Eq.(5)with a iÀ1,we have:L1i¼a iÀ1L2;iÀ1:ð6ÞFrom Eqs.(4c)and(6),we have:L2i¼L2;iÀ1a iÀ1ðQ1iÀQ3iÞQ2iþQiQ2iL i:ð7ÞDefine:a iÀ1¼a iÀ1ðQ1iÀQ3iÞQ2i;b i¼QiQ2i:A matrix expression can thus be used to replace Eq.(7):A~L2¼B~Lþ~g;ð8Þwhere A and B are n·n matrixes,and~g is an n-dimensional vector.From Eq.(8),we have: ~L2¼AÀ1B~LþAÀ1~g:ð9ÞEq.(9)defines the relationship between BOD input and output at each river section,where~L2is the simulated BOD level,and~L is the observed BOD discharge concentration.However,when Eq.(9)is applied in optimi-zation models,~L2generally represents as the desired BOD-discharge criteria,and~L is the variable that needs to be determined.The DO equilibrium can be written as follows:O2i Q2i¼O1iðQ1iÀQ3iÞþO i Q i:ð10ÞAccording to Eq.(3b),the DO level can be given by:O1i¼O2;iÀ1eÀk a;iÀ1t iÀ1Àk d;iÀ1L2;iÀ1k a;iÀ1Àk d;iÀ1ðeÀk d;iÀ1t iÀ1ÀeÀk a;iÀ1t iÀ1ÞþO sð1ÀeÀk a;iÀ1t iÀ1Þ:ð11ÞDefine c i¼eÀk a i t i;b i¼k d iða iÀc iÞk a iÀk d iand d i=O s(1Àc i).Then function(10)becomes:O2i¼Q1iÀQ3iQ2iðQ2;iÀ1c iÀ1ÀL2;iÀ1b iÀ1þd iÀ1ÞþQiQ2iO i:ð12Þ1334X.S.Qin et al./European Journal of Operational Research180(2007)1331–1357Definec iÀ1¼Q1iÀQ3iQ2ic iÀ1;d iÀ1¼Q1iÀQ3iQ2ib iÀ1and fiÀ1¼Q1iÀQ3iQ2id iÀ1:Then we have:O2i¼c i O2;iÀ1Àd iÀ1L2;iÀ1þf iÀ1þb i O i:ð13ÞA matrix function could be written as follows:C~O2¼ÀD~L2þB~Oþ~fþ~h:ð14ÞEq.(14)can be further converted into:~O2¼ÀCÀ1DAÀ1B~LþCÀ1B~OþCÀ1ð~fþ~hÞÀCÀ1DAÀ1~g:ð15ÞThus,Eqs.(9)and(15)can be used for predicting water quality at each river segment.3.Optimization of water quality management system3.1.Deterministic modeling formulationBased on the definition of b i,if the volumeflow of the wastewater is negligible compared with the volume flow of the river(i.e.Q i(Q2i such that b i!0),then Eq.(15)becomes:~O2¼ÀCÀ1DAÀ1B~LþCÀ1ð~fþ~hÞÀCÀ1DAÀ1~g:ð16ÞThus,the BOD and DO models can be simplified into:~L2¼U~Lþ~m;ð17aÞ~O2¼V~Lþ~n;ð17bÞwhere U and V are coefficient matrixes defined by Eqs.(9)and(16).The main objective of the water quality management is to minimize net operating cost of wastewater treatment under the constraints of water quality requirements and river conditions.The operational cost function for each wastewater treatment unit can be written as(Cheng and Cheng,1990):C¼k1Q k2þk3Q k2g k4;ð18Þwhere C is the treatment cost for a specific pollutant,Q is the wastewaterflow rate,g is the treatment effi-ciency,k1and k3are the cost-function coefficients,k2is the economy-of-scale index of the treatment cost, and k4is the scale index of treatment efficiency(k1,k3>0,0<k2<1,and k4>1).The operating cost functions for handling different types of wastewater generally imply the relevant eco-nomical consequences in terms of both scale and efficiency.They were normally expressed in various empirical formulations by researchers from different countries(Thomann,1972;Rinaldi et al.,1979;Lee and Wen, 1997).In China,the most widely used expression is written as Eq.(18).When the treatment efficiency is con-sidered as a constant,the equation becomes C=aQ k2,where a=K1+K3g K4.The equation of C=aQ k2can also be written in a differential style as d C/d Q=ak2Q k2À1.According to the results from a number of researchers,K2ranges from0.7to0.8(Fu and Cheng,1985;Cheng and Cheng,1990).Thus,the unit operating cost(d C/d Q)will decrease with the increase of the treatment scale(Q).Such a relationship between opera-tional cost and treatment scale is normally called economy-of-scale for wastewater treatment.This reveals X.S.Qin et al./European Journal of Operational Research180(2007)1331–13571335the economic advantages of large-scale wastewater treatment.When the scale of treatment(Q)is a constant, only the treatment efficiency becomes a variable.Since K4in Eq.(18)is always greater than1(Cheng and Cheng,1990;Qin and Zeng,2000),the treatment cost per unit of pollutant loading will increase with the treat-ment efficiency.Such a relationship is normally named economy-of-efficiency.The optimization model for water quality management can thus be established as follows: Min f¼X ni¼1C iðgÞ;ð19aÞs:t:U~Lþ~m6~L c;V~Lþ~n P~O c;ð19bÞ~L P0;g1 i 6gi6g2i;where C i(g i)is the wastewater treatment cost at segment i;g i is the treatment efficiency at segment i;U and V are the relational matrixes of BOD and DO;~L c and~O c are the environmental criteria of BOD and DO con-centrations,respectively;~m and~n are the constant vectors as delineated in(17);g1i and g2iare the technicalconstraints of treatment efficiencies.The relationship between the pollutant concentration in dischargedflow (L i)and the treatment efficiency(g i)can be given by:g i ¼1ÀL iL0i;ð20Þwhere L0iis the initial BOD concentration at segment i.3.2.Modeling formation under uncertaintyKothandaraman and Ewing(1969)proposed that both deoxygenation and reaeration rates in river systems have uncertain and dynamic characteristics.The uncertainties characteristics of k a and k d can be represented as intervals since their distributional information is often unavailable.Thus,Eqs.(3a)and(3b)become:LƼL0eÀkÆd t;ð21aÞOƼO sÀðO sÀO0ÞeÀkÆd tÀkÆdL0kaÀkdðeÀkÆd tÀeÀkÆa tÞ:ð21bÞTherefore,Eqs.(17a)and(17b)can be transformed into:~LÆ2¼UÆ~LÆþ~mÆ;ð22aÞ~OÆ2¼VÆ~LÆþ~nÆ:ð22bÞUncertainties of wastewater discharge rates can also be presented as intervals,leading to the following cost function as converted from Eq.(18):CƼk1QÆk2þk3QÆk2gÆk4:ð23ÞGenerally,pollutant concentrations would increase under low stream-flow rates.This occurs typically in win-ter for rivers in the central-south of China.This is due to the low dilution from the streamflow.Thus the low-estflow rate is applied in formulating the optimization model since it represents the most serious pollution conditions.Generally,since the data of the lower and upper bounds of water quality related parameters are easier to be estimated,uncertainties associated with water quality simulations are represented as intervals. The water quality criteria that present as the constraint right-hand sides are quantified as fuzzy sets due to its 1336X.S.Qin et al./European Journal of Operational Research180(2007)1331–1357effectiveness in avoiding or mitigating violation of model solutions.These uncertainty-manipulation settings result in an interval-parameter fuzzy nonlinear programming(IPFNP)model as follows:Min fƼX ni¼1CÆiðgÆÞ;ð24aÞs:t:UÆ~LÆP$~LcÀ~mÆ;ð24bÞVÆ~LÆP$~OcÀ~nÆ;ð24cÞ~LÆP0;ð24dÞg1 i 6gÆi6g2i;ð24eÞgÆi ¼1ÀLÆiL0i;ð24fÞwhere CÆi ðgÆiÞis the wastewater treatment cost at the segment i;the symbol P$represents that the right-hand-side of the constraint is treated as fuzzy-valued parameters.From the cost function as shown in Eq.(23),the objective function(24a)is mostly nonlinear.4.Solution method4.1.Interval-fuzzy linear programmingConsider an interval linear programming(ILP)problem as follows:Min fƼCÆXÆ;ð25aÞSubject to AÆXÆ6BÆ;ð25bÞXÆP0;where A±2{R±}m·n,B±2{R±}m·1,C±2{R±}1·n and X±2{R±}n·1(R±denotes a set of interval parameters).When the system’s goal and constraints are fuzzy,model(1)can be converted into an interval fuzzy linear programming(IFLP)problem,through incorporating the fuzzy-programming concept within the ILP frame-work(Huang et al.,1993,1995).The IFLP model can be formulated as follows(Huang et al.,1993): Max k;ð26aÞSubject to CÆXÆ6fþopt1ÀkÁ½fþopt1ÀfÀopt1;AÆXÆ6BÆÀkÁ½BþÀBÀ ;ð26bÞ06k61;XÆP0;where fÀopt1and fþopt1denote the least and most desirable system objectives,respectively;k is a control variablethat corresponds to the degree(membership grade)to which the model’s solution fulfills the fuzzy goal or con-straints.Specifically,flexibility in the constraints and fuzziness in the system objective,which are represented by fuzzy sets and denoted as‘fuzzy constraints’and a‘fuzzy goal’respectively,are expressed as a membership grade(k)corresponding to the degrees of satisfaction for the constraints and the objective.An interactive two-step solution algorithm was provided by Huang et al.(1993)for solving model Eqs. (26a)and(26b).In detail,thefirst submodel that corresponds to the desired lower-bound objective functionvalue can be formulated as follows(in water quality planning problems,the water quality criteria bÆi >0,andthe cost f±>0):X.S.Qin et al./European Journal of Operational Research180(2007)1331–13571337Max k;ð27aÞSubject toX k1j¼1cÀjxÀjþX nj¼k1þ1cÀjxþj6fþÀkðfþÀfÀÞ;X k1 j¼1j a ij jþSignðaþijÞxÀjþX nj¼k1þ1j a ij jÀSignðaÀijÞxþj6bþiÀkðbþiÀbÀiÞ;8ið27bÞ06k61;XÆjP0;8j;where j a ij j is the absolute value of a ij,and SignðaÆij Þis the sign of a ij.Thus,the solutions of xÀj optðj¼1;2;...;k1Þand xþj opt ðj¼k1þ1;k1þ2;...;nÞcan be obtained through submodel(27);fÀoptcan then be obtained from(25a).In the second step,the following submodel corresponding to the upper-bound objective function value can then be formulated:Max k;ð28aÞSubject toX k1j¼1cþjxþjþX nj¼k1þ1cþjxÀj6fþÀkðfþÀfÀÞ;X k1 j¼1j a ij jÀSignðaÀijÞxþjþX nj¼k1þ1j a ij jþSignðaþijÞxÀj6bþiÀkðbþiÀbÀiÞ;8ið28bÞ06k61;XÆjP0;8j;xþj P xÀj opt;j¼1;2;...;k1;xÀj 6xþj opt;j¼k1þ1;k2þ2;...;n:The solutions of xþj opt ðj¼1;2;...;k1Þ,xÀj optðj¼k1þ1;k1þ2;...;nÞand fþoptcan thus be obtained.Theresulting solution can be presented as xÆj opt ¼½xÀj opt;xþj opt,and fÆj opt¼½fÀj opt;fþj opt.4.2.Linearization of nonlinear cost functionThe cost function as shown in Eq.(18)is nonlinear(exponential).A linearization approach is proposed to solve such a problem.According to Cheng and Cheng(1990),cost function(18)could be simplified into: C¼aþb g k4;ð29Þwhere g is the treatment efficiency of BOD as defined in Eq.(20).The presentation of this equation is shown in pared with Eq.(18),we have a=K1Q k2and b=K3Q k2.When the scale of treatment(Q)is a con-stant,a and b become constant coefficients.In such a case,there will be no effect of scale economy.According to Cheng and Cheng(1990)and Fu and Cheng(1985)(1990)and Fu and Cheng(1985),the exponent index k4 in Eq.(29)is greater than1.Thus,Eq.(20)is always presented in a convex shape.The basic idea of the linearization is to cut the nonlinear function into several segments,and then use linear functions formulations to approximate them.The obtained linear models can thus be solved through the tra-ditional ILP method(Huang et al.,1993,1995).4.2.1.Piecewise conversion of the nonlinear function into multiple segmentsAssume that there exist t0segments within the interval of06g<1.The segmental values are g¼0;g 1;g2;...;gt0,and the relevant slopes are s1;s2;...;s t(Fig.2).In each segment,the linearized functions couldbe written as follows:1338X.S.Qin et al./European Journal of Operational Research180(2007)1331–1357C 1¼a þs 1ðg Àg 0Þ;g 06g <g 1;C 2¼a þs 1ðg 1Àg 0Þþs 2ðg Àg 1Þ;g 16g <g 2;...C t 0¼a þX i À1j ¼1s j ðg j Àg j À1Þþs t 0ðg Àg t 0À1Þ;g t 0À16g <g t 0:4.2.2.Minimization of linearization errorsThe sum of squared errors at all segments should be minimized (Fig.2):Min Z g i g i À1d E ¼Z g i g i À1a þX i À1j ¼1s j ðg j Àg j À1Þþs i ðg Àg i À1ÞÀða þb g k 4Þ"#2d g ;ð30ÞwhereR g ig i À1d E represents the integration approximation of errors in segmentsg i À1to g i .4.2.3.Slope at each segment To minimize R g i g i À1d E ,let d R g ig i À1d E =d s i ¼0.Thus,we can derive the slope of each linearized function as follows:s i ¼b 4g k 4þ2i Àg k 4þ2i À1ÀÁÀb g i À14g k 4þ1i Àg k 4þ1i À1ÀÁÀðg i Àg i À1Þ2P i À1j ¼1s j ðg j Àg j À1Þ13ðg i Àg i À1Þ:ð31Þ4.2.4.Formulation of piecewise linear objective functionThrough the linearization effort,Eq.(29)becomes:C ¼a þX t 0j ¼1s j g j :ð32ÞThus,cost function (19a)could be converted into:Min f ¼X n i ¼1a i þXt 0j ¼1s ij g ij "#;ð33Þwhere f is the total cost,n is the number of polluting sources,and t 0is the number of linearizedsegments.X.S.Qin et al./European Journal of Operational Research 180(2007)1331–135713394.3.Solution through linearizationBased on Zeng et al.(1994),the inexact water quality management problems can be generalized into:Min fƼX nj¼1½pÆjþcÆjðyÆjÞk j ;ð34aÞs:t:AÆi YÆ6$B;AÆi2AÆ;i¼1;2;...;m;ð34bÞwhere y j is the treatment efficiency,y i2Y.Suppose that the defined range of y j is cut into t0segments.Then, we have additional technical constraints as follows:yðtÞj6yjt 6yðtþ1Þj;t¼1;2;...;t0:ð35ÞThrough the linearization effort,we have:pÆj þcÆjðxÆjÞk j¼pÆjþX t0t¼1sÆjtðxÆjtÀxðtÞjÞ:ð36ÞBased on Section4.2,we have:s jt¼c jÁu jtðyð0Þj;yð1Þj;...;yðtÞjÞ¼c jÁu jtðÁÞ;ð37Þwhere u jt(Æ)is defined in Eq.(31).Therefore,we have:pÆj þcÆjðxÆjÞk j¼pÆjþcÆjX t0t¼1u jtðÁÞðxÆjtÀxðtÞjÞ:ð38ÞBecause the cost function is monotonically increasing(Fig.2),we have:ðyÆj Þk j¼X t0t¼1u jtðÁÞðyÆjtÀyðtÞjÞP0;ð39Þwhere u jt(Æ)>0when g0P0.The u jt(Æ)is also a monotonically increasing function.Based on the definition of yÆj,we then have:yÆj ¼X t0t¼1ðyÆjtÀyðtÞjÞ:ð40ÞFor the sake of simplification,define zÆjt ¼yÆjtÀyðtÞj within the following scope:06zÆjt 6yðtþ1ÞjÀyðtÞj;t¼1;2;...;t0:ð41ÞThus,model(34a)can be written as:Min fƼX nj¼1pÆjþcÆjX t0t¼1u jtðÁÞzÆjt"#;ð42Þs:t:X nj¼1aÆijÁX t0t¼1zÆjt6$b i;i¼1;2;...;m;ð43Þwhere zÆjt P0and u jt(Æ)P0.This is a conventional IFLP problem that can be solved through the followingtwo submodels(Huang et al.,1993):Max kð44aÞs:t:X nj¼1pþjþcþjX t0t¼1ujtðÁÞzÀjt"#þkðfþÀfÀÞ6fþ;ð44bÞ1340X.S.Qin et al./European Journal of Operational Research180(2007)1331–1357。