A FREILP Approach for Long-Term Planning of MSW Management System in HRM, Canada

Transcription

1 A FREILP Approach for Long-Term Plannng of MSW Management System n HRM, Canada by Wenwen Pe Submtted n partal fulfllment of the requrements for the degree of Master of Appled Scence at Dalhouse Unversty Halfax, Nova Scota August 2011 Copyrght by Wenwen Pe, 2011

2 DALHOUSIE UNIVERSITY DEPARTMENT OF CIVIL AND RESOURCE ENGINEERING The undersgned hereby certfy that they have read and recommend to the Faculty of Graduate Studes for acceptance a thess enttled A FREILP Approach for Long-Term Plannng of MSW Management System n HRM, Canada by Wenwen Pe n partal fulflment of the requrements for the degree of Master of Appled Scence. Date of Defence: August 26 th 2011 Supervsor: Readers:

3 DALHOUSIE UNIVERSITY DATE: August 26 th 2011 AUTHOR: TITLE: Wenwen Pe A FREILP Approach for Long-Term Plannng of MSW Management System n HRM, Canada DEPARTMENT OR SCHOOL: Department of Cvl and Resource Engneerng DEGREE: MASc. CONVOCATION: October YEAR: 2011 Permsson s herewth granted to Dalhouse Unversty to crculate and to have coped for non-commercal purposes, at ts dscreton, the above ttle upon the request of ndvduals or nsttutons. I understand that my thess wll be electroncally avalable to the publc. The author reserves other publcaton rghts, and nether the thess nor extensve extracts from t may be prnted or otherwse reproduced wthout the author s wrtten permsson. The author attests that permsson has been obtaned for the use of any copyrghted materal appearng n the thess (other than the bref excerpts requrng only proper acknowledgement n scholarly wrtng), and that all such use s clearly acknowledged. Sgnature of Author

4 Table of Contents Lst of Tables... v Lst of Fgures... x Abstract... x Lst of Abbrevatons and Symbols Used... x Acknowledgement... xv CHAPTER 1 INTRODUCTION Statement of the Problem Muncpal Sold Waste (MSW) Management System Optmzaton Model Development for MSW Plannng MSW Management System n HRM Obectve Structure of the Thess... 7 CHAPTER 2 LITERATURE REVIEW Conventonal MSW Management and Plannng Methods Optmzaton Approaches that Deal wth Uncertantes Fuzzy Mathematcal Programmng Stochastc Mathematcal Programmng Interval-Parameter Mathematcal Programmng Hybrd and Mxed Mathematcal Programmng Prevous Studes on Sold Waste Management n HRM CHAPTER 3 VALIDITY CHECKING FOR ILP ALGORITHMS Exstng ILP Soluton Algorthms Monte Carlo Smulaton Algorthm Two-Step Algorthm Best-Worst Case Algorthm v

5 Comparson of Two-Step and BWC Algorthms Valdty Checkng for Two-Step and BWC Algorthms A Numercal Example for Valdty Checkng Result Interpretaton and Valdty Checkng CHAPTER 4 FREILP MODEL DEVELOPMENT Rsk Explct Interval Lnear Programmng (REILP) REILP Modelng Approach Dscusson of REILP An Illustratve Example of REILP FREILP Model Development Fuzzy Set Theory Fuzzy Nature of the Aspraton Level ( 0 ) Fuzzy Rsk Explct Interval Lnear Programmng (FREILP) Soluton Process for the FREILP Model CHAPTER 5 FREILP MODEL APPLICATION TO MSW SYSTEM IN HRM Overvew of MSW System n HRM Waste Generaton Recyclng Compostng Landfll Incnerator Transfer Staton Model Input Data Dscount Factor Waste Collecton and Transportaton Cost Waste Faclty Operatng Cost Revenues Captal Costs for Faclty Expanson and Development Resdue Rate v

6 5.3. Model Development for MSW System n HRM Obectve Functon Constrants CHAPTER 6 RESULTS AND DISCUSSIONS REILP Results FREILP Results under Dfferent -cuts Scenaro 1: -cut = Scenaro 2: -cut = Scenaro 3: -cut = Dscusson Conservatve-Rsk Decson Support Medum-Rsk Decson Support Aggressve-Rsk Decson Support Comparson of Three Decson Supports CHAPTER 7 SUMMARY AND CONCLUSION Summary Research Achevements Recommendaton for Future Research REFERENCES v

7 Lst of Tables Table 3.1 Results obtaned from three ILP algorthms Table 4.1 The optmal solutons of the example problem Table 4.2 The optmal solutons of the model (4.1.31) Table 4.3 The optmal solutons when the rsk functon s (4.1.33) Table 5.1 MSW management programs mplemented by HRM Table 5.2 Waste generaton n Canada, Table 5.3 Yearly waste generaton n HRM from Aprl 2009 to March Table 5.4 Estmate of populaton and waste generaton n HRM for the plannng horzon Table 5.5 Capacty of the Otter Lake Landfll by March 31, Table 5.6 Dscount factors Table 5.7 HRM waste collecton areas and the number of the households served Table 5.8 Waste collecton and transportaton data Table 5.9 Waste collecton and transportaton costs for dfferent types of wastes Table 5.10 Operatng cost of dfferent facltes n HRM Table 5.11 Revenues estmaton for waste management facltes Table 5.12 Costs and evenues estmaton Table 5.13 Transportaton ost, operaton cost, and revenue for waste management facltes Table 5.14 The number of compostng facltes n each provnce of Canada and estmated captal costs Table 5.15 Compostng faclty expanson optons and the estmated costs Table 5.16 MRF facltes n each provnce of Canada and ther annual capactes Table 5.17 Recyclng faclty expanson optons and the estmated costs Table 5.18 Captal ost for waste management facltes v

8 Table 6.1 REILP soluton for the obectve functon under dfferent aspraton levels Table 6.2 REILP wastes flow allocaton and the values of rsk functon Table 6. 3 Dstrbuton of total system cost from the REILP model Table 6.4 FREILP soluton when -cut = Table 6.5 FREILP soluton when -cut = Table 6.6 FREILP soluton when -cut = Table 6.7 Waste allocaton solutons Table 6.8 Faclty expanson solutons v

9 Lst of Fgures Fgure 3.1 The PDF curve of a parameter Fgure 3.2 The CDF curve of a parameter converted from PDF Fgure 3.3 An example of randomly settng a value for a parameter Fgure 3.4 Two-step soluton space and feasble space Fgure 3.5 BWC soluton space and feasble space Fgure 4.1 example of the membershp functon of the fuzzy set Fgure 4.2 The membershp functon of real numbers close to 10 under -cut = Fgure 4.3 The membershp functon of the fuzzy set young people Fgure 4.4. The membershp functon of conservatve aspraton level Fgure 4.5 The membershp functon of medum aspraton level Fgure 4.6 The membershp functon of aggressve aspraton level Fgure 5.1 HRM waste collecton areas Fgure 5.2 The waste allocaton flow chart Fgure 6.1 Relatonshp between aspraton level wth rsk functon and system cost Fgure 6.2 Waste flow allocated to the landfll (REILP) Fgure 6.3 Waste flow allocated to the recyclng facltes (REILP) Fgure 6.4 Waste flow allocated to the compostng facltes (REILP) Fgure 6.5 Determnaton of the nterval aspraton levels under -cut levels of 0.5, 0.6 and 0.7 for conservatve scenaros Fgure 6.6 Waste allocaton and desgn capacty of recyclng facltes under conservatve-rsk decsons Fgure 6.7 Waste allocaton and desgned capacty of compostng facltes under conservatve-rsk decsons Fgure 6.8 Waste allocaton to landfll under conservatve-rsk decsons Fgure 6.9 Avalable landfll capacty under conservatve-rsk decsons x

10 Fgure 6.10 Waste allocaton and desgned capacty of recyclng facltes under medum-rsk decsons Fgure 6.11 Waste allocaton and desgned capacty of compostng facltes under medum-rsk decsons Fgure 6.12 Waste flow allocaton to landfll under medum-rsk decsons Fgure 6.13 Avalable capacty of landfll at the end of each perod under Fgure 6.14 Waste allocaton and desgned capacty of recyclng facltes Fgure 6.15 Waste allocaton and desgned capacty of compostng facltes under aggressve-rsk decsons Fgure 6.16 Waste flow allocaton to the landfll under aggressve-rsk decsons Fgure 6.17 Avalable landfll capacty at the end of each perod under aggressve-rsk decsons Fgure 6.18 Costs comparson Fgure 6.19 Waste allocaton comparson Fgure 6.20 Landfll waste flows comparson Fgure 6.21 Recyclng waste flows comparson Fgure 6.22 Compostng waste flows comparson Fgure 6.23 Compostng waste flows comparson x

11 Abstract The muncpal sold waste (MSW) management system s conssted of plannng, development, executon of captal works, and so on. Too many factors n the system make the decson makng process plagued wth uncertantes, vagueness and complcaton. Interval-parameter Lnear Programmng (ILP) s wdely used to deal wth uncertantes exsted n the MSW system and to assst optmal decson makng. However, the exstng ILP soluton algorthms,.e., best-worst case algorthm and 2-step algorthm, are found to be neffectve through a valdty checkng process. Moreover, the results from ILP cannotreflect the lnkage between decson rsks and the system return. In ths study, a fuzzy rsk explct nterval-parameter lnear programmng (FREILP) model s developed and appled to the long-term plannng of the MSW management system n Halfax Regonal Muncpalty (HRM). Ths method s specfcally desgned to deal wth extensve uncertantes exsted n the MSW management system and to provde decson supports to HRM planners. In the model, ILP s used to reflect uncertantes exsted n both obectve functon and constrants. Based on the basc ILP, a rsk functon s defned to assst n fndng solutons wth mnmum system cost whle mnmzng the system rsk, under certan aspraton levels. The aspraton level could be conservatve, medum or aggressve, and can thus be presented as a fuzzy set to reflect the preference of decson makers. Three sets of solutons are obtaned accordngly. Besdes, the model was also solved under the aspraton level from 0 to 1, wth a step of 0.1, for provdng a comprehensve decson support. Ths approach can effectvely reflect dynamc, nteractve, uncertan characterstcs, as well as the nteractons between overall cost and rsk level of the MSW management system, thus provde valuable nformaton to support the decson-makng process, such as waste allocaton pattern, tmng and expanson capactes of the muncpal sold waste management actvtes. The result can drectly reflect the tradeoff between decson rsks and the system return. x

12 Lst of Abbrevatons and Symbols Used BWC C&D CCP CDF DF DP FEP FFP FLP FMP FPP FREILP HHW HRM ILP IPMP IMSW LP MILP MLP MSW NRL PDF RDF REILP SMP Best-worst case algorthm Constructon and demolton Chance-constraned programmng Cumulatve dstrbuton Functon Dscount factor Dynamc programmng Front end processor Fuzzy flexblty programmng Fuzzy lnear programmng Fuzzy mathematcal programmng Fuzzy possblty programmng Fuzzy rsk-explct nterval lnear programmng Household hazardous waste Halfax regonal muncpalty Interval-parameter lnear programmng Interval-parameter mathematcal programmng Integrated muncpal sold waste Lnear programmng Mx-nteger lnear programmng Mult-obectve mathematcal programmng Muncpal sold waste Normalzed rsk level Probablty dstrbutons functon Resdual dsposal faclty Rsk explct nterval-parameter lnear programmng Stochastc mathematcal programmng x

13 A ± A set of nterval parameters of ILP constrants B ± A set of rght-hand sde of parameters of ILP C ± A set of nterval parameters of ILP obectve functon CA k Captal cost (expanson cost) for faclty under expanson opton k ($) CP DF t Current capacty of faclty (tonnes/year when =1 and 2; tonnes when =3) Dscount factor n perod t EP kt The expandng capacty of opton k (tonnes/year when =1 and 2; tonnes when =3) HWG t Wastes dsposed by HRM (tonnes/5 years) P Feasble decson space of ILP solved by 2-step algorthm P l Smallest decson space of ILP solved by 2-step algorthm P u Largest decson space of ILP solved by 2-step algorthm Q Decson space of ILP solved by BWC algorthm Q l Smallest decson space of ILP solved by BWC algorthm Q u Largest decson space of ILP solved by BWC algorthm RR T TWG t UC Resdue rate from both recyclng and compostng facltes A partcular tme perod Total wastes generated n HRM (tonnes/ 5 years) The unt collecton and transportaton cost for faclty x

14 UO UR Unt operatng cost for the faclty ($/tonne) Unt revenue from faclty ($/tonne) X ± A set of decson varables of ILP X ± opt Optmum decson varable values X - opt Lower bound of optmum decson varable values X + opt Upper bound of optmum decson varable values X t Waste flow allocated to the faclty n perod t (tonnes/5 years) Bnary varable: Y kt Y kt = 1 when faclty (wth capacty opton k) needs to be developed n perod t Y kt = 0 when there s no expanson a Interval parameters of ILP constrants b Rght-hand sde of parameters of ILP c Interval parameters of ILP obectve functon f ± Value of obectve functon f ± opt Optmum obectve functon nterval f + opt Upper bound of optmum obectve functon f - opt Lower bound of optmum obectve functon f 2-stepopt Optmum obectve functon solved by 2-step algorthm f BWCopt Optmum obectve functon solved by BWC algorthm k 1 Number of postve c parameters xv

15 0 Aspraton level Rsk level varables Rsk functon Dfferent waste processng facltes: I =1 for recyclng faclty =2 for compostng faclty =3 for landfll T The plannng tme perod, t=1,2,,6 x Decson varables of ILP x ± opt Optmum decson varable ntervals of x x + opt Upper bound of optmum decson varable x x - opt Lower bound of optmum decson varable x - - superscrpt represents the lower bound of an nterval-parameter or varable + + superscrpts represents the upper bound of an nterval-parameter or varable xv

16 Acknowledgement It s a pleasure to thank those who have helped and nspred me to make ths thess possble. Frstly, I owe my deepest grattude to my supervsor, Dr. Le Lu, whose encouragement, gudance and support from the ntal to the fnal level enabled me to develop an understandng of the subect. I apprecate hs contrbutons of tme, deas, and fundng to make my study experence productve. I could not have magned havng a better advsor and mentor for my study. Besdes my advsor, I would lke to thank my supervsory commttee members, Dr. Graham Gagnon and Dr. Uday Venkatadr, for ther great advce for my thess. I would also lke to express my apprecaton to Kara Weatherbee, the admnstratve assstant from Sold Waste Resources department of HRM, who answered all my questons and provded valuable data to me. I am heartly thankful to my Englsh tutor and frend, Elane Dowlng, for her great help on my wrtng sklls. She helped me to edt the thess, and her opnons are always encouragng and nsprng. My specal thanks are due to my husband, Shuntan Wang, who stands by my sde all the tme supportng and encouragng me, mentally and fnancally. Thanks for the numerous dnners he prepared for me, so that I had the energy and tme to work. The thess would not be possble wthout hs lovng support. Last but not least, I would lke to thank my parents Zhenguo Pe and Mn Zhang, and my sster Lnln Pe, for ther unflaggng love and support throughout my lfe. Thank you. xv

17 CHAPTER 1 INTRODUCTION 1.1. Statement of the Problem Muncpal Sold Waste (MSW) Management System Muncpal sold waste (MSW) ncludes garbage, refuse, sludge, rubbsh, talngs, debrs, ltter and other dscarded materals resultng from resdental, commercal, nsttutonal and ndustral actvtes whch are commonly accepted at a muncpal sold waste management faclty. (Nova Scota Envronment and Labour, 2004) Generally, three maor components exst n a MSW management system. They nclude waste generaton dstrcts, waste processng facltes, and landfll. Transfer statons are also ncluded n the system n some bg ctes lke New York Cty (USEPA, 2000). In 1996, Envronment Canada defned the Integrated Muncpal Sold Waste (IMSW) system as a management system that contans programs whch can reduce or reuse the wastes produced, and/or dvert wastes from tradtonal dsposal facltes to recyclng, compostng, and/or ncneraton processes (Envronment Canada, 1996). Based on ths defnton, a MSW management system must be substantally modfed to perform the far more complex tasks to ensure that wastes are ntally reduced, certan products are reused, and wastes are separated at ther sources from recyclables and compostables whch can then be processed, stored, marketed and transported to buyers (Baetz et al., 1989). As a consequence, many factors must be consdered n plannng such an ntegrated management system. Examples nclude collecton technques to be adopted, how to allocate the trash, when to expand the exstng facltes, where to develop a new faclty, and how to control the total system cost (Wlson, 1985). Interactons among these factors make the management system more complcated. Besdes, conflcts may exst among dfferent decson-makers and nterest groups, such as local government offcals, faclty 1

18 owners and operators, consultants, regulatory agency specalsts, and resdents. Multple optons may lead to dfferent levels of satsfacton for each of the stakeholders (Barlshen, 1996). Ths complexty makes t dffcult to dentfy the optmal MSW management optons among decson makers. Thus, optmzaton technques that could consder and ncorporate factors wthn a general framework, rather than examne them n solaton, would be desrable for provdng a holstc analyss of the factors, and for a comprehensve evaluaton of the related actvtes and polcy responses (Baetz, 1990b; Huang et al., 2001) Optmzaton Model Development for MSW Plannng Optmzaton technques for waste management were frst proposed by Anderson n Snce then, a number of waste-related plannng models have been developed. They nclude the determnstc optmzaton models, such as lnear, mxed-nteger lnear, dynamc, and mult-obectve programmng (Anderson, 1968; Jenkns, 1982; Baetz et al., 1989; Baetz, 1990a&1990b; Thomas and Baetz, 1990). However, t has been recognzed that determnstc optmzaton technques are not suffcent to model such a complex problem, partcular ts uncertan features. To better reflect uncertantes n waste management systems, several optmzaton technques were developed. They nclude fuzzy mathematcal programmng (FMP) (Zmmermann, 1985; Huang, 1994; Lembach, 1996; Chang, 1997; Lee and Wen, 1997; Chanas, 2000; Seongwon et al., 2003), stochastc mathematcal programmng (SMP) (Zare and Daneshmand, 1995; Schwarm, 1999; Schultz, 2003), nterval-parameter mathematcal programmng (IPMP) (Moore, 1979; Zou et al., 1999; Rocha and Krenovch, 2003) and some hybrd or ntegrated programmng methods (Huang et al., 1993b; Huang et al., 1995; Infanger and Morton, 1996; Xa et al., 1997; Huang et al., 2001; Huang et al., 2002; L et al., 2009a; Sun et al., 2009; Ne et al., 2009; Guo and Huang, 2009a&2009b; L and Huang, 2010a;). Mxed outcomes have been obtaned when dfferent methods were used to reflect system uncertantes n hypothetcal or real-world case studes. Compared to fuzzy and stochastc programmng, n terms of data qualty and requrements, IPMP does not need the nformaton of membershp functons or dstrbuton of parameters whch may be hard to 2

19 obtan n practcal applcatons. Moreover, fuzzy and stochastc methods often lead to more complcated sub-models and may not be practcal for many real lfe stuatons. Interval-parameter lnear programmng (ILP) s one of the IPMP whch can effectvely deal wth uncertantes wthout leadng to more complcated sub-models. Because of ths characterstc, ILP has been wdely used n many envronmental and cvl modelng felds (Ben-Israel and Robers, 1970; Rommelfanger et al., 1989; Huang and Moore, 1993; Tong, 1994; Hansen and Walster, 2004; Maqsood et al., 2004; L et al, 2008&2009b; Guo et al, 2010; Wu et al., 2010; Dong et al., 2011), ncludng MSW system plannng and management ssues. Three soluton algorthms have developed to facltate the use of ILP, ncludng Monte-Carlo smulaton, best-worst case analyss (BWC) and 2-step nteractve algorthm. The Monte-Carlo smulaton method randomly sets values for each parameter wthn ther nterval range to form a classc Lnear Programmng (LP) model. Ths method could be most relable n smulatng a real model stuaton f the modeler sets values for parameters and runs the model mllons of tmes. However, t s not always realstc to run the models mllons of tmes, especally when facng a real envronmental systems plannng problem wth hundreds or thousands of decson varables and constrants. As more realstc soluton methods, BWC analyss and 2-step algorthm were developed (Huang and Moore. 1993; Tong, 1994; Chnneck and Ramadan, 2000). Both approaches reformulate the orgnal model usng extreme constrants to represent the most conservatve and the most aggressve condtons. The man dfference between the two algorthms s that the 2-step algorthm dfferentates the selecton of extreme parameter values wth dfferent sgns after the obectve functon s fxed, whle the BWC treats all the parameters wthout dscrmnaton. These two algorthms have been wdely used n ILP applcatons. Both algorthms provde an nterval soluton space, and each pont n the nterval soluton space becomes a potental soluton to form a decson alternatve for mplementaton. However, n practcal decson makng process, when the decson makers need to pck a pont from the nterval soluton space for an mplementaton scheme, there wll be problems of feasblty and optmalty, whch mght be resulted from the flaws n the algorthm development. Therefore, valdty checkng seems to be 3

20 absolutely desred for the ILP algorthms n order to mprove the applcablty of the ILP modelng results. Moreover, the result of the ILP lacks of a lnkage between decson rsks and system return. Dfferent decson makers may expect dvergent system returns under varous acceptable rsk levels. Thus, a modelng framework whch could not only be effectvely and effcently solved but also can reflect the lnkage between system return and decson rsks became desrable. Ths modelng framework was prelmnarly examned by Zou et al. (2010) through developng a rsk explct nterval-parameter lnear programmng model (REILP). However, several defcences stll exst n the REILP, ncludng the nfeasblty problem, the rsk functon ssue, and the dffculty of the aspraton level selecton. These defcences wll be addressed n ths study as an evoluton of the proposed REILP approach MSW Management System n HRM Nova Scota s the frst provnce n Canada that has acheved the target of a 50% sold waste dverson rate, accordng to the crtera establshed by the Canadan Councl of Mnsters of the Envronment (Walker et al., 2004). The Halfax Regonal Muncpalty (HRM), as the captal of Nova Scota, provded an outstandng contrbuton to ths hgh dverson rate achevement. HRM reported n January 2011 that the current dverson rate was rght on target at 60% (CMC, 2010 Newsletter). Along wth the dverson target, HRM developed several long-term goals for effectvely managng the MSW management system of HRM, ncludng maxmzng the 3Rs (Reducton, Reuse and Recyclng), maxmzng envronmental sustanablty and mnmzng costs, and fosterng stewardshp and values of a conserver socety (FCM, 2000). To reach the dverson rate target and the long-term envronmental goals, HRM has strved to establsh a new MSW management system snce 1999, along wth many programs ncludng source separaton program, new landfll ste constructon, and materals recovery faclty expanson program (Walker et al., 2004). Durng the 4

21 mplementaton of these programs and contnuous mprovement of the MSW dverson rate, cost s a key factor that must be consdered by decson makers. Ths new MSW management system has a gross annual budget of 32 mllon Canadan dollars. Captal costs nclude $8.5 mllon for the carts, mn bns and trackng system; $24 mllon for the mxed waste processng system and the back-end stablzaton plant; $20 mllon for the new landfll (whch opened n January 1999) constructon and access roads; and $500,000 for the upgrade to the Materals Recovery Facltes (MRF). The mxed waste processng and stablzaton operaton employed 85 people (Goldsten and Gray, 1999). Ths s a sgnfcant cost for a cty lke the sze of HRM. Furthermore, f a hgher dverson rate s desred, or f the facltes need to be expanded or relocated, addton nvestment would be needed. There s always a tradeoff between the system nvestment and envronment effects. Economc growth s usually sacrfced for the good of the envronment. The deal and best soluton for envronmental protecton s usually unaffordable. It s more realstc to fnd an optmal soluton whch s economcally feasble and envronmentally acceptable. In HRM, a problem that decson makers have to consder and fnd the answer s what the least-cost strategy for achevng the waste dverson rate target as well as the long-term envronmental goals s. Prevously, very few studes have reported a comprehensve MSW management system study for HRM. Ghadr (2004) developed a fuzzy-stochastc mxed nteger LP model for the long-term plannng and management of the MSW system of HRM. Ths study focused on addressng system uncertantes assocated wth the MSW management processes and t has provded a very valuable decson support tool for local decson makers and been apprecated by them. However, the dverson rate target was not ncluded n the study and the model was solved by the 2-step algorthm. No valdty checkng for the ILP method was conducted and the connectons between decson rsks and system benefts were not examned as well. All the relevant ssues wll be addressed n our study for provdng better and more relable decson support nformaton for the HRM MSW managers. 5

22 1.2. Obectve As an extenson of prevous efforts on MSW management system modelng, ths study wll focus on the development of a fuzzy rsk explct nterval lnear programmng model and ts applcaton to the HRM case study. Ths study entals the followng obectves: Valdty checkng of two ILP soluton algorthms,.e., Best-Worse Case algorthm and the 2-step nteractve algorthm. A numercal example wll be formulated and solved by the Monte-Carlo smulaton, BWC and 2-Step algorthm respectvely for nvestgatng the valdtes of BWC and 2-Step algorthm, and the focus wll be on the feasblty and optmalty of the nterval solutons. Development of a Fuzzy Rsk Explct Interval Lnear Programmng (FREILP) model for reflectng complex connectons between system return and decson rsk. The proposed FREILP model s desgned to mnmze the decson rsks whle the total system cost s mantaned at a mnmum level and the aspraton level s preferably selected by the decson makers. In addton, problems of model nfeasblty and rsk functon formulaton wll also be dscussed. Applcaton of the developed FREILP model to the long-term plannng of the MSW management system n HRM. The modelng results could provde HRM managers scentfc bass for generatng practcal MSW management mplementaton schemes. In ths applcaton, three optons based on the dfferent aspraton levels of decson makers wll be provded, ncludng aggressve schemes, medum schemes, and conservatve schemes. 6

23 1.3. Structure of the Thess The structure of ths thess s organzed as follows: Chapter 1 ntroduces the MSW system and the applcaton of mathematcal programmng to the plannng and management of the MSW system. Problems assocated wth the current modelng studes are dscussed, leadng to the need of a new approach that could provde decson support for effectve and effcent MSW plannng and management. Chapter 2 provdes a comprehensve lterature revew wth respect to prevous efforts n usng optmzaton technques to solve MSW plannng and management ssues. The focuses have been placed on dscussng uncertanty-handlng optmzaton technques and ther applcatons to MSW management system. A summary has been provded for dentfyng a few knowledge gaps that need to be addressed n future studes. Chapter 3 uses a numercal example to conduct the valdty checkng of exstng algorthms to ILP models wth focuses on checkng the feasblty and optmalty of the model solutons. Chapter 4 dscusses the detals of FREILP model formulaton and development. The soluton process for solvng the FREILP model s also provded. Chapter 5 presents a detaled descrpton of the MSW management system of HRM and the FREILP-based model formulaton. Chapter 6 provdes the modelng results obtaned from the REILP model and FREILP model. The mplcatons of the modelng results to the plannng and management of the HRM waste system s also dscussed. Chapter 7 s a summary of ths study along wth some conclusons. Recommendaton for further work s also provded n ths chapter. 7

24 CHAPTER 2 LITERATURE REVIEW 2.1. Conventonal MSW Management and Plannng Methods Snce the 1960 s, much efforts have been devoted to develop mathematcal programmng models to support the decson makers of MSW management system and to evaluate the relevant operatonal and nvestment polces. Anderson frstly proposed an economc optmzaton method for the plannng of MSW n 1968 (Anderson, 1968). After that, a number of waste-related determnstc optmzaton models have been developed. Champ et al.(1982) appled the lnear programmng (LP) technques to nvestgate the relatve costs of hazardous waste management schemes. Hseh and Ho (1993) also used the LP to optmze the sold waste dsposal and recyclng system. Jenkns developed a mx-nteger lnear programmng (MILP) method and appled t to the plannng of the muncpal waste reclamaton system for Southeastern Ontaro (Jenkns, 1982). Huang et al. appled the MILP n the MSW management system n Chna (Huang et al., 1997). Baetz et al. appled the dynamc programmng (DP) model to determne the optmal szng and tmng of faclty expansons for Mecklenburg County, North Carolna, US, the Long Island Communty n New York, US, and the Regonal Muncpalty of Hamlton, Ontaro, Canada (Beatz et al., 1989); Chang and Wang (1995) appled a mult-obectve lnear programmng model (MLP) to a MSW system. They consdered both economc and envronmental obectves whle establshng and selectng the long term optmal management alternatves. Solano et al. (2002) used LP and developed an ntegrated sold waste-management model to assst n dentfyng desred sold waste management strateges that could meet cost, energy and envronmental emsson obectves. As envronmental polces have become more ntegrated and strct, t has been recognzed that the above determnstc optmzaton technques are not suffcent to model complex 8

25 MSW management problems. In MSW management system, many processes need to be consdered by decson makers, ncludng waste collecton, allocaton, transportaton, treatment and dsposal (Wlson, 1985). These processes contan many factors that nteract wth each other wth mult-perod, mult-layer, and mult-obectve features (Thomas et al., 1990). The spatal and temporal varatons of many system components may further multply the uncertantes n the system (Thompson and Tanapat, 2005). Therefore, these factors are assocated wth uncertantes and hard to be evaluated n precse terms. Snce the determnstc optmzaton technques requre determnstc data and crsp model constrants, optmzaton technques that can reflect uncertantes became desrable. Varous approaches that can deal wth uncertantes have been developed and appled to MSW management study. When parameters n the model are fuzzy sets, the optmal problem becomes the fuzzy mathematcal programmng (FMP). Comparng to FMP problems, stochastc mathematcal programmng (SMP) models are smlar n style but probablty dstrbutons are governng the data. When the parameters are known only wthn certan bounds, the approach to tacklng such problems s called nterval-parameter mathematcal programmng (IPMP) or robust optmzaton Optmzaton Approaches that Deal wth Uncertantes Fuzzy Mathematcal Programmng Fuzzy mathematcal programmng (FMP) s based on the fuzzy set theory formalzed by Professor Loft Zadeh n Comparng to the classcal sets (membershp can only be 0 or 1), the member of fuzzy sets could be descrbed wth the ad of a membershp functon valued n the real unt nterval [0, 1]. Parameters n both the obectve functon and the constrants could be fuzzy sets. FMP has been appled to many optmzaton problems, waste management ncluded. Koo et al. (1991) accomplshed the locaton plannng of a regonal hazardous waste treatment 9

26 center by usng FMP. In the same year, Lee et al. (1991) appled FMP to a dredged materal management problem, usng the envronmental rsk and cost as fuzzy parameters. Chang and Wang (1997) appled a fuzzy goal programmng approach for the optmal plannng of sold waste management system n a metropoltan regon n Tawan. It demonstrated how fuzzy obectves of the decson maker can be quantfed under varous types of sold waste management alternatves. More recently, Fan et al. (2009) developed a fuzzy lnear programmng (FLP) model for the long term plannng of MSW management system. It can deal wth uncertantes expressed as fuzzy sets that exst n the constrant s left-hand and rght-hand sdes and obectve functon. FMP can be categorzed nto two maor streams: fuzzy flexblty programmng (FFP) and fuzzy possblty programmng (FPP) (Inuguch and Sakawa, 1994). In FFP, the flexblty n the constrants and fuzzness n the obectve are represented by fuzzy sets and denoted as fuzzy constrants and fuzzy goal respectvely, whch can be expressed as membershp grades. However, FFP could hardly tackle uncertantes expressed as ambguous coeffcents n the obectve functon and constrants (Inuguch and Tanno, 2000; Inuguch and Ramk, 2000; Inuguch et al., 2003). In FPP, fuzzy parameters are ntroduced nto the modelng framework, and these parameters are presented as fuzzy sets wth possblty dstrbutons. The lmtaton of FPP s that when many uncertan parameters are expressed as fuzzy sets n a model, nteractons among these uncertantes may lead to serous complextes, partcularly for large-scale practcal problems (Huang et al., 1993a). Also, a key factor of FMP, whch s also the central concept of fuzzy set theory, s the membershp functon of fuzzy sets, numercally representng the degree to whch an element belongs to a set. But the membershp functons of parameters are not so easy to defne. An naccurate membershp functon may lead to undesrable results Stochastc Mathematcal Programmng Stochastc mathematcal programmng (SMP) s derved from probablty theory. In SMP, random elements are ntroduced to account for probablstc uncertanty n the coeffcents. Wlson and Baetz (2001) developed a derved probablty model for curbsde 10

27 waste collecton actvtes that allowed for analyzng stochastc nformaton n the MSW management. The nherent uncertantes n a model can be expressed as stochastc elements n the constrant matrx, the rght-hand sde stpulatons, or the obectve functons (Sengupta, 1972). However, f all parameters are expressed as random varables, the model would be extremely hard to solve and often leads to nfeasblty problems. Chance-constraned programmng (CCP) s one of the SMP methods that contan only the rght-hand sde parameters (b ) as random dstrbutons. The CCP approaches do not requre all the constrants to be satsfed. Instead, t allows a certan level of volaton of constrans wth random b under some crcumstance (Loucks et al., 1981). As descrbed n the followng equaton P[ g ( x) b ] p, the probablty that the th constrant s satsfed s p, where p s greater than 0 and less than 1. The random b can be determned by ts dstrbuton and the possblty p. As many factors other than the rght-hand sde stpulatons exst as uncertantes n the system, CCP s often combned wth other uncertanty reflectng methods lke FMP and IPMP. For example, Maqsood et al. (2004) developed an nexact two-stage stochastc mxed-nteger lnear programmng model for waste management. L and Huang (2006) appled the nexact two-stage stochastc mxed-nteger lnear programmng to sold waste management n the Cty of Regna. Guo and Huang developed an nexact fuzzy-stochastc mxed-nteger programmng approach and appled the approach to the long term plannng of MSW management system n the cty of Regna, Canada (Guo and Huang, 2009a& 2009b). L et al. (2009a) developed an nexact fuzzy-stochastc constrant-softened programmng method for waste-management systems plannng by ntroducng FFP nto an nexact multstage stochastc programmng framework, where a number of volaton varables for the constrants are ntroduced, allowng n-depth analyses of tradeoffs among economc obectve, satsfacton degree, and constrant-volaton rsk. The maor strength of SMP method s that t does not smply reduce the complexty of the problems; t allows decson makers to have a complete vew of the effects of 11

28 uncertantes and the relatonshps between uncertan nputs and resultng solutons (Huang, 1994). The maor problem of SMP s that no suffcent data are avalable to obtan the probablty dstrbuton functons (PDF) for random parameters. In addton, even f these functons are avalable, t s extremely hard to solve a large scale stochastc management system plannng problem wth all uncertan data beng expressed as PDFs (Brge and Louveaux, 1997; Luo et al., 2007) Interval-Parameter Mathematcal Programmng The FMP and SMP can effectvely reflect uncertantes n the model system. However, n many practcal problems, the avalable data s often not enough to be presented as dstrbuton functons or membershp functons. Instead, the obtaned data may often show a robust dstrbuton or no dstrbuton nformaton at all. It s more dffcult for planners or decson makers to specfy dstrbutons than to defne a fluctuaton nterval for uncertantes (Huang, et al., 1992). For example, the daly waste generaton rate often fluctuates wthn a certan nterval, but t may be dffcult to state a relable probablty dstrbuton for ths varaton (L and Huang, 2010b). Interval-parameter mathematcal programmng (IPMP) s an alternatve for handlng uncertantes n the model's constrants and obectves (Huang et al., 1992). Unlke FMP and SMP, IPMP does not need the membershp functons or dstrbutons of nexact parameters. It s based on the nterval analyss, and needs only the lower and upper bounds of uncertan parameters. The nterval analyss was proposed by Moore (1979) and then extended nto IPMP. The IPMP method emphaszes the ntrnsc vagueness of ts nformatonal characterstcs durng parameter estmaton (Change et al., 1997). Due to the smplcty of IPMP, the last three decades have seen a wde applcaton of t to deal wth uncertantes n many felds (Ben-Israel and Robers, 1970; Rommelfanger et al., 1989; Huang and Moore, 1993; Tong, 1994; Hansen and Walster, 2004; Maqsood et al., 2004; L et al, 2008&2009b; Guo et al, 2010; Wu et al., 2010; Dong et al., 2011). However, smlar to FPP, when many uncertan parameters are expressed as ntervals, nteractons among these uncertantes may lead to serous complcated problems that are 12

29 hard to solve, partcularly for large-scale practcal problems (Huang et al., 1993). Interval-parameter lnear programmng (ILP) s a key member of the IPMP famly and s a method that can deal wth uncertantes expressed as ntervals wthout any dstrbutonal nformaton and wthout leadng to more complcated problems. The ILP allows the nterval nformaton to be drectly communcated nto the optmzaton process and resultng soluton. Early applcatons of ILP ncorporated nterval numbers nto the obectve functon (Ishbuch and Tanaka, 1990), constrant matrx (Huang and Moore, 1993; Tong, 1994), rght-hand sdes of constrants, and all of the above (Huang, 1996; Huang et al., 1992& 1995). Case studes of ILP nclude hypothetcal examples for sold waste management (Huang et al., 1992&1995; Maqsood and Huang, 2003; Maqsood et al., 2004; L and Huang, 2006; L et al., 2006a), water resources allocaton (Huang and Loucks, 2000; L et al., 2006b; Maqsood et al., 2005), and flood dverson plannng (L et al., 2007). Practcal examples nclude water qualty management n Chna (Huang, 1998), sold waste management for the cty of Regna (L and Huang, 2006), and water system plannng n Amman, Jordan (Rosenberg and Lund, 2009). The reason that ILP has been wdely used n the past decades s that t can effectvely reflect the uncertantes n the modelng system, the nformaton requrement s low, and the soluton algorthms are easy to use. Three algorthms have been used to solve ILP models, ncludng Monte Carlo smulaton algorthm, two-step nteractve algorthm, and best-worst case (BWC) algorthm. Monte Carlo smulaton method reles on repeated random samplng to compute the results (Rubnsten, 1981). It has a very hgh computatonal requrement, so t s unrealstc to solve complcated models wth a large number of uncertan parameters and varables. Two-step nteractve algorthm and BWC algorthm were developed by Huang and Moor (1992) and Tong (1994) respectvely. Both algorthms solve the model by generatng two determnstc sub-models that correspond to the lower bound and the upper bound of the obectve functons. Wth regards to the two algorthms, Rosenberg (2009) revewed several ILP models and summarzed that the two-step algorthm sometmes cannotprovde a good performance. No valdty checkng for the two algorthms has been conducted prevously n term of the feasblty and optmalty of the obtaned solutons. 13

30 Hybrd and Mxed Mathematcal Programmng Combnng advantages of the above optmzaton methods, many hybrd and mxed programmng approaches have been developed. Due to the flexblty of nterval numbers, most approaches combne ILP wth other optmzaton methods. For example, Huang et al. (1993b) proposed a grey fuzzy flexble programmng method and appled t to MSW management systems to tackle uncertantes that presented as fuzzy sets and ntervals. Chang et al. (1997) proposed a fuzzy nterval mult-obectve mxed nteger programmng model to evaluate sold waste management strateges. The study demonstrated how uncertan factors could be quantfed by specfc membershp functons and nterval numbers n a mult-obectve model. Maqsood I. et al. (2004) developed a hybrd optmzaton approach, nexact two-stage mxed nteger lnear programmng model for MSW management. The model mproved upon the mxed nteger, two-stage stochastc and ILP approaches by allowng uncertan parameters presented as random dstrbutons and dscrete ntervals. L and Huang (2010b) developed an nterval-based possblstc programmng method for the plannng of waste management, wth mnmzed system cost and envronmental mpact. Ths model was a combnaton of FMP and SMP These hybrd and mxed approaches can combne the advantages and avod the lmtaton of sngle methods, and flexbly use the avalablty of nput data. However, all these nterval-based methods need to be solved by ILP algorthms eventually. As a result, f the exstng ILP algorthms prove to have fundamental flaws n ts soluton process, the valdty of these prevous ILP-based hybrd studes wll be problematc. It s then n a pressng need of re-examnng the exstng ILP algorthms to overcome ther lmtatons. Ths has been proven to be very dffcult (Zhou et al., 2009). Otherwse, rsks of the decsons generated by these flawed optmzaton processes need to be evaluated f they have to be mplemented. 14

31 2.3. Prevous Studes on Sold Waste Management n HRM The Halfax Regonal Muncpalty (HRM) was founded n 1996 wth the combnaton of four communtes: Halfax, Dartmouth, Bedford, and Halfax County (HRM, 2010a). It s the largest populaton center of the Canadan east coast and the captal of the provnce of Nova Scota. HRM s also a muncpalty that commtted to envronmental sustanablty. It was the frst wnner of FCM-CH2M HILL Sustanable Communty Awards for the communty-based waste resource management strategy (FCM, 2000). By mplementng many envronmental programs (Walker et al., 2004), HRM now has a very hgh waste dverson rate, whch s 60% (HRM, 2010c). Ghadr (2004) developed an nterval-parameter fuzzy-stochastc mxed nteger lnear programmng model for the long term plannng of MSW management system n HRM. The modelng results can help to answer questons related to types, tmng and stes of the MSW management actvtes, as well as reflect dynamc, nteractve, and uncertan characterstcs of the MSW system n HRM. However, the dverson rate target was not consdered n the developed model and also the model was solved by the 2-step algorthm. Both facts lead to obvous lmtatons n Ghadr s study. The method proposed n ths thess attempts to address these ssues n a desrable fashon. 15

32 CHAPTER 3 VALIDITY CHECKING FOR ILP ALGORITHMS 3.1. Exstng ILP Soluton Algorthms Defnton 3.1.1: An nterval-parameter lnear programmng (ILP) model (for maxmzed problems) s defned as (Huang et al. 1992): Max f C X (3.1.1) s.t. A X B (3.1.2) X 0 (3.1.3) where, C [ c1, c2,..., c n ] [ 1 2 x n X x, x,..., ] T [ 1 2 b m B b, b,..., ] T A { a }, =1,,m; =1,2,,n. For mnmze problems, the ILP model s as the followng: Mn f C X (3.1.4) s.t. A X B (3.1.5) X 0 (3.1.6) Where the - and + superscrpts represent lower and upper bounds of an nterval-parameter or varable respectvely. 16

33 Snce nterval parameters exst n the obectve functon and constrants to reflect uncertantes n the model, the optmal solutons of the ILP model are also ntervals: f [ f, f ] opt opt opt X [ x, x,..., x ] opt 1opt 2opt nopt (3.1.7) (3.1.8) xopt [ xopt, xopt ], 1,2,..., n (3.1.9) As ntervals exst n ILP models, and no software can drectly solve the model wth ntervals n t, the ILP models need to be converted nto some formats that software can recognze and solve. Presently, three algorthms have been developed and used for solvng ILP problems, ncludng Monte Carlo smulaton algorthm, 2-step algorthm, and best-worst case (BWC) algorthm Monte Carlo Smulaton Algorthm Monte Carlo smulaton method s a computatonal algorthm that reles on repeated random samplng to compute the results (Rubnsten, 1981). It s useful for modelng phenomena wth sgnfcant uncertanty n nputs. To solve an ILP problem, t randomly sets values for parameters wthn ther ranges to form a classc Lnear Programmng model (Sabelfeld, 1991). The process of Monte Carlo smulaton algorthm (Robert and Casella, 2004.) for ILP models s descrbed n the followng steps: [Step 1] Assgn random feature to each nterval parameter (.e., a, b, and c ) n the ILP model wth probablty dstrbuton functons (PDFs), and then convert each PDF nto ts cumulatve dstrbuton functon (CDF). Fgure 3.1 gves an example of the PDF of a normally dstrbuted parameter and t could be converted nto ts CDF as shown n Fgure 3.2. Normal dstrbuton s the most commonly used dstrbuton of nterval parameters, although other dstrbutons lke unform dstrbuton and exponental dstrbuton could also descrbe the dstrbutng characterstc of parameters. 17

34 0.1 PDF of a parameter 0.08 Probablty Parameter value Fgure 3.1 The PDF curve of a parameter 1 CDF curve of a parameter P(X<=x) Parameter value Fgure 3.2 The CDF curve of a parameter converted from PDF 18

35 [Step 2] Use random number generator (computed code avalable n most programmng software) to generate a random number between 0 and 1, denoted as r. [Step 3] Relate the generated random number r to the CDF curve for each parameter, and then get a set of determnstc values for parameters n a, b, c. An example of how to randomly set a value for a parameter s shown n Fgure 3.3, n whch the parameter s found to be CDF curve of a parameter 0.8 P(X<=x) r Parameter value Fgure 3.3 An example of randomly settng a value for a parameter [Step 4] Use the set of determnstc parameter values to replace the nterval a, b, c and thus form a classc determnstc lnear programmng model. Then solve the classc determnstc LP model and produce a set of determnstc soluton. [Step5] Repeat Steps 2, 3 and 4 for a suffcent number of runs untl a dstrbuton of the soluton for each specfc decson varable s obtaned. 19

36 Monte Carlo smulaton method could be a very successful algorthm for ILP problems (Jahanshahloo et al, 2008) because t can generate the sound results through smulatng the real-world stuaton. However, because of ts relance on large numbers of repeated computaton of random or pseudo-random numbers, Monte Carlo smulaton method has a very hgh computatonal requrement. Mllons of tmes, or even bllons of smulaton need to be conducted n order to obtan a dstrbuton for a specfc soluton. Ths makes the method unrealstc to solve complcated practcal models wth a large number of uncertan parameters and varables. In addton, n real-world problems, t s almost mpossble to obtan suffcent data to formulate dstrbutons for nterval parameters (Nawrock, 2001). Therefore, the best-worst case algorthm and the 2-step algorthm were developed as alternatves to solve the ILP models Two-Step Algorthm Two-step algorthm was ntroduced by Huang and Moore n It s a 2-step nteractve method, whch at frst formulates a sub-model to solve for the upper bound of obectve functon when t s a maxmzaton problem, and then solve the sub-model correspondng to the lower bound of obectve functon (Huang and Moore, 1993). For the n nterval coeffcents c ( = 1, 2,, n) n the orgnal obectve functon f C X n the ILP model, f k 1 of these coeffcents are non-negatve, and the rest are negatve, let the n coeffcents be rearranged such that c 0 ( = 1, 2,, k 1 ) and 0 (= k 1 +1, k 1 +2,, n). The sub-model correspondng to the upper bound of the obectve functon (when the obectve s to be maxmzed) s formulated as: c Max f k1 n x 1 k1 1 c c x (3.1.10) s. t. k1 n a Sgn( a ) x a Sgn( a ) x b, 1 k11 (3.1.11) 20

37 x 0, (3.1.12) The sub-model (3.1.10)-(3.1.12) s a classc lnear programmng model, whch can be solved wth any exstng algorthm such as the Smplex method. After solvng the above sub-model correspondng to the upper bound of the obectve functon, we can then formulate the sub-model correspondng to the lower bound: Max f k1 n x 1 k1 1 c c x (3.1.13) s. t. k1 n a Sgn( a ) x a Sgn( a ) x b, 1 k1 1 (3.1.14) x 0, (3.1.15) x x opt, 1,2,..., k 1 (3.1.16) x x opt k 1, k 2,..., n, 1 1 (3.1.17) where x opt ( = 1, 2,, k 1 ) and x opt (= k 1 +1, k 1 +2,, n) are the optmal solutons generated from sub-model (3.1.10)-(3.1.12). The sub-model (3.1.13)-(3.1.17) s also a classc lnear programmng model that can be easly solved to obtan the optmal solutons. Thus, after mplementng the above two-step method, we can obtan optmal nterval solutons of the ILP model as x [ x, x ] opt opt opt and f [ f, f ]. opt opt opt For mnmzaton problems, the sequence of solvng the sub-models s opposte. The frst sub-model could be formulated as (3.1.18) to (3.1.20): Mn f k1 n c x 1 k11 c x (3.1.18) s. t. k1 n a Sgn( a ) x a Sgn( a ) x b, 1 k1 1 (3.1.19) 21

38 x 0, (3.1.20) The second sub-model could then be formulated as equatons (3.1.21) to (3.1.25). Mn f k1 n c x 1 k1 1 c x (3.1.21) s. t. k1 n a Sgn( a ) x a Sgn( a ) x b, 1 k1 1 (3.1.22) x x opt, 1,2,..., k 1 (3.1.23) x x opt k 1, k 2,..., n, 1 1 (3.1.24) x 0, (3.1.25) Smlar as maxmzaton models, after solvng the above two sub-models, we can obtan optmal nterval solutons of the orgnal ILP model Best-Worst Case Algorthm The best-worst case algorthm (BWC) (Tong, 1994; Chnneck and Ramadan, 2000) s smlar as the 2-step algorthm n solvng two sub-models. The maor dfference between them les n that the 2-step algorthm dfferentates the selecton of extreme parameter values (.e., lower or upper bounds of coeffcents) for decson varables n the obectve functon based on ther dfferent sgns (.e., negatve or postve coeffcents n c )., whle the BWC treats all the parameters wthout dscrmnaton. The soluton process of the BWC algorthm s provded below for a maxmzaton problem: The frst step of the BWC algorthm s to formulate the Best-Case sub-model correspondng to the upper bound of the obectve functon as follows: Max f c x (3.1.26), s. t. a x b (3.1.27) 22

39 x 0, (3.1.28) The second step s to formulate and solve the sub-model correspondng to the worst-case stuaton: Max f c x (3.1.29), s. t. a x b (3.1.30) x 0, (3.1.31) For mnmzaton problems, the BWC algorthm reformulates the orgnal ILP model nto followng two sub-models correspondng to best- and worst-case, respectvely. Best-Case Sub-model: Mn f c x (3.1.32), s. t. a x b (3.1.33) x 0, (3.1.34) Worst case Sub-model: Mn f c x (3.1.35), s. t. a x b (3.1.36) x 0, (3.1.37) In the BWC algorthm, best case or worst case refers to a stuaton that both obectve functon and constrants of the best case or worst case sub-model represent the best or worst extremtes, respectvely, of the orgnal ILP model. Specfcally, for a maxmzaton problem as ndcated n the model ( ), the obectve functon of the best-case sub-model represents the upper bound of the orgnal ILP model (model ), and ts constrants delmt a largest decson space for the optmal soluton to be searched from t (model ); whle the obectve functon of the worst-case sub-model 23

40 gves the lower bound of the orgnal ILP model (model ) and ts constrants bound a smallest decson space. However, for a mnmzaton problem as ndcated n the model ( ), the obectve functon of the best case sub-model represents the lower bound of the orgnal ILP model (model ), and ts constrants delmt a smallest decson space for the optmal soluton to be searched from t (model ); whle the obectve functon of the worst-case sub-model gves the upper bound of the orgnal ILP model (model ) and ts constrants bound a largest decson space Comparson of Two-Step and BWC Algorthms (1) Model Equvalence Theorem 3.1.1: The 2-step algorthm s equvalent to the BWC algorthm for an ILP problem f and only f, Sgn( a ) Sgn( c ), : Max f k1 n x 1 k1 1 c c x (3.1.38) s. t. k1 n a x a x b, 1 k11 (3.1.39) x 0, (3.1.40). Proof: Model ( ) gves a general ILP model whch s used as an llustratve example n the proof. Assume c ± > 0 for =1, 2,,k 1 and c ± < 0 for =k 1 +1, k 1 +2,,n. Also, let ( Sgn ) 1 for =1, 2,,k 1 and Sgn ( ) 1 for =k 1 +1, k 1 +2,,n. Thus, c accordng to the condton of Theorem 3.1.1, we have Sgn ( a ) Sgn( ) 1 for =1, c c 2,,k 1 and Sgn ( a ) Sgn( ) 1 for =k 1 +1, k 1 +2,,n. The proof of Theorem s provded as follows: c 24

41 The frst step we need to prove s that the obectves functons of two sub-models reformulated by the two algorthms are equvalent to each other. The 2-step algorthm reformulates the obectve functon of the orgnal ILP model n a way that the decson varables are regrouped accordng to the sgns of the correspondng coeffcents (c ± ), as ndcated n model ( and ). It s qute obvous that the upper-bound obectve functon (.e., f + ) reformulated by the 2-step algorthm s equvalent to the obectve functon of the best-case sub-model from the BWC algorthm, as ndcated n model (3.1.41), and the lower-bound obectve functon (.e., f - ) from the 2-step algorthm s equvalent to that of the worst-case sub-model as seen n model (3.1.42). The only treatment s to gnore the upper- or lower-bound sgn assocated wth the decson varables and the sgn could be gnored mathematcally wthout changng the model formulaton. Sub-model #1(best-case sub-model obectve functon): Max f k1 n c x 1 k11 c x (3.1.41) k1 n x 1 k11 c c x n 1 c x Sub-model #2(worst-case sub-model obectve functon): Max f k 1 c x n 1 1 k 1 c x (3.1.42) k 1 n 1 c x n 1 1 c x k 1 c x Wth equvalent obectve functons beng proved, we also need to prove that the correspondng decson spaces delmted by the ther own constrants are also dentcal to each other under the condton of Sgn ( a ) Sgn( c ). 25

42 For 2-step algorthm, the feasble decson space for the sub-model #1 (f + ) s defned as: u X k 1 a Sgn( a ) x n k a Sgn( a ) x b, x, x X, X 0, (3.1.43) For BWC algorthm, the feasble decson space for the best-case sub-model s defned as: Q u X k 1 a x n k a x b, x, x X, X 0, (3.1.44) a Snce Sgn ( a ) Sgn( c ), we have Sgn ( ) 1 for =1, 2,,k 1 and Sgn ( ) 1 a for =k 1 +1, k 1 +2,,n. In addton, a a for =1, 2,,k 1 because they are all postve coeffcents, and a a for = k 1, k 1 +1,,n because they are all negatve coeffcents. Therefore, we have: k k a Sgn( a ) x ( a )(1) x a x, 1 1 k 1 (3.1.45) n n a Sgn( a ) x ( a )( 1) x a x, k k 1 n k (3.1.46) Combne (3.1.45) and (3.1.46), we have: P u X k 1 a x n k a x b, x, x X, X 0, (3.1.47) If we gnore the upper- or lower-bound sgn assocated wth the decson varables, t s apparent that u P = u Q. It ndcates that the feasble decson spaces delmted respectvely by two algorthms for sub-model #1 and best-case sub-model are dentcal to each other. It can then be concluded that the 2-step algorthm s equvalent to the BWC algorthm f the condton Sgn ( a ) Sgn( c ) exsts. 26

43 Now we need to prove that, only f Sgn ( a ) Sgn( c ), ths theorem stll holds. Let s assume that there exsts an a (where =p, = q), Sgn ( a ) Sgn( c ). a pq If p k1, t s obvous that Sgn ( ) 1 and pq a a. Followng the same pq procedure as shown n equatons (3.1.) to (3.1.), t s easy to see that u u P Q, ndcatng that the 2-step algorthm s not equvalent to the BWC algorthm. Smlarly, t s straghtforward to see that ths condton also holds for p k1. Therefore, t has been proved that only when Sgn( a ) Sgn( c ),,, the 2-step algorthm s equvalent to the BWC algorthm. Theorem gves the condtons for 2-step algorthm to be equvalent to the BWC, algorthm. If an ILP problem satsfes the condton of Sgn( a ) Sgn( c ),, the solutons obtaned by these two algorthms wll be same; otherwse the solutons wll dffer from each other. (2) Feasble Decson Space Theorem 3.1.2: Suppose an ILP problem has nterval nequaltes as Q X n 1 a x b, x X, X 0, (3.1.48) the largest and smallest feasble decson space correspondng to the upper bound and lower bound of the obectve functon soluton can be presented as: Q u X n 1 a x b, x X, X 0, (3.1.49) 27

44 Q l X n 1 a x b, x X, X 0, (3.1.50) Ths theorem was frstly stated by Tong (1994) wthout provdng a proof, and was then proved by Chnneck and Ramadan (2000) for the form of mnmzaton problems. Theorem gves the largest and smallest feasble decson space of an ILP model. A practcal nterpretaton for an ILP model s that the nterval parameters can potentally take any value between ts prescrbed lower and upper bound. When each parameter takes a value wthn ts range, the ILP becomes a classc LP problem, and the feasble decson space of ths classc LP s found to be located between the smallest and largest feasble decson space of the orgnal ILP problem, and we defne ths classc LP as an event model of the ILP problem. Defnton 3.1.2: An event model of an ILP s defned as a classc LP model where the nterval parameters n A, B and respectve lower and upper bounds. C take a specfc set of crsp values wthn ther Based on ths defnton, apparently, two sub-models reformulated by the 2-step algorthm and BWC algorthm are two specfc event models representng two opposte extreme condtons of the orgnal model, respectvely. As the matter of fact, represents the constrants of the best-case sub-model by BWC and u Q n model l Q n model represents the constrants of the worst-case sub-model by BWC. In other words, the feasble decson spaces bounded by the BWC algorthm represent the largest and smallest feasble spaces of the orgnal ILP model. From ths defnton, t becomes understandable that the feasble decson spaces provded by the 2-step algorthm only represent two general event model stuatons, and are enclosed by the BWC feasble spaces. It can then be concluded that the feasble decson space of an ILP delmted by BWC s larger than or, equal to (when Sgn( a ) Sgn( c ), bounded by 2-step algorthm. accordng to Theorem 3.1.1) the feasble space 28

45 (3) Optmal Soluton Theorem 3.1.3: The optmal solutons obtaned by the 2-step algorthm can be dfferent from that obtaned by the BWC algorthm n that: stepopt f BWCopt f 2 (3.1.51) stepopt f BWCopt f 2 (3.1.52) Proof: In Theorem 3.1.1, t has been proved that when Sgn( a ) Sgn( c ),,, two sub-models reformulated by the 2-step algorthm and BWC algorthm are equvalent, and thus the solutons obtaned from two algorthms wll be same. Therefore, we have stepopt f BWCopt f 2 and stepopt f BWCopt f 2. For a general stuaton when,, Sgn( a ) Sgn( c ), ths theorem also holds. Theorem tells that the feasble decson space for solvng the upper bound obectve functon of BWC algorthm (.e. best-case sub-model) s gven asq n model Accordng to Defnton 3.1.2, the feasble decson space for solvng the sub-model #1 from the 2-step u algorthm s smaller than Q and s enclosed by u u Q when,, Sgn( a ) Sgn( c ). Based on the fundamental lnear programmng theory, the maxmum obectve functon value obtaned by the BWC algorthm (.e., by the 2-step algorthm. Thus, we have f BWCopt ) s equal to or greater than that obtaned f 2 stepopt f BWCopt. Smlarly, the mnmum obectve functon value obtaned by the BWC algorthm s equal to or less than that from the 2-step algorthm,.e., stepopt f BWCopt f 2. Remark 3.1.1: Both 2-step and BWC algorthms were proposed to account for system uncertantes n an ILP problem, accordng to Theorems and 3.1.3, the 2-step algorthm searches for the optmal solutons n a smaller feasble decson space, and moreover, the decrease of feasble decson space s caused smply by the way the left-hand 29

46 sdes of the sub-model constrants are formulated. Due to ths space decrease, the 2-step algorthm arbtrarly gnores some system uncertantes to a certan degree and ths gnorance was not theoretcally or mathematcally ustfed. In ths sense, the BWC algorthm seems to be a better method n handlng the uncertantes presented as ntervals. Remark 3.1.2: In terms of the optmal solutons, both algorthms use the optmal solutons generated from each sub-model to form an nterval optmal soluton for each decson varable as well as an nterval obectve functon value. For example, when the 2-step algorthm s used for solvng a maxmzaton ILP problem, ts upper bound sub-model s solved frst to obtan the upper bound solutons for decson varables x opt, where 1,2,..., k 1, and the lower bound soluton for decson varables x opt, where k 1, k1 2,..., n. Then, ts lower bound sub-model s solved to obtan the lower bound 1 solutons for decson varables x opt, where 1,2,..., k1, and the upper bound soluton for decson varables x, where k 1, k1 2,..., n. The obtaned upper bound and opt 1 lower bound solutons wll then be combned to form the fnal nterval optmal soluton for the orgnal ILP problem,.e., x [ x, x ],, 1,2,..., n. opt opt opt The optmal value of the obectve functon of the orgnal ILP model s also an nterval: f [ f, f ], where f opt and opt opt opt f opt represent the upper- and lower- bound values of the orgnal obectve functon, respectvely, as gven n model (3.1.44) and (3.1.45). opt ( 1opt 2opt k1opt ( k1 1) opt ( k12) opt ( n) opt f f x, x,..., x, x, x,..., x ) (3.1.53) opt ( 1opt 2opt k1opt ( k11) opt ( k1 2) opt ( n) opt f f x, x,..., x, x, x,..., x ) (3.1.54) In the practcal decson-makng process, the decson makers can choose any values wthn the nterval ranges for each decson varable to develop a practcal mplementaton scheme, dependng on ther preferences to dfferent polces and ther nterpretaton of system rsk and economc return (Huang et al, 1993; Zou et al, 2000; Yeh and Tung, 2003). 30

47 3.2. Valdty Checkng for Two-Step and BWC Algorthms As ILP models are commonly formulated for decson problems n many felds (Ben-Israel and Robers, 1970; Rommelfanger et al. 1989; Huang and Moore 1993; Tong 1994; Hansen and Walster 2004), the 2-step and BWC algorthms are also wdely employed to solve these models (Huang and Moore 1993; Olvera and Antunes 2007; Qn et al. 2007). As explaned n Remark 3.1.1, 2-step algorthm gnores some of the system uncertantes when reformulatng the sub-model constrants and ths treatment could be a potental flaw of ths algorthm and could very possbly lead to feasblty and optmalty concerns towards the generated nterval optmal solutons. Ths concern has trggered off a desre to check the valdty of both algorthms whch has not been conducted prevously. In ths study, a numercal example s desgned to llustrate the valdty checkng process for both algorthms, and the focus of the valdty checkng s on the nvestgaton of any nfeasble solutons exsted n the generated nterval optmal soluton and any optmal solutons mssng from t A Numercal Example for Valdty Checkng In order to perform the valdty checkng for both algorthms, a mnmzed ILP model wth two decson varables and two nterval constrants was desgned as follows and used as the numercal example: Mn f [ 2,3] x x (3.2.1) 1 2 s t. x [1.2,1.4] x [3,4] (3.2.2). 1 2 x 1.5,2.0] x [5,6] (3.2.3) x 1 [ 2 1, x2 0 (3.2.4) Before solvng ths ILP model by 2-step and BWC algorthms, the frst step s to generate a large number of event models usng the Monte-Carlo Smulaton method as descrbed n Secton Each event model s a classc determnstc LP model whch can be easly solved. By solvng these event models, a large number of soluton sets for decson 31

48 varables can be produced and the soluton ranges of the obectve functon and decson varables can then be obtaned. The larger the numbers of the event models are solved, the better the soluton resoluton and accuracy could be. In ths study, 50 mllon event models were generated and solved by the Monte-Carlo smulaton method, and the obtaned nterval solutons are: f = [8.18, 15.50], x 1 = [3.76, 4.95], x 2 = [0.30, 1.11]. (1) 2-Step Algorthm Soluton Based on the 2-step nteractve algorthm descrbed n Secton 3.1.2, two sub-models correspondng to f - and f + could be formulated as follows: Sub-model #1: 2x1 x2 Mn f (3.2.5). 1 2 s t. x 1.4x 3 (3.2.6) x x 5 (3.2.7) 1, x 2 x 0 (3.2.8) Sub-model #2: 3x1 x2 Mn f (3.2.9). 1 2 s t. x 1.2x 4 (3.2.10) x.5x 6 (3.2.11) 1, x 2 x 0 (3.2.12) x x1 opt 1 (3.2.13) x x2opt 2 (3.2.14) In Sub-model #2, x 1 opt n constrant (3.2.13) and x 2 opt n constrant (3.2.14) are the optmal solutons of decson varables from Sub-model #1. Both sub-models are classc 32

49 determnstc LP models and could be solved easly. The optmal nterval solutons obtaned by usng 2-step algorthm are f = [8.24, 15.41], x 1 = [3.82, 4.89], x 2 = [0.59, 0.74]. (2) BWC Algorthm Soluton Accordng to the BWC algorthm explaned n Secton 3.1.3, two sub-models correspondng to the best-case and worst-case stuatons can be formulated as follows: Best-Case Sub-model: 2x1 x2 Mn f (3.2.15). 1 2 s t. x 1.2x 3 (3.2.16) x x 5 (3.2.17) 1, x 2 x 0 (3.2.18) Worst-Case Sub-model: 3x1 x2 Mn f (3.2.19). 1 2 s t. x 1.4x 4 (3.2.20) x.5x 6 (3.2.21) 1, x 2 x 0 (3.2.22) Both sub-models are classc determnstc LP models and could be solved easly. The optmal nterval solutons obtaned by the BWC algorthm are f = [8.13, 15.58], x 1 = [3.75, 4.97], x 2 = [0.63, 0.69]. 33

50 Result Interpretaton and Valdty Checkng (1) Optmalty Checkng Accordng to the prncple of Monte Carlo smulaton algorthm, although a large number of event model runs (50 mllon tmes) were mplemented, the optmal soluton space provded by the Monte-Carlo smulaton should be narrower than the real soluton space of the orgnal example model, at most get very close to t. Before checkng the valdty of the optmal nterval solutons obtaned by 2-step and BWC algorthms, two facts should be noted: (1) every optmal soluton generated by solvng Monte-Carlo smulaton even model represents a subset of true optmal soluton sets of the orgnal model, and soluton nfeasblty s not an ssue; (2) the optmal soluton spaces provded by both 2-step and BWC algorthm should completely nclude the optmal soluton space from the Mont-Carlo smulaton method. Mathematcally, t yelds: x1opt x1opt 3, x2opt x2opt 0, and opt f f opt Based on these two facts, f the optmal soluton space provded by 2-step or BWC algorthm does not cover the nterval ranges of Monte-Carlo smulaton results,.e., the above relatonshp cannotbe satsfed, ths could lead to two sgnfcant consequences: (1) some optmal soluton pars are mssng from the 2-step algorthm or BWC algorthm; (2) the optmal solutons produced by both algorthms mght nclude some par ponts whch are nfeasble. The results obtaned from three algorthms are summarzed n Table 3.1 for soluton comparson purpose. From Table 3.1, t can be seen that nterval ranges of the optmal solutons provded by 2-step algorthm for both decson varable x 1, x 2 and obectve functon f are all smaller than that provded by the Monte-Carlo smulaton. It s obvous that some optmal soluton pars (x 1, x 2 ) are mssng from the 2-step algorthm, and ths algorthm fals the valdty checkng n terms of optmalty valdty. Ths result s also n lne wth the prevous observaton that the sub-models reformulated by 2-step algorthm do not represent the actual extreme stuatons. 34

51 Comparng to the 2-step algorthm, BWC has a better performance on decson varable x 1 whch covers ts full nterval range produced by the Monte-Carlo smulaton; however, ts performance on decson varable x 2 s worse wth an even smaller nterval range than 2-step algorthm. It s obvous that, smlar as the 2-step algorthm, some optmal soluton pars (x 1, x 2 ) are mssng from the BWC algorthm as well, and ths algorthm fals the checkng n terms of optmalty valdty. Accordng to the fundamental prncples of BWC algorthm, reformulated best-case and worst-case sub-models represent two extreme stuatons of the orgnal model, and the mssng optmal solutons are not supposed to occur. Ths needs further nvestgaton and s out of the scope of ths study. As to the obectve functon, the BWC provdes the largest ranges among three algorthms to cover the most optmstc and most pessmstc decson spaces, whch allows the system to have a more flexble performance. Table 3.1 Results obtaned from three ILP algorthms Algorthms x 1 - x 1 x 2 Obectve functon x x 2 2 f - f + Monte Carlo smulaton step algorthm BWC (2) Feasblty Checkng Fgure 3.4 gves the feasble decson space and optmal soluton space for the numercal example provded by the 2-step algorthm. The lnes BJIG and CKLF represent the boundares of the feasble decson space delmted by two sub-models constrants (3.2.6) and (3.2.10), respectvely; whle the dotted blue lnes besde each of them represent two BWC sub-models constrants reformulated by the same orgnal constrants. The lnes AJKD and HILE represent the boundares of the feasble decson space delmted by two sub-models constrants (3.2.7) and (3.2.11), respectvely; these two lnes are n 35

52 concdence wth BWC sub-models constrants (3.2.17) and (3.2.21) snce all ther coeffcents are postve. Fgure 3.4 gves the feasble decson space delmted by the 2-step algorthm whch s under the lnes of BJIG and CKLF and above the lnes of AJKD and HILE. We can bascally dvde the entre feasble decson space nto several regons and categores: the trangle CDK represents the absolute feasble regon whch satsfes all the constrants; the bg trangle above lne BJIG, and the bg trangle below lne HILE represent nfeasble regon as they volate at least one orgnal constrant; the space bounded by ponts n sequence B, J, I, E, D, K, C represents softly feasble regon whch means the soluton pars (x 1, x 2 ) are not guaranteed to satsfy all the constrants; Quadrangle IJKL s the feasble optmal soluton space provded by 2-step algorthm. The rectangular grey area n Fgure 3.4 s the optmal soluton regon for decson varable par (x 1, x 2 ) obtaned by the 2-step algorthm. Ths plot can help explan the nfeasblty checkng results. Ths fgure show that most of the 2-step optmal solutons are located n the softly feasble regon. There s a small trangular area (MOP) rght above the dotted blue lne whch s located n an nfeasble regon. Any optmal soluton pars obtaned by the 2-step algorthm and located n ths small trangular area are nfeasble solutons for the orgnal example model. For example, when x 1 =3.82 and x 2 =0.74, whch s the left upper pont of the soluton rectangle (pont M), the constrant (3.2.2) would be volated. It s obvous that nfeasble solutons have been generated by the 2-step algorthm. In addton, the lttle trangular area (RNQ) under the blue dotted blue lne s the non-optmal soluton space. Solutons n ths area are vald but not optmal. 36

53 1.5 A 2-Step soluton space and feasble space B C X2 1 H Infeasble soluton space M O P Q R J K 0.5 G I L N Non-optmal soluton space D F E X1 Fgure 3.4 Two-step soluton space and feasble space 1.5 A' BWC soluton space and feasble space B' J' 1 C' X2 H' Infeasble soluton space M' P' K' 0.5 G' I' R' N' Non-optmal soluton space L' D' F' E' X1 Fgure 3.5 BWC soluton space and feasble space 37

54 Fgure 3.5 presents the feasble decson space and optmal soluton space provded by the BWC algorthm. Lne B J I G represents the constrant of (3.2.16); Lne C K L F represents the constrant of (3.2.20); Lne A J K D represents the constrant of (3.2.17); and Lne H I L E represents the constrant of (3.2.21). The two dotted blue lnes represent the two correspondng constrants from the 2-step algorthm. The trangle C D K s an absolutely feasble regon as t satsfes all the constrants; the bg trangle above Lne B J I G, and the bg trangle below Lne H I L E are the nfeasble regons as they volate at least one constrant; the space bounded by ponts n sequence B, J, I, E, D, K, C represents softly feasble regon whch means the soluton pars (x 1, x 2 ) are not guaranteed to satsfy all the constrants; the quadrangle I J K L s the feasble optmal soluton space. The rectangular grey area n Fgure 3.5 represents the optmal nterval soluton obtaned by the BWC algorthm. Once agan, t shows that most of the BWC optmal solutons are located n the softly feasble regon. The small trangular grey area located rght above Lne B J I G ncludes all the nfeasble soluton pars whch are generated by the BWC algorthm. The observatons from Fgures 3.4 and 3.5 ndcate that both 2-step and BWC algorthm fals the soluton feasblty checkng, and the optmal solutons provded by them are not always vald. From the decson-makng standpont, the generaton of nfeasble solutons mght lead to a rsky even faled decson. Fgures 3.4 and 3.5 can also help explan the mssng optmal solutons from both 2-step and BWC algorthms. For example, for the mnmzed problem (3.2.15), the optmzaton soluton exsts when both x 1 and x 2 take the values as small as the constrants permt. In Fgure 3.5, a soluton at pont L s obvously a feasble soluton and can provde a lower obectve functon than pont K, snce both x 1 and x 2 take smaller values at pont L than pont K. However, pont K s ncluded n the optmal soluton generated by the BWC algorthm whle pont L s mssng from the soluton. 38

55 In addton, the soluton spaces from both algorthms are mostly located wthn the softly feasble regons. It ndcates that the soluton provded cannotbe guaranteed to be feasble n all stuatons. There exsts the rsk of volatng some constrants n the softly feasble regon. If we have to use the optmal solutons generated by two algorthms to develop practcal mplementaton schemes, the rsk level of volatng the constrants needs to be consdered. However, the exstng ILP models cannotncorporate ths rsk nto ts decson-makng process. The results from ths numercal example can help get the followng conclusons n terms of valdty checkng for both 2-step and BWC algorthms: (a) the optmal solutons are not always vald and part of the results mght be nfeasble; (b) some optmal solutons are mssng and not ncluded n the obtaned nterval solutons. Two possble solutons may be helpful for dealng wth ths dlemma: (1) mprovng or redevelopng both algorthms, and (2) ncorporatng the decson rsks nto the decson-makng process for helpng develop practcal mplementaton polces. 39

56 CHAPTER 4 FREILP MODEL DEVELOPMENT 4.1. Rsk Explct Interval Lnear Programmng (REILP) REILP Modelng Approach From the ILP valdty checkng conducted n Chapter 3, t has proved that the optmal solutons provded by both 2-step and BWC algorthms are not always vald. The ILP solutons are mostly located n the softly-feasble decson regon, ndcatng that some solutons would have rsks of volatng some of the constrants. Moreover, both algorthms tend to generate nfeasble or suboptmal mplementaton schemes. If the solutons are used for actual decson makng, the decson makers need to acknowledge the potental rsks assocated wth the generated decsons for makng good use of them. However, the exstng ILP soluton algorthms are ncapable of reflectng the lnkage between decson rsks and system performance. It s desred to develop new approaches. To overcome the lmtatons of ILP algorthms whle mantanng the strengths of ILP, a Rsk Explct Interval Lnear Programmng (REILP) was recently proposed. The development of a rsk explct ILP (REILP) model s presented as followng (Zou et al., 2010). Based on Defnton 3.1.1, an event model of a general ILP model can be formulated as: Max f n 1 [ c x (4.1.1) 0 ( c c )] s. t. n [ a ( a a )] x [ b ( b b )] 0, (4.1.2) 1 x 0, (4.1.3) (4.1.4) 0 1, (4.1.5), 40

57 0 1, (4.1.6) Apparently, the model represented by equatons (4.1.1) to (4.1.6) s a classc LP model, whch corresponds to a specfc set of crsp value of each coeffcent gven 0, and. By re-arrangng terms n equatons (4.1.1) to (4.1.6), the model becomes: Max f n [ c x 0 ( c c ) x ] (4.1.7) 1 s. t. n n a x b ( a a ) x ( b b ), (4.1.8) 1 1 x 0, (4.1.9) (4.1.10) 0 1, (4.1.11), 0 1, (4.1.12) Let x, and ( a a ) x ( b b ), where 1,2,..., m, 0 (c c ) n 1 the model can be reformatted as: n Max f ( c x ) (4.1.13) 1 s. t. n a x b, (4.1.14) 1 x 0, (4.1.15) When and equal to 0, the model (4.1.13) to (4.1.15) becomes the worst-case sub-model of the BWC algorthm and the worst-case sub-model represents a most pessmstc stuaton. In an nterval decson envronment, the soluton obtaned for the most pessmstc sub-model would have no rsk of volatng the constrants snce the formulaton has guaranteed satsfyng the tghtest constrants (.e., rsk = 0 when and 41

58 equal to 0). In cases where takes values greater than 0, the constrants are relaxed by a level of to obtan optmal solutons for achevng hgher system return; n the meantme, the soluton tself would be subected to certan level of rsk of volatng the constrants. Obvously, the larger the, the hgher the rsk would be assocated wth the solutons untl reaches ts maxmum values when both 1(, ) and 1 ( ), whch represents the most optmstc stuaton. Therefore, ( ) s qualfed to evaluate the rsk level of a decson, representng the possblty of a decson volatng the constrants. n Defnton 4.1.1: Functon ( a a ) x ( b b ) s defned as the rsk 1 functon for constrant n an ILP problem. From equatons (4.1.13) to (4.1.15), we know that (a) when = 0, the decson based on the optmal soluton has no rsk of volatng the correspondng constrant; and (b) when > 0, the decson based on the optmal soluton wll have a level of rsk of volatng the correspondng constrant n proporton to the value of. The orgnal ILP model s to maxmze the obectve functon (.e., system return). Snce the system return and decson rsk represent two conflctng factors n practcal decson makng process, a sound and satsfactory decson can be obtaned only through mnmzng the rsk functon whle maxmzng the system return. Ths leads to a mult-obectve optmzaton problem: Max f c n 1 x u (4.1.16) Mn n ( a a ) x ( b b ) (4.1.17) 1 42

59 s. t. n a x b, (4.1.18) 1 x 0, (4.1.19) Where s a general arthmetc operator whch can be a smple addton, a weghted addton, smple arthmetc mean, weghted arthmetc mean, or a max operator. The subscrpt for,, suggests that the operator be appled across constrants to obtan a unfed rsk functon for the entre optmzaton problem. For each ndvdual constrant, the n constrant-wse rsk functon ( a a ) x ( b b ) can dffer from that of 1 another constrant by order of magntude due to dfferent categores of b as well as the ncorporaton of nteractons among, a, a, x,, b and b n the functon. Therefore, t s necessary to convert the constrant-wse rsk functon nto comparable magntude. A smple method through scalng each constrant-wse rsk functon by 1 can be a feasble b choce, whch essentally represents a fractonal rsk factor from the most pessmstc case. In applcaton, more refned approaches can be developed to better reflect the decson envronment for the specfc case. To solve the mult-obectve programmng problem, the model (add the model numberng here) can be re-formulated as: Mn n ( a a ) x ( b b ) (4.1.20) 1 n ( opt 0 1 s.t. c x ) f ( f f ) (4.1.21) n opt opt a x b, (4.1.22) 1 43

60 0 pre (4.1.23) 0 1 (4.1.24) x 0, (4.1.25) 0 1, (4.1.26) Defnton 4.1.2: The optmzaton model (model numberng) s derved from the orgnal ILP model and ncludes a rsk-mnmzaton obectve functon. Ths model s defned as a Rsk Explct ILP (REILP) model. Here, 0 s pre-defned by the decson makers, representng the degree of aggressveness, or alternatvely, the aspraton level of decson makers gven the uncertantes n the optmzaton model. When 0 = 0, the model s correspondng to the least aggressve case where the most conservatve and safe soluton s expected. On the other hand, when 0 = 1, the model s correspondng to the most aggressve case where the most optmstc solutons but rsky solutons wll be generated. Apparently, n most real-world stuatons, decson makers would prefer balanced solutons wth 0 < 0 < 1 to the extreme solutons represented by 0 = 0 or 0 = 1 as the bass of practcal decson makng. Therefore, the task s to fnd the optmal solutons wth least rsk level for a desred degree of aggressveness. The rsk optmzaton model REILP (4.1.20) to (4.1.26) s a nonlnear model. The non-lnearty s generated by the ntroducton of rsk level varables (.e., 0 and ) to represent the complex non-lnear nteractons of uncertantes between dfferent varables and terms n a constrant. It s apparent that for a specfc constrant, f a large s assocated wth a small x the large would have small contrbuton to the rsk n the decson. On the other hand, f the s assocated wth a large x, t would result n sgnfcant contrbuton to the overall rsk of decson makng. 44

61 Dscusson of REILP (1) Aspraton Level 0 For respondng to the ssues assocated wth the ILP soluton, the REILP approach was developed attemptng to provde decson-makers more satsfactory and practcal mplementaton schemes through mnmzng the decson rsks whle maxmzng the system return. The mprovement over the exstng 2-step and BWC solutons s that the rsks assocated wth the possble optmal solutons and decsons derved from them could be explctly ncorporated nto the decson-makng process. However, one potental problem assocated wth the aspraton level 0 needs to be further dscussed. In the formulaton of a REILP model, ts orgnal obectve was converted nto a constrant wth the followng format (Zou et al., 2010): n ( c x opt 0 1 opt opt ) f ( f f ), where, 0 (c c ) x, So we get: n 1 [( c 0 ( c c )) x ] fopt 0 ( fopt fopt ) (4.1.27) Where, 0 appears n both rght-hand sde and left-hand sde of ths constrant, representng the aspraton level that reflects the decson-maker s preference and needs to be preset. A maor assumpton used behnd ths formulaton s that the system return coeffcent c has the same changng rate (.e., 0 ) from ts lower bound as f opt. However, ths s not always true n a real-world decson makng problem, and c and f opt mght take 45

62 dfferent rates changng wth ther own ntervals. A better formulaton for the nequalty (4.1.27) should be: n 1 [( c ( c c )) x ] fopt 0 ( fopt fopt ) (4.1.28) Where, s the changng rate for c, = 1,, n, and t s not necessarly equal to the aspraton level 0 n most cases. (2) Rsk Functon In the REILP formulaton, the rsk functon was defned as (Zou et al., 2011): n 1 ( a a ) x ( b b ) By ths equaton, the rsk functon can reflect the rsk of volatng the correspondng constrants a and b, but cannotreflect the rsk of volatng the constrants of c, and t cannotdrectly reflect the relaton between the rsk functon and the aspraton level. When the orgnal obectve functon s converted nto a constrant as (4.1.28), the rsk of volatng ths secondary constrant should also be consdered. Ths fts the common sense that a hgher aspraton level, whch means an aggressve decson and a hgher system return, wll affect the rsk level of the whole system. The rsk functon wthout consderng the rsk level of parameters c and aspraton level 0 s not suffcent. (3) Preset of Aspraton Level It mght be smple for some decson makers to preset an aspraton level and then obtan a serous of results. However, n some stuatons, decson makers may not have clear understandng of the aspraton level and cannot select a sound aspraton level that 46

63 represents ther professonal udgment of the stuaton. Blndly selectng an aspraton level may lead to undesrable results An Illustratve Example of REILP (1) The Example and Results In the development of the REILP approach, an example was gven to llustrate the applcablty of the approach (Zou et al., 2010). The example was a land-use management problem for nutrent loadng control and maxmum profts gan. In ths hypothetcal case, there s 1,200 acre of lands n a watershed that s avalable for two types of crop producton. It s known that crop 1 can reach unt productvty of 4,326-4,920 kg/acre, wth a net proft of $0.26 to 0.3/kg, and the producton of crop 2 can reach 3,480 4,120 kg/acre, whch can realze a net proft of 0.22 to 0.29 dollar/kg. To produce crop 1, the unt area ntrogen and phosphorus loadng dscharged to a lake n the watershed s 4.3 to 5.2 kg/acre/yr and 0.42 to 0.48 kg/acre/yr, respectvely. For crop 2, the loadng rates are 3.2 to 3.6 kg/acre/yr for ntrogen and 0.27 to 0.32 kg/acre/yr for phosphorus. It s known from a Total Maxmum Daly Load (TMDL) study that the total loadng of ntrogen and phosphorus dscharged nto the lake cannot be greater than 4,144 and 379 kg/yr, respectvely, wthout consderng an explct margn of safety. However, when a 10% margn of safety s mposed, the maxmum allowable loadng for ntrogen and phosphorus are 3,730 and 341 kg/yr, respectvely. The watershed authortes need a land-use plannng scheme to optmally allocate lands to dfferent crops n order to maxmze the crop producton proft whle satsfyng the envronmental requrements n terms of ntrogen and phosphorus dscharges. Ths land-use ILP model was frstly formulated as follows: Max f [ 0.26,0.3]*[4326,4920]* X X (4.1.29) 1 [0.22,0.29]*[3480,4120]* 2 Max f [ 1125,1476]* X X 1 [765,1194.8]* 2 s. t. X X

64 [ 4.3,5.2]* X [3.2,3.6]* X 2 1 [ 0.42,0.48]* X [0.27,0.32]* X 2 X 1, X [3730,4144] [341,379] Through the BWC algorthm, an nterval maxmzed system beneft could be obtaned: f = [803250, ]. Accordng to the REILP approach, ths orgnal land-use ILP model can be converted nto a rsk explct ILP model: Mn r3 ( ) X1 / 3730 r4 ( ) X 2 / 3730 r5 ( ) / 3730 r ) X / 341 r ( ) X / 341 r ( ) / 341 (4.1.30) 6 ( s.t. ( 1125 r0 ( )) X1 (765 r0 ( )) X r0 ( ) X 1 X X1 3.6X r3 ( ) X1 r4 ( ) X 2 r ( ) X1 0.32X r6 ( ) X1 r7 ( ) X 2 r ( ) 0 8 X 1, X r, r3, r4, r5, r6, r7, r8 0 1 Where, r 0 s the aspraton level that needs to be preset by decson makers. The solutons of the example problem are gven n Table 4.1 (Zou et al., 2010). In the table, NRL refers to the Normalzed Rsk Level, whch s calculated by multplyng the rsk functon value by a number to make the smallest NRL value close to 0 and the greatest close to 1. 48

65 Table 4.1 The optmal solutons of the example problem r Proft (10 5 $) X 1 (acre) X 2 (acre) NRL Source: Zou et al., 2010 (2) Revst of the Example Model and Results As dscussed n the prevous secton, when the orgnal obectve functon s transformed n nto a new constrant (4.1.27), [( c 0 ( c c )) x ] fopt 0 ( fopt fopt ), 0 1 appears n both rght-hand sde and left-hand sde of ths constrant. It s assumed that the system return coeffcent c has the same changng rate (.e., 0 ) from ts lower bound as f opt. However, n a real-world decson makng problem, the changng rate for c ( ) s not necessarly same as the changng rate for f opt (.e., 0 ). Usng 0 n both hand sdes may lead to nfeasble problems. A better formulaton for the nequalty (4.1.27) should be: n 1 [( c ( c c )) x ] fopt 0 ( f opt f opt ) Under ths crcumstance, we use r 1 for X 1, r 2 for X 2, and stll 0 for f opt. The example model (4.1.29) could then be reformulated as (4.1.31), whch s dfferent from the model (4.1.30): Mn RISK r3 ( ) X1 / 3730 r4 ( ) X 2 / 3730 r5 ( ) / 3730 r6 ( ) X1 / 341 r7 ( ) X 2 / 342 r8 ( ) / 341 (4.1.31) 49

66 s.t. ( 1125 r1 ( )) X1 (765 r2 ( )) X r0 ( ) X 1 X X1 3.6X r3 ( ) X1 r4 ( ) X 2 r ( ) X1 0.32X r6 ( ) X1 r7 ( ) X 2 r ( ) 0 8 X 1, X 2 0 r0, r1, r2, r3, r4, r5, r6, r7, r, where r 0 s the aspraton level gven the decson-makers. Table 4.2 The optmal solutons of the model (4.1.31) Aspraton level r r r r r r r r X 1 (acre) X 2 (acre) Proft (10 3 $) Rsk functon

67 The model (4.1.31) was solved by LINGO, and the solutons was obtaned and presented n Table 4.2. The results ndcate that r 1, r 2, r 3, r 4, r 5, r 6, r 7 and r 8 all satsfy the range of [0, 1], and also a reasonable ncreasng trend between system beneft and aspraton level s observed. However, the values of rsk functon all take 0 for r from r 1 to r 6, ndcatng that there s no lnkage between the decson rsks (n terms of the aspraton levels) and system return. As dscussed n secton 4.1.2, the rsk functon defned by Zou et al. (2010) can reflect the rsk of volatng the correspondng constrants a and b, but cannotreflect the rsk of volatng the constrants of c, and t cannotdrectly reflect the relaton between the rsk functon and the aspraton level. When the orgnal obectve functon s converted nto a constrant as (4.1.28), the rsk of volatng ths secondary constrant should also be consdered. Thus, Based on the defnton and formulaton of the rsk functon: n 1 ( a a ) x ( b b ) n 1 ( a a ) x ( b And the secondary constrants that: b ) n 1 ( c ( c c ) x ) fopt 0 ( f opt f opt ) A comprehensve rsk functon could be formulated as: n 1 n ( a a ) x ( b b ) k ( c c ) x 0 ( f 1 opt f opt ) (4.1.32) By choosng the general operator and k as 2 b b and 2 f opt f opt, the rsk functon then becomes: 51

68 m 2 n 2 n [ ( ) ( )] [ ( ) 0( )] a a x b b c c x fopt fopt 1b b 1 f f 1 opt opt (4.1.33) The equaton (4.1.33) s the rsk functon that wll be used n the model (4.1.31) to replace ts obectve functon. Solvng ths new model and the obtaned results are gven n Table 4.3. The results ndcate that the rsk functon s drectly related wth the aspraton level, and the system beneft ncreases wth the ncreasng rsk functon. Table 4.3 The optmal solutons when the rsk functon s (4.1.33) Aspraton level Rsk functon r r r r r r r r X 1 (acre) X 2 (acre) Proft (10 3 $)

69 4.2. FREILP Model Development In the REILP, the aspraton level, 0, represents the degree of aggressveness and the aspraton level of decson makers under an uncertan decson-makng envronment. When 0 = 0, the model corresponds to the least aggressve case and the most conservatve and safe solutons wll be obtaned; whle when 0 = 1, the model corresponds to the most aggressve case and the most optmstc but rsky solutons wll be obtaned. The aspraton level needs to be gven and preset by the decson makers before runnng the model. However, when facng many practcal decson-makng problems, the decson makers may not be able to determne exactly ther specfc aspraton levels; and more mportantly, dfferent decson makers and stakeholders may have dfferent opnons about ther preferences to dfferent decsons and polces n terms of ther aggressveness or conservatveness. In ths sense, the aspraton levels exst n the human thnkng, beng fuzzy n nature. As an extenson to prevous efforts, ths study attempts to develop a fuzzy based REILP approach to account for the fuzzness assocated wth the aspraton levels, and thus mprove the applcablty of the proposed method Fuzzy Set Theory Fuzzy Set Theory (Zadeh, 1965; Zadeh, 1968) was formalzed as an extenson of the classcal noton of set. In classcal set theory, the membershp of elements n a set s assessed n bnary terms accordng to a bvalent condton an element ether belongs or does not belong to the set. By contrast, fuzzy set theory permts the gradual assessment of the membershp of elements n a set; ths s descrbed wth the ad of a membershp functon valued n the real unt nterval [0, 1]. Fuzzy sets generalze classcal sets, snce the ndcator functons of classcal sets are specal cases of the membershp functons of fuzzy sets, f the latter only take values 0 or 1. The fuzzy set theory can be used n a wde range of domans n whch nformaton s ncomplete or mprecse. Defnton 4.2.1: If X s a collecton of obects denoted genercally by x, then a fuzzy set à n X s a set of ordered pars: 53

70 A ~ x ~ A x, ~ x A x s called the membershp functon or grade of membershp of x n Ã, whch maps X to the membershp space M. When M contans only the two ponts, 0 and 1, Ã s non-fuzzy. The range of the membershp functon s a subset of the nonnegatve real numbers whose supremum s fnte. Elements wth a zero degree of membershp are normally not lsted. X For example, a fuzzy set of real numbers close to 10 (Zmmermann, 1991) could be expressed as the followng expresson and the correspondng curve s gven n Fgure 4.1: ~ A x, ~ ~ A x x 1 x 10 A 2 1 y 1 y=(1+(x-10) 2 ) x Fgure 4.1 An example of the membershp functon of the fuzzy set real number close to 10 For any gven number, x, the membershp grade could be calculated by the above expresson x A ~ 10 s For example, when x = 7, the membershp grade of number 7 close to ~ A ; and when x = 10, the membershp grade s 54

71 ~ A of a real number belongng to ths fuzzy set.. The membershp grade ndcates the possblty or lkelhood Defnton 4.2.2: The crsp set of elements that belong to the fuzzy set à at least to the degree s called the -level set (Zadeh, 1975; Zmmermann, 1991). x X x A A ~ Gven the -level, the -level set could be obtaned smply by drawng a lne of -level n the fgure of the membershp functon. The elements above the lne would be the -level set. So the degree s also called -cut. For the fuzzy set real numbers close to 10, when -cut = 0.5, the lne of -cut = 0.5 cuts the membershp functon nto two parts (above and below the lne_ and two ntersecton ponts were obtaned wth x = 9 and x = 11, respectvely (see Fgure 4.2). So when -cut = 0.5, the elements of the -level set nclude all the real numbers between 9 and 11 wth ther membershp grades all beng greater than 0.5. The -cut concept has been wdely used n the practcal decson-makng problems to deal wth system fuzzness. y 1 y=(1+(x-10) 2 ) x Fgure 4.2 The membershp functon of real numbers close to 10 under -cut =

72 Loft Zadeh also developed a membershp functon for young people (Zadeh, 1972) as followng: 1 ~ A( x) x 50 [1 ( ) 5 2 ] 1 0 x x 100 Ths s based on the assumpton that people s ages are from 0 to 100. The plot of membershp functon of young people s shown n Fgure 4.3. Under -cut of 0.7, the -cut subset of the fuzzy set young people would nclude all people aged from 0 to 28. Young person membershp functon Membershp grade Age Fgure 4.3 The membershp functon of the fuzzy set young people and the subset of =0.7 56

73 Fuzzy Nature of the Aspraton Level ( 0 ) As explaned earler, the aspraton level, 0, has fuzzy characterstcs n nature. Dfferent values of aspraton levels represent the degree of aggressveness or conservatveness of decson makers. In ths study, three levels of 0 are consdered, ncludng aggressve aspraton level, medum aspraton level, and conservatve aspraton level. Ther membershp functons are developed based on the fuzzy set theory of Zadeh and hs young person membershp functon. Fgure 4.4 gves the membershp functon curve of the fuzzy conservatve aspraton level. Its x-axs represents the aspraton level nstead of age (Fgure 4.3) and ts ranges change from [0, 100] of age to [0, 1] of the aspraton level. The membershp functon of the aggressve aspraton level s the opposte of the conservatve one, as shown n Fgure 4.6. The membershp functon of the medum aspraton level s a combnaton of conservatve and aspraton level functons, as gven n Fgure 4.5. The equatons and fgures of the membershp functons for three stuatons are provded below. 57

74 (1) Conservatve aspraton level membershp functon: ~ 1 [1 (40 1) 2 1 ] ; ; 0 (4.2.1) Conservatve Membershp grade Aspraton level Fgure 4.4. The membershp functon of conservatve aspraton level 58

75 (2) Medum aspraton level membershp functon [1 (15 40 ) ~ 0 1 [1 (400 15) 2 1 ] 2 1 ] ; ; ; 0 (4.2.2) Medum Membershp grade Aspraton level Fgure 4.5 The membershp functon of medum aspraton level 59

76 (3) Aggressve aspraton level membershp functon ~ [1 ( ) ] ; ; 0 (4.2.3) Aggressve Membershp grade Aspraton level Fgure 4.6 The membershp functon of aggressve aspraton level Fuzzy Rsk Explct Interval Lnear Programmng (FREILP) Wth the aspraton level beng handled as the fuzzy numbers, a Fuzzy Rsk Explct Interval Lnear Programmng (FREILP) can then be formulated as follows. For a maxmzed ILP problem, we have: Max f n 1 c x (4.2.4) 60

77 61 b x a t s n,.. 1 x 0, The obectve functon of ts FREILP formulaton s to mnmze the rsk functon: )] ( ) ( [ 2 )] ( ) ( [ opt opt n opt opt m n f f x c c f f b b x a a b b (4.2.5) Constrants nclude: ) ( ) ) ( ( 0 1 opt opt opt n f f f x c c c (4.2.6) b b x a a b x a n n ), ( ) ( 1 1 (4.2.7) 1,, 0 (4.2.8) x, 0 (4.2.9) In the above formulaton from (4.2.5) to (4.2.6), the aspraton level 0 s treated as a fuzzy set. For a mnmzed ILP problem, we have: n c x f Mn 1 (4.2.10) b x a t s n,.. 1 x 0,

78 The obectve functon of ts FREILP s also to mnmze the rsk functon, whch s the same as the model (4.2.5). Constrants nclude n ( c opt 0 1 opt opt ( c c ) x ) f ( f f ) (4.2.11) b n n a x ( a a ) x ( b b ), (4.2.12) 1 1 0,, 1 (4.2.13) x 0, (4.2.14) Smlarly, the aspraton level 0 s also treated as the fuzzy set Soluton Process for the FREILP Model For solvng the models formulated n Secton 4.2.3, the soluton process ncludes the followng steps: [Step 1] Use the BWC algorthm to covert the orgnal ILP model nto two sub-models and solve both sub-models to fnd the solutons of the lower bound and the upper bound of the obectve functon of the orgnal ILP model. [Step 2] Use the solutons of obectve functon obtaned n Step 1 to formulate a fuzzy REILP model as gven n equatons (4.2.4) to (4.2.14). [Step 3] Accordng to the preference of the decson makers, defne the aspraton level as conservatve, medum, or aggressve, accordngly, and also choose an -cut level to cut the membershp functon curve. Two -cut crsp values obtaned would be used as the aspraton level nput for solvng the formulated FREILP model. 62

79 [Step 4] Run the FREILP model by LINGO and fnd out the solutons. 63

80 CHAPTER 5 FREILP MODEL APPLICATION TO MSW SYSTEM IN HRM In ths study, the developed approach s appled to the Muncpal Sold Waste (MSW) management system n Halfax Regonal Muncpalty (HRM), Canada, not only for testng ts applcablty to real-world problems, but also for provdng the muncpal waste managers wth a more practcal decson support tool. A FREILP model s developed for the long-term plannng for the MSW management system n HRM. The plannng horzon s 30 years startng from the year of 2011, and s dvded nto 6 plannng perods wth 5 years n each perod. The latest data recorded for modelng purposes nclude a perod from Aprl 2009 to March of 2010 whch s used to represent the nput data for the year of All the monetary values wll be converted to the value of 2010 Canadan dollar consderng the annual nflaton factor n the comng years Overvew of MSW System n HRM The Halfax Regonal Muncpalty (HRM) s the largest populaton center of Canadan East Coast and the captal cty of the provnce of Nova Scota. The muncpalty was founded n 1996 through combnng four communtes: Halfax, Dartmouth, Bedford, and Halfax County (HRM, 2010a). It s the economc and cultural centre of Canada s East Coast, accountng for 40% of Nova Scota s populaton, and s one of Canada s prme tourst locatons (HRM, 2010b). The HRM s commtted to envronmental sustanablty. It s one of the hghest waste dverson rate muncpaltes n Canada (Walker, et al., 2004). By August 2009, the waste dverson rate n HRM has reached 59%, whch s n the 4 th place across Canada followng the Regonal Dstrct of NaNamo, Brtsh Columba (64%), the cty of Vctoravll, Quebec (64%), and Charlottetown, Prnce Edward Island (60%) (FCM, 2009). 64

81 In 1995, the HRM Waste Resource Management Strategy was developed to meet three long-term goals (CSC, 1995): To maxmze the 3 Rs (reducton, reuse and recyclng) of MSW. To maxmze envronmental sustanablty and mnmze costs. To foster stewardshp and values of a conserver socety. The research toward ths strategy began n the early 1990 s, when the local raw waste landfll was reachng ts desgned capacty and began causng odour and other envronmental problems (Goldsten and Gray, 1999; HRM 2007a). To acheve these goals, the HRM, along wth publc and prvate partners, has mplemented many programs snce then, as shown n Table 5.1 below. In 2000, HRM became the frst wnner of FCM-CH2M HILL Sustanable Communty Awards for the communty-based waste resource management strategy (FCM, 2000). Currently, the MSW system n HRM has 3 components (HRM, 2002): landfll, recyclng, and compostng. HRM operates one landfll wthout relyng on ncneraton facltes. All the refuses are delvered to the Otter Lake Waste Processng and Dsposal Facltes. The desgned capacty s 150,000 tonnes per year for 25 years (from 1998 to 2023) (Goldsten and Gray, 1999). HRM also has one recyclng faclty n operaton, located at 50 Chan Lake Drve, wth the capacty of 28,000 tonnes per year (Goldsten and Gray, 1999). Two compostng facltes are n use n HRM for compostable materals: New Era Farms faclty, and Mller Compostng faclty. The nomnal capacty of each faclty s 25,000 tonnes per year (Fresen, 1999; Fresen 2000). There s no sold waste transfer staton beng n use n HRM. Only one type of vehcle, large garbage truck, s beng used n HRM. 65

82 Table 5.1 MSW management programs mplemented by HRM Program name Descrpton Source separaton Waste of organcs, recyclables and trash were separated at source, wth bweekly collecton of organcs and trash, weekly collecton of recyclables n most areas and bweekly recyclables collecton n the rural areas of the county. Collecton zones creaton Creaton of eght collecton zones (from 25 before amalgamaton) wth sx haulers. Aerated carts Use of aerated carts for organcs collecton. New landfll ste constructon One ste that ncludes a mxed waste processng faclty desgned to handle 119,000 tonnes/year of MSW; a 13 channel agtated bed compostng system to process the mxed waste after recyclables are removed; and a landfll for stablzed waste. HRM owns these facltes. However, the desgn, buldng and operaton of these facltes are the responsblty of Mrror Nova Scota. Compostng facltes mprovement Two separate compostng facltes wth total processng capacty of 50,000 tonnes/year. Both facltes are prvately owned and operated, each wth put or pay guarantees by HRM of 20,000 tonnes/year. Materals recovery faclty expanson Expanson of an exstng materals recovery faclty. Source: Goldsten and Gray, Waste Generaton In 2008, Canadans produced over 1,031 klograms of resdental waste per person, vrtually the same per capta producton as n 2006 (Statstcs Canada, 2008). In Aprl 2006, Natural Resources Canada and Envronment Canada summarzed the waste generaton data for year 2002 and part of the data s shown n Table 5.2. Comparng to other provnces, Nova Scota has the lowest waste generaton rate across Canada. 66

83 Table 5.2 Waste generaton n Canada, 2002 Provnce/Terrtory Populaton Resdental generaton (tonnes) Generaton rate (kg/capta) Newfoundland and Labrador 519, , Prnce Edward Island 136, Nova Scota 934, , New Brunswck 750, , Quebec 7,443,491 3,471, Ontaro 12,096,627 4,388, Mantoba 1,155, , Saskatchewan 995, , Alberta 3,114,390 1,159, Brtsh Columba 4,114,981 1,354, Yukon Terrtory, Northwest Terrtores and Nunavut 111, Canada Total 31,361,611 12,008, Source: Natural Resources Canada and Envronment Canada, As HRM s the largest and the most developed cty n Nova Scota, the muncpal waste generaton n HRM would be hgher than the average rate n Nova Scota. Here for HRM, the MSW ncludes both resdental and commercal sold wastes. By the latest approachable statstc data provded by Sold Waste Resources of HRM, the sold waste generaton and collecton data are shown n Table 5.3. A total of 373,989 tonnes of sold waste were generated from Aprl 1 st 2009 to March 31 st 2010, n whch 34.7% were from resdental sources and 65.3% were from commercal sources. Among all the wastes, refuse, organcs, recyclng and drop-off materals accounts for 59.9% of total wastes and were collected and dsposed by HRM. All the other wastes ncludng fbers prvate recyclng, backyard compostng, constructon and demolton (C&D) waste, and household hazardous waste (HHW) were collected and dsposed by n-person or prvate companes (HRM By-Law S600, 2007; HRM By-Law L200, 2002), and these wastes are not ncluded 67

84 n the plannng. As for the rato of 59.9%, n ths study, the rato of total wastes beng dsposed by HRM s estmated to be a lttle lower wth a range of [57%, 58.5%] due to the exstence of waste resdues whch are not collected and dsposed by HRM. Table 5.3 Yearly waste generaton n HRM from Aprl 2009 to March 2010 Waste category Resdental waste (tonne) Commercal waste (tonne) Total waste (tonne) Resdental Rato of the total Refuse 63,703 85, , % Organcs 35,806 16,291 52, % Recyclng 17,420 5,312 22, % Fbers prvate recyclng (Est) N/A 43,000 43, % Backyard compostng (Est) 5,000 N/A 5, % Drop-off materals (Est) 7,500 N/A 7, % C&D N/A 93,920 93, % HHW(Est) 500 N/A % Totals 129, , , % Dverson rate 50.97% 64.95% 60% -- Source: HRM Sold Waste Resources, 2010c Total waste generaton s drectly related to the populaton of a cty. In 2006, the populaton of HRM has reached 372,679 (2006 census, Statstcs Canada) and spreads 2,224 square mles and ranges from hgh-densty urban settngs to rural communtes (HRM, 2010a). Comparng to the populaton n 2001, whch was 359,111, the populaton n HRM has ncreases 3.8% n 5 years. If HRM keeps ths populaton growth rate, the populaton wll be around 386,760 n Accordng to the waste generaton data (373,989 tonnes n 2010), the waste generaton rate would be tonne per capta per year. In ths study, an nterval waste generaton rate wth a range of [0.95, 0.98] tonne per capta per year was used to reflect ts uncertan feature. Based on avalable nformaton, the populaton and waste 68

85 generaton data n HRM for the next 30 years can be estmated and are provded n Table

86 Table 5.4 Estmate of populaton and waste generaton n HRM for the plannng horzon Year Populaton Waste generaton(tonne/year) Waste generaton(tonne/5years) lower level upper level lower level upper level

87 Recyclng Recyclng turns materals that would otherwse become waste nto valuable resources. Many benefts (USEPA, 2008) can be expected from recyclng: t saves energy; t reduces the need for landfllng and ncneraton; t prevents polluton caused by the manufacturng of products from vrgn materals; t decreases emssons of greenhouse gases; t conserves natural resources such as tmber, water, and mnerals; t helps sustan the envronment for future generatons; t can also protects and expands employment opportuntes. Many recyclng programs (HRM, 2007b; HRM, 2011b) are beng mplemented n HRM. They nclude Blue Bag Program, Paper Recyclng Program, Corrugated Cardboard Program, and some other provncal programs lke Pant Recyclng Program, Household Hazardous Waste (HHW) program, and Used Tre Management Program, Derelct Vehcle Program, and Safe Sharps Brng-Back Program. Blue Bag program requests resdents to put recyclables nto a clear blue bag for curbsde collecton. The recyclables n the blue bag nclude plastc bottles and contaners, all plastc bags used for grocery, retal, bread, dry cleanng and frozen food, and bubble wrap, glass bottles and ars, steel and alumnum cans, clean alumnum fol and plates, paper mlk cartons, mn sps and tetra uce packages. Paper recyclables nclude dry and clean paper, newspapers and flyers, glossy magaznes and catalogues, envelopes, paper egg cartons, paperbacks and phone books. These paper recyclables should be placed n a grocery bag, retal carry-out bag or a clear bag, and be kept separately from Blue Bag recyclables. Corrugated cardboard program encourages people to fold boxes flat and te them n bundles no more than 2 ft x 3 ft x 8 nches n sze. Corrugated cardboard s waffled between the layers such as applance boxes and pzza boxes. And the bundles should be placed besde the blue bag. 71

88 Nova Scotans purchase more than 3 mllon contaners of pant every year, and up to 25 % of ths pant s not used after the purchase (NS Envronment and Labour, 2003). In the past, most of ths pant was ether burned n dumps or bured n landflls. The recent pant-recyclng program recovers thousands of lters of pant and pant cans. It allows consumers to return surplus pant to any one of the provnce's 85 recyclng depots at no charge. It apples to all latex, ol and solvent-based pants, ncludng aerosol pant cans, but does not apply to specally formulate ndustral, automotve or marne coatngs. New recycled pants wll be manufactured from the recovered waste pant. By February 2009, the provncal government has banned most common electronc equpment from landflls. The banned electronc equpments nclude televsons, computers, audo and vdeo playback and recordng systems, telephones (corded and cordless), fax and answerng machnes, computer scanners, cell Phones and other wreless devces. Snce the ban, HRM has encouraged ts resdents to brng ther old electronc equpment to ther local Atlantc Canada Electroncs Stewardshp (ACES) recyclng centre (HRM, 2010d). Currently, there s one recyclng faclty n operaton n HRM, located at 50 Chan Lake Drve, wth a capacty of 28,000 tonnes per year. The Materals Recyclng Faclty (MRF) of HRM processes all blue bag and fber (paper and cardboard) collected n resdental curbsde program and the small amount of materal deposted n the onste publc drop off bns (HRM, 2011a). Recyclable materals nclude newsprnts, fber, cardboard, glass and steel contaners, alumnum etc (see Table 5.4). Besdes that, 22 recyclng centers are separately located around HRM openng 7 days a week to collect recyclables (RRFB, 2011). Resdents can send ther used bottles or cans to the collecton center and get the depost. As gven n Table 5.2, 6.1% of generated wastes were recycled by the recyclng facltes n HRM and 11.5% were recycled by prvate recyclng companes. Two recyclng streams contrbuted a 20% of recycle rate n HRM. Accordng to the documented report of USEPA (USEPA, 2008), the avalablty of recyclables could be as hgh as 35% of the total sold 72

89 wastes generated. So n ths study, the rato of recyclables s bounded between 6% and 35% Compostng Compostng s the natural breakdown of organc materals by lvng organsms, ncludng bactera, fung, worms and small nsects. Any materal from a lvng source, plant or anmal, s called "organc". The end product s a dark, earthy, sol-lke substance called compost. Compost has ts market value because t can be used as a sol amendment or as a medum to grow plants. It serves as a marketable commodty and s a low-cost alternatve to standard landfll cover and artfcal sol amendments. Compostng also extends muncpal landfll lfe by dvertng organc materals from landflls and provdes a less costly alternatve to conventonal methods of remedatng (cleanng) contamnated sol (USEPA, 1994). Presently, two compostng facltes are servng HRM: the New Era Farm compostng faclty and the Mller compostng faclty, wth a total capacty of 50,000 tonnes/year. The New Era Farms compost faclty (Fresen, 1999; Fresen, 2000), located n 61 Evergreen Place, Ragged Lake, has a capacty of 25,000 tonnes/year. It was szed to allow expanson by an addtonal 10,000 tonnes. It has three man areas: a recevng and preprocessng buldng, a compostng pad and a curng structure. The Mller Compostng (Fresen, 1999; Fresen, 2000) utlzes the Ebara technology to compost waste organcs. Same as the New Era, the Mller ste has a capacty of 25,000 tonnes/year. It s located at 80 Glora McClusky Avenue, Burnsde, Nova Scota s largest busness park. The buldng footprnt s 55,000 sq. ft., set on a 20 acre lot. The property could accommodate a second plant drectly besde the orgnal one f needed n the future. The man buldng has three areas: recevng and preprocessng, compostng and curng. From Aprl 1 st 2009 to March 31 st 2010, 52,097 tonnes of organcs were composted by the two compostng facltes n HRM, n whch, 35,806 tonnes were from resdental sources and 16,291 tonnes were from commercal sources. It dverted 13.9% of total waste 73

90 generated from landfll. Ths percentage could be mproved to about 26%. As documented n US Envronmental Protecton Agency (USEPA, 1994), yard trmmngs and food resduals together consttute 26 percent of muncpal sold waste stream. The compostng wastes are bounded as 12% to 26% of total generated wastes n ths study. Backyard compostng (HRM, 2011) s also promoted n HRM. Backyard or onste compostng can be conducted by resdents and other small-quantty generators of organc waste on ther own property. By compostng these materals onste, select busnesses can sgnfcantly reduce the amount of waste that needs to be dsposed of and thereby save money from avodng dsposal costs, and home owners can generate natural fertlzer that can be appled to lawns and gardens to help condton the sol and replensh nutrents. The household practcng backyard compostng dverts approxmately 5,000 to 7,5000 tonnes per year of wastes from landfll and t contrbuted 12.25% of total organc materal generated n HRM n 2010 (see Table 5.2) Landfll Currently, the Otter Lake Landfll s the only landfll beng operated n HRM. It opened for full operatons n 1999 to replace the closed Sackvlle Landfll, wth a nomnal capacty of 150,000 tonnes/year and a desgned lfe tme of 25 years from year1998 to The ste s 200 acres n sze and employs over 100 workers. Ths faclty ncludes three maor components: (1) Front End Processor (FEP) Ths s the frst stage of Otter Lake Faclty where garbage arrves and bags are opened and nspected. It conssts of a system of conveyors, bag breaker, sortng platforms and mechancal screenng operatons. The FEP allows for dentfcaton and removal of materal that should not be gong to landfll. Clean recyclable paper, metals and contaners are removed durng the sortng process. Scrap metal (.e., applances) s separated for recyclng. By FEP, the Otter Lake landfll makes some avenue from recyclable wastes. In the 2010 budget of sold waste resources n HRM, approxmately $80,000 revenue was receved from the FEP. 74

91 (2) Waste Stablzaton Faclty (WSF) - Mxed wastes contanng organc materal leftover from the FEP s brought by conveyor and undergoes processng n the waste stablzaton faclty. The process s much the same as compostng, where after a perod of 18 to 21 days, the materals leavng ths process s a dry-lke fluff whch then goes to the landfll. Ths process could sgnfcantly reduce not only the volume of wastes enterng the landfll cells (about 40%), but also the strength of the landfll leachate snce a large amount of the organcs have been degraded n ths process. (3) Resdual Dsposal Faclty (RDF) - Ths s the place to landfll or bury the resdues from the WSF process. The landfll contans a lner system, a cover system, a leak detecton system, a leachate and gas management system, surface dranage control and envronmental montorng controls. In the Otter Lake landfll, 9 cells were desgned 4 of them have been flled before Table 5.5 presents the capactes of unflled cells. Cell 5 was started to use from January The orgnal capacty s 546,000 tonnes and t s estmated that an approxmate 167,500 tonnes of garbage has been placed n Cell 5 by March 31 st, he total capacty of Otter Lake Landfll by March 31 st 2010 s 2,421,500 tonnes. Table 5.5 Capacty of the Otter Lake Landfll by March 31, Cell # Capacty unflled Cell 5 378,500 Cell 6 503,000 Cell 7 520,000 Cell 8 520,000 Cell 9 500,000 Total 2,589,000 75

92 Incnerator Incneraton, as a type of thermal treatment, s recognzed as an effectve and envronmentally sound dsposal method for a wde range of wastes. In the early 1990s when the Sackvlle Landfll was constructed and stll n operaton was causng severe envronmental damages to ts nearby communtes. The Muncpalty was then plannng to buld an ncnerator close to the urban center of the regon. However, the plan was eventually reected due to varous envronmental and economc concerns by the Nova Scota Mnstry of Envronment. There s no ncnerator faclty servng HRM Transfer Staton Waste transfer statons are facltes where muncpal sold wastes are unloaded from the collecton vehcles and brefly held before they are reloaded onto larger long-dstance transport vehcles for shpment to landflls or other treatment or dsposal facltes (USEPA, 2002). The HRM used to own a sold waste transfer staton n Lady Hammond Road for the past few decades. But t has been closed snce May 2000 (HRM, 2000). Accordng to HRM s Integrated Sold Waste/Resource Management System, HRM no longer needs a sold waste transfer staton. Waste materals could be collected for a drect shpment to the Otter Lake landfll and be separated, sorted and processed there as well (HRM, 1999) Model Input Data Dscount Factor Snce the plannng problem under consderaton ncludes long multple plannng perods, dscount factors have to be consdered for each plannng perod to obtan the total present value for the obectve functon. The dscount factor (DF) s defned as the coeffcent whch a future dollar should multply by to be converted to the present value. The dscount factor could be calculated through the followng equaton: 76

93 1 DF (1 r) Where r s a fxed dscount rate and s determned by the nterest rate and nflaton rate. It s a fxed dscount rate and s determned by the nterest rate and nflaton rate. t s the tme factor. In ths study, a dscount rate of 3.18% was used for the dscount factor calculaton, based on the predctons of Canada s nflaton and nterest rates n the comng years (IndexMund, 2010; Scota Bank, 2011). t In ths study, the 30-year long plannng horzon s dvded nto 6 plannng perod, wth 5 years n each perod. All the costs and revenues n the comng plannng years are calculated as 2011 dollars. The dscount factors for each year and perod average are calculated usng the above equaton and are provded n Table 5.6. For the net present value of operaton and transportaton costs, an average dscount factor s chosen for each tme perod. For the captal costs used for faclty development and expanson, t s assumed that, f the MSW management system requres addtonal capacty at the begnnng of a partcular perod, the development or expanson of that faclty has to be completed by the end of the prevous perod. Thus, the dscount factor for faclty development and expanson would be the regular dscount factor from the prevous perod. In addton, t s assumed that all garbage streams can be handled by the exstng facltes n the frst plannng perod, and no new facltes are needed. 77

94 Table 5.6 Dscount factors Perod Year Dscount factor perod average DF Waste Collecton and Transportaton Cost HRM serves sngle famly households, row houses, duplexes, sem-detached, small apartments (up to sx unts) wth curbsde collecton, to collect the waste. MSW collecton 78

95 n HRM s equpped to serve about 120,500 households, ncludng 5,500 condomnums (HRM, 2002). Accordng to HRM Sold Waste Resource Collecton and Dsposal By-law (By-law No. S600, 2007), resdents n HRM are requred to source-separate all collectble waste generated from elgble premses at the pont of generaton so as to comply wth the provncal dsposal bans and to facltate ther recyclng, compostng or dsposal n accordance wth the Muncpalty s waste resource management system (HRM, 2007c). After the waste source separaton, organcs and trash are collected bweekly and recyclables are collected weekly (bweekly n the rural areas of the county). Aerated carts are used for organcs collecton. Eght collecton zones were created for the purpose of waste collecton (from 25 before amalgamaton). The zones and approxmate number of the served households are presented n Table 5.7. Fgure 5.1 shows the spatal dstrbuton of the 8 collecton areas n HRM. Table 5.7 HRM waste collecton areas and the number of the households served. Area No. Area No. of the served households (1998) (2002) 1 Halfax 27,750 28,861 2 Dartmouth 19,806 20,815 3 Bedford, Hammond Plans, Pockwock & Area 7,355 9,230 4 Beechvlle, Lakesde, Tmberlea, Prospect & West 12,104 13,668 5 Sackvlle, Shubenacade Lakes & Area 17,712 19,387 6 Cole Harbour, Westphal, Eastern assage & Area 11,361 12,637 7 and East Preston, Lake Maor, Lake Loon, Cherrybrook, Lawrencetown & Area 6,507 7,227 8 Elderbank, Musquodobot & all Eastern Shore 6,758 7,163 Total 109, ,988 Source: HRM sold waste resources,

96 Fgure 5.1 HRM waste collecton areas Waste collecton and transportaton cost s usually estmated based on the vehcle type and the worker s salary. Mulolland has calculated the cost n 1997 as shown n Table 5.8. In HRM, snce most wastes are collected by automated truck wth cart lfter and packer, the collecton and transportaton cost would be around [21, 37] dollar per tonne n 1997, whch s around [32, 56] dollar per tonne currently. Table 5.8 Waste collecton and transportaton data Waste collecton type Automated Salary per hour ($/h) Cost of vehcle per hour ($/h) 27 Volumetrc capacty (m 3 ) 22 length of workng day (h) 8 Daly loads (tonnes) [12 20] Total collecton and transportaton costs ($/tonne) [21 37] 80