Programming problems on time scales: Theory and computation

Transcription

1 Scholars' Mne Doctoral Dssertatons Student Theses and Dssertatons Summer 2017 Programmng problems on tme scales: Theory and computaton Rasheed Basheer Al-Salh Follow ths and addtonal works at: Part of the Mathematcs Commons, and the Operatons Research, Systems Engneerng and Industral Engneerng Commons Department: Mathematcs and Statstcs Recommended Ctaton Al-Salh, Rasheed Basheer, "Programmng problems on tme scales: Theory and computaton" (2017). Doctoral Dssertatons Ths Dssertaton - Open Access s brought to you for free and open access by Scholars' Mne. It has been accepted for ncluson n Doctoral Dssertatons by an authorzed admnstrator of Scholars' Mne. Ths work s protected by U. S. Copyrght Law. Unauthorzed use ncludng reproducton for redstrbuton requres the permsson of the copyrght holder. For more nformaton, please contact scholarsmne@mst.edu.

2 ECONOMIC AND ENVIRONMENTAL COMPARISON OF DIFFERENT ORDERING POLICIES FOR AN INTEGRATED INVENTORY CONTROL AND SUPPLIER SELECTION PROBLEM by SEPIDEH ALMASI MONFARED A THESIS Presented to the Faculty of the Graduate School of the MISSOURI UNIVERSITY OF SCIENCE AND TECHNOLOGY In Partal Fulfllment of the Requrements for the Degree MASTER OF SCIENCE IN ENGINEERING MANAGEMENT 2014 Approved by Dr. Dncer Konur, Advsor Dr. Suzanna Long Dr. Ruwen Qn

3

4 PUBLICATION THESIS OPTION Ths thess conssts of the followng artcle that has been submtted for publcaton and s formatted accordng to the unversty format. Page 1-37 are ntended for submsson to the EUROPEAN JOURNAL OF OPERATION RESEARCH.

5 v ABSTRACT Ths study analyzes an ntegrated nventory control and suppler selecton problem n stochastc demand envronment under carbon emssons regulatons. In partcular, a contnuous revew nventory model wth multple supplers s nvestgated under carbon taxng and carbon tradng regulatons. We analyze and compare the optmal suppler selecton and order splttng decsons wth sngle sourcng and two alternatve delvery structures for mult-sourcng, namely, sequental orderng and sequental delvery. For each of the three orderng polces, a soluton method s proposed and these polces are compared n terms of ther economc as well as envronmental performances. A numercal study s conducted to demonstrate the effcences of the soluton methods proposed. Further numercal studes analyze how the economc and envronmental performances of dfferent orderng polces vary as the suppler capactes and lead tmes change. Keywords: Carbon emssons, Contnuous Revew Inventory

6 v ACKNOWLEDGMENTS I would lke to thank my advsor, Dr. Dncer Konur, for all of hs support durng my graduate career. I would lke to thank hm for not only hs support durng my graduate educaton, but also for all of the freedom and tme that he gave me along the way to conduct the work on my own tme. It was a pleasure learnng from hm and hs style of teachng. I also owe thanks to Dr. Suzanna long and Dr. Ruwen Qn, as they are my commttee and provded me helpful comments. Dr. James Campbell provded much needed support n preparaton of ths thess. Hs experence n optmzaton was very valuable and much thanks goes to hm. However wthout the support of my parents there was no way I would be able to contnue my educaton and they provded me countless opportuntes, so I am very grateful. Unversty of Mssour System Interdscplnary Intercampus Research Program also provded support for ths research.

7 v TABLE OF CONTENTS Page PUBLICATION THESIS OPTION... ABSTRACT... v ACKNOWLEDGMENTS... v LIST OF ILLUSTRATIONS... v SECTION 1. INTRODUCTION LITERATURE REVIEW... 3 PAPER I. TITLE... 6 Abstract INTRODUCTION LITERATURE REVIEW PROBLEM FORMULATION SINGLE SOURCING SEQUENTIAL ORDERING SEQUENTIAL DELIVERY SOLUTION ANALYSIS SOLUTION OF SINGLE SOURCING SOLUTION OF SEQUENTIAL ORDERING SOLUTION OF SEQUENTIAL DELIVERY COMPARISION OF THE ORDERING POLICIES NUMERICAL STUDIES EFFICIENCY OF THE ALGORITHMS EFFECTS OF SUPPLIERS CONCLUSIONS SECTION

8 v 3. CONCLUSIONS APPENDIX A. NOTATION AND POSSIBLE METRICS B.PROOFS OF PROPERTIES IN SECTION C. TABLES OF SECTION REFERENCES VITA... 65

9 v LIST OF ILLUSTRATIONS Fgure Page PAPER 1. Inventory vs. Tme... 13

10 1. INTRODUCTION Global warmng s a growng concern and carbon emssons are a leadng contrbutor to global clmate change whch was created ncreasng pressure around the world to enact legslaton to curb these emssons. Carbon emsson regulatons have emerged to address these ssues and ncentvze frms to curb greenhouse gas (GHG) emssons, prmarly carbon-doxde (other GHG emssons can be measured n terms of carbon-doxde, see e.g., EPA 2014). Furthermore, the ncreased envronmental awareness of consumers enforces frms to green ther operatons to stay compettve. Industry and transportaton sectors are the largest contrbutors to GHG emssons. For nstance, ndustry and transportaton sectors generated 29% and 15% of the global GHG emssons n 2010 (ECOFYS, 2010). The U.S. Envronmental Protecton Agency (EPA) reports that ndustral and transportaton sectors contrbuted 20% and 28%, respectvely, to natonal GHG emssons n 2012 (EPA, 2014). Thus, a very large fracton of carbon emssons are due to supply chan actvtes ncludng nventory holdng, freght transportaton, and logstcs and warehousng actvtes. Inventory management s partcularly mportant for a company as ths determnes not only the level of nventory carred and warehousng actvtes but also the amount and the frequency of freght shpments and logstcal operatons. The nventory control polcy of a company, therefore, s nextrcably lnked wth ts envronmental performance. There s a growng body of lterature that analyzes nventory control models wth envronmental consderatons. As wll be revewed n Secton 2, these studes nclude envronmental aspects of the nventory related operatons by ether assocatng drect costs wth the envronmental damage due to the nventory related operatons or consderng envronmental objectves such as emssons mnmzaton along wth the classcal economc objectves such as cost mnmzaton (proft maxmzaton) or modelng the nventory control polces under envronmental regulatons such as carbon cap, carbon tax, carbon tradng, or carbon offsettng. In ths study, we ncorporate the envronmental aspects of nventory related operatons by formulatng an nventory control model under carbon taxng and

11 2 carbon tradng polces. Specfcally, under carbon taxng, a company pays taxes for the emssons t generates. The tax per unt carbon emssons s defned by governmental agences. European countres Denmark, Fnland, Sweden, Netherlands, and Norway are among the frst countres that mplemented carbon taxng (Ln and L, 2011). Under carbon tradng, on the other hand, a company s subject a carbon emssons lmt per unt tme, whch s known as carbon cap, and carbon emssons are tradable through an emssons tradng system such as European and New Zealand Emssons Tradng systems. That s, the company can buy extra carbon allowances or sell ts excess carbon emssons. Partcularly, our focus s on a retaler s ntegrated nventory control and suppler selecton problem under the aforementoned envronmental regulatons. We consder the case of stochastc demand and assume a contnuous revew nventory control system. The retaler can splt hs/her order among an arbtrary number of heterogeneous supplers.

12 3 2. LITERATURE REVIEW Sustanablty has been consdered n varous operatons and supply chan management settngs (see, e.g., the revews by Corbett and Klendorfer, 2001a, b, Lnton et al., 2007, Srvastava, 2007). In ths study, we ntegrate sustanablty n an nventory control model wth multple sources of supply. In case of multple sources of supply, the suppler selecton models have been ntroduced for companes to choose the supplers to buld relatonshps wth. Suppler evaluaton and selecton models have been ntensvely studed n the lterature. One may refer to Ho et al. (2010) for a revew of suppler evaluaton and selecton studes. Generally, suppler selecton models consttute multattrbute decson makng problems and varous methods such as data envelopment analyss, mathematcal programmng, analytc herarchy process, fuzzy set theory, and rankng methods have been utlzed to help companes evaluate and select supplers (Ho et al., 2010). Wth ncreasng sustanablty concerns along supply chans, envronmental consderatons have also been consdered n suppler selecton models. In partcular, green suppler selecton models take nto account not only the suppler attrbutes consdered n the classcal suppler evaluaton and selecton models but also envronmental/sustanablty attrbutes of the supplers. Igarash et al. (2013) note that product- and company-related envronmental attrbutes are manly ntroduced n green supplers selecton models. We refer the reader to Genovese et al. (2010), Govndan et al. (2013), and Igarash et al. (2013) for revews of the green suppler selecton models. Our study does not consder a mult-attrbute suppler selecton model wth envronmental consderatons. We rather consder an nventory control model wth multple possble source of supply under envronmental regulatons. Therefore, n the followng revew, our focus s on the nventory control studes that account for envronmental aspects of the nventory related operatons. We dstngush such studes based on the demand characterstcs (determnstc vs. stochastc demand), sourcng characterstcs (sngle vs. multple supply sources), and model characterstcs. Most of the studes that ntegrate envronmental aspects nto nventory control models focus on well-known nventory control models such as the economc order quantty model,

13 4 economc lot-szng model, and sngle-perod stochastc demand model, and ther varatons. Furthermore, envronmental aspects of the nventory related operatons are ntegrated nto these models through ether modelng envronmental regulatons such as carbon cap, taxng, tradng, and offsettng or assocatng drect costs wth envronmental polluton generated from nventory control related operatons or regardng envronmental objectves along wth the classcal economc objectves. Ths study consders a stochastc demand contnuous revew nventory control model wth multple supply sources under envronmental regulatons.. In partcular, Benjaafar et al. (2012), Abs et al. (2013), Palak et al. (2014), and Helmrch et al. (2015) study ELS problems under envronmental regulatons and, Mafakher et al. (2011) and Azadna et al. (2014) formulate a mult-objectve EL wth envronmental consderatons. Among these studes, whle Abs et al. (2013) and Palak et al. (2014) account for dfferent sources of supply by consderng dfferent transportaton modes, Mafakher et al. (2011) and Azadna et al. (2014) drectly ntegrate suppler selecton decsons wth ELS model and assess the suppler s envronmental performance n the selecton. Unlke these studes, we consder a stochastc nventory control model over a long plannng horzon nstead of mult-perod determnstc demand model. Furthermore, we model dfferent delvery structures n case of multple sourcng. Most of the stochastc nventory control models wth envronmental consderatons revst the classcal sngle-perod stochastc demand model,.e., the Newsvendor model. The Newsvendor model maxmzes the expected profts due to a sngle order by consderng the costs assocated wth unsold tems n case of overage and unmet demand n case of underage. Song and Leng (2012), Zhang and Xu (2013), Cho (2013a,b), Lu et al. (2013), Rosc and Jammernegg (2013), Hoen et al. (2014), and Arkan and Jammernegg (2014) study the Newsvendor model and ts varatons (ncludng dual sourcng and mult-tem settngs) under envronmental regulatons. Among these studes, Hoen et al. (2014) consder dfferent modes of transportaton and Cho (2013a,b), Rosc and Jammernegg (2013), and Arkan and Jammernegg (2014) ntegrate dfferent sourcng channels (dual sourcng wth a local and an off-shore suppler) as alternatve optons to order from.

14 5 The most related study to ours s the one by Arkan et al. (2013). Arkan et al. (2013) numercally demonstrate how the costs and carbon emssons generated change wth dfferent transportaton modes and delvery lead tmes when a cost- or emssons-mnmzng order quantty-reorder pont polcy s used for orderng decsons. That s, they do not consder order splttng and envronmental regulatons.

15 6 PAPER Economc and Envronmental Comparson of Dfferent Orderng Polces for An Integrated Inventory Control and Suppler Selecton Problem October 30, 2014 ABSTRACT Ths study analyzes an ntegrated nventory control and suppler selecton problem n stochastc demand envronment under carbon emssons regulatons. In partcular, a contnuous revew nventory model wth multple supplers s nvestgated under carbon taxng and carbon tradng regulatons. We analyze and compare the optmal suppler selecton and order splttng decsons wth sngle sourcng and two alternatve delvery structures for mult-sourcng, namely, sequental orderng and sequental delvery. For each of the three orderng polces, a soluton method s proposed and these polces are compared n terms of ther economc as well as envronmental performances. A numercal study s conducted to demonstrate the effcences of the soluton methods proposed. Further numercal studes analyze how the economc and envronmental performances of dfferent orderng polces vary as the suppler capactes and lead tmes change. Keywords: Carbon emssons, Contnuous Revew Inventory

16 7 1. INTRODUCTION There s a growng consensus that carbon emssons are a leadng contrbutor to global clmate change whch was created ncreasng pressure around the world to enact legslaton to curb these emssons. Carbon emsson regulatons have emerged to address these ssues and ncentvze frms to curb greenhouse gas (GHG) emssons, prmarly carbon-doxde (other GHG emssons can be measured n terms of carbondoxde, see e.g., EPA 2014). Furthermore, the ncreased envronmental awareness of consumers enforces frms to green ther operatons to stay compettve. Industry and transportaton sectors are the largest contrbutors to GHG emssons. For nstance, ndustry and transportaton sectors generated 29% and 15% of the global GHG emssons n 2010 (ECOFYS, 2010). The U.S. Envronmental Protecton Agency (EPA) reports that ndustral and transportaton sectors contrbuted 20% and 28%, respectvely, to natonal GHG emssons n 2012 (EPA, 2014). Thus, a very large fracton of carbon emssons are due to supply chan actvtes ncludng nventory holdng, freght transportaton, and logstcs and warehousng actvtes. Inventory management s partcularly mportant for a company as ths determnes not only the level of nventory carred and warehousng actvtes but also the amount and the frequency of freght shpments and logstcal operatons. The nventory control polcy of a company, therefore, s nextrcably lnked wth ts envronmental performance. There s a growng body of lterature that analyzes nventory control models wth envronmental consderatons. As wll be revewed n Secton 2, these studes nclude envronmental aspects of the nventory related operatons by ether assocatng drect costs wth the envronmental damage due to the nventory related operatons or consderng envronmental objectves such as emssons mnmzaton along wth the classcal economc objectves such as cost mnmzaton (proft maxmzaton) or modelng the nventory control polces under envronmental regulatons such as carbon cap, carbon tax, carbon tradng, or carbon offsettng. In ths study, we ncorporate the envronmental aspects of nventory related operatons by formulatng an nventory control model under

17 8 carbon taxng and carbon tradng polces. Specfcally, under carbon taxng, a company pays taxes for the emssons t generates. The tax per unt carbon emssons s defned by governmental agences. European countres Denmark, Fnland, Sweden, Netherlands, and Norway are among the frst countres that mplemented carbon taxng (Ln and L, 2011). Under carbon tradng, on the other hand, a company s subject a carbon emssons lmt per unt tme, whch s known as carbon cap, and carbon emssons are tradable through an emssons tradng system such as European and New Zealand Emssons Tradng systems. That s, the company can buy extra carbon allowances or sell ts excess carbon emssons. Partcularly, our focus s on a retaler s ntegrated nventory control and suppler selecton problem under the aforementoned envronmental regulatons. We consder the case of stochastc demand and assume a contnuous revew nventory control system. The retaler can splt hs/her order among an arbtrary number of heterogeneous supplers. We note that nventory control models wth order splttng among multple sources of supply have been studed n the lterature. In ths study, the sources of the supply are defned as supplers; hence, order splttng decsons also determne the suppler selecton decsons. Nevertheless, the sources of supply can be not only dfferent supplers (dstrbuton centers, manufacturers) but also dfferent transportaton modes avalable for shpment, or even dfferent carrers of the same transportaton mode such as dfferent truck/vehcle types (see, e.g., Konur, 2014 and?) or truckload and less-than-truckload carrers (see, e.g., Konur and Schaefer, 2014). The models and soluton methods dscussed n ths paper, therefore, apply to the ntegrated stochastc nventory control and transportaton mode selecton and/or ntegrated stochastc nventory control and carrer selecton problems. One may refer Mnner (2003) for a revew of nventory control models wth suppler selecton. Our study consders stochastc demand and the nventory control models wth suppler selecton under stochastc demand are grouped nto two classes: models wth determnstc and stochastc lead tmes (Mnner, 2003). Smlar to Monzadeh and Nahmas (1988), Monzadeh and Schmdt (1991), Zhang (1996), Chang and Guterrez (1996), and Jan et al. (2010), we assume that

18 9 supplers have determnstc lead tmes,.e., they are relable. One may refer to Thomas and Tyworth (2006) for a revew of stochastc nventory control models wth order splttng n case of stochastc lead tmes. The supplers consdered heren vary n ther shppng specfcatons (delvery lead tmes and freght mnmums) and shppng costs (unt procurement and fxed delvery setup costs) as well as envronmental characterstcs (per unt and fxed emssons generaton from order shpments). Dfferent supplers can have dfferent delvery lead tmes due to dstnct ponts of orgn or transportaton modes used for delvery. Due to the same reasons, the supplers mght have varyng unt procurement costs (whch can nclude the unt purchasng/manufacturng and unt shppng cost) and fxed delvery costs as well as carbon emsson generaton characterstcs. Therefore, smlar to the most of the studes ntegratng nventory control and suppler selecton, we consder heterogeneous supplers. Furthermore, smlar to Burke et al. (2007), Da and Q (2007), Awasth et al. (2009), and Zhang and Zhang (2011), we account for suppler capactes and assume that dfferent supplers have dfferent capactes. For nstance, dfferent transportaton modes have dfferent capactes or dfferent vehcle types of the same transportaton mode can have dfferent capactes (varous freght trucks have dfferent volume/weght lmts, see, e.g., Konur, 2014). In most of the ntegrated nventory control and mult-sourcng models, the splt orders are assumed to be delvered to the retaler sequentally. That s, after the retaler places the orders, the suppler wth the lowest lead tme (or the lowest realzed lead tme n case of stochastc lead tmes) delvers frst, then the suppler wth the second lowest lead tme delvers second, and so on (n case of stochastc lead tmes, t s possble that dfferent supplers delver smultaneously). As noted by Glock (2012) as well, delvery structure of the orders affects the nventory related costs. Furthermore, as s dscussed n ths study, dfferent delvery structures have dfferent envronmental performances. Therefore, t s mportant to consder dfferent delvery structures n ntegrated nventory control and suppler selecton models.we note that dfferent delvery structures are generally modeled for the suppler (or manufacturer) n two-

19 10 echelon supply chans n the context of shpment consoldaton (see, e.g., C etnkaya (2005) for a revew of consoldaton polces) or mult-tem nventory systems n the context of jont replenshment problem (see, e.g., Khouja and Goyal (2008) for a revew of jont replenshment problems). Unlke shpment consoldaton and jont replenshment problems, ths study analyzes a sngle-echelon (retaler) and sngletem nventory system wth multple supply sources (supplers) n a stochastc demand envronment. We consder three dfferent orderng polces, namely sngle sourcng, sequental orderng, and sequental delvery, for the ntegrated nventory control and suppler selecton problem of nterest n ths study under carbon taxng and carbon tradng regulatons. In partcular, under sngle sourcng, the retaler does not consder order splttng; hence, he/she chooses the sngle suppler to order from. Gven the selected suppler, the retaler s problem s then to determne the reorder pont R (the on-hand nventory level to place an order) and the order quantty, q, f suppler s the sngle selected suppler. On the other hand, n the case order splttng s consdered as an opton, the retaler can control the delveres from dfferent supplers by changng the order release tmes to the supplers. For nstance, Km and Goyal (2009) consder two dfferent delvery optons, whch they refer to as lumpy and phased delveres, n a sngle buyer-multple supplers settng. In case of lumpy delveres, the orders from dfferent supplers are delvered smultaneously whle dfferent supplers orders are delvered alternately n case of phased delveres. Glock (2012) defnes sx dfferent delvery structures regardng the producton cycles of two manufacturers and the delvery at the sngle buyer. Both of these studes consder the two-echelons (buyer and vendor) of the supply chan smultaneously and they assume determnstc demand. In ths study, our focus s on the retaler only and the retaler s subject to stochastc demand. We, therefore, consder two structures for order splttng: sequental orderng and sequental delvery. Under sequental orderng, the retaler starts orderng from the selected supplers such that the orders from dfferent supplers are receved smultaneously. Specfcally, n the case the retaler enjoys less frequent warehousng actvtes such as unloadng operatons and nventory placement, sequental orderng can be preferred. Furthermore, all of the orders are delvered at once under sequental orderng; however, the

20 11 retaler needs to carefully montor the tmng to release splt orders to the supplers. On the other hand, under sequental delvery, the retaler places the orders from selected supplers smultaneously, thus, receves the orders from dfferent supplers sequentally due to dstnct lead tmes. Therefore, compared to sequental orderng, order placement s smpler under sequental delvery; however, there are more frequent shpments,.e., more frequent warehousng operatons and nventory placements can be requred. Fgure 1 llustrates the retaler s nventory over tme wth sngle sourcng when suppler 2 s selected, sequental orderng, and sequental delvery when an order s splt among three supplers such that τ1 < τ2 < τ3, where τ s the lead tme of suppler. Ths study contrbutes to the body of lterature on nventory control models wth envronmental consderatons by () ntegratng suppler selecton decsons n contnuous revew nventory systems and () regardng dfferent delvery structures. To the best knowledge of the authors, ntegrated contnuous revew nventory control and suppler selecton models under stochastc demand wth envronmental consderatons have not been analyzed n the lterature. Actually, as wll be dscussed n our lterature revew, whle there s a growng body of lterature on envronmental nventory control models, most of these studes assume determnstc demand or stochastc demand n the sngle perod. Furthermore, whle ntegrated stochastc nventory control and suppler selecton models have been analyzed extensvely (see the revews cted above and the references cted n those revews), dfferent delvery structures are not consdered n such models. Most of the ntegrated stochastc nventory control and suppler selecton studes adopt sequental delvery and focus on the economc comparson of sngle sourcng and order splttng. In ths study, we compare not only sngle sourcng to order splttng but also two dfferent delvery structures for orderng splttng. And, our comparson evaluates economc as well as envronmental performance of the dfferent orderng polces consdered. Specfcally, we formulate the retaler s suppler selecton and nventory control model under carbon tradng regulaton wth the three orderng polces (t s dscussed

21 12 that carbon taxng s a specal case of carbon tradng regulaton). For each model, a soluton method s developed. Then, we compare these three orderng polces not only n terms of economc but also envronmental aspects. It s noted that whle a retaler can prefer order splttng to mnmze costs under carbon tradng, sngle sourcng can be a more envronmental alternatve. Also, when the two delvery structures for order splttng are compared, we note that there s no pure domnance between them n terms of economc objectves. Ths observaton suggests that sequental orderng can be a better alternatve n terms of costs compared to sequental delvery, whch, as aforementoned, s the delvery structure commonly assumed n ntegrated stochastc nventory control and suppler selecton models. Furthermore, when sequental orderng (sequental delvery) s a better polcy n terms of economc performances, sequental delvery (sequental orderng) can be a better polcy n terms of envronmental performance. Thus, the retaler s preference for a delvery structure wll depend on hs/her economc as well as envronmental goals. The tools provded n ths study enable comparng dfferent delvery structures for multple sourcng and sngle sourcng from both economc and envronmental aspects. Fnally, we conduct a set of numercal studes to demonstrate the effcency of the proposed soluton methods. Further numercal studes are presented to llustrate the effects of suppler characterstcs on the economc and envronmental performances of the orderng polces.

22 13

23 14 The rest of the paper s organzed as follows. In Secton 2, we revew the nventory control models wth envronmental consderatons. Secton 3 dscusses the settngs of the problem and formulates the mathematcal models of the retaler s optmzaton problems under sngle sourcng, sequental orderng, and sequental delvery polces. A soluton method for each model s proposed n Secton 4. Secton 5 economcally and envronmentally compares the orderng polces. Numercal studes are presented n Secton 6 and concludng remarks, summary of contrbutons, and future research drectons are gven n Secton 7.

24 15 2. LITERATURE REVIEW Sustanablty has been consdered n varous operatons and supply chan management settngs (see, e.g., the revews by Corbett and Klendorfer, 2001a, b, Lnton et al., 2007, Srvastava, 2007). In ths study, we ntegrate sustanablty n an nventory control model wth multple sources of supply. In case of multple sources of supply, the suppler selecton models have been ntroduced for companes to choose the supplers to buld relatonshps wth. Suppler evaluaton and selecton models have been ntensvely studed n the lterature. One may refer to Ho et al. (2010) for a revew of suppler evaluaton and selecton studes. Generally, suppler selecton models consttute mult-attrbute decson makng problems and varous methods such as data envelopment analyss, mathematcal programmng, analytc herarchy process, fuzzy set theory, and rankng methods have been utlzed to help companes evaluate and select supplers (Ho et al., 2010). Wth ncreasng sustanablty concerns along supply chans, envronmental consderatons have also been consdered n suppler selecton models. In partcular, green suppler selecton models take nto account not only the suppler attrbutes consdered n the classcal suppler evaluaton and selecton models but also envronmental/sustanablty attrbutes of the supplers. Igarash et al. (2013) note that product- and company-related envronmental attrbutes are manly ntroduced n green supplers selecton models. We refer the reader to Genovese et al. (2010), Govndan et al. (2013), and Igarash et al. (2013) for revews of the green suppler selecton models. Our study does not consder a mult-attrbute suppler selecton model wth envronmental consderatons. We rather consder an nventory control model wth multple possble source of supply under envronmental regulatons. Therefore, n the followng revew, our focus s on the nventory control studes that account for envronmental aspects of the nventory related operatons. We dstngush such studes based on the demand characterstcs (determnstc vs. stochastc demand), sourcng

25 16 characterstcs (sngle vs. multple supply sources), and model characterstcs. Most of the studes that ntegrate envronmental aspects nto nventory control models focus on well-known nventory control models such as the economc order quantty model, economc lot-szng model, and sngle-perod stochastc demand model, and ther varatons. Furthermore, envronmental aspects of the nventory related operatons are ntegrated nto these models through ether modelng envronmental regulatons such as carbon cap, taxng, tradng, and offsettng or assocatng drect costs wth envronmental polluton generated from nventory control related operatons or regardng envronmental objectves along wth the classcal economc objectves. Ths study consders a stochastc demand contnuous revew nventory control model wth multple supply sources under envronmental regulatons. Most of the determnstc nventory control models wth envronmental consderatons revst the classc Economc Order Quantty (EOQ) model. The EOQ model analyzes the trade-off between nventory holdng and order setup costs for a product that has determnstc demand. Hua et al. (2011), Jaber et al. (2013), Arslan and Turkay (2013), Chen et al. (2013), Toptal et al. (2014), Konur and Schaefer (2014), Konur (2014), and He et al. (2014) study the EOQ model and/or ts extensons (to addtonal decson varables or mult-tem/mult-echelon settngs) under carbon regulaton polces such as carbon cap, taxng, tradng, and offsettng. Among these studes, only Konur and Schaefer (2014) and Konur (2014) consder multple sources of supply. In partcular, whle Konur and Schaefer (2014) model the EOQ model under four dfferent carbon emssons regulatons wth less-than-truckload and truckload carrers, Konur (2014) consders dfferent freght trucks for shpments under carbon cap regulaton. On the other hand, Bonney and Jaber (2011), Wahab et al. (2011), Rtha and Martn (2012), Dges et al. (2012), Rtha and Vnolne (2013), and Battn et al. (2014) analyze nventory control models smlar to the EOQ model by drectly assocatng costs to the envronmental polluton/carbon emssons generated from the nventory control related operatons. Among these studes, Dges et al. (2012) and Battn et al. (2014) consder dfferent sources of supply by ncludng dfferent modes of transportaton n ther models. Fnally, Bouchery et al. (2012), Chan et al. (2013), and Bozorg et al. (2014) ntegrate envronmental aspects nto the EOQ

26 17 model and/or ts extensons by consderng envronmental objectves n addton to the economc objectves and these studes consder sngle source of supply. Other than the EOQ model, the economc lot-szng (ELS) models wth determnstc demand have been recently analyzed wth envronmental consderatons. In partcular, Benjaafar et al. (2012), Abs et al. (2013), Palak et al. (2014), and Helmrch et al. (2015) study ELS problems under envronmental regulatons and, Mafakher et al. (2011) and Azadna et al. (2014) formulate a mult-objectve EL wth envronmental consderatons. Among these studes, whle Abs et al. (2013) and Palak et al. (2014) account for dfferent sources of supply by consderng dfferent transportaton modes, Mafakher et al. (2011) and Azadna et al. (2014) drectly ntegrate suppler selecton decsons wth ELS model and assess the suppler s envronmental performance n the selecton. Unlke these studes, we consder a stochastc nventory control model over a long plannng horzon nstead of multperod determnstc demand model. Furthermore, we model dfferent delvery structures n case of multple sourcng. Most of the stochastc nventory control models wth envronmental consderatons revst the classcal sngle-perod stochastc demand model,.e., the Newsvendor model. The Newsvendor model maxmzes the expected profts due to a sngle order by consderng the costs assocated wth unsold tems n case of overage and unmet demand n case of underage. Song and Leng (2012), Zhang and Xu (2013), Cho (2013a,b), Lu et al. (2013), Rosc and Jammernegg (2013), Hoen et al. (2014), and Arkan and Jammernegg (2014) study the Newsvendor model and ts varatons (ncludng dual sourcng and mult-tem settngs) under envronmental regulatons. Among these studes, Hoen et al. (2014) consder dfferent modes of transportaton and Cho (2013a,b), Rosc and Jammernegg (2013), and Arkan and Jammernegg (2014) ntegrate dfferent sourcng channels (dual sourcng wth a local and an off-shore suppler) as alternatve optons to order from. Brto and de Almeda (2012) model a mult-objectve Newsvendor model wth a sngle supply source, where one of the objectves s to mnmze the envronmental damage due to salvaged products n case of overage. In a recent study, Carrllo et al.

27 18 (2014) study envronmental mplcatons of dfferent retal channels (such as classcal channels and onlne channels) such that the retaler s decson n each channel s defned under the settngs of the Newsvendor model. They assocate a cost value, whch can represent the unt envronmental savngs or premums, for the onlne retalng channel. Smlar to these studes, we consder multple optons for sourcng; however, we do consder an arbtrary number of optons as the supply sources nstead of dual sourcng. Furthermore, we drectly ntegrate sourcng decsons wth order decsons nstead of analyzng the orderng decsons under each source and compare them. That s, the models we formulate jontly determne the optmal sourcng and orderng decsons under envronmental regulatons. Also, we consder a contnuous revew nventory control model nstead of a sngle-perod stochastc demand model. To the best knowledge of the authors, envronmental consderatons are not drectly ntegrated wthn contnuous revew nventory control models. The most related study to ours s the one by Arkan et al. (2013). Arkan et al. (2013) numercally demonstrate how the costs and carbon emssons generated change wth dfferent transportaton modes and delvery lead tmes when a cost- or emssons-mnmzng order quantty-reorder pont polcy s used for orderng decsons. That s, they do not consder order splttng and envronmental regulatons. In ths study, we formulate and analyze a contnuous revew nventory control model under envronmental regulatons and we ntegrate suppler selecton decsons n ths model. Furthermore, dfferent delvery structures are consdered n case of order splttng. In the next secton, we explan the detals of the settngs and formulaton of the models analyzed n ths study.

28 19 3. PROBLEM FORMULATION We consder a retaler s nventory control problem for a sngle tem whch has stochastc demand. Let the demand per unt tme for the tem be a normally dstrbuted random varable wth mean λ and standard devaton υ. We therefore assume that the demand durng a tme perod of t s normally dstrbuted wth a mean of λt and standard devaton of υ t (see, e.g., Nahmas, 2009). Let f t (y) and F t (y) denote the probablty densty and cumulatve probablty functons, respectvely, of the normally dstrbuted random varable y wth mean λt and standard devaton υ t. Due to the stochastc demand, there mght be shortages and be the expected number of shortages and let n(r, t) be the expected number of shortages over a tme perod t when the startng nventory s r. It then follows that n(r, t) = (y r)f r t (y)dy. It s assumed that the nventory s contnuously revewed,.e., the retaler knows the nventory level at any moment. In case of contnuous nventory revew, a common nventory control polcy adopted s (Q, R) model, where Q denotes the order quantty and R denotes the re-order pont to place an order. That s, whenever the nventory on hand s R, an order of Q unts s placed. In the settngs of the classcal (Q, R) model, the retaler s subject to nventory holdng, penalty, procurement, and order setup costs. Let h denote the retaler s per unt per unt tme nventory holdng cost. It s assumed that all of the shortages are backordered and there s a penalty cost p backordered. Furthermore, let A be the setup cost per order. In ths study, we assume that the retaler can partally order hs/her order quantty from a set of n supplers, ndexed by such that S wheres = {1,2,, n},.e., we allow order splttng. As dfferent supplers mght have dstnct characterstcs wth regards to ther locatons, wholesale prces, and shpment requrements, we defne ĉ as the retaler s unt procurement cost from suppler. Furthermore n addton to the retaler s major setup cost per order, we assume that the retaler s subject to fxed order setup cost â, when an order s placed from suppler S. Note that ĉ can be defned to nclude suppler s unt transportaton cost n addton to the unt procurement cost; and, â can nclude the fxed transportaton or delvery cost such as the truck drver s cost or loadng/unloadng charges for an order from suppler. Furthermore, we assume that each suppler has a shpment capacty of w unts per order due to lmted supply or the capacty of the transportaton mode used

29 20 by suppler. We defne τ as the delvery lead tme of suppler and t s assumed that dfferent supplers mght have dfferent lead tmes due to dfferent ponts of orgn or transportaton modes used. As noted n Secton 1, there s a sgnfcant amount of carbon emssons generated from nventory holdng, freght transportaton, and warehousng actvtes. Smlar to, Hua et al. (2011), Chen et al. (2013), Toptal et al. (2014), Konur (2014), and Konur and Schaefer (2014), we assume that ĥ unts of carbon emssons generated from holdng one unt nventory per unt tme due to electrcty used n the warehouse for coolng/heatng/lghtng operatons and  as the emssons generated from each nventory replenshment due to materal handlng and unloadng/loadng operatons. We also consder that p unts of carbon emssons are generated from backordered shortages as the retaler mght need to shp the backordered unt to the customer (see, e.g., Anderson et al., 2012) or the customer mght need to re-travel to the retaler s store to pck the backordered unt (see, e.g., Cachon, 2014). A substantal amount of carbon emssons are due to freght transportaton and the transportaton emssons depend on the transportaton mode selected, type of vehcles used, the load carred, and the shpment dstance (Konur, 2014, Konur and Schaefer, 2014). As dfferent supplers can use dfferent transportaton modes, or even dfferent vehcle types of the same transportaton mode (such as dfferent truck types or ral cars), we consder that each suppler s delvery to the retaler has dfferent carbon emssons generaton characterstcs. In partcular, we let ĉ be the carbon emssons generated per unt shpped and â denote the fxed carbon emssons generated per shpment made by suppler S. For nstance, â can be consdered as the carbon emssons generated due to the empty weght of the transportaton unt (e.g., a truck) and ĉ s the carbon emssons generated from each addtonal unt loaded to the truck (smlar parameters are also defned n Hua et al., 2011, Chen et al., 2013, Konur, 2014, Konur and Shaefer, 2014). In ths study, we assume that the retaler s subject to one of the two mostcommon envronmental regulatons: carbon taxng and carbon tradng. Under carbon taxng, the retaler s charged per unt of carbon emssons generated and let α denote

30 21 the carbon tax per unt of carbon emssons generated. On the other hand, under carbon tradng, the retaler s subject to a carbon cap and carbon emssons are tradable. As mentoned prevously, f the retaler s carbon emssons per unt tme s below the carbon cap, the retaler can sell hs/her excess carbon emssons; whereas, f the retaler s carbon emssons per unt tme s above the carbon cap, the retaler needs to buy the extra carbon allowances. Let β denote the carbon tradng prce per unt of carbon emssons and Φ be the carbon cap per unt tme. Smlar to Hua et al., 2011 and Toptal et al. (2014), we assume that there are suffcent demand and supply for carbon tradng n the market; hence, the retaler can sell all of hs excess carbon credts or buy unlmted carbon allowances. One can note that when Φ = 0, carbon taxng and carbon tradng regulatons are dentcal f β = α. Therefore, n the mathematcal formulaton and the soluton analyss, we wll only focus on carbon tradng regulaton as carbon taxng s the aforementoned specal case of carbon tradng. The retaler s objectve s to mnmze hs/her total expected costs per unt tme by determnng whch supplers to select, how much to shp from each suppler, and when to start orderng from the supplers. Let 1 s suppler s selected, x ={ 0 otherwse and x be the bnary n-vector of x values. Furthermore, let q be the quantty ordered from suppler at each replenshment and q denote the n-vector of q values. Note that f x = 0 then q = 0 and f x = 1 then q w. As s defned prevously, R s the re-order pont. We assume that the suppler can use one of the three polces for order splttng among the selected supplers: () sngle sourcng, () sequental orderng, and () sequental delvery. In case of sngle sourcng, the retaler selects a sngle suppler to order from; hence, there s no need for order splttng. On the other hand, when multsourcng s allowed, we consder two dfferent polces for order splttng, whch are

31 22 sequental orderng and sequental delvery. In sequental orderng, the retaler splts hs/her order among dfferent supplers sequentally consderng ther lead tmes such that the splt orders from dfferent supplers are receved by the retaler at the same tme. In sequental delvery, the retaler splts the order among dfferent supplers at the same tme and the splt orders from dfferent supplers are receved by the retaler at dfferent tmes due to varyng suppler lead tmes. In sequental delvery, we assume that the next order wll not be placed untl the partal order of the last suppler (suppler 3 n Fgure 1) has been delvered. In what follows, we mathematcally formulate the retaler s nventory control and suppler selecton problem wth each order splttng polcy. A table summarzng the notaton and possble metrcs s noted n the Appendx. Addtonal notaton wll be defned as needed.

32 SINGLE SOURCING In the case the retaler adopts sngle sourcng polcy, for any selected suppler, the retaler s nventory control polcy s the classcal (Q, R) model wth an addtonal upper bound constrant on the order quantty due to supply lmt. Suppose that suppler s selected to be ordered from; hence, the lead tme sτ. Then, the retaler s cost functon s the cost functon of the classcal (Q, R) model. partcular, assumng that only suppler s used under the settngs of the classcal (Q, R) model, one can derved that C λ expected procurement cost per unt tme and In h (R λτ q ) s the expected nventory holdng cost per unt tme. Also, as the expected cycle length (the tme between recevng two consecutve orders from suppler ) s equal to q λ penalty cost per unt tme amount to the expected order setup cost per unt tme and expected (Ã+a )λ q and p λn(r,τ q respectvely. It then follows that the retalers expected cost per unt tme under sngle sourcng as a functon of the decson varables R, q, and x, denoted by C 1 (R,q,x) s C 1 (R, q, x) = S x )λ [c λ + h (R λτ + 1 q 2 ) + (A +a q + p λn(r,τ ) q ] (1) The expected carbon emssons generated from nventory related operatons under sngle sourcng can be defned smlar to the expected nventory related costs gven n Equaton (1). Partcularly, t can be shown that the retaler s carbon emssons per unt tme under sngle sourcng as a functon of the decson varables R, q, and x, denoted by E1 (R, q, x), reads E 1 (R, q, x) = S x )λ [c λ + ĥ (R λτ + 1 q 2 ) + (A +a )λ q + p λn(r,τ ) q ] (2) Where c λ, ĥ (R λτ + 1 q 2 ), (A +a and P λn(r,τ ) defne the expected carbon q emssons generated per unt tme from transportaton, nventory holdng, order setup and background operaton respectvely, when suppler s selected. q

33 24 Under a carbon tradng polcy wth carbon cap of Φ, the total amount of traded carbon emssons s equal to E 1 (R, q, x) Φ. Note that f E 1 (R, q, x) Φ > 0, the retaler s buyng extra carbon allowances at a cost of β per unt; and, f E1(R, q, x) Φ < 0, the retaler s sellng hs/her excess carbon emssons at a prce of β per unt. The retaler s optmzaton problem wth sngle sourcng under carbon tradng then can be formulated as follows: 2 (P1): mn (R, q, x) = C 2 (R, q, x) + β(e 2 (R, q, x) ϕ) s.t P λn(r,τ q 0 q x w S x = 1 x {0,1} S R > 0. Π1(R, q, x) defnes the total expected costs per unt tme under sngle sourcng and the frst constrant ensures that only a sngle suppler s selected. The second set of constrants guarantees that the retaler can only order from the selected suppler and the order quantty s less than or equal to the selected suppler s capacty. The thrd set of constrants s the bnary defntons of x values and the fourth constrant s the non-negatvty of the re-order pont. Let (R1, q1, x1) denote an optmal soluton of P1.

34 SEQUENTIAL ORDERING In the case retaler adopts sequental orderng polcy, the effectve lead tme,.e., the tme between the retaler starts orderng from the supplers untl the orders are smultaneously receved, s the maxmum of the lead tmes of the selected supplers. Let τ (x) denote the effectve lead tme when suppler selecton decson s gven by x. It then follows that τ(x) = max εs {τ x } (3) The expected nventory level wth sequental orderng s defned smlar to the classcal (Q,R) model and one can derve that h (R λτ(x) εs q ) s the expected nventory holdng cost per unt tme. Smlarly t can be argued that the expected cycle length s S c q S q 1 q λ ϵs and λ(ã+ S a ) q ; thus the expected procurement cost per unt tme S the expected order setup cost per unt tme amount to and respectvely. Fnally shortage can occur durng the effectve lead tme and the expected number of shortages per cycle sn(r, τ(x)) t then follows that the expected penalty cost per unt tme s equal to p λn(r,τ. These mply that the retalers expected cost per unt tme ϵs q under sequental orderng as a functon of the decson varables R, q and x denoted by C 2 (R,q,x), s C 2 (R, q, x) = λ S c q + h (R λτ(x) + 1 q S q 2 S )+ λ(a + S a S q x )λ + p λn(r,τ ) (4) S q Where the frst, second, thrd and the last terms are expected procurement. Inventory holdng, order setup and penalty cost per unt tme, respectvely, such that τ(x) s defned n equaton (3). The expected carbon generaton from nventory related operatons could be defned smlar to the expected nventory related costs gven n equaton (4). Partcularly t can be shown that retalers carbon emsson put unt tme under sequental orderng as functon of the decson varables R, q and x denoted by E 2 (R,q,x), reads

35 26 E 2 (R, q, x) = λ S c q + ĥ (R λτ(x) + 1 q S q 2 S )+ λ(a + S a S q x )λ + p λn(r,τ ) (5) S q Where the frst, second, thrd and the last terms are expected procurement. Inventory holdng, order setup and penalty cost per unt tme, respectvely, such that τ(x) s defned n equaton (3). Smlar to P1, the retaler s optmzaton problem wth sequental orderng under carbon tradng such that carbon cap s ϕand carbon tradng prce s β, then can be formulated as follows: 2 (P2): mn (R, q, x) = C 2 (R, q, x) + β(e 2 (R, q, x) ϕ) s.t P λn(r,τ q 0 q x w S x = 1 x {0,1} S R > 0. Π2 (R, q, x) defnes the total expected costs per unt tme under sequental orderng and the frst set of constrants guarantees that the retaler can only order from the selected supplers and the order quantty from each selected suppler s less than or equal to the suppler s capacty. The second set of constrants s the bnary defntons of x values and the thrd constrant s the non-negatvty of the re-order pont. Let (R2, q2, x2) denote an optmal soluton of P2.

36 SEQUENTIAL DELIVERY In the case the retaler adopts sequental delvery polcy, we defne a cycle as the tme between recevng two consecutve orders from the same suppler; therefore, the expected cycle length can be defned smlar to the classcal (Q, R) model. That s expected cycle length s 1 λ tme s equal to to λ(ã+ S a ) S q S c q S q ϵs q. It then follows that the expected procurement per unt and the expected order setup cost per unt tme s equal. Defnng the expected nventory holdng cost and expected penalty cost per unt tme, on the other hand s dfferent than the sequental orderng polcy. To do so, wthout loss of generalty, let us assume that the supplers are sorted such that τ1 < τ2 <... < τn. Gven that x = 1 for k and x = 0 for k+1such that k+1 n, one can show that the expected nventory held durng one cycle amounts to R + k =1 q (τ n+1 2 τ ) λ/2 τ n+1, whch does not depend on x values. It then can be concluded that the expected nventory holdng cost per unt tme for any x s equal to h(r λ S τ q + 1 S q 2 S q ). Notce that we wll guarantee that q = 0 f x = 0 by addng constrants n formulatng the retaler s optmzaton problem. Now, let us focus on defnng the expected penalty cost per unt tme. To do so, we frst calculate the expected number of shortages wthn one cycle. Shortages can occur durng the tme perods from the moment orders placed untl the frst order receved, from the moment frst order receved untl the second order receved, and so on. Let e be the random varable defnng the nventory rght before recevng suppler s order. Furthermore, let us defne z j = max {0, (τ τj )/ τ τj }. That s, z j = { 1 fτ > τ j 0 otherwse Such that z =0. Then, one can show that e s a normally dstrbuted random varable wth mean μ (R, q, x) = x (R + εs z j q j λτ ) andσ (x) = x θ τ. By

37 28 defnton of e, t follows that the expected number of shortages rght after the moment the prevous suppler s order receved untl rght before the moment suppler I s order receved s n (R, q, x) = 0 e f (e )de, where f (e ) s the normal densty functon wth mean μ (R, q, x) and standard devaton σ (x). It then follows that (6) n (R, q, x) = μ (R, q, x) + σ (x)l ( μ (R,q,x) ), σ (x) Where L(z) s the standard loss functon. That s, the expected number of total shortages wthn one replenshment cycle s n (R, q, x) S. The above dscusson leads that the retaler s expected cost per unt tme wth sequental delvery as a functon of the decson varables R, q, and x, denoted byc 3 (R, q, x),s (7) C 3 (R, q, x) = λ S c q + h (R λ S τ q + 1 q S q S q 2 S )+ λ(a + S a x ) S q + p λn (R,q,x) S q Where the frst, second, thrd, and the last terms are the expected procurement, nventory holdng, order setup, and penalty cost per unt tme, respectvely, such that n (R, q, x) s defned n Equaton (6).The expected carbon emssons generated from nventory related operatons can be defned smlar to the expected nventory related costs gven n Equaton (7). Partcularly, t can be shown that the retaler s expected carbon emssons per unt wth sequental delvery as a functon of the decson varables R, q, and x denoted by (8) E 3 (R, q, x) = λ S c q + ĥ (R λ S τ q + 1 q S q S q 2 S )+ λ(a + S a x ) S q + p λn (R,q,x) S q Where the frst, second, thrd, and the last terms are the expected carbon emssons generated per unt tme from transportaton, nventory holdng, order

38 29 setup, and backorderng operatons, respectvely, such that n (R, q, x) s defned n Equaton (6). Smlar to P2, the retaler s optmzaton problem wth sequental delvery under carbon tradng, such that carbon cap s Φ and carbon tradng prce s β, can be formulated as follows: (P3): mn F 3 (R, q, x) = C 3 (R, q, x) + β(e 3 (R, q, x) ϕ) s.t 0 q x w S x {0,1} S R > 0. F 3 (R, q, x) defnes the total expected costs per unt tme under sequental delvery. The constrants are defned smlar to P2. Let (R3, q3, x3) denote an optmal soluton of P3.

39 30 4. SOLUTION ANALYSIS In ths secton, we analyze models P1, P2, and P3, and propose a soluton method for each model. We note that each model has dfferent settngs; hence, we analyze underlyng characterstcs of the models and develop soluton methods accordngly. Pror to the analyss of each model, we next note a stran forward property of the optmal solutons of models P1, P2 and P3. j j j Property 1 For (R, q, x ) for j = 1,2,3, f x 1, then 0 < q w. Property 1 states that the retaler wll order a postve amount from each selected suppler n optmal solutons of P1, P2, and P3. Ths s ntutve as the retaler wll nether pay extra setup costs nor generate unnecessary carbon emsson unless the order quantty from a selected suppler s postve. In the remnder of ths secton, let c = c + βc, c = c + βc, h = h + βĥ, A = A + βa a = a + βa, and p = p + βp. j j 4.1 SOLUTION OF SINGLE SOURCING Suppose that the retaler order from suppler I,.e., x 1 and x 0 j ; j hence 0 q w and q 0 j ;. In ths case, the retaler s total expected cost per j unt tme under carbon tradng s equal 1 ( A a ) p n( R, ) ( R, q ) c h( R q ). 2 q q Therefore gven x 1 and x 0 j ; P1 reduce to (P1- ) : mn p (R,q ) s.t 0 q w R>0. j * * Let ( R, ) be the optmum soluton of P1-I. Note that s a constant; thus, q ( ) ( ) R, q ) s the expected cost functon of the classcal (Q,R) model. Let ( R, q ) be a ( mnmzer of R, q ). An effcent heurstc method commonly used to approxmate the (

40 31 mnmzer of R, q ) s the teratve method proposed by Hadley and Whtn (1963). ( Partcularly, Hadley and Whtn (1963) method, startng wth the EOQ formula for teratvely solves the followng frst order condtons of equaton (9) untl two consecutve R and q values are close to each other wthn a specfed value. q, q = 2l [ A + a + pn(r,t )] h 1- F t (R) = q h pl. Ths method s an heurstc approach as the convexty of R, q ) s condtonal ( (see, e.g., Brooks and Lu, 1969); however n most cases, Hadley and Whtn (1963) method s able to fnd the mnmzer of R, q ) (see, e.g., XXX). Therefore n our ( analyss we accept the output of Hadley and Whtn (1963) method as (R(),q () ). Note that f q () w then (R *,q * ) = (R (),q () ) ; on the other hand (R (),q () ) s not feasble for P1-I f q() > w. P1-I s a nonlnear programmng model and nteror pont method (IPM) s a common method used to solve such models ( see, e.g, Forsgren et.al., 2002). Nevertheless we utlze the Hadley and Whtn(1963) method n solvng P1-I as detaled n the followng algorthm. Algorthm1 solvng P1-1. Determne (R(),q () ) usng Hadley and Whtn (1963) method. 2. If q ( ) w, let q ( ) w and calculate * * ( ) ( ) 3. Return ( R, q ) ( R, q ). () R usng equaton (11). Upon comparng algorthm 1 to IPM through a numercal study we observe that algorthm 1 fnds the same soluton wth IPM and requres less computatonal tme. The

41 32 detals of the numercal compresson can be seen n secton 5. Therefore we use * * * * algorthm 1 to fnd ( R, ). Once ( R, ) s found for each suppler, ( R, q, x ) can q q 1 * * be easly determned. Partcularly let j argmn (( R, q ). S then R1 * = R j1, q 1 * j1= q j1 and 1 1 q 0 j and j x and x 1 j SOLUTION OF SEQUENTIAL ORDERING In ths secton we frst analyze the retalers order quantty decson gven the suppler selecton decsons. Then usng the order quantty analyss, we develop a local search method to fnd the suppler selecton decsons. Gven x, let (R 2 x, q 2 x )denote a mnmzer of Π 2 (R, q, x x) subject to q w x S. One can use IPM to determne(r 2 x, q 2 x ). However, n what follows, we use the propertes of (R 2, q 2, x 2 ) n determnng (R 2 x, q 2 x ) and then, develop a local search heurstc to fnd the retaler s suppler selecton decson. Now, suppose that the suppler selecton decsons are known,.e., x s gven. Let S(x) and S (x) denote the set of selected and unselected supplers, respectvely, as ndcated by x. That s, fx = 1, then S(x); else, S (x)(note thats = S(x) S (x)). Furthermore, let us defnej x2 = argmax S(x) {c },.e., j x2 s the suppler wth the maxmum per unt purchase cost among the selected supplers ndcated by x. Next, we characterze an mportant property of (R 2, q 2, x 2 ). Property 2 q 2 = 0 S (x 2 ) {j x2 2 }, and 0 < q j x 2 < w j x 2. LetQ 2 = q 2,. e., S Q 2 s the total order quantty n the optmal soluton of P2. Property 2 mples that S(x 2 ) w w j x 2 < Q2 < S(x 2 ) w Partcularly, once x 2 and Q 2 are known, one can determne q 2 usng Property 2. It further follows from Property 2 that, gvenx = x 2, the retaler s total expected costs per unt tme under carbon tradng s equal to

42 33 g x (R, Q) = c j xλ + h (R λτ (x) + 1 ) + λ( (c 2Q S(x) c 2)w j x + A + S(x) a x ) /Q + pλn(r, τ(x))/q βϕ (12) Therefore, assumng that x = x 2, P2 reduces to s. t. (P2-x): mn g x (R,Q) S(x 2 ) w w j x 2 < Q2 < w S(x 2 ) R > 0. Let (R 2 x, q 2 x ) be an optmal soluton of P2-x. One can notce that g x (R, Q) s defned smlar to π x (R, q ) when S(x) (c c 2)w j x + A + a x 0. S(x) Note that the condtonal jont convexty of the expected cost functon of the (Q,R) model assumes the order setup cost to be non-negatve. Specfcally, for non-negatve order setup cost, the expected cost functon of the (Q,R)model s convex n Q for a gven R, convex n R for a gven Q, and jontly convex n Q and R gven that R s greater than or equal to the expected lead tme demand (.e., safety stock s non-negatve). Therefore Hadley and Whtn (1963) method can be used to determne a mnmzer of g x (R, Q), denoted by (R (x), Q (x) ), when S(x) (c c 2)w j x + A + S(x) a x 0. Smlar to Equatons (10) and (11), the frst order condtons od equatons 12 read as follows: Q = 2λ [ S(x) (c c 2)w j x + A + S(x) a x + pn(r, τ(x)) ] /h (13) 1 F τ(x) (R) = Qh pλ. (14)

43 34 Let ( R (x), Q (x) ) be defned as the output of Hadley and Whtn (1963) method when (c S(x) c j x 2)w + A + a x 0. S(x) If w S(x 2 ) w j x 2 < Q2 < S(x) w, (R x 2, q x 2 ) = ( R (x), Q (x) ). On the other hand, ( R (x), Q (x) ) can be feasble for P2-x n two cases: () Q (x) < S(x 2 ) w w j x and () f Q (x) > S(x) w. Smlar to Algorthm 1, Algorthm 2, stated below, frst utlzes the Hadley and Whtn (1963) n to fnd (R 2 x, Q 2 x ) when S(x) (c c 2)w j x + A + S(x) a x 0. For the cases when S(x) (c c 2)w j x + A + S(x) a x < 0, Algorthm 2 uses IPM to determne (R x 2, Q x 2 ). Algorthm2 Solvng P2-x: 1. If S(x) (c c 2)w j x + A + S(x) a x 0 2. Determne ( R (x), Q (x) ) usng Hadley and Whtn (1963) method. 3. If Q (x) < S(x 2 ) w w j x, let Q (x) = lm Q S(x 2 ) w calculate R (x) usng equaton (14). w j x and (14). 4. f Q (x) >, let Q (x) = S(x) w 5. Return(R x 2, Q x 2 )= ( R (x), Q (x) ). 6. If S(x) (c c 2)w j x + A + S(x) a x < 0 and calculate R (x) usng equaton 7. Return (R 2 x, Q 2 x ) usng IPM. Upon comparng Algorthm 2 to IPM through a numercal study, we observe that Algorthm 2 fnds the same solutons wth IPM and requres less computatonal tme. The detals of the numercal comparson can be seen n secton6. Therefore, we use Algorthm 2 to fnd(r 2 x, Q 2 x ). Then, one can use Property 2 to determne(r 2 x, q 2 x ). Partcularly, gven x andq 2 x, let q 2 x = 0 s (x), q (x) =w s(x) {j x }, and (x) = Q x 2 q j x S(x) w w j x. Then q x 2 = [q 1 (x), q 2 (x),, q n (x)]. Once (R 2 x, q 2 x ) s determned for all possble bnary x vectors, x 2 can be determned by comparng Π 2 (R 2 x, q 2 x, x) values. However, there are 2 n 1 bnary x vectors; hence, total enumeraton can be computatonally cumbersome. Therefore, we

44 35 next develop a local search heurstc to fnd a good selecton vector. Pror to the detals of the local search heurstc, we note another property ofx 2. Property 3 f Q 2 x = lm Q ( S(x 2 ) w w j x), thenx x 2. Property 3 elmnates those bnary x vectors where Q x 2 converges to the lowest cumulatve capacty of the selected supplers from the search of x 2. The local search heurstc that we explan next, therefore, dsregards such vectors. The local search heurstc method for solvng P2 works as follows. Suppose that x s gven. Frst, usng Algorthm 2, we determne (R x 2, Q x 2 ). If Q x 2 = S(x 2 ) w w j x + ε, where ε s a very small number, we letπ 2 (R x 2, q x 2, x) snce x cannot be optmum for P2 n ths case based on Property 3. Else usng Property 2 we determne q x 2 and calculateπ 2 (R x 2, q x 2, x). After that we seek the best neghbor of x, where a neghbor of x s another bnary n vectors that dffers from x wth a sngle entry. Partcularly, x has n neghbors and we defne the th neghbor of x, denoted by x [] by lettng x [] = 1 f x = 0 otherwse, and x j [j] = x j j s {}. Smlar to the calculaton of Π 2 (R 2 x, q 2 x, x)we calculate Π 2 (R x [], q x [], x [] ) for each usng Algorthm 2 and Propertes 2 and 3. If the best neghbor of x s worse than x,.e., f Π 2 (R 2 x, q 2 x, x) mn {F 2 (R x [], q x [], x [] )} x defnes a local optmum and we stop the search. On the other hand, f Π 2 (R 2 x, q 2 x, x) > mn {F 2 (R x [] defne a new x, such that x = argmn {F 2 (R x [], q x [], x [] )}, we, q x [], x [] )} and contnuo the local search process. Now let x be the local optmum reached va the local search when the local search starts wth x. we repeat the local search startng wth m dfferent x vectors to avod returnng a bad qualty local optmum. The best local optmum returned s accepted as the soluton of P2. Algorthm 3 states ths local search heurstc wth multple startng solutons.

45 36 Algorthm 3 solvng P2 0. Let x 1, x 2,, x m be m gven startng x vectors. 1. For l = 1: m 2. Let x = x l 3. Calculate (R x 2, Q x 2 ) usng Algorthm 2 4. If Q 2 x = S(x 2 ) w w j x + ε, Π 2 (R x 2, q x 2, x) 5. Else, determne q x 2 usng Property 2 and calculate Π 2 (R x 2, q x 2, x) 6. For = 1: n 7. Let x [] = x and x [] = 1 f x = 0 and x [] = 0 x = 1 8. Calculate (R 2 x [], Q 2 x []) usng Algorthm If Q x [] = w S(x [] ) w j x [] + ε, Π2 (R 2 x [], q 2 x [], x [] ) Else, determne q x [] usng Property 2 and calculate Π 2 (R 2 x [], q 2 x [], x [] ) 11. End 12. IfΠ 2 (R 2 x, q 2 x, x) > mn {Π 2 (R 2 x [], q 2 x [], x [] )},x = argmn {Π 2 (R 2 x [], q 2 x [], x [] )} and go to Else x l = x 14. End 15. Return (R 2, q 2, x 2 ) = argmn {Π 2 (R 2 x l, q 2 x l, x l )} 4.3 SOLUTION OF SEQUENTIAL DELIVERY In ths secton, we propose an algorthm smlar to Algorthm 3 to solve Model P3. However, due to the defnton of the expected number of shortages n each shortage perod, determnng re-order pont and order quanttes from the selected supplers s more complex. Partcularly, gven x let (R 3 x, q 3 x ) denote a mnmzer of Π 3 (R, q, x x) subject to q x w s. We use IPM to determne (R 3 x, q 3 x ) for a gven x. then, smlar to Algorthm 3, a local search s used to fnd

46 37 the selected supplers. Algorthm 4 states ths local search heurstc wth multple startng solutons, where IPM s used for solvng the subproblems. Algorthm 4 solvng P3 0. Let x 1, x 2,, x m be m gven startng x vectors. 1. For l = 1: m 2. Let x = x l 3. Determne (R x 3, q x 3 ) usng IPM 4. For = 1: n 5. Let x [] = x and x [] = 1 f x = 0 and x [] = 0 x = 1 6. Determne (R 3 x [] 7. End, Q 3 x []) usng IPM 8. IfΠ 3 (R 3 x, q 3 x, x) > mn {Π 3 (R 3 x [], q 3 x [], x [] )},x = argmn {Π 3 (R 3 x [], q 3 x [], x [] )} and go to Else x l = x 10. End 11. Return (R 3, q 3, x 3 ) = argmn {Π 3 (R 3 x l, q 3 x l, x l )}

47 38 5. COMPARISONS OF THE ORDERING POLICIES In ths secton, our focus s to dscuss how the three orderng polces modeled compare to each other n terms of not only expected total costs but also expected carbon emssons per unt tme. In partcular, whle envronmental regulatons are becomng more common worldwde, there are stll many countres that do not have natonally legslated envronmental regulatons. For nstance, there s no federal envronmental regulaton n the U.S. However, envronmental regulatons are not the only motvaton for ompanes to green ther operatons. As t dscussed n surveys by Loebch et.al. (2011) and Kron et.al. (2012), recent motvaton for companes become greener s rather to stay compettve n the market consderng the ncreasng awareness of consumers on envronment and/or brand mage. Therefore, we next compare the three orderng polces n terms of not only expected total cost per unt tme after carbon tradng (P j (R j,q j, x j ), denoted as P j and the expected costs per unt tme (C j (R j,q j, x j ), denoted as C j )but also expected carbon emsson per unt tme (E j (R j,q j, x j ), denoted as E j )where j =1,2,3defnes sngle sourcng (SS), sequental orderng (SO), and sequental delvery (SD), respectvely. Based on the comparson of the total expected costs per unt tme after carbon tradng (.e., the expected costs per unt tme plus the expected costs due to buyng carbon allowances or mnus the expected revenues due to sellng excess carbon emsson), one can note that P 1 ³ P 2 and P 1 ³ P 3. Ths smply follows from the fact that the optmal soluton of model P1 s a feasble soluton for model P2 and P3 for any gven settng. Therefore, under carbon tradng polcy, the retaler wll not prefer sngle sourcng unless other crtera are regarded. Furthermore, t can be notced that when 2 3 å = å x = å x =1,.e., the retaler chooses to order from a sngle suppler x 1 ÎS ÎS ÎS even f order splttng s allowed P 1 = P 2 = P 3,. Nevertheless, f C j and E j are also consdered n comparng sngle sourcng to sequental orderng and sequental delvery, the followng cases are possble:

48 39 SS vs. SO SS vs. SD Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 P 1 > P 2 P 1 > P 2 P 1 > P 2 P 1 > P 3 P 1 > P 3 P 1 > P 3 C 1 > C 2 C 1 > C 2 C 1 < C 2 C 1 > C 3 C 1 > C 3 C 1 < C 3 E 1 > E 2 E 1 < E 2 E 1 > E 2 E 1 > E 3 E 1 < E 3 E 1 > E 3 Specally, n cases 1 and 3, sequental orderng not only reduces expected costs after carbon tradng but also expected carbon emsson compared to sngle sourcng. Smlarly, n case 4 and 6, sequental delvery not only reduces expected costs after carbon tradng but also expected carbon emsson compared to sngle sourcng. That s, multple sourcng can result n cheaper as well as greener nventory control for a company. On the other hand, n case 2 and 5, whle the retaler would prefer sequental orderng and sequental delvery, respectvely based on the expected costs after carbon tradng, expected carbon emsson are lower wth sngle sourcng. The nsghts of these cases are as follows. In absence of carbon tradng, f the retaler tres to mnmze not only expected costs but also carbon emsson (.e., a mult-objectve nventory control model smlar to the one gven n Bouchery et.al., 2012 s used by the retaler), dependng on the retalors cost and emsson targets, the retaler can prefer SS over SO and SS over SD or vce versa. On the other hand, when the two delvery structures n case of order splttng are compared n terms of total expected costs per unt tme after carbon tradng, one cannot guarantee that the retaler wll prefer one polcy over the other for any gven settng. That s, t s the both possble to have ( R, q, x ) ( R, q, x ) and ( R, q, x ) ( R, q, x ) dependng on demand, retaler, and supplers characterstcs as well as regulaton parameters. Ths then mples that, under carbon tradng, sequental orderng can be a better polcy compared to sequental delvery, whch s the delvery structure generally assumed n the ntegrated stochastc nventory control and suppler selecton models. We note that ths result readly apples for the case when the retaler does not operate under any envronmental regulaton (.e., when b = 0 );

49 40 hence, consderng sequental orderng as an alternatve to sequental delvery can result n substantal cost savngs for a retaler. Nevertheless, f C j and E j are also consdered n comparng sequental orderng to sequental delvery, the followng case are possble: SO vs. SD SD vs. SO Case 7 Case 8 Case 9 Case 10 Case 11 Case 12 P 2 < P 3 P 2 < P 3 P 2 < P 3 P 3 < P 2 P 3 < P 2 P 3 < P 2 C 2 < C 3 C 2 < C 3 C 2 > C 3 C 3 < C 2 C 3 < C 2 C 3 > C 2 E 2 < E 3 E 2 > E 3 E 2 < E 3 E 3 < E 2 E 3 > E 2 E 3 < E 2 In case 7 and 9, sequental orderng not only reduces costs after carbon tradng but also expected carbon emsson compared to sequental delvery. Smlarly, n case 10 and 12, sequental delvery not only reduce1s` expected costs after carbon tradng but also expected carbon emssons compared to sequental orderng. That s, by consderng dfferent delvery structures n case of order splttng, the retaler can lower hs/her costs as well as carbon emssons. On the other hand, n case 8, whle the retaler would prefer sequental orderng over sequental delvery based on the expected costs after carbon tradng, expected carbon emsson are lower wth sequental delvery. Smlarly, n case 11, whle the retaler would prefer sequental delvery over sequental orderng based on the expected costs after carbon tradng, expected carbon emssons are lower wth sequental orderng. Those observatons suggest that, n case there s no envronmental regulaton n place, the retalers preference for delvery structure depends on the retalers cost and emsson targets.

50 41 6. NUMERICAL STUDIES Ths secton focus on the two sets of numercal studes: () effcency of the algorthms proposed and () effects of the changes n suppler capactes and suppler lead tmes on the retalers expected costs, carbon emssons, and total costs. We do not evaluate how the changes n the carbon tradng prce and carbon cap wll affect the retalers expected costs and carbon emssons per unt tme one can easly dscuss 21that the models presented n ths study wll mply observatons smlar to the ones gven for the EOQ model n Hua et.al. (2011) and Chen et.al. (2013). Our focus s rather on the effects of multple sourcng and delvery structures. The tools provded n ths study can be used for analyzng the effects of regulaton parameters as well as the retaler parameters such as nventory related costs and emssons and demand characterstcs. All of the algorthms are coded n Matlab2014a and the problem nstances solved usng a personal computer wth 8GB RAM and 3.30GHz processor. The tables referred n ths secton are gven n the Appendx. In the followng analyss we assume that the retaler operates under a carbon tradng regulaton wth carbon tradng prce b = 0.1and F = 20,000 carbon cap. Unless stated otherwse, the followng values are used for the other problem parameters to generate problem nstances (smlar values are used for nventory control models wth envronmental consderatons, see, e.g., Hua et al., 2011, Chen et al., 2013, Toptal et al., 2014, Konur, 2014, and Konur and Schaefer, 2014):Retaler parameters: the retalers demand per unt tme (year) s normally dstrbuted wth mean l =10,000 and standard devaton u =1,000 and t assumed ~ ˆ ~ h U[2,4], h U[0.5,1], A U[50,100], and  U[25,50], where U[a, b] defnes a contnuous unform dstrbuton wthn the range [a, b]. Suppler parameters: Gven supplers, t s assumed that s rounded to the nearest multpler of 10 for practcal purposes.

51 EFFICIENCY OF THE ALGORITHMS Recall that algorthms 1 and 2 are stated as alternatves to IPM for solvng problems P1-I and P2-x, respectvely, whch are the subproblems analyzed n models P1 and P2. We, therefore, frst compare algorthms 1 and 2 to IPM. To compare algorthm 1 to IPM, for each n {3,6,9,12,15}, we randomly generate 10 problem nstances and solve each problem nstance n tmes, one wth each n for R *,q * and p x (R x *,q x * ) suppler as the sngle source of supply, usng both methods. Table 1 documents the averages over all problems solved wth each values along wth the computatonal tmes n seconds (denoted as CPU). As can be seen n table 1, algorthm 1 and IPM fnd the same soluton for all problem nstances solved. Furthermore, algorthm 1 s more effcent computatonally. Thus, we use algorthm 1 to solve the retalers orderng decsons for a gven suppler n case of sngle sourcng. To compare algorthm 2 to IPM, for each n = {3,6,9,12,15}, we randomly generate 10 problem nstances and solve each problem nstance wth n randomly generated n-bnary x vectors. Smlar to table 1, table 2 documents the average over all problems solved wth each R 2* x,q 2* x and g x (R 2* x,q 2* x ) values along wth the computatonal tmes n seconds. One can observe from table 2 that algorthm 2 to IPM fnd the same solutons for all the problem nstances solved and algorthm 2 requres less than half of the soluton tme requred by IPM on average. Therefore, we used algorthm 2 then property 2 to determne the retalers orderng decson for gven suppler selectons n case of sequental orderng. Recall that algorthm 3 and 4 are local search heurstc methods proposed for models P2 and P3 respectvely (for model P1, we solve each of the n optons wth algorthm 1). Total enumeraton, where each of the possble bnary n-vector s evaluated, can be used as an alternatve method to algorthms 3 and 4 for solvng problems P2 and P3. Therefore, we compare algorthms 3 and 4 for solvng problems P2 and P3. Therefore, we compare algorthms 3 and 4 to total enumeraton.

52 43 To compare algorthm 3 to total enumeraton, for each n {3,6,9,12,15}, we randomly generate 10 problem nstances and solve each problem nstance usng both methods. Table 3 documents the averages over all problems solved wth each n n for s 2 x for (.e., number of select supplers wth sequental orderng), s R, and ( R, q, x ) (denoted as q 2 2 ) values along wth the computatonal tmes n seconds. It can be seen n table 3 that algorthm 3 s able to fnd the optmal soluton n all of the problem nstances solved. Furthermore, whle Algorthm 3 requres less than a second to solve the problem nstances, total enumeraton requres more than 800 seconds on average. That s, algorthm 3 can fnd the optmal solutons very effcently. Therefore, n analyss (), we use algorthm 3 to solve model P2. To compare algorthm 4 to total enumeraton, for each n = {3,6,9,12,15} We randomly generate 10 problem nstances and solve each problem nstance usng both methods. Smlar to table 3, table 4 documents the average over all problems solved wth each n for s 3 x (.e., number of selected supplers wth sequental delvery), s the total order quantty wth sequental delvery), R3,and P 3 (R 3,q 3, x 3 ) (denoted as P 3 ) values along wth the computatonal tmes n seconds. One can observe that algorthm 4 fnds the same solutons wth total enumeraton. Furthermore, whle for smaller n values (when n=3 and n-6) algorthm 4 takes longer tme to solve the problem nstances on average (specfcally, due to evaluatng same x vectors more than once), for larger n values, total enumeraton requres longer computatonal tmes on average. In partcular, for n=12 and n=15, algorthm 4 s drastcally more effcent n terms of computaton tmes compared to total enumeraton. Based on these observatons, n analyss (), we use algorthm 4 to solve model P3 nstead of total enumeraton. q 3 (.e.,

53 EFFECTS OF SUPPLIERS In ths secton, we numercally analyze how the multple sourcng affects ther retalers nventory control and suppler selecton decsons as well as hs/her expected costs, carbon emssons, and total costs per unt tme wth carbon tradng under each of the orderng polces consdered. Specfcally, we focus on llustratng the changes n the number of selected supplers, the total order quantty, and the re-order pont ( R j ) as well as the expected costs per unt tme ( (C j (R j,q j, x j ), denoted as C j ),expected carbon emsson per unt tme ( (E j (R j,q j, x j ), denoted as E j )and expected total cost per unt tme after carbon tradng ( (P j (R j,q j, x j ) denoted as P j ) as the suppler capactes ( w ) S and lead tmes ( ) ncrease for j =1,2,3. Note that under sngle sourcng x 1 1n all of the problem nstances solved. To analyze the effect of suppler capactes, wth each n={3,6,9}, we randomly generate 10 problem nstances wth the range gven for w values n table 5 and 6. Table 5 documents the averages overall 30 problem nstances solved wth each w range for S j x, S q, and R j for j 1,2, 3. Smlarly table 6 documents the average over all j 30 problem nstances solved wthn each w range for C j, E j and Õ j for j =1,2,3. We have the followng observatons based on table 5 and 6. As expected and can observed n Table 5, the number of selected supplers (except wth sngle sourcng) and the re-order pont tend to decrease whle the total order quantty tends to ncrease wth an ncrease n the supplers capactes wth any orderng polcy. Partcularly, the retaler wll prefer to use fewer supplers n case the supplers capactes are larger. Furthermore, snce the supplers have larger capactes, the retaler can ncrease hs/her order quantty whle avodng the extra setup costs and carbon emssons (t s even possble to decrease setup costs and carbon emssons whle the order quantty ncreases as the retaler mght prefer fewer supplers wth larger cumulatve capacty). Ths ncrease n the order quanttes, n turn, leads to lower re-order

54 45 ponts. We note that the retaler wll not contnuously ncrease hs/her order quantty wth ncreasng suppler capactes snce t wll not be costly justfable to order more than needed. As can be observed n Table 6, wth any orderng polcy, the retaler s expected costs, carbon emssons, and total costs per unt tme after carbon tradng decrease wth an ncrease n the supplers capactes. These observatons are expected snce an ncrease n the suppler capactes wthout an ncrease n the suppler setup costs and carbon emssons mply cheaper and cleaner transportaton capacty; hence, both expected costs and carbon emssons per unt tme decrease. Ths then leads to decreased total expected costs per unt tme after carbon tradng. To analyze the effects of suppler lead tmes, wth each n={3,6,9}, we randomly generate 10 problem nstances wth the ranges gven for t values n tables 7 and 8. Table 7 documents the averages over all 30 problem nstances solved wthn each t range for j j å x ÎS q, å ÎS, and R j for j =1,2,3. Smlarly table 8 documents the averages over all 30 problem nstances solved wthn each range for each t range for C j, E j and Õ j for j =1,2,3. We have the followng observatons based on table 7 and 8. As expected and can observed n Table 7, the retaler s re-order pont ncreases whle the number of selected supplers (except sngle sourcng) and the total order quantty do not follow an ncreasng or decreasng pattern as the supplers lead tmes ncrease wth any orderng polcy. As can be seen n Table 8, the retaler s expected costs, carbon emssons, and total costs after carbon tradng per unt tme show nether an ncreasng nor a decreasng trend wth ncreased suppler lead tmes on average wth any orderng polcy. Ths follows from the fact that by ncreasng hs/her re-order pont, and selectng supplers and order quanttes accordngly, the retaler can avod the drawbacks of longer lead tmes.

55 46 7. CONCLUSIONS Ths paper studes an ntegrated stochastc nventory control and suppler selecton model under envronmental regulatons. In partcular, we formulate and analyze a contnuous revew nventory control model under carbon tradng regulaton wth three orderng polces: sngle sourcng, sequental orderng, and sequental delvery. A soluton method s dscussed for each polcy. A comparson of these polces In terms of ther economc and envronmental performances s provded. A set of numercal studes s conducted to demonstrate the effcency of the soluton methods proposed. Further numercal studes llustrate the effects of suppler capactes and lead tmes on the retaler s orderng and suppler selecton decsons as well as costs and carbon emssons. The followng results are documented. In case the retaler solely has economc objectves, preferrng multple sourcng nstead of sngle sourcng wll reduce the total expected costs after carbon tradng. Furthermore, t s also possble that multple sourcng wll reduce expected carbon emssons. However, t mght be the case that expected carbon emssons are lower wth sngle sourcng; therefore, n case the retaler has economc as well as envronmental objectves, sngle sourcng can be preferred over multple sourcng dependng on the retaler s economc and envronmental targets. Furthermore, n case the retaler solely has economc objectves, any of the delvery structures consdered for order splttng can be preferred dependng on the settngs. It s possble that sequental orderng (sequental delvery) reduces not only expected costs but also carbon emssons compared to sequental delvery (sequental orderng). Nevertheless, t mght be the case that whle one delvery structure outperforms the other economcally, t can be outperformed by the other envronmentally. The contrbutons of ths study are as follows. An ntegrated contnuous revew nventory control and suppler selecton model s analyzed under envronmental regulatons wth three orderng polces. We economcally and envronmentally compare sngle sourcng to multple sourcng and, sequental orderng to sequental delvery. Even

56 47 wthout envronmental aspects of the models consdered, t s a contrbuton of ths study that sequental orderng s dscussed to be a potentally better delvery structure. The models enable numercal analyss of the suppler capactes and delvery lead tmes on a retaler s orderng decsons, suppler selecton decsons, expected costs, and expected carbon emssons wth each orderng polcy. Future research drectons nclude consderng smlar models wth stochastc delvery lead tmes and analyze the effects of the varablty of the lead tmes economcally and envronmentally. Furthermore, the lterature revew reveals that there are a lmted number of studes that nvestgate mult-tem nventory control systems wth envronmental consderatons. Economc and envronmental analyses of mult-tem nventory systems under determnstc and stochastc demand wth dfferent delvery structures reman as future research questons.

57 48 SECTION 3. CONCLUSIONS In case the retaler solely has economc objectves, preferrng multple sourcng nstead of sngle sourcng wll reduce the total expected costs after carbon tradng. Furthermore, t s also possble that multple sourcng wll reduce expected carbon emssons. However, t mght be the case that expected carbon emssons are lower wth sngle sourcng; therefore, n case the retaler has economc as well as envronmental objectves, sngle sourcng can be preferred over multple sourcng dependng on the retaler s economc and envronmental targets. Furthermore, n case the retaler solely has economc objectves, any of the delvery structures consdered for order splttng can be preferred dependng on the settngs. It s possble that sequental orderng (sequental delvery) reduces not only expected costs but also carbon emssons compared to sequental delvery (sequental orderng). Nevertheless, t mght be the case that whle one delvery structure outperforms the other economcally, t can be outperformed by the other envronmentally. These are some of the contrbutons of ths study. An ntegrated contnuous revew nventory control and suppler selecton model s analyzed under envronmental regulatons wth three orderng polces. We economcally and envronmentally compare sngle sourcng to multple sourcng and, sequental orderng to sequental delvery. Even wthout envronmental aspects of the models consdered, t s a contrbuton of ths study that sequental orderng s dscussed to be a potentally better delvery structure. The models enable numercal analyss of the suppler capactes and delvery lead tmes on a retaler s orderng decsons, suppler selecton decsons, expected costs, and expected carbon emssons wth each orderng polcy. Future research drectons nclude consderng smlar models wth stochastc delvery lead tmes and analyze the effects of the varablty of the lead tmes economcally and envronmentally. Furthermore, the lterature revew reveals that there are a lmted number of studes that nvestgate mult-tem nventory control systems wth

58 49 envronmental consderatons. Economc and envronmental analyses of mult-tem nventory systems under determnstc and stochastc demand wth dfferent delvery structures reman as future research questons.

59 APPENDIX A. NOTATION AND POSSIBLE METRICS

60 51

61 APPENDIX B. PROOFS FOR PROPERTIES IN SECTION 4

62 53

63 APPENDIX C. TABLES IN SECTION 6

64 55

65 56

66 57

67 58