Non-stationary stochastic inventory lot-sizing with emission and service level constraints in a carbon cap-and-trade system

Transcription

1 Loughborough Unversty Insttutonal Repostory Non-statonary stochastc nventory lot-szng wth emsson and servce level constrants n a carbon cap-and-trade system Ths tem was submtted to Loughborough Unversty's Insttutonal Repostory by the/an author. Ctaton: PUROHIT, A.K.... et al., Non-statonary stochastc nventory lot-szng wth emsson and servce level constrants n a carbon cap-and-trade system. Journal of Cleaner Producton, 113, pp Addtonal Informaton: Ths paper was accepted for publcaton n the journal Journal of Cleaner Producton and the defntve publshed verson s avalable at Metadata Record: Verson: Accepted for publcaton Publsher: c Elsever Rghts: Ths work s made avalable accordng to the condtons of the Creatve Commons Attrbuton-NonCommercal-NoDervatves 4.0 Internatonal (CC BY-NC-ND 4.0) lcence. Full detals of ths lcence are avalable at: Please cte the publshed verson.

2 Non-Statonary Stochastc Inventory Lot-Szng wth Emsson and Servce Level Constrants n a Carbon Cap-and-Trade System Arun Kr. Puroht a *, Rav Shankar a, Prasanta Kumar Dey b, Alok Choudhary c a Department of Management Studes, Indan Insttute of Technology Delh, New Delh, Inda b Aston Busness School, Operatons & Informaton Management Group, Aston Unversty, Brmngham, UK c School of Busness and Economcs, Management Scence and Operatons Management Group, Loughborough Unversty, Lecestershre, UK Abstract Frms worldwde are takng major ntatves to reduce the carbon footprnt of ther supply chans n response to the growng governmental and consumer pressures. In real lfe, these supply chans face stochastc and non-statonary demand but most of the studes on nventory lot-szng problem wth emsson concerns consder determnstc demand. In ths paper, we study the nventory lot-szng problem under non-statonary stochastc demand condton wth emsson and cycle servce level constrants consderng carbon cap-and-trade regulatory mechansm. Usng a mxed nteger lnear programmng model, ths paper ams to nvestgate the effects of emsson parameters, product- and system-related features on the supply chan performance through extensve computatonal experments to cover general type busness settngs and not a specfc scenaro. Results show that cycle servce level and demand coeffcent of varaton have sgnfcant mpacts on total cost and emsson rrespectve of level of demand varablty whle the mpact of product s demand pattern s sgnfcant only at lower level of demand varablty. Fnally, results also show that ncreasng value of carbon prce reduces total cost, total emsson and total nventory and the scope of emsson reducton by ncreasng carbon prce s greater at hgher levels of cycle servce level and demand coeffcent of varaton.the analyss of results helps supply chan managers to take rght decson n dfferent demand and servce level stuatons. Keywords: Carbon emsson; Lot-szng; Non-statonary stochastc demand; Cycle servce level; Carbon Cap-and-Trade System. 1

3 1. Introducton Across the world, frms are under ntense pressure from ther major stakeholders ncludng governments and customers, to cut carbon emssons generated durng conduct of ther busness. Consderng the detrmental mpacts of carbon emsson such as global warmng and clmate change, many countres are enforcng dfferent carbon regulatory mechansms such as carbon cap-and-trade and carbon tax or envronmental standards such as ISO 14000, to control the carbon emsson. Employng more energy effcent machnes/equpments and facltes, and usng energy generated by renewable sources are also commonly used but costler solutons to reduce emssons.but, gven the mpact of supply chan decsons on carbon emssons, a potental soluton could be acheved by ncorporatng emsson concerns n the models to optmze these decsons (Husngh et al., 2015; Govndan et al., 2014; Toptal et al., 2014; Benjaafar et al., 2013).In addton, modelng of these supply chans consderng of stochastcty and dynamc nature of demand process would provde results applcable to more realstc stuatons. Ths paper deals wth the nventory lot-szng problem of a frm under non-statonary stochastc demand wth carbon emsson constrants. We consder cycle servce level as customer servce measure snce the demand process s not determnstc. Wth the help of a mxed nteger lnear programmng model we determne the optmal replenshment schedule that mnmzes the system-wde cost, n advance of the plannng horzon. The model explctly accounts for the emssons generated due to purchasng, orderng and storage actvtes along wth correspondng costs. Under the carbon cap-and-trade regulatory mechansm, the present study analyses the mpacts of emsson parameters such as orderng emsson, carbon prce and carbon cap, and product- and system- related features such as cycle servce level, orderng cost, demand pattern, level of demand varablty and demand coeffcent of varaton through extensve computatonal expermentatons on the model. We consder carbon emsson cap-and-trade polcy snce t s a market-based mechansm and generally accepted as an effectve soluton n curbng emsson (Hua et al., 2011). Moreover, the other regulatory mechansms such as carbon tax or envronmental standard are not market controlled, but government controlled (Dobos, 2005). The European Unon Emssons Tradng System (EU-ETS) s one of the frst and bggest emsson tradng systems whch may lower carbon emssons by 21% than the 2005 levels by 2020 (Jaber et al., 2013). The results of the study cover general type busness settngs and not a specfc scenaro. Furthermore, we analyze the results to help supply chan manager to take rght decson under a gven busness settng. 2

4 The rest of the paper s organzed as follows. In Secton 2 we present the extant lterature revew followed by Secton 3 n whch we gve detals of methodology, problem statement and mathematcal formulaton. In Secton 4 we present expermental desgn followed by detaled analyss of expermental results. Fnally we dscuss the results n Secton 5 and conclude the paper wth future research scopen Secton Lterature Revew The extant lterature on sustanable supply chan management mostly focuses on product recyclng or re-use. The ssue of sustanablty has been addressed n several studes n the context of reverse or closed-loop supply chan, but only few papers apply the sustanablty concerns n forward supply chans usng quanttatve modelng (Seurng, 2013; Gonzalez et al., 2013).Many studes n ths area conclude that supply chan decsons have sgnfcant mpact on carbon emssons(husngh et al., 2015; Govndan et al., 2014; Toptal et al., 2014; Benjaafar et al., 2013). The study of Plambeck (2012) suggests that tremendous changes n supply chan desgn and operaton are requred to avert clmate change and explans through case studes that how carbon emssons can be reduced proftably n supply chans. Furthermore, a few studes ntegrate carbon emssons concerns n forward supply chan problems such as network desgn, producton plannng, product mx and suppler selecton(jn et al., 2014; Zhang and Xu, 2013; Chaabane et al., 2012; Shaw et al., 2012; Ba and Sarks, 2010; Letmathe and Balakrshnan, 2005; Gong and Zhou, 2013) and others nvestgate ts mpact on supply chan structures and transportaton mode selecton (Cachon, 2011; Hoen et al., 2014). Some studes ncorporate emsson concerns n nventory lot-szng decson whchnot only nfluences total system-wde cost,customer servce levels and carbon emssons sgnfcantly, but also affects other decsons such as packagng, waste and locaton (Bonney and Jaber, 2011). Most of the studes n the area of nventory lot-szng wth carbon emssons concerns, consder contnuous demand (He et al., 2014; Hua et al., 2011; Wahab et al., 2011; Arslan and Turkay, 2010; Bouchery et al., 2012; Toptal et al., 2013; Chen et al., 2013; Battn et. al., 2014) or tmevaryng determnstc demand (Kantas et al., 2015; Abs et al., 2013; Helmrch et al., 2012). Only the work of Song and Leng, (2011) consders stochastc contnuous demand. In real lfe, demand process s non-statonary stochastc type due to shrnkng product lfe cycles, frequent new product launches and use of promoton schemes (Martel et al., 1995; Neale and Wllems, 2009; Choudhary and Shankar, 2014; Choudhary and Shankar, 2015). But t has been gnored whle addressng the emsson concerns n nventory lot-szng decson. 3

5 In ths lne, our research ams to model the nventory lot-szng problem wth emsson and cycle servce level constrants. Unlke Benjaafar et al. (2013) who address a smple lot-szng problem wth determnstc demand, ths paper attempts to address sngle tem nventory lot-szng problem under non-statonary stochastc demand. 3. Methodology 3.1 Problem statement We consder a supply chan as shown n Fgure 1, n whch a buyer frm fulflls the dynamc and uncertan demands of a sngle product over a plannng horzon of T perods wth a cycle servce level of α. The plannng horzon wth all possble order cycles has been depcted n Fgure 1 (nsde the box) where tme lne shows startng and endng of adjacent perods wth thck dark crcles. An arc represents an order cycle where startng of the frst perod s connected to the endng of last perod of the cycle. The random demands d n dscrete tme perods {1,,T}aremutually ndependent and normal dstrbuted wth known means and standard devatonsthat may vary over tme. We assume that these quanttes are the outcome of a forecastng procedure. Moreover, we account for carbon emssons generated by dfferent actvtes of the frm such as orderng (e.g. transportaton emsson), holdng (emssons due to energy spent on storage) and purchasng (emssons due to handlng). We consder a carbon capand-trade emsson regulatory mechansm n whch the total emssons due to all actvtes over the plannng horzon cannot exceed a carbon cap as mposed by the regulator. The frm can buy or sell the balance carbon credts from the open carbon trade market at a gven carbon prce f the generated emssons are greater or less than the carbon cap. [Insert Fgure 1] The frm needs to decde the replenshment schedule that mnmzes the total costs whch nclude orderng and holdng costs along wth the cost (revenue) of purchasng (sellng) balance carbon credts. As the quantfcaton of shortage penalty cost s very dffcult, we assume that the frm meets target cycle servce level and backorders the unflled demand. The cycle servce level specfes a mnmum probablty α whch ensures that at the end of every perod the net nventory wll not be negatve. The near optmal soluton for stochastc dynamc lot-sze problem has been gven n Bookbnder and Tan (1988). They consder three strateges of lot-szng decson, named statc uncertanty, dynamc uncertanty and statc-dynamc uncertanty. The replenshment tmngs and order szes are decded at the begnnng of the plannng horzon n case of statc uncertanty strategy whereas these decsons are taken n every perod n case of dynamc 4

6 uncertanty. We assume statc-dynamc uncertanty strategy n ths paper where replenshment tmngs and correspondng stock levels are fxed at begnnng of the plannng horzon and the order szes for comng perods are determned after realzaton of the demands of prevous perods. The replenshment tmngs and order szes are crucal n achevng desred cycle servce level and determnng the total costs and emssons generated. Followng the mxed nteger programmng formulaton proposed by Tarm and Kngsman (2004) we develop the non-statonary stochastc lot-szng model by ntegratng carbon emsson and cycle servce level constrants, as descrbed n the followng secton. The model selects the seres of those order cycles from all possble T ( T +1) / 2 order cycles n a T perods plannng horzon problem whch mnmzes the total cost and satsfes the carbon emsson constrant.furthermore, we use full factoral expermental desgn consderng emsson parameters such as orderng emsson, carbon prce and carbon cap, and product- and system- related features such as cycle servce level, orderng cost, demand pattern, level of demand varablty and demand coeffcent of varaton. Through extensve computatonal expermentatons usng the developed model, we produce results for all expermental settngs. For each expermental set-up, t s also requred to calculate some quanttes offlne, as descrbed later n the llustraton secton. The results are further analyzed to dentfy the mpacts of problem parameters on supply chan performance n terms of costs and emssons. 3.2 Model development In the followng paragraphs, we explan the detals of objectve functon and constrants of the model. The model determnes the optmum replenshment schedule, defned by a seres of adjacent replenshment order cycles whch mnmzes the total cost. 1. Objectve functon The objectve functon n Equaton (1) mnmzes the total cost over the plannng horzon of T perods. The frst term represents the orderng cost where o s orderng cost per order and Z s a bnary varable ndcatng whether the order n perod s placed or not. The second term ndcates holdng cost of the expected net nventory E(I ) n each perod where h s holdng cost per tem per perod. The purchasng cost s calculated n the thrd term where v and E(X ) are unt purchasng costand expected lot-sze ordered n perod, respectvely. The fourth term represents cost or revenue related to carbon credt purchased (ee pp ) orsold (ee nn ) at a carbon prce p n the open carbon trade market. 5

7 Mn = T p n [ oz + he I ) + ve( X ) + p( e e )] = 1 ( (1) 2. Inventory balance constrants The Equatons (2) calculate expected end-perod nventory by subtractng expected demand from the expected order-up-to-levele(r ) of the perod. The Equatons (3) calculate expected lotszee(x ) n any perod whch s the dfference of order-up-to-level of the present perod and net nventory level of the prevous perod. The Constrants (4) ensure that the expected order-upto-level of a perod must be greater than the endng nventory of prevous perods. E ) = E( R ) E( d ), = 1,..., T (2) ( I E X ) E( R ) E( I ), = 1,..., T (3) ( = 1 E R ) E( I ) = 1,..., T (4) ( 1 3. Orderng cost chargng constrants The Constrants (5) ensure chargng of orderng cost f an order s placed n perod where M s a large number. E( X ) MZ, = 1,..., T (5) 4.Cycle servce level constrants The Constrants (6) ensure achevng a cycle servce level (α) n every order cycle, endng n 1 perod and has a cycle length of j perods. The term G ( ) s the nverse of d j+ + j+ 1 d d α cumulatve probablty dstrbuton functon and represents order-up-to-level at the begnnng of the order cycle. The bnary varable P j equals one f the order cycle endng n perod and havng a cycle length of j perods s carred out, otherwse t s zero. E 1 ( I ) Gd d d E dk P j j ( α ) ( ) j, 1,..., T 1 2 j= 1 k= j+ 1 + = (6) 5. Constrants to unquely dentfy optmum replenshment schedule Wth the help of Equatons (7) and Constrants (8), the bnary varables P j dentfy the optmum replenshment schedule unquely. They select the adjacent order cycles from T ( T +1) / 2 number of possble order cycles for a T perods plannng horzon so that total costs are mnmzed. The Equatons (7) ensure termnaton of only one replenshment cycle n a perod whle t may start from any perod before perod. 6

8 j= 1 P = 1, = 1,..., T (7) j P j Z j+ 1 Z, = 1,..., T j = 1,..., (8) k k= j Emsson constrants Constrant (9) controls carbon emssons under carbon cap-and-trade regulatory mechansm appled over the plannng horzon. The frst term n Constrant (9) accounts for carbon emssons produced due to orderng where o e s the orderng related emsson per order.the second and thrd terms account for varable and storage related emssons where v e and h e denote varable emsson per tem and holdng emsson per tem per perod respectvely. On the rght hand sde of the Constrant (9), the frst term denotes carbon cap CAP horzon appled over the horzon whereas the second term ncludes ee pp and ee nn denotng the amount of carbon credts the organzaton buys or sells, respectvely n any perod. Buyng or sellng carbon credts help n relaxng the carbon cap mposed over the horzon. Constrants (10) correspond to bnary and non-negatvty constrants T T p n ( oez + vee X ) + he E( I )) CAPhorzon + ( e e ) = 1 (, (9) = 1 P j, Z ϵ {0, 1}, E(I ), E(R ), E(X ) 0 ϵ [1, T], jϵ [1, ], (10) 3.3An llustraton The soluton of nteger lnear programmng model, as descrbed by expressons (1) - (10), 1 requres offlne calculaton ofnverse cumulatve dstrbuton functon G ( ) for all d j+ 1+ d α j+ d possble order cycles. In ths secton, ths offlne calculaton procedure s explaned. Through a numercal example, as detaled n next paragraph, we also explan the steps nvolved n solvng the nteger lnear programmng model along wth ts theoretcal framework. A buyer frm faces random demands of a product over a plannng horzon of sx perods and wants to fulfll t wth a cycle servce level of 90%. The demand n each perod s assumed to be mutually ndependent and normally dstrbuted wth mean d and a coeffcent of varaton (CV) of 0.3. Table 1 shows the expected mean demand and ts standard devaton n each perod. We assume holdng cost and orderng cost to be 1 and200respectvely. Replenshment lead tme and unt purchasng cost are gnored. Let emsson parameter such as orderng emsson, holdng 2 7

9 emsson and untary carbon emsson are 400, 1 and 2 respectvely. We also assume carbon prce to be 5 and carbon cap to be [Insert Table1] Let an order cycle endng n perod and havng a cycle length of j perods s carred out, therefore t starts n perod ( j + 1), as shown n Fgure 1. It s requred to fnd the order-up-tolevel R j + 1 at the begnnng of the cycle so as to acheve targeted cycle servce level of α. As suggested n Tarm and Kngsman (2004), the order-up-to-level R j + 1 s gven by thenverse 1 cumulatve dstrbuton functon Gd... ( α) j+ 1+ d j d whch ensure that P α R j+ 1 dk +,.e. k = j 1 probablty of havng order-up-to-level R j + 1 greater than expected total demand n the cycle, to be greater than cycle servce level α. Ths mples that the cumulatve probablty of havng total expected demand of the cycle less than the startng order-up-to-level, s greater than cycle servce level,.e. G d ( + 1) 1 d 2 d R j j j α. Wth the help of followng steps, we can calculate 1 nverse cumulatve dstrbuton functon G ( ) for all possble cycles. d j + + j + 1 d d α Step 1: The nverse cumulatve dstrbuton functon s gven as, G ( α ) = E{ d k } + Z * CV * E k = j+ 1 k = j d + + d d d j j... { k } 1 2 α For an order cycle coverng the demands of perods 2 to 5, the value of = 5 and the length of cycle j = 4. Therefore, G 1 d2+ d3+ d4+ d (0.90) = E{ dk} *0.3* E { d k= 2 k= 2 k } ( ) G d (0.90) = ( ) *0.3* 2+ d3+ d4+ d 5 1 For all possble cycles the G ( ) values are shown n Table 2. d 1 d d α j + + j + [Insert Table2] 1 Step 2: Based on G ( ) values calculated offlneand the parameters values assumed d j + 1+ d α + d j earler, the optmum replenshment schedule can be determned by solvng the model as defned n Equatons (1) (10) on any commercal solver. For the gven example, we obtan order up-to- 8

10 levelsrr 1 = 413, RR 3 = 490 and RR 5 = 566 wth total expected costs equal to and total emsson equal to 4980, as detaled n Table 3. [Insert Table3] Expected total emsson over the plannng horzon = Orderng emsson + Holdng emsson + Untary emsson = 4980 Carbon credt purchased = = 1980 Expected total cost = Total order cost + Expected total holdng cost + cost of carbon credts purchased = 3* * *1980 = Numercal Experments In ths secton, we conduct extensve computatonal experments to nvestgate the mpacts ofemsson parameters, product- and system-related features on supply chan performance measures such as reducton n emssons, costs and nventory levels. The emsson parameters nclude orderng emsson, carbon prce and carbon cap whereas product- and system- related features nclude cycle servce level, orderng cost, demand pattern, level of demand varablty and demand coeffcent of varaton. 4.1 Expermental desgn To cover general type busness settngs and not a specfc scenaro, we desgn a test set nvolvng a varety of demand patterns, system-and emsson- related parameters. We carry out full factoral expermentsusng the model descrbed n Secton 2.3 and the followng sets of parameters: demand patterns DP ϵ {STAT, RAND, SIN1, SIN2, LCY1, LCY2}, orderng cost o ϵ {200, 400, 900}, cycle servce level α ϵ {0.9, 0.95, 0.99}, demand coeffcent of varaton CV ϵ {0.1, 0.4, 0.7}, orderng emsson o e ϵ {200, 400, 900}, carbon cap CAP horzon ϵ {10000, 25000} and carbon prce p ϵ {1, 5}. In all the experments, we set nventory holdng cost h = 1 and holdng related emsson h e = 1. We assume zero ntal nventory and unt purchasng cost, as these values do not affect the soluton. The plannng horzon conssts of 18 perods of equal tme duraton. Fgure2 descrbes the demand patterns, where each pont represents the mean demand of a perod for whch the actual demand may assume dfferent values for dfferent plannng horzons. The demand patterns consdered nclude statonary (STAT),random (RAND), snusodal (SIN) and lfe cycle (LCY) patterns as dfferent knd of products exhbt dfferent demand patterns. Whle the STAT and the RAND patterns are the extremes of statonary and dynamc dchotomy, the 9

11 SIN1 and SIN2 patterns as well as the LCY1 and LCY2 patterns are at lower and hgher levels of demand varablty coeffcent respectvely. Demand varablty coeffcent s a measure of the varablty of a demand pattern and defned as the rato of varance of demand per perod to the square of average demand per perod.the demand patterns are so desgned that the average demand s same for each pattern thus avodng any effect that may occur because of varaton n total demand. The total mean demand s 3600 for each demand pattern. [Insert Fgure 2] 4.2 Analyzng the mpact of varaton n problem parameters Wth the help of full-factoral expermental desgn we create 1944 test nstances by varyng problem parameters such as DP, o, α, CV, o e, CAP horzon and p. The results produced after conductng experments on these test nstances are shown n Fgure 3-6. [Insert Fgure 3] [Insert Fgure 4] Fgure 3-4 reveal that the demand pattern strongly nfluences the amount of emsson produced and total cost n a supply chan. Under the RANDOM demand pattern, the emssons and costs are always hgher as compared to SIN and LCY patterns. Moreover, at a lower level of demand varablty, the LCY pattern gves the lowest value of emsson and costs whle at hgher levels, the SIN pattern gves mnmum costs and emssons. Wth the ncrease n demand varablty, the gap n the values of total emssons and total costs among dfferent demand patterns would go on decreasng. Meanng thereby, a product s demand pattern plays a sgnfcant role n determnng total cost and emsson when demand varablty s low. [Insert Fgure 5] In Fgure 5, t s evdent that the cycle servce level also has sgnfcant mpacts on both the costs and emssons. The ncrease n cycle servce level would ncrease the requred level of nventory n the system accompaned wth an ncrease n frequency of replenshment as shown n Fgure 7. It results n ncrease n orderng costs and emssons as well as ncrease n nventory holdng costs and emssons. Therefore, when cycle servce level ncreases from 0.90 to 0.95 total cost, total nventory and total emsson would ncrease sgnfcantly. Moreover, these ncrements are even steeper when cycle servce level ncreases from 0.95 to 0.99 as the rate of ncrease n nventory levels s hgher. 10

12 [Insert Fgure 6] [Insert Fgure 7] [Insert Fgure 8] Lkewse, the demand coeffcent of varaton nfluences total costs and emssons sgnfcantly. The varatons n total nventory and order frequency wth demand coeffcent of varaton are shown n Fgure 8. It reveals that total nventory as well as order frequency ncrease wth ncrease n coeffcent of varaton. Therefore, total costs and emssons would also ncrease wth the ncrease n demand coeffcent of varaton snce the costs and emssons related to nventory holdng would ncrease at a hgher level of demand uncertanty. But at a lower level of demand uncertanty, the requred nventory s low and the frm would make revenue by sellng extra emsson credts especally when the emsson cap s hgher. Therefore, total costs are shown negatve n Fgure 6. [Insert Fgure 9] [Insert Fgure 10] The mpacts of orderng cost and orderng emsson are shown n Fgure 9 and Fgure 10. The total cost ncreases wth both the orderng parameters, though the rate of ncrease s hgher at hgher values of orderng cost and emsson. But the change n total emsson s not undrectonal wth the change n orderng cost, though t keeps on ncreasng wth orderng emsson. The results show that the varaton n carbon cap has no effect on total emsson and nventory level but t affects total cost sgnfcantly. Ths result s n lne wth Benjaafar et al. (2013). 4.3 Analyzng effects of carbon prce The expermental results are summarzed n Table 4 for the cases when carbon prce s ncreased from p=1 to p=5. The average reducton n total cost, total nventory and total emsson are , and respectvely as shown n the last row of Table 4. In order to mnmze cost, the frm tends to save carbon emsson wth the ncrease n carbon prce by reducng the requred nventory level. Though t ncreases orderng frequency, but the correspondng ncrease n orderng cost and orderng emsson s lesser than the savngs acheved. The decrease n total holdng cost and reducton n total emssonand correspondng cost result nto net decrease n total cost. The savng of emsson credts acheved by ncreasng carbon prce at the hgher carbon cap can be sold n the open carbon trade marketto earn revenue. [Insert Table4] 11

13 The ncrease n carbon prce would reduce total cost, nventory level and emsson, but ts mpact on these reductons vares whle havng nteracton wth other problem parameters. The reductons n performance measures along wth nteracton wth other problem parameters are shown n Table 4. For example, when cycle servce level (α)or demandcvncreases along wth rse n carbon prce, the reducton n total cost would go on decreasng and becomes negatve, but the reducton n emsson behaves contrary.e. t ncreases wth cycle servce level (α) /demand CV. Total cost s at α=0.99and at demand CV=0.70. Therefore, total cost ncreases wth an ncrease n carbon prce at hgher levels of cyclc servce level or demand CV but t decreases n ther lower levels. The scope of emsson reducton by ncreasng carbon prce s greater at hgher levels of cycle servce level (α) and demand CV. Moreover, the mpact of carbon prce rse on total cost reducton s greater when the cycle servce level (α) and demand CV are at lower levels. Furthermore, the reducton n total nventory decreases wth ncreasng value of cycle servce level (α) but t s not undrectonal wth demand CV varaton. The carbon prce rse would acheve greater total nventory reducton at hgher level of α and md range values of demand CV. We observe from Table 4 that the varaton n total cost reducton wth carbon prce rse s more sensble to varaton n orderng emsson than to orderng cost. It decreases wth ncreasng values of orderng cost/emsson and even becomes negatve when orderng emsson s 900. The reducton n total emsson keeps decreasng wth ncrease n orderng emsson but t s not undrectonal n case of ncrease n orderng cost. The total cost reducton as well as total emsson reducton wth the rse n carbon prce s greater at lower levels of orderng cost/emsson.the reducton n nventory wth an ncrease n carbon prce keeps decreasng wth ncrease n orderng emsson level, but keeps ncreasng wth ncrease n orderng cost. Even at the hghest level of orderng emsson (= 900) or at the lowest level of orderng cost (=200), the nventory level ncreases wth carbon prce as extra emssons generated by ncreased nventory are lesser than the saved emsson due to less frequent orderng. The demand patterns also have sgnfcant mpacts on performance measures wth the rse n carbon prce. As shown n Table 4, at low demand varablty when carbon prce ncreases, the statonary demand pattern gves lowest reductons n both total cost and total emsson whle these reductons are hghest for lfe cycle demand pattern. Furthermore, at hgher demand varablty these reductons are lowest for random demand pattern whereashghestreducton n total cost happens for lfe cycle demand pattern and hghest reducton n total emsson happensfor snusodal demand pattern, as shown n Table 4. 12

14 5. Dscusson The results of a detaled numercal study suggest that at a lower level of demand varablty coeffcentthe demand pattern factor has greater mpact. Total costs and emsson are lower for lfe cycle (LCY) demand pattern at low level of demand varabltybut are lower for snusodal pattern at hgher level. However, random (RAND) demand pattern results n hgher costs and emssons for both the levels. The results also show that both cycle servce level and demand coeffcent of varaton affect total cost and emsson sgnfcantly. But the changes are even steeper for ther hgher levels.we also observe that the ncreasng value of carbon prce reduces total cost, total emsson and total nventory. But the ncreasng value of carbon prce along wth an ncrease n cycle servce level (α) or demand CV would decrease the total cost reducton and ncrease the total emsson reducton. The scope of emsson reducton by ncreasng carbon prce s greater at hgher levels of cycle servce level (α) and demand CV. The effect of carbon cap on total emsson reducton s neglgble, whch s n lne wth Benjaafar et al. (2013). The results also show that the varaton n total cost reducton wth carbon prce rse s more sensble to varaton n orderng emsson than to orderng cost. 6. Conclusons The ncreasng pressure of governments and other stakeholders has been forcng the frms worldwde to revew the supply chan decsons and ncorporate emsson concerns. Ths study addresses the nventory lot-szng problem under non-statonary stochastc demand wth emsson and cycle servce level constrants. It explctly accounts for emssons due to orderng and storage actvtes along wth emssonper unt purchased.we am to nvestgate the mpacts of product features, system- and emsson-related parameters on the emssons, costs and nventory levels of a supply chan. In practce, frms need to acheve several objectves such as servce level, total cost and total emssons. Our study would help managers as t establshes how problem parameters affect the performance measures and t would be easer to make decsons accordng to ther specfc busness envronment. Ths paper evaluatesthe mpacts of dfferent problem parameters on costs and emssons. The results of ths study are applcabe to general type busness settngs not to any specfc scenaro. It s evdent n the analyss that the demand pattern of a product has a sgnfcant mpact on costs and emssons generatedwhen demand varablty s low. Moreover, a product wth random demand pattern would always have hgher costs and emssons as compared to a productwth snusodal or lfe cycle demand pattern. Lkewse, f a frm rases ts servce level, t would ncur hgher costs and emssons but the rate of ncrease are hgher at hgher cycle servce levels. 13

15 Therefore, t s requred to strke a trade-off among servce level, total costs and emssons. Smlarly, the analyss alsoreveals that meetng demand of a product wth hgher demand uncertanty s costler and more emsson ntensve as compared to meetng demand of a product wth stable demand. The ncrease n carbon prce always motvates frms to reduce emssons as reflected n the results. The scope of emsson reducton s greater for those organzatons who keeps very hgh servcelevel or deal wth products wth hgh demand uncertanty. Except for such organzatons or for those whch ncur hgh orderng emsson, the rse n carbon prce would also get reducton n total cost. Smlarly, the products wth lfe cycle demand pattern would have greater reducton n total costs wth ncreasng carbon prce whereas the tems wth snusodal or lfe cycle demand pattern would seehgher reducton n total emssons. The present study assumes carbon prce to be constant. Future research may consder varaton n carbon prce wth other factors to nvestgate ts mpact on emsson and cost reducton. Addtonal research, based on emprcal data drawn from real lfe cases, s worthwhle. Future studes wth dfferent supply chan structures may further help n understandng the mpact of system parameters on cost and emsson reducton objectves. References: Abs, N., Dauzère-Pérès, S., Kedad-Sdhoum, S., Penz, B., Rapne, C., Lot szng wth carbon emsson constrants. European Journal of Operatonal Research, 227(1), Arslan, M., Turkay M.,2013. EOQ revsted wth sustanablty consderatons. Foundatons of Computng and Decson Scences, 38(4), Ba, C., Sarks, J., Green suppler development: analytcal evaluaton usng rough set theory. Journal of Cleaner Producton, 18(12), Battn, D., Persona, A., Sgarbossa, F., A sustanable EOQ model: Theoretcal formulaton and applcatons. Internatonal Journal of Producton Economcs, 149, Benjaafar, S., L, Y., Daskn, M., Carbon Footprnt and the Management of Supply Chans: Insghts From Smple Models. IEEE Transactons on Automaton Scence and Engneerng,10(1), Bonney, M., Jaber, M. Y., Envronmentally responsble nventory models: Non-classcal models for a non-classcal era. Internatonal Journal of Producton Economcs, 133(1), Bookbnder, J.H., Tan, J.-Y., Strateges for the probablstc lot-szng problem wth servce-level constrants. Management Scence, 34(9), Bouchery, Y., Ghaffar, A., Jema, Z., Dallery, Y., Includng sustanablty crtera nto nventory models. European Journal of Operatonal Research, 222(2), Cachon G.P., Supply chan desgn and the cost of greenhouse gas emssons, Workng Paper, Unversty of Pennsylvana, Pennsylvana. 14

16 Chaabane, A., Ramudhn, A., Paquet, M., Desgn of sustanable supply chans under the emsson tradng scheme. Internatonal Journal of Producton Economcs, 135(1), Chen, X., Benjaafar, S., Elomr, A., The carbon-constraned EOQ. Operatons Research Letters, 41(2), Choudhary, D., Shankar, R., The value of VMI beyond nformaton sharng under tmevaryng stochastc demand. Internatonal Journal of Producton Research, 53(5), Choudhary, D., Shankar, R., The value of VMI beyond nformaton sharng n a sngle suppler multple retalers supply chan under a non-statonary (Rn, Sn) polcy. Omega, 51, Dobos, I., The effects of emsson tradng on producton and nventores n the Arrow Karln model. Internatonal Journal of Producton Economcs, 93-94, Gong, X., Zhou, S. X., Optmal Producton Plannng wth Emssons Tradng. Operatons Research, 61(4), Gonzalez, E. D., Maculan, N., Huatuco, L. H., Montoya-Torres, J. R., Dabat, A., de Almeda, C. M., Gannett, B., Husngh, D., Decson-support models and tools for helpng to make real progress to more sustanable socetes. Journal of Cleaner Producton, 30, 1e2. Govndan, K., Azevedo, S. G., Carvalho, H., Cruz-Machado, V., Impact of supply chan management practces on sustanablty. Journal of Cleaner Producton, 85, He, P., Zhang, W., Xu, X., Ban, Y., Producton lot-szng and carbon emssons under cap-and-trade and carbon tax regulatons. Journal of Cleaner Producton 1-8. Hoen, K. M. R., Tan, T., Fransoo, J. C., Houtum, G. J., Effect of carbon emsson regulatons on transport mode selecton under stochastc demand. Flexble Servces and Manufacturng Journal, 26(1-2), Hua, G., Cheng, T. C. E., Wang, S., Managng carbon footprnts n nventory management. Internatonal Journal of Producton Economcs, 132(2), Husngh, D., Zhang, Z., Moore, J. C., Qao, Q., L, Q., Recent advances n carbon emssons reducton: polces, technologes, montorng, assessment and modelng. Journal of Cleaner Producton, 103, Jaber, M. Y., Glock, C. H., El Saadany, A. M. A., Supply chan coordnaton wth emssons reducton ncentves. Internatonal Journal of Producton Research, 51(1), Jn, M., Granda-Marulanda, N. A., Down, I., The mpact of carbon polces on supply chan desgn and logstcs of a major retaler. Journal of Cleaner Producton, 85, Kantas, A. B., Cobuloglu, H. I., Büyüktahtakιn, İ. E., Mult-source capactated lot-szng for economcally vable and clean bofuel producton. Journal of Cleaner Producton Letmathe, P., Balakrshnan, N., Envronmental consderatons on the optmal product mx. European Journal of Operatonal Research, 167(2), Martel, A., Daby, M., Boctor, F., Multple tems procurement under non statonary stochastc demands. European Journal of Operatonal Research, 87,

17 Neale, J. J., Wllems, S. P., Managng Inventory n Supply Chans wth Nonstatonary Demand. Interfaces, 39(5), Plambeck, E. L., Reducng greenhouse gas emssons through operatons and supply chan management. Energy Economcs, 34, S64 S74 Retel Helmrch, M., Jans, R.F., van den Heuvel, W., Wagelmans, A.P.M., The Economc Lot-Szng Problem wth an Emsson Constrant (No. EI ). Report. Econometrc Insttute, Erasmus Unversty Rotterdam, pp Seurng, S., A revew of modelng approaches for sustanable supply chan management. Decson Support Systems, 54(4), Shaw, K., Shankar, R., Yadav, S. S., Thakur, L. S., Suppler selecton usng fuzzy AHP and fuzzy mult-objectve lnear programmng for developng low carbon supply chan. Expert Systems wth Applcatons, 39(9), Song J, Leng M., Analyss of the sngle-perod problem under carbon emsson polces. In: Cho, T.M. (Ed.), Handbook of Newsvendor Problems: Models, Extensons and Applcatons. Sprnger, New York, 2012, pp Tarm, S. A., Kngsman, B. G., The stochastc dynamc producton/nventory lot-szng problem wth servce-level constrants. Internatonal Journal of Producton Economcs, 88(1), Toptal, A., Özlü, H., Konur, D., Jont decsons on nventory replenshment and emsson reducton nvestment under dfferent emsson regulatons. Internatonal Journal of Producton Research, 52(1), Wahab, M. I. M., Mamun, S. M. H., Ongkunaruk, P., EOQ models for a coordnated twolevel nternatonal supply chan consderng mperfect tems and envronmental mpact. Internatonal Journal of Producton Economcs, 134(1), Zhang, B., Xu, L., Mult-tem producton plannng wth carbon cap and trade mechansm. Internatonal Journal of Producton Economcs, 144(1),

18 REGULATOR CARBON CAP SUPPLIERS BUYER FIRM 1 2 j + 1 T DEMAND N (µ, σ) α CV DP CUSTOMERS C PRICE CARBON MARKET Fgure 1 System at a glance wth graphcal representaton of all possble order cycles Demand STAT RAND SIN1 SIN2 LCY1 LCY Tme Perod Fgure 2 End-customer demand patterns 17

19 17400 LOW HIGH LOW HIGH Total emsson Total cost RAND SIN LCY 0 RAND SIN LCY Demand patterns Demand patterns Fgure 3 Varaton n total emsson wth demand varablty and demand pattern Fgure 4 Varaton n total cost wth demand varablty and demand pattern 18

20 total cost Total emsson Total Cost Total Emsson Cycle Servce Level Fgure 5 Effect of cycle servce level on total cost and total emsson Total cost Total emsson Total revenue Total cost Coeffcent of Varaton Total Emsson Fgure 6 Effect of coeffcent of varaton on total cost and total emsson 19

21 Total nventory Order frequency Total nventory Order frequency Total nventory Cycle servce level Order frequency Total nventory Coeffcent of varaton Order frequency Fgure 7 Effects of cycle servce level on total nventory and order frequency Fgure 8 Effects of demand CV on total nventory and order frequency Total Cost Total cost Total emsson Orderng cost Total Emsson Total revenue Total Cost Total cost Total emsson Orderng emsson Total Emsson Fgure 9 Effects of orderng cost on total cost and emsson Fgure 10 Effects of orderng emsson on total cost and emsson 20

22 Table 1 Demand data for the numercal example Perod () Demand (d ) σ = CV * d Table 2 Inverse cumulatve dstrbuton functon G ( ) d α j + d j + d /j Table 3 Optmal replenshment schedule Perod () Replenshment-up-to level Expected openng nventory Expected demand Expected closng nventory Expected order quantty Orderng emsson Holdng emsson Untary emsson

23 Table 4 Summary of expermental results when the carbon prce ncreases from p=1 to p=5 Independent Varables Total cost reducton Total nventory reducton Order frequency reducton Total emssons reducton Cycle servce level (α) Demand coeffcent of varaton (CV) Orderng emsson Orderng cost Demand Pattern (DP) STAT RAND SIN SIN LCY LCY Average