Sustainable transportation and order quantity: insights from multiobjective optimization Bouchery, Y.; Ghaffari, A.; Jemai, Z.; Fransoo, J.C.

Transcription

1 Sustainable transportation and order quantity: insigts from multiobjective optimization Boucery, Y.; Gaffari, A.; Jemai, Z.; Fransoo, J.. Publised in: Flexible Services and Manufacturing Journal DOI: 0.007/s z Publised: 5/03/06 Document Version Accepted manuscript including canges made at te peer-review stage Please ceck te document version of tis publication: A submitted manuscript is te autor's version of te article upon submission and before peer-review. Tere can be important differences between te submitted version and te official publised version of record. People interested in te researc are advised to contact te autor for te final version of te publication, or visit te DOI to te publiser's website. Te final autor version and te galley proof are versions of te publication after peer review. Te final publised version features te final layout of te paper including te volume, issue and page numbers. Link to publication itation for publised version (APA: Boucery, Y., Gaffari, A., Jemai, Z., & Fransoo, J.. (06. Sustainable transportation and order quantity: insigts from multiobjective optimization. Flexible Services and Manufacturing Journal, 8(3, DOI: 0.007/s z General rigts opyrigt and moral rigts for te publications made accessible in te public portal are retained by te autors and/or oter copyrigt owners and it is a condition of accessing publications tat users recognise and abide by te legal requirements associated wit tese rigts. Users may download and print one copy of any publication from te public portal for te purpose of private study or researc. You may not furter distribute te material or use it for any profit-making activity or commercial gain You may freely distribute te URL identifying te publication in te public portal? Take down policy If you believe tat tis document breaces copyrigt please contact us providing details, and we will remove access to te work immediately and investigate your claim. Download date:. Oct. 08

2 Sustainable transportation and order quantity: Insigts from multiobjective optimization Yann BOUHRY (, Asma GHAFFARI (, Zied JMAI (,(3, Jan FRANSOO (4 ( cole de Management de Normandie, Axe Logistique Terre Mer Risque 30 Rue de Ricelieu, Le Havre, France ( Laboratoire Génie Industriel, cole entrale Paris Grande Voie des Vignes, 990 atenay-malabry, France (3 OASIS-NIT, University of Tunis l Manar BP 37, Le Belvedere 00 Tunis, Tunisia (4 indoven University of Tecnology, Scool of Industrial ngineering P.O. Box 53, 5600 MB indoven, te Neterlands Marc 4, 06 Abstract: Tis article applies multiobjective optimization to sow ow te tradeoffs between cost and carbon emissions may be obtained in te context of sustainable operations. We formulate a model were transportation mode selection and order quantity decisions are considered jointly. We derive structural properties of te model and develop several insigts. First, we sow tat switcing to a greener mode of transportation wile continuing to optimize te total logistics costs function may lead to a dominated solution. Second, we prove tat te modal sift occurs only under strong carbon emissions reduction requirements. Tird, we sow tat te efficient frontier is non-convex and we analyze some implications. Finally, we analyze te impacts of an increase in truck capacity. Te results are illustrated troug an example of a Frenc retailer. orresponding autor: Yann Boucery, yboucery@em-normandie, Tel:

3 . Introduction nvironmental and social awareness as considerably increased since te Brundtland report s publication (World ommission on nvironment and Development, 987. Nowadays, many leading companies worldwide are committed in creating value for a broader set of stakeolders instead of focusing solely on creating profits for sareolders or owners. In line wit tis trend, te number of articles on sustainable operations as drastically increased tese last years. We refer to Linton et al. (007, Srivastava (007, Seuring and Müller (008, Kleindorfer et al. (009, Dekker et al. (0, Tang and Zou (0 and Govindan et al. (04 for reviews. By acknowledging te different dimensions of sustainability, sustainable operations aim at optimizing several objectives. In tis context, companies may identify situations were te objectives may be improved simultaneously, i.e. win-win situations. However, tese situations may become more difficult to find as more sustainable practices are deployed. For instance, in an article focusing on cost and carbon emissions objectives, aro et al. (03 assume tat firms ave exausted all teir carbon abatement initiatives tat are profitable in te absence of external incentives, i.e. tat win-win situations are unavailable. An increasing number of companies also start tinking tat sustainability can only be attained by optimizing seemingly conflicting targets (DHL, 00. In tis case, a company interested in sustainable operations seeks to identify te most favorable trade-off between te considered objectives. Tis leads Fransoo et al. (04 to consider tat tis naturally leads to multiobjective considerations in order to analyze te tradeoff between economic and environmental performance measures (p.. Accordingly, tis article applies multiobjective optimization to a decision problem related to sustainable operations in order to sow ow to obtain te tradeoffs between different types of objectives. Te aim of multiobjective optimization is to identify particular solutions suc tat, wen attempting to improve an objective furter, oter objectives suffer as a result (rgott, 005. Tese solutions are called efficient or Pareto optimal and tey correspond to te tradeoffs tat are of interest for companies interested in sustainable operations. Te model developed consists of a joint transportation mode selection and order quantity optimization problem. We indeed acknowledge tat inventory and transportation decisions are strongly interrelated. However, tere is a lack of articles on sustainable operations explicitly addressing tis issue. We focus on two objective functions, i.e., cost and carbon emissions. Our multiobjective optimization results enable sowing tat switcing to a greener mode of transportation wile continuing to optimize te total logistics costs function may lead to a dominated solution (suc tat it is possible to reac te same level of carbon emissions wit a lower cost by solely adjusting te order quantity. Second, we prove tat a sift towards

4 a more carbon-efficient mode of transportation is interesting only for strong carbon emissions reduction requirements (i.e., for ig carbon emissions reduction target or for ig carbon price. Oterwise, order quantity adjustments may enable efficiently greening te supply cain. Tird, we sow tat te efficient frontier (i.e., te set of efficient solutions is non-convex. Tis structural property is of great importance as tis implies tat some non-supported solutions exist (see e.g. Geoffrion (968 for more details about non-supported solutions. Tis type of solutions cannot be generated by using a linear combination of te objectives. Tus, using carbon pricing wen facing a non-convex efficient frontier may provide a misleading impression to te decision maker as te non-supported solutions would be idden. Te article is organized as follows. Te related literature is reviewed in Section. Section 3 is ten devoted to te presentation of te model, to te multiobjective optimization results and to te presentation of an example. We also analyze in Section 3 te impacts of using carbon pricing and te impacts of extending veicle capacity. Finally, Section 4 is devoted to te conclusion and to future researc directions.. Literature review and contribution Tis article is directly connected to two main fields of te literature on sustainable supply cains, i.e., order quantity models and transportation mode selection. First, several articles on order quantity decisions wit carbon emissions considerations ave recently been proposed. Te classical conomic Order uantity (O model as been revisited by considering veicle emissions costs (Bonney and Jaber, 0, a constraint on carbon emissions (en et al., 03, te emissions trading sceme (Hua et al., 0 and four different regulatory policies (Arslan and Turkay, 03. Boucery et al. (0 include sustainability criteria into single and multiecelon O models by using multiobjective optimization. Jaber et al. (03 revisit te joint economic lot size problem by considering tat te companies face different types of emissions trading scemes. Benjaafar et al. (03 include carbon emissions constraints on single and multi-stage lot-sizing models wit a cost minimization objective and consider four regulatory policy settings. Battini et al. (04 extend te O model by explicitly including transportation modeling. Tey compare te results obtained wit a pure cost model and te results obtained wen additionally including environmental costs. Konur and Scaefer (05 include carbon emissions into te joint replenisment problem. Tey apply multiobjective optimization and identify te set of efficient solutions for two types of strategies (namely direct and indirect grouping. Based on a set of numerical examples, tey identify conditions suc tat a strategy outperforms te oter one bot in terms of costs and carbon emissions. Te dynamic lot-sizing model as 3

5 also been revisited by Absi et al. (03, Velázquez-Martínez et al. (04 and Retel Helmric et al. (05 by considering an additional constraint on carbon emissions. xcept from Velázquez-Martínez et al. (04 and Battini et al. (04, te articles cited above do not take explicitly transportation features into account. As a result, tese articles do not account for te impact of order quantity decisions on te load factor even if tis could strongly impacts te transportation emissions as sown by Velázquez-Martínez et al. (04. On te oter and, Velázquez-Martínez et al. (04 and Battini et al. (04 do not consider transportation mode selection issues. Te second field of literature related to tis work focuses on including carbon emissions concerns into freigt transportation mode selection problems. Winebrake et al. (008 present an energy and environmental analysis model to explore te tradeoffs in an intermodal transportation network. Bauer et al. (009 especially focus on determining te optimal planning for intermodal rail transportation in order to minimize te carbon emissions from transportation. olette and Venkat (009 present a case study were several modes of transportation are available in a wine supply cain context. Teir analysis takes cost, carbon emissions and energy consumption into account. Pan et al. (03 investigate ow freigt consolidation may elp in decreasing te carbon emissions from transportation. Tey formulate a carbon emissions minimization model were bot road and rail transportation are available. Tey apply teir model for optimizing te carbon emissions of two large retail cains. Leal and D Agosto (0 consider te transportation mode selection decision in a case study based on a bio-etanol supply cain. Socio-environmental considerations are included into te model. Finally, Kopfer et al. (04 analyze te benefits of considering a eterogeneous fleet of veicles for veicle routing problems. Te autors sow tat te fuel consumption (and terefore te carbon emissions can be reduced by coosing veicles of an adequate size for transport fulfillment. Te impact of inventory decisions suc as order quantity decisions are not considered in tis stream of literature, as te order quantity are assumed to be given exogenously. To our knowledge, six articles jointly consider transportation mode selection and inventory optimization problem wile explicitly accounting for transportation features. Rosič and Jammernegg (03 extend te dual sourcing model based on te newsvendor framework by considering te environmental impact of transportation. Tey analyze two types of regulatory policies, i.e. te carbon tax and te carbon cap-and-trade mecanism. Tey prove tat it is possible to reduce te carbon emissions from transportation witout substantially affecting te economic performance of te system if te capand-trade mecanism is applied wit appropriate carbon cap setting. In Hoen et al. (04, a stocastic inventory model is extended to incorporate transport emissions costs. Te transport mode and order-up-to level of a base-stock inventory policy are jointly optimized in a single product setting. Hoen et al. (03 4

6 extend te previous model to a multiple products settings and study te portfolio effect of setting a global emissions target instead of a per product target. Tese tree articles mainly focus on te relationsip between transportation leadtime and safety stock levels by arguing tat greener modes of transportation are often slower (and/or less flexible and tat tis results in increasing te inventory levels. Our paper focuses on a different aspect of te interaction between transportation mode selection and inventory decisions. Indeed, we consider tat order quantity decisions ave a strong impact on te transportation mode selection decisions and on te carbon emissions levels. First, greener modes of transportation are often able to carry bigger volumes. Second, te order quantity decisions ave an effect on load factors and tus on carbon emissions from transportation. Our model enables investigating suc relationsip and may be viewed as complementary to te literature focusing on te relationsip between transportation leadtime and safety stock levels. To our knowledge, tree articles focus on suc interaction wile explicitly accounting for transportation features. Konur (04 extends te economic order quantity model to account for full truckload cost structure. Te autor proposes a carbon constrained model and focuses more particularly on te situations for wic te optimal order quantity exceeds one full truckload. Te article sows tat considering a eterogeneous fleet of trucks may elp reducing bot cost and carbon emissions. Konur and Scaefer (04 additionally consider less tan truckload tariffs and tey analyze te impact of four types of regulatory policies. Finally, Scaefer and Konur (05 propose an integrated continuous review inventory model wit explicit transportation decisions. Te model accounts for bot truckload and less tan truckload transportation. Te autors consider two objectives, i.e., te total cost and te total carbon emissions and tey propose to approximate te efficient frontier by using linear combinations of te two objectives. Te results sow tat an increase in demand variance and/or in lead time negatively affects bot cost and emissions. Unlike Konur (04 and Konur and Scaefer (04, we focus ere on situations for wic te optimal order quantity does not exceed one full veicle load as tis is te case in a large variety of industrial situations. Scaefer and Konur (05 is seminal in te sense tat tis is te first article tat simultaneously considers te relationsip between transportation leadtime and safety stock levels as well as te effect of order quantity levels on load factors. Due to te complexity of te problem, te autors approximate te efficient frontier by applying te weigted sum metod. We focus ere solely on te effect of order quantity levels on load factors. Tis enables us to analytically identify te efficient frontier. In addition, Scaefer and Konur (05 solve te problem for eac mode separately. Tey propose some examples wit two modes of transportation and solve te integrated inventory and transportation mode selection problem numerically. We propose ere analytical results tat 5

7 enable us identifying te efficient frontier wen multiple transportation modes are available. Our model may consequently be viewed as complementary to te existing literature. Our contribution is treefold. First, from a modeling perspective, tis article is one of te first studying explicitly te relationsip between order quantity and transportation mode decisions wit cost and carbon emissions concerns. Our model may be applied wit multiple modes of transportation including air, water, rail, road and any type of intermodal combinations. Te model may also be used to study te effect of speed reduction as speed is recognized to ave some major impacts on costs, fuel consumption (orbett et al., 009; Fransoo and Lee, 03 and safety. Te model is flexible enoug to account for realistic transportation costs and carbon emissions structures as any type of piecewise linear functions may be considered. Second, from a teoretical perspective, we apply te concept of multiobjective optimization in order to identify te existing tradeoffs tat a company can face wen jointly optimizing transportation mode selection and order quantity decisions. Multiobjective optimization elps te decision maker to build a conviction of wat is possible and to use tis knowledge to identify te most valuable trade-off. We identify analytically te set of efficient solutions for te problem. Our tird contribution consists in providing new insigts to decision makers. We sow tat switcing to a greener mode of transportation wile continuing to optimize te total logistics costs function may lead to a dominated solution. We prove tat te modal sift occurs only under strong carbon emissions reduction requirements. We sow tat te efficient frontier is non-convex and we analyze some implications. Finally, we analyze te impacts of an increase in truck capacity. 3. Model description and multiobjective optimization results 3. Model description In tis article, we consider tat several modes of transport are available for inbound transportation. ac mode is caracterized by a cost function in te form of a fixed cost (per veicle and a variable cost (per product unit. Moreover, a fixed lead time is associated to eac mode. Te lead time as an effect on te average in-transit inventory level. We also take te capacity of eac mode into account. Te per sipment transportation cost function is defined as follows ( x represents te nearest integer x : wit: T ( TV TF T L, ( max 6

8 = order quantity, max T V = transportation mode capacity, = variable transportation cost per product unit, T F = fixed transportation costs per veicle, T = in-transit inventory olding costs per product unit and time unit, L = transportation lead time. wit: Te total inventory olding and transportation cost function per time unit is defined as follows: D = demand per time unit, D D Z ( O T ( P D, ( = inventory olding costs per product unit and time unit, O = fixed order costs, P = purcase costs per product unit. Assume tat is te order quantity tat minimizes te total cost function (i.e. Z. ( Z Lemma provides a sufficient condition ensuring tat te optimal order quantity does not exceed one veicle load. Lemma. Let be te order quantity minimizing xpression (. Assume tat O c max D, ten: max. All proofs may be found in Appendix A. In te following, we assume tat te retailer as no control over te production sceme of te manufacturer (tat certainly supplies several oter retailers as in most practical situations. In tis case, O represents te fixed costs for order forms, autorization, receiving, inspection and/or andling of invoice from te supplier (Axsäter, 006 as transport costs are accounted for separately. Tese costs are usually small comparing to te costs of requiring a second veicle for inbound transportation as well as te costs of olding extra inventory. Tese costs ave also been reduced by new tecnologies suc as electronic data intercange or radio frequency identification. onsequently, 7

9 we assume tat in wat follows, i.e., tat tere is no incentive to order more tan a full veicle. max In tis case, te cost function to be considered may be expressed as follows: Z D ( ( O TF ( P TV T L D, max, (3 We refer to Konur (04 and references terein for an analysis of te complexity and for providing algoritms to solve te problem in te cases for wic te optimal order quantity consists in ordering more tan a full veicle load. In practice, several transportation tariffs are often available, depending on te sipment quantity. We focus on eavy duty truck transport as an example. Te truck capacity is 33 euro pallets. Assume tat te logistics provider offers tree different tariffs. Te first one is a less tan truckload tariff wit T 30 per pallet and T 0 per truck. A second less tan truckload tariff is available if te sipment V F max size is at least pallets. In tis case, T 0 per pallet and T 0 per truck. Finally, a full truckload V F tariff is available wit T 600 per truck and T 0 per pallet. Assume tat te retailer decides to F V order 5 pallets. Te tariff leads to 450 per sipment. On te oter and, to retailer may over-declare its sipment to pallets to take advantage of te all unit discount less tan truckload tariff leading to 40 per sipment. onsequently, te transportation costs function follows a modified all-unit discount structure. Tis cost structure is commonly used in te literature (see e.g. an et al., 00; roxton et al., 003; Rieksts and Ventura, 008. Te per sipment transportation cost as a function of te order quantity (up to te truck capacity is sown in Figure. We intend to develop a model tat enables accounting for realistic transportation costs structures suc as te modified all-unit discount cost structure introduced above. To do so, let minimum amount per sipment to make a mode available. By making use of be te min max min and max, any type of piecewise linear transportation cost function may be taken into account by considering eac piecewise linear segment of te transportation cost function as a separate mode caracterized by a fixed cost (per veicle and a variable cost (per product unit. Indeed, te minimum and maximum capacity limits enable making te mode available only for te quantities related to te segment under consideration. onsequently, we consider 4 different transportation modes wit different cost parameters and minimum quantities for te example above. 8

10 Figure : Transportation cost In te cases for wic min max, te order quantity minimizing quation 3 is obtained easily as Z is convex on ; ]. First, te optimal order quantity witout taking te minimum and [ min max maximum quantity constraints into account can be calculated. Second, te feasible order quantity wic is te closest from tis teoretical quantity is selected: max min ;min max, ( O T F D. (4 Modeling carbon emissions across te supply cain is attracting more and more researc (see e.g. Scipioni et al., 0; Sundarakani et al., 00. In our model, tree main sources of carbon emissions may be identified. First, carbon emissions are generated by producing te item. Tis amount of carbon emissions is independent of te decisions taken by te retailer as we assume tat te retailer as no control over te production sceme of te manufacturer. A fixed amount of carbon emissions is tus associated to eac item due to production. Second, carbon emissions are generated by inbound transportation. Te transportation mode selection as well as te order quantity decisions may affect te amount of carbon emissions generated by inbound transportation. For a given transportation mode, te generated carbon emissions are modeled wit a fixed term (due to emissions generated by te veicle if running empty and a linear term as a function of te order quantity (due to te extra energy consumption generated by transporting te items. Note tat tis modeling is commonly used in te transportation literature (see e.g. Pan et al. (03 for rail and road transportation and is in accordance wit te available metodologies 9

11 for estimating carbon emissions from transportation (see e.g., Demir et al., 04; NTM, 008; Hickman et al., 999. Tird, an amount of carbon emissions is associated wit te storage of eac product unit per time unit. Tis amount is mainly due to indirect carbon emissions from energy consumption (mainly electricity in te wareouse. Tis amount may become important in case of refrigeration. Te total carbon emissions as a function of te order quantity may be expressed as follows for all min : wit: P = purcase emissions per product unit, Z D ( P D TF TV D, (5 max = inventory olding emissions per product unit and time unit, T F = fixed amount of carbon emissions per sipment, T V = variable amount of carbon emissions per product unit. Assume tat We can observe tat capacity. It follows tat: is te order quantity tat minimizes te carbon emissions function (i.e. Z. ( Z (tere is no incentive to order more tan te maximum transportation max max min ;min max; T F D. (6 3. Multiobjective optimization results We consider costs and carbon emissions as two distinct objective functions tat ave to be minimized. An alternative a is tus said to be dominated if tere exists anoter alternative b tat performs at least as good as a on one objective and tat performs better tan a on te oter objective. Multiobjective optimization consists in identifying all te non-dominated alternatives called efficient solutions. Te results presented in tis section enable identifying te set of efficient solutions in te situations for wic multiple modes of transportation are available. 3.. A single mode of transportation As a first step, we identify te set of efficient solutions wen considering a single mode of transport. In tis case, te only decision variable for te problem is te order quantity and te set of possible values for is min ; max A. Let A Z :, Z( a Z ( a; Z ( a, for all a A, wit Z defined by 0

12 Formula 3 representing te total costs and emissions. Z( A Z ( ; Z ( Z defined by Formula 5 representing te total carbon A is te image of A in te criterion space (evaluation space. Te set of efficient solutions is a subset of A noted. Its image in te criterion space referred as te efficient frontier is Z (. Proposition enables identifying te set of efficient solutions in te situations for wic a single mode is available. Tis one can be expressed as a function of order quantities defined by Formulas 4 and 6 respectively. and, te optimal Proposition : Let be te set of efficient solutions wen considering a single mode of transportation, ten: [min( ; ;max( ; ]. Note tat in many practical cases, tus [ ; ]. Indeed, O T F is often lower tan T F as te olding cost includes te opportunity cost of te capital tied up into inventory (meaning tat is ig and as transportation is recognized as a major source of carbon emissions (meaning tat T F is ig. Accordingly, we focus on te situations for wic of relevance and clarity. O T F T F in wat follows for te sake We define now te convexity property for te efficient frontier (wic corresponds to a subset of and we prove tat te efficient frontier is convex wen considering a single mode of transportation. Let S be a subset of, onv(s is te convex ull of S, i.e. te set of all convex combinations of points in S. Let ff (S be te efficient frontier of S, ten ff (S is convex if and only if ff ( onv ( S ff ( S. Proposition : Let Z ( be te efficient frontier wen considering a single mode of transportation, ten: Z ( is convex. Proposition implies tat te problem beaves nicely wen considering a single mode of transportation. Indeed, all te elements of a convex efficient frontier may be generated by minimizing a weigted sum of objectives. We prove later tat te efficient frontier is non-convex wen more tan one mode of transportation is considered (see Proposition 6.

13 3.. Two modes of transportation onsider now tat a second mode of transportation (mode is available. In wat follows, subscript refers to mode and subscript refers to mode. As for mode, and may be obtained by using Formula 4 and Formula 6 respectively. Moreover, let Z Z ( and Z Z (. Proposition and Proposition are also valid for mode. Witout loss of generality, we assume tat Z Z (mode is less costly tan mode. Propositions 3 and Lemma restrict te possible number of intersection between Z and Z. Note tat S corresponds to te cardinality of te set S. ( ( Proposition 3. Let Z and Z be te efficient frontiers for transportation mode and ( ( transportation mode respectively, ten: Z Z (. ( Lemma. Let Z and Z be te efficient frontiers for transportation mode and transportation ( ( mode respectively, ten: If Z Z, ten Z Z (, ( lse Z ( 0; Z. ( Figure is an illustration of te possible outcomes wen two modes are considered. Figures a to d illustrate te outcomes wen Z Z and Figures e to g illustrates te outcomes wen Z Z. Te results we present in Proposition 3 and Lemma enable providing additional insigts compared to a classical single objective analysis. If Z Z, mode is more expensive but greener. In tis case, tere exists at most one intersection between te efficient frontiers of te two modes considered separately. In tis case, mode will be preferred from a pure cost perspective, and mode two will be preferred if te requirements to green te operations reac a given tresold. Tis result implies tat adjusting te order quantity is te most efficient way of greening operations until a given tresold. Tis also implies tat switcing to a greener mode of transportation wile continuing to optimize te total logistic cost function may lead to a dominated solution (see Figures a and c. In tis case, te same level of carbon emissions may be obtained wit a lower cost by only increasing te order quantity. Tis result proves tat poor decisions may be taken wen ignoring te strong interrelationsip between inventory control and transportation mode selection.

14 arbon emissions Figure a arbon emissions Figure b Z ( ( c ; e Z ( Z ( Z ( arbon emissions Figure c ost arbon emissions Figure d ost Z ( Z ( Z ( Z ( arbon emissions Figure e ost arbon emissions Figure f ost Z ( Z ( Z ( ( c ; e ( c ; e Z ( arbon emissions Figure g ost ost Z ( Z ( ost Figure : Te seven situations wen considering two modes of transportation 3

15 If Z Z and Z Z, we may expect mode to be out of interest as tis is possible to obtain a ceaper solution as well as a greener solution wit mode. However, our results proves tat some solutions obtained wit mode may still be efficient in case Z Z ( (see Figure f. Mode ( may indeed be considered in some situations as a better compromise between costs and carbon emissions. Te efficient frontier of te problem wit two modes of transportation may be identified by applying Lemma and by acknowledging tat Z Z as sown in Propositions 4 and Proposition 5. Proposition 4. Let Z ( be te efficient frontier for te problem wit two modes of transportation wit Z Z and Z Z ten: If Z ( Z, ten te intersection point is noted ( lse Z Z ( 0 : ( If Z ( Z(, ten: lse, c ; and: e ( c ; e Z ( c c ( c ; e Z ( e e Z (. Z ( ( Z ( c; e Z ( e Z, ( c ; e Z ( c Z Z ( Z(. Proposition 5. Let Z ( be te efficient frontier for te problem wit two modes of transportation wit Z Z and Z Z ten: If Z Z ( 0 ten: ( If Z ( is efficient ten: lse: Z ( lse Z ( e ( ( c ; e Z( c Z Z c ; e Z ( e Z ( Z Z (. ( Z. Let c e and e e, ten: ; ; ( c be te two intersection points and assume tat Z ( c ; e Z ( e e c ; e Z ( e e e ( c ; e Z ( e e ( ( 4

16 Propositions 4 and 5 enable identifying analytically te efficient frontier wen two modes of transport are considered. We additionally provide a sufficient condition ensuring tat te efficient frontier of te problem wit two modes of transportation is non-convex in Proposition 6. Proposition 6: Let Z ( be te set of efficient solutions for te problem wit two modes of transportation, ten: If Z Z ( ten Z ( is non-convex. ( Proposition 6 proves tat te efficient frontier is non-convex in most of practical situations. Te fact tat te efficient frontier is non-convex as soon a more tan one mode of transportation is available implies tat some efficient solutions are non- supported ( see e.g. Geoffrion (968 for more details about nonsupported solutions. Tese solutions, even if tey migt be of interest for te decision maker, would not be generated by single objective approaces suc as te weigted sum metod (pricing emissions is an example of a weigted sum. Te multiobjective approac developed in tis article enables identifying suc non-supported solutions and tus provides a olistic view of te efficient frontier. We provide a discussion on te implications of using a weigted sum metod for te problem in Section More tan two modes of transportation Te study of te global problem wit n modes of transportation may be conducted as follows. First, te set of efficient solutions for mode k denoted as k (wit corresponding efficient frontier denoted as Z may be identified by using Proposition for all k [ ; n]. We assume witout loss of generality k ( k tat for all k [ ; n ], Z. Propositions 4 and 5 may be applied to compare mode k [ ; n] to k Z k all te oter available modes j [ ; n] (suc tat j k. Z k, j ( is te corresponding efficient frontier (by extension, we also consider Z ( Z (. Let define n Z ( Z k, k k k k k j k, j (. Note tat Z k ( k may be an empty set. Te efficient frontier of te global problem may be identified by applying Proposition 7. 5

17 Proposition 7. Let Z ( be te set of efficient solutions for te global problem wit n modes of transportation, ten: n k Z( Z k ( k. Tis results enables identifying all te efficient solutions in te case wit multiple transportation modes. We provide an example of application in te next section. 3.3 xample and insigts We present ere an application tat illustrates te type of outcome generated by te multiobjective optimization analysis. We decide to focus on a single application as we provide analytical results tat enable understanding te beavior of te model for any set of parameters. Te application is followed by an illustration of te main insigts igligted in Section 3.. Te application is based on real data in order to ensure te practical relevance of insigts xample of te Frenc retailer We consider a Frenc retailer wo orders bottles of wine from an external supplier. Te bottles are delivered on pallets and te retailer as to order an integer amount of pallets. Terefore, we consider tat one product unit equals to one pallet. We assume tat te assumptions of te O model are fulfilled. We exclude purcase costs and purcase emissions from te analysis as tey do not affect te decisions we consider. Te data relative to te problem may be found in Table. time unit mont quantity unit pallet unit weigt 500 kg inbound distance 500 km D 0 pallets/mont 75 euro/pallet.mont T 50 euro/pallet.mont O 00 euro/order.65 kg O/pallet.mont Table : Application s data Assume first tat te Frenc retailer decides to use eavy duty trucks for inbound transportation. Te truck capacity is 33 euro pallets and te transportation leadtime is L 0.07 mont (i.e., 0.5 day. max 6

18 At first, we assume tat te truck transportation cost is linear in te order quantity wit TV 30 per pallet and TF 0 per truck (i.e., following a less tan truckload tariff and we consider tat te minimum capacity for getting tis tariff consists in ordering 0 pallets. min Te Network for Transport and nvironment metodology (NTM, 008 is used to evaluate te carbon emissions related to transportation. Tis metodology developed by a Swedis not-for-profit organization provides estimates for emissions generated by different modes of transportation in urope. Te NTM metodology may be applied at te aggregate level for less tan truckload transport by computing a per product amount of carbon emissions by considering a given average load factor. However, Velázquez- Martínez et al. (03 ave sown tat suc aggregate approac may lead to substantial errors as te effects of lot sizing decisions on te load factor are disregarded. Te NTM metodology also enables a more detailed estimation of carbon emissions by considering te fixed emissions generated by te truck wen running empty and te per pallet emissions due to te extra energy consumption. Te detailed model provided by te NTM metodology is used in our application as te load factor depends on te order quantity decision. Indeed, we assume tat te retailer does not allow te logistics provider for including oter type of cargo for inbound transportation and consequently as a direct control on te load factor. Tis situation is classical for retail distribution and tis explains wy te minimum sipment size is 0 pallets. According to te NTM metodology, te fixed carbon emissions are equal to T 34 kg O for te proposed example. Te maximum load of te truck is 6 tons of cargo. Te truck is tus fully loaded in volume wit 33 pallets for a corresponding load factor of 0.63 in tis application. Tis leads to a variable amount of carbon emissions T kg O per pallet. V F Te conditions stated in Lemma are satisfied for te example so we can conclude tat max. ( O By applying Formula 3 wit tis example, we obtain tat TF D 0 pallets as T. By applying Formula 5, we obtain tat F D 33 pallets as Note tat in te example as stated in Section 3... Te results are summarized in Table. min max T F T F (O TF D TF D Truck transportation Table : Parameters for truck transportation Z Z 7

19 By applying Proposition, we obtain tat [0;33]. Figure 3 displays te efficient frontier as well as its convex ull. Te x-axis corresponds to te costs and te y-axis corresponds to te carbon emissions. Figure sows tat Z ( is convex as stated in Proposition. We also include in Figure 3 some dominated solutions obtained wen ordering more tan a full truckload (dased line. We can observe tat ordering more tan a full truckload is not effective as tis implies an increase in cost (as stated in Lemma as well as an increase in carbon emissions (as emissions cannot be decreased by ordering more tan a full veicle according to xpression (5. Figure 3: Te efficient frontier wit a single mode of transportation onsider now tat rail transportation is also available for inbound transportation to te Frenc retailer. A train includes 6 freigt cars, eac of tem fully loaded in volume wit 36 pallets. Te rail transportation leadtime is mont (i.e., days. A fixed transportation cost T 449 per freigt car is considered. In opposition to truck transportation, several types of cargo (from several retailers may be included into te same train. In tis case, te carbon emissions associated wit te train wen running empty may be split between te different users. A fixed amount of carbon emissions per freigt car is ten derived from te average utilization rate of te train. Moreover, a variable amount of carbon emissions is associated to eac pallet. By using average values provided by te NTM metodology, we obtain tat T 333 kg O per freigt car and T. 30 kg O per pallet. Due to bot costs and carbon emissions structure, V F F 8

20 tere is no incentive to order more tan one full freigt car. Note tat te results of Section 3. and 3.. may be applied to train transportation. Te results are summarized in Table 3. min max T F T F (O TF D TF D Train transportation Table 3: Parameters for train transportation Z Z From Table 3 and Table, we can notice tat Z and Z tus Proposition 4 can be truck Z train truck Z train applied. By solving te system of equations leading to an intersection between Z and Z train ( train, we obtain tat Ztruck( truck Ztrain( train truck ( truck Ø. Moreover, we can notice tat Z ( ; 33.9 is dominated by Z ( 7.79 ; truck truck train train applying Proposition 4, we obtain tat Z( ( c ; e Z ( c Z Z ( truck truck train train train. Ten, by. Te results are illustrated in Figure 4. We can also notice tat te efficient frontier Z ( is non-continuous tus nonconvex as sown by Proposition 6. Figure 4: Truck and train transportation efficient frontiers Finally, assume tat te logistics provider offer two additional tariffs for truck transport. One tariff is an all unit discount less tan truckload tariff wit TV 0 per pallet if te quantity ordered is iger or equal to pallets. Te second additional tariff is a full truckload tariff wit TF 600 per truck and 9

21 TV 0 per pallet. As noticed in Section 3., transportation costs follow a modified all-unit discount structure wit one discount rate and one full truckload rate. As stated in Section 3., truck transportation is considered in te model as 4 different transportation modes as te cost parameters and minimum quantities differ for eac tariff. Te global problem consists of deciding between 5 different modes of transport and in deciding on te optimal order quantity. We refer to tese modes as mode i [;5 ]. Table 4 provides a syntesis of te main parameters for mode i [;5 ] as well as te results obtained from Section 3... Train transportation is referred to as mode 4 as tis ensure tat for all k [;4 ], Z k Z k. Wen focusing on mode, max is set to 4 pallets as a better tariff (wit te same level of carbon emissions may be obtained wit mode wen 4. Figure 5 illustrates te results obtained in Table 4 in te criterion space. min max T F T F (O TF D TF D mode mode mode mode mode Table 4: Syntesis of te different modes Z Z Te results of Section 3..3 are applied to identify te set of efficient solution for te global problem. Wit 5 different modes, 4 pairwise comparisons are teoretically required, but we can notice tat te analysis is totally disjointed for mode and mode and tat Z ( 5 by comparing to mode 4. Te results of Section 3..3 boil down to apply te results of Section 3.. to mode and mode and to mode 3 and mode 4 separately. Te final results are presented in Figure

22 Figure 5: fficient frontiers for te different modes Figure 6: Joint efficient frontier of te different modes 3.3. Insigts from multiobjective optimization Te new multiobjective optimization results applied to te example of te Frenc retailer enables to exemplify te insigts igligted in Section 3... First, assume tat te retailer currently orders 0 pallets (i.e. tat te retailer minimizes its total logistics costs. Figure 5 sows tat a carbon emissions reduction of 57% can be acieved by increasing

23 te order quantity up to 33 pallets witout switcing to a greener mode of transportation. Tis feature illustrates tat increasing te order quantity (i.e., increasing te load factor for transportation is very efficient for reducing te supply cain emissions. Moreover, te required financial effort first remains reasonable wen decreasing carbon emissions. For instance, coosing truck transportation wit 6. 7 enables a 34% reduction in carbon emissions for a 6% costs increase. On te opposite, te financial effort will increase as is getting closer to te order quantity tat minimizes te amount of carbon emissions for truck transportation. Tis feature is commonly igligted in te literature on inventory control wit carbon emissions concerns (Boucery et al., 0; en et al., 03 due to te relative insensitivity of te costs to a variation in te order quantity. We prove ere tat te results remain valid wen explicitly including transportation features into te economic order quantity model. Tis implies tat switcing to a greener mode of transportation is efficient only in case of a strong carbon emissions reduction target. For te example proposed, sifting from truck to train becomes interesting if te carbon emissions reduction target is a least 50% or for a carbon price greater tan 670 /ton O (tis result is obtained by identifying te common tangent between truck and train transportation. In oter situations, adjusting te order quantity wile continuing to use truck transportation would be more efficient to green te supply cain. Te example of te Frenc retailer also illustrates tat switcing to a greener mode of transportation wile continuing to optimize te total logistic cost function may lead to a dominated solution (we refer to Figure 5. Witout performing te multiobjective optimization analysis provided in tis article, te retailer may decide to switc to rail for inbound transportation wile continuing to minimize te total cost function in order to decrease te carbon emissions of te supply cain. Tis solution leads to 350 and 44 kg O per mont and may be perceived as appropriate as tis leads to a 40% decrease in carbon emissions for a 3% increase in costs, wen compared to minimizing costs wit truck transportation. However, Figure 5 clearly sows tat te solution consisting in switcing to rail wile continuing to minimize te cost function is a dominated solution. Te same decrease in carbon emissions may be acieved wit a cost reduced by more tan 5% by continuing to use truck for inbound transportation and by coosing 9 (leading to 77 and 440 kg O per mont. Tis example clearly sows tat switcing to a greener mode of transportation wile continuing to optimize te cost function may not be te best option to green te supply cain. Tis igligts te necessity of taking an integrated inventory control and transportation mode selection perspective wen intending to green te supply cain.

24 3.4 ost model wit a carbon price In tis section, we aim at comparing our results to te ones obtained if a price is associated to carbon emissions. We indeed argued tat te weigted sum metod was not appropriate to generate all te efficient solutions if te efficient frontier is non-convex, and we concluded tat carbon pricing migt not be a proper way of generating efficient solutions for te problem we consider as we proved tat te efficient frontier is not convex in many cases. We consider tat a price is associated to te company s carbon emissions. Tis price can be imposed to te company in te case of a carbon tax. However, tis price can also come from an internal evaluation from te company, by considering te cost of te energy used or te cost obtained wit an environmental accounting analysis. Tis price per amount of carbon emissions is noted 0;. We perform te analysis by considering two transportation modes. In tis context, tere exists a value 0; allows deciding wic mode is te most interesting as we sow in Proposition 8. L tat Proposition 8. Assume tat a price is associated to te company s carbon emissions and assume tat te company can decide between modes and wit L suc tat: Z Z and Z Z. Ten tere exists a value - if L, ten min( Z( ( min(( Z( (, - if L, ten min( Z( ( min(( Z( (. Te value of L is unique and may be found by identifying a value suc tat min( Z( ( min(( Z( ( (i.e. L and by applying te bisection metod on te interval 0; (at eac iteration, we select te interval suc tat i; i min( Z( i ( min(( Z ( i ( and min( Z( i ( min(( Z ( i (. We apply te metod to te example proposed in Figure 4 and we illustrate te results in Figure 7. Te common tangent as well as te efficient solutions not identified by te weigted sum approaced (te nonsupported efficient solutions are displayed (dased lines in addition to te efficient solutions obtained wit te weigted sum metod. 3

25 Figure 7: Te efficient frontier obtained wit te weigted sum metod Figure 7 clearly igligts tat te weigted sum approac is not appropriate for identifying all te efficient solutions of te problem. Tis could provide misleading impression to te decision maker. For instance, te decision maker could conclude tat te maximum reduction in carbon emissions acievable by efficiently using truck transport is 4% (from 735 kg O per mont to 634 kg O per mont, instead of 33% (from 735 kg O per mont to 495 kg O per mont, if we include te non-supported efficient solutions. In addition, te emissions may be reduced by 0% (from 735 kg O per mont to 590 kg O per mont, for an increase in cost of 6% (from 9 per mont to 58 per mont if we include te non-supported solutions. In opposite, te cost as to be increased by 4% (from 9 per mont to 36 per mont to reduce te emissions by at least 0% if we ignore te non-supported solutions. Moreover, a marginal cange in carbon price around L would ave a uge effect on te amount of carbon emissions generated. Assume tat raises from L to L wit being arbitrarily small. Ten, te optimal level of carbon emissions for te example presented above would drop from 634 kg O per mont to 395 kg O per mont (i.e., 38% reduction. Tis implies tat a small cange in carbon price may ave uge impact on carbon emissions. Te case wit L is also very interesting. In tis case, bot transportation modes give te same overall result (operational costs + carbon tax. However, te costs and te carbon emissions are different for bot options. In te example igligted above, wen 0. 54, using truck would lead to =575 per mont wile using train would lead to =575 per mont. 4

26 onsequently, te company may decide among te two options. Tis operational flexibility implies tat te total amount of carbon emissions is ardly controllable by setting a carbon price as te amount of carbon emissions for a given carbon price is not necessarily unique and may widely differ. We provide additional insigts on te impact of veicle capacity in te next section. 3.5 Impact of veicle capacity Veicle capacity is often limited by regulation. For instance, te maximum lengt of a truck in France is limited to 8.5 meters (33 euro pallets wile te maximum weigt eligible for truck transportation is 6 tons of goods. Logistics providers often argue tat increasing veicle capacities could strongly impact te performance of deliveries bot in terms of costs and carbon emissions. In tis section, we intend to analyze te impact of increasing trucks capacity in France, based on our results and on te example we provide in Section 3.3. Most of te existing studies on tis topic only focus on transportation and only compare te performances of existing trucks wit te performances of longer trucks. Our results enable a more extensive study by including te effects of te variation in inventory and by also comparing to a sift to train. We focus on te impact of allowing uropean Modular System (MS in France. MS consists of combining existing loading units to form a longer truck. Tese longer trucks are currently tested in many uropean countries wit good results in terms of carbon emissions. An MS truck can be up to 5.5 meters long and carries up to 5 pallets. We investigate te impact of allowing for MS trucks in France by using te example presented in Section 3.3. We assume tat te trucks can now be loaded wit 5 pallets and tat te full truckload tariff does not cange (i.e., TF 600 per truck. We also take te optimistic assumption tat te emissions due to empty running are similar to a conventional eavy duty truck, i.e., T 34 kg O. Finally, we assume tat te variable emissions from transportation remain F uncanged, i.e., T kg O per pallet. Te results we obtain in tis case are sown in Figure 8. V Figure 8 igligts tat te increase in truck capacity does not affect te results. Indeed, te new solutions available are dominated by train transportation. Tis may be explained as follows by comparing te results obtained wen 5 pallets are transported wit an MS truck wit te results obtained wen 36 pallets are transported by rail. MS trucks perform better tan rail transport bot in terms of costs and emissions for te example we propose. Indeed, transportation costs (respectively emissions equal 47 per mont (respectively 00 kg O per mont wit a full MS truck as compared to 36 per mont (respectively kg O per mont for a full freigt car. However, inventory olding costs equal to 950 per mont if te order quantity is 5 pallets, instead of 350 per mont if te order quantity is 5